j. differential geometry 49 (1998) 265-410

PU(2) MONOPOLES. I: REGULARITY, UHLENBECK COMPACTNESS, AND TRANSVERSALITY

PAUL M. N. FEEHAN & THOMAS G. LENESS

1. Introduction

At seminars at Harvard and MIT, during October 1994, Edward Witten introduced the U(1) monopole equations and the Seiberg-Witten invariants to smooth four-manifold topology and conjectured their relationship with Donaldson invariants on the basis of new developments in quantum field theory [19], [98]. The conjecture, recently extended in [60], has been verified for all four-manifolds whose Donaldson and Seiberg-Witten invariants have been independently computed. Within two months of Witten’s announcement, a program was outlined by V. Pidstrigach and A. Tyurin and others, which should lead to a mathematical proof of the relationship between these two invariants [68], [71], [74]. This approach is unrelated to the quantum field-theoretic arguments of [60], [98] and uses a moduli space of PU(2) monopoles to construct a cobordism between links of Seiberg-Witten moduli spaces of U(1) monopoles and the Donaldson moduli space of anti-self-dual connections, which appear as singularities in this larger stratified moduli space.

It was soon recognized, however, that despite the appeal and elegance of the PU(2) monopole program, its implementation involves substantial technical difficulties due to the contributions of moduli spaces of U(1) monopoles in the lower levels of the Uhlenbeck compactification of the moduli space of PU(2) monopoles. Many of these difficulties had never been resolved even in the case of Donaldson theory where similar problems arise, albeit in a rather simpler form, in attempts to prove the Kotschick-Morgan conjecture for Donaldson invariants of four-manifolds X with b^{+}(X) = 1. That conjecture asserts that the Donaldson invariants computed using metrics lying in different chambers of the positive cone of H^{2}(X; ℝ)/ℝ^{∗} differ only by homotopy-invariant terms [45]. In the case of the Kotschick-Morgan conjecture, the heart of the problem lies in describing the links of the lower-level reducibles via gluing and then in calculating the pairings of the Donaldson cohomology classes with those links. Thus far, such links have been described and their pairings with cohomology classes computed only in certain special cases [14], [15], [16], [20], [21], [32], [58], [99]. The methods used to obtain these special cases fall very far short of the kind of general analysis needed to prove the Kotschick-Morgan conjecture. On the other hand, by assuming the Kotschick-Morgan conjecture, L. Göttsche computed the coefficients of the wall-crossing formula in [45] in terms of modular forms by exploiting the presumed homotopy invariance of the coefficients [35]. A related approach to the Witten conjecture was proposed by Pidstrigach and Tyurin [72]. Certain aspects of the PU(2) monopole program have been considered from a quantum-field theoretic viewpoint in [13], [38], [39], [50], [51], [52], [53].

Received September 11, 1996, and, in revised form, May 5, 1997. The first author was supported in part by an NSF Mathematical Sciences Postdoctoral Fellowship under grant DMS 9306061.

In the present article and its sequels [25], [26], [27] we address the analytical problems associated with constructing the links of lower-level Seiberg-Witten moduli spaces and in establishing the analogues of the Kotschick-Morgan conjecture for PU(2) monopoles needed to compute the pairings of cohomology classes with these links. We hope to return to the actual computations and a verification of Witten’s conjecture [60],

[98] in a subsequent paper. In this article we describe the basic regularity, Uhlenbeck compactness, and transversality results we need for the moduli space of PU(2) monopoles, and in the sequels [26], [27] we develop the gluing theory required to construct the links of the lower-level Seiberg-Witten moduli spaces. An announcement of the main results of this article appeared in [24].

1.1. Statement of results

1.1.1. PU(2) monopoles and holonomy perturbations. We consider Hermitian two-plane bundles E over X whose determinant line bundles det E are isomorphic to a fixed Hermitian line bundle over X endowed with a fixed C^{∞}, unitary connection. Let (ρ, W^{+}, W^{−}) be a spin^{c} structure on X, where ρ : T^{∗}X → End W is the Clifford map, and the Hermitian four-plane bundle W = W^{+} ⊕W^{−} is endowed with a C^{∞} spin^{c} connection.

Let k ≥ 3 be an integer and let A_{E} be the space of L_{k}^{2} connections A on the U(2) bundle E all inducing the fixed determinant connection on det E. Equivalently, following [48, §2(i)], we may view A_{E} as the space of L_{k}^{2} connections A on the PU(2) = SO(3) bundle 𝔰𝔲(E). We pass back and forth between these viewpoints, via the fixed connection on det E, and rely on the context to make the distinction clear. Given a connection A on 𝔰𝔲(E) with curvature F_{A} ∈ L_{k−1}^{2}(Λ^{2} ⊗ 𝔰𝔬(𝔰𝔲(E))), then ad^{−1}(F_{A}^{+}) ∈ L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) is its self-dual component, viewed as a section of Λ^{+} ⊗𝔰𝔲(E) via the isomorphism ad : 𝔰𝔲(E) → 𝔰𝔬(𝔰𝔲(E)). When no confusion can arise, the isomorphism ad : 𝔰𝔲(E) → 𝔰𝔬(𝔰𝔲(E)) will be implicit, and so we regard F_{A} as a section of Λ^{+} ⊗ 𝔰𝔲(E) when A is a connection on 𝔰𝔲(E). Let D_{A} : L_{k}^{2}(W^{+} ⊗ E) → L_{k−1}^{2}(W^{−} ⊗ E) be the corresponding Dirac operator.

For an L_{k}^{2} section Φ of W^{+} ⊗ E, let Φ^{∗} = ⟨·, Φ⟩ be its pointwise Hermitian dual and let (Φ ⊗ Φ^{∗})_{00} be the component of the Hermitian endomorphism Φ ⊗ Φ^{∗} of W^{+} ⊗ E which lies in 𝔰𝔲(W^{+}) ⊗ 𝔰𝔲(E). The Clifford multiplication ρ defines an isomorphism ρ : Λ^{+} → 𝔰𝔲(W^{+}) and thus an isomorphism ρ = ρ⊗id_{𝔰𝔲(E)} of Λ^{+}⊗𝔰𝔲(E) with 𝔰𝔲(W^{+})⊗𝔰𝔲(E). Then

(1.1)  F_{A}^{+} − ρ^{−1}(Φ ⊗ Φ^{∗})_{00} = 0,
D_{A}Φ = 0,

are the unperturbed PU(2) monopole equations considered in [67], [68], [71], [74], with a slightly differing trace conditions (see below), for a pair (A, Φ) consisting of a connection on 𝔰𝔲(E) and a section Φ of W^{+} ⊗ E.

Donaldson’s proof of the connected-sum theorem for his polynomial invariants [18, Theorem B] makes use of certain ‘extended anti-self-dual equations’ [18, Equation (4.24)] to which the Freed-Uhlenbeck generic metrics theorem does not apply [18, §4(v)]. These extended equations model a neighborhood of the product connection appearing in the Uhlenbeck compactification of the moduli space of anti-self-dual SU(2) connections. As the zero locus of the extended equations may not be transverse, Donaldson employs holonomy perturbations which give gaugeequivariant C^{∞} maps A_{E}^{∗}(X) → Ω^{+}(𝔰𝔲(E)) and thus perturbations of the extended anti-self-dual equations [17, §2], [18, pp. 282–287]. These perturbations are continuous with respect to Uhlenbeck limits and yield transversality not only for the top-level moduli space, but also for all lower-level moduli spaces and all intersections of the geometric representatives defining the Donaldson invariants.

In §2.5.2 and the Appendix we describe a generalization of Donaldson’s idea which we use to prove transversality for the moduli space of solutions to a perturbed version of the PU(2) monopole equations (1.1). Unfortunately, in the case of of PU(2) monopoles, the analysis is considerably more intricate and the method we employ here is rather different from the one developed in [18]. We use an infinite sequence of holonomy sections defined on the infinite-dimensional configuration space of pairs; when restricted to small enough open balls in the configuration space, away from reducible connections, only finitely many of these perturbing sections are non-zero and they vanish at reducible connections.

Let G_{E} be the Hilbert Lie group of L_{k+1}^{2} unitary gauge transformations of E with determinant one. Let S_{Z}^{1} denote the center of U(2) and set ^{∘}G_{E} := S_{Z}^{1} ×_{{±id_{E}}} G_{E}, which we may view as the group of L_{k+1}^{2} unitary gauge transformations of E with constant determinant. The stabilizer of a unitary connection A on E in ^{∘}G_{E} (which coincides with its stabilizer in the full group Aut E of unitary automorphisms of E) always contains the center S_{Z}^{1} ⊂ U(2), corresponding to the constant, central, unitary automorphisms of E. We call A irreducible if its stabilizer is exactly S_{Z}^{1} and reducible otherwise.

It is also possible, as in [71], [74], to fix a smooth representative ω ∈ Ω^{2}(X, ℝ) for c_{1}(E) and instead consider the space of unitary connections A on E satisfying the trace condition tr F_{A} = −2πiω, modulo the action of the full group Aut E of unitary automorphisms of E. The resulting moduli space of nonabelian monopoles is then a torus bundle over the moduli space which we define below, with fibers H^{1}(X; ℝ)/H^{1}(X; ℤ). These tori complicate the analysis of the links of singularities when b^{1}(X) > 0 and do not contain any additional information, so we choose to eliminate them by instead imposing the stronger fixed-determinant-connection trace condition and working with a compatible group of gauge transformations. A similar framework is used in [68], [92].

We refer to §2.5 for a detailed account of the construction of our holonomy perturbations. The large number of technical points involving regularity and uniform estimates for these perturbations (which still allow us to obtain an Uhlenbeck compactification) are discussed in the Appendix. We fix r ≥ k +1 and define gauge-equivariant C^{∞} maps (see §2.5),

\begin{matrix}A_{E}(X) → L_{k+1}^{2}(X, 𝔤𝔩(Λ^{+}) ⊗_{ℝ} 𝔰𝔬(𝔰𝔲(E))), \\ A ↦ τ⃗ · 𝔪⃗(A), \\ A_{E}(X) → L_{k+1}^{2}(X, Hom(W^{+}, W^{−}) ⊗_{ℂ} 𝔰𝔩(E)), \\ A ↦ ϑ⃗ · 𝔪⃗(A), \end{matrix} (1.2)

where τ⃗ := (τ_{j,l,α}) is a suitably convergent sequence in C^{r}(X, 𝔤𝔩(Λ^{+})), and ϑ⃗ := (ϑ_{j,l,α}) is a suitably convergent sequence in C^{r}(X, Λ^{1} ⊗ ℂ), while 𝔪⃗(A) := (𝔪_{j,l,α}(A)) is a sequence in L_{k+1}^{2}(X, 𝔰𝔲(E)) of holonomy sections constructed by extending the method of [17], [18], and

τ⃗ · 𝔪⃗(A) := ∑_{j,l,α} τ_{j,l,α} ⊗_{ℝ} ad(𝔪_{j,l,α}(A)),
ϑ⃗ · 𝔪⃗(A) := ∑_{j,l,α} ρ(ϑ_{j,l,α}) ⊗_{ℂ} 𝔪_{j,l,α}(A).

We call a point (A, Φ) in the pre-configuration space of L_{k}^{2} pairs C̃_{W,E} := A_{E} × L_{k}^{2}(W^{+} ⊗ E) a PU(2) monopole if it solves

F_{A}^{+} − (id + τ_{0} ⊗ id_{𝔰𝔲(E)} + τ⃗ · 𝔪⃗(A))ρ^{−1}(Φ ⊗ Φ^{∗})_{00} = 0,
D_{A}Φ + ρ(ϑ_{0})Φ + ϑ⃗ · 𝔪⃗(A)Φ = 0.

(1.3)

We let M_{W,E} be the moduli space of solutions cut out of the configuration space of pairs C_{W,E} := C̃_{W,E}/^{∘}G_{E} by the equations (1.3). We let M_{W,E}^{∗,0} ⊂ M_{W,E} be the subspace of pairs [A, Φ] such that A is irreducible and the section Φ is not identically zero. The sections τ⃗ · 𝔪⃗(A) and ϑ⃗ · 𝔪⃗(A) vanish at reducible connections A by construction; plainly, the terms in (1.3) involving the perturbations τ⃗ · 𝔪⃗(A) and ϑ⃗ · 𝔪⃗(A) are zero when Φ is zero.

1.1.2. Uhlenbeck compactness. The holonomy-perturbation maps in (1.2) are continuous with respect to the Uhlenbeck topology (see §4.5), just as those of [18]. Suppose {A_{β}} is a sequence in A_{E}(X) which converges in the Uhlenbeck topology to a limit (A, x) in A_{E_{−ℓ}}(X) × Sym^{ℓ}(X), where E_{−ℓ} is a Hermitian two-plane bundle on X with det(E_{−ℓ}) = det E and c_{2}(E_{−ℓ}) = c_{2}(E)−ℓ, and ℓ ≥ 0 is an integer. The sections τ⃗ · 𝔪⃗(A_{β}) and ϑ⃗ · 𝔪⃗(A_{β}) then converge in L_{k+1}^{2}(X) to a section τ⃗ · 𝔪⃗(A, x) of 𝔤𝔩(Λ^{+}) ⊗_{ℝ} 𝔰𝔬(𝔰𝔲(E_{−ℓ})) and a section ϑ⃗ · 𝔪⃗(A, x) of Hom(W^{+}, W^{−}) ⊗_{ℂ} 𝔰𝔩(E_{−ℓ}), respectively. For each ℓ ≥ 0, the maps of (1.2) extend continuously to gauge-equivariant maps

A_{E_{−ℓ}}(X) × Sym^{ℓ}(X) → L_{k+1}^{2}(X, 𝔤𝔩(Λ^{+}) ⊗_{ℝ} 𝔰𝔬(𝔰𝔲(E_{−ℓ}))),

A_{E_{−ℓ}}(X) × Sym^{ℓ}(X) → L_{k+1}^{2}(X, Hom(W^{+}, W^{−}) ⊗_{ℂ} 𝔰𝔩(E_{−ℓ})), (1.4)

given by (A, x) ↦ τ⃗ · 𝔪⃗(A, x) and (A, x) ↦ ϑ⃗ · 𝔪⃗(A, x), which are C^{∞} on each C^{∞} stratum determined by Sym^{ℓ}(X).

Our construction of the Uhlenbeck compactification for M_{W,E} requires us to consider moduli spaces

M_{W,E_{−ℓ}} ⊂ C_{W,E_{−ℓ}} × Sym^{ℓ}(X)

of triples [A, Φ, x] given by the zero locus of the ^{∘}G_{E_{−ℓ}}-equivariant map

𝔖 : C̃_{W,E_{−ℓ}} × Sym^{ℓ}(X) → L_{k}^{2}(Λ^{+} ⊗ 𝔰𝔲(E_{−ℓ})) ⊕ L_{k}^{2}(W^{−} ⊗ E_{−ℓ})

defined as in (1.3) except using the perturbing sections τ⃗ · 𝔪⃗ and ϑ⃗ · 𝔪⃗ in (1.4) instead of those in (1.2). We call M_{W,E_{−ℓ}} a lower-level moduli space if ℓ > 0 and call M_{W,E_{−0}} = M_{W,E} the top or highest level.

In the more familiar case of the unperturbed PU(2) monopole equations (1.1), the spaces M_{W,E_{−ℓ}} would simply be products M_{W,E_{−ℓ}} × Sym^{ℓ}(X). In general, though, the spaces M_{W,E_{−ℓ}} are not products when ℓ > 0 due to the slight dependence of the section 𝔖(A, Φ, x) on the points x ∈ Sym^{ℓ}(X) through the perturbations τ⃗ · 𝔪⃗ and ϑ⃗ · 𝔪⃗. A similar phenomenon is encountered in [18, §4(iv)–(vi)] for the case of the extended anti-self-dual equations, where holonomy perturbations are also employed in order to achieve transversality.

We define M̄_{W,E} to be the Uhlenbeck closure of M_{W,E} in the space of ideal PU(2) monopoles,

⋃_{ℓ=0}^{N} M_{W,E_{−ℓ}} ⊂ ⋃_{ℓ=0}^{N} (C_{W,E_{−ℓ}} × Sym^{ℓ}(X)) ,

for any integer N ≥ N_{p}, where N_{p} is a sufficiently large integer to be specified below.

Theorem 1.1. Let X be a closed, oriented, smooth four-manifold with C^{∞} Riemannian metric, spin^{c} structure (ρ, W^{+}, W^{−}) with spin^{c} connection on W = W^{+} ⊕ W^{−}, and a Hermitian two-plane bundle E with unitary connection on det E. Then there is a positive integer N_{p}, depending at most on the curvatures of the fixed connections on W and det E together with c_{2}(E), such that for all N ≥ N_{p} the topological space M̄_{W,E} is compact, second countable, Hausdorff, and is given by the closure of M_{W,E} in ∪_{ℓ=0}^{N}M_{W,E_{−ℓ}}.

Remark 1.2. The existence of an Uhlenbeck compactification for the moduli space of solutions to the unperturbed PU(2) monopole equations (1.1) was announced by Pidstrigach [71] and an argument was outlined in [74]. A similar argument for equations (1.1) was outlined by Okonek and Teleman in [68]. Theorem 1.1 yields the standard Uhlenbeck compactification for the system (1.1) and the perturbations of (1.1) described in [23], [93] — see Remark 1.4. An independent proof of Uhlenbeck compactness for (1.1) and certain perturbations of these equations is given in [92], [93].

1.1.3. Transversality. The space Sym^{ℓ}(X) is smoothly stratified, the strata being enumerated by partitions of ℓ. If Σ ⊂ Sym^{ℓ}(X) is a smooth stratum, we define

M_{W,E_{−ℓ}}|_{Σ} := {[A, Φ, x] ∈ M_{W,E_{−ℓ}} : x ∈ Σ},

with M_{W,E_{−0}} := M_{W,E} when ℓ = 0. We then have the following transversality result for equations (1.3), at least away from the solutions where the connection is reducible or the spinor vanishes. Let M_{E}^{asd} denote the moduli space of anti-self-dual connections on 𝔰𝔲(E).

Theorem 1.3. Let X be a closed, oriented, smooth four-manifold with C^{∞} Riemannian metric, spin^{c} structure (ρ, W^{+}, W^{−}) with spin^{c} connection on W = W^{+} ⊕ W^{−}, and a Hermitian line bundle det E with unitary connection. Then there is a first-category subset of the space of C^{∞} perturbation parameters such that the following holds. For each 4-tuple (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) in the complement of this first-category subset, the corresponding moduli space M_{W,E}^{∗,0} is a smooth manifold of the expected dimension,

dim M_{W,E}^{∗,0} = dim M_{E}^{asd,∗} + 2 Ind_{ℂ} D_{A} − 1
= −2p_{1}(𝔰𝔲(E_{−ℓ})) − \frac{3}{2}(e(X) + σ(X))
+ \frac{1}{2}p_{1}(𝔰𝔲(E_{−ℓ})) + \frac{1}{2}((c_{1}(W^{+}) + c_{1}(E))^{2} − σ(X)) − 1.

Moreover, for each integer ℓ ≥ 0, and smooth stratum Σ ⊂ Sym^{ℓ}(X), the moduli space M_{W,E_{−ℓ}}^{∗,0}|_{Σ} is a smooth manifold of the expected dimension, dim M_{W,E_{−ℓ}}^{∗,0}|_{Σ} = dim M_{W,E_{−ℓ}}^{∗,0} + dim Σ.

Remark 1.4. Different approaches to the question of transversality for the PU(2) monopole equations (1.1) with generic perturbation parameters have been considered by Pidstrigach and Tyurin in [74] and Teleman in [92]. More recently, a new approach to transversality for (1.1) has been discovered independently by the first author [23] and Teleman [93]: the method uses only the perturbations (τ_{0}, ϑ_{0}) together with perturbations of the Riemannian metric g on X and compatible Clifford map ρ.

A choice of generic Riemannian metric on X ensures that the moduli space M_{E}^{asd,∗} of irreducible anti-self-dual connections on 𝔰𝔲(E) is smooth and of the expected dimension [20], [30], although the points of M_{E}^{asd,∗} need not be regular points of M_{W,E}^{∗} as the linearization of (1.3) may not be surjective there. A choice of generic parameter τ_{0} ensures that the moduli spaces M_{W,E,L_{1}}^{red,0} of non-zero-section solutions to (1.3) which are reducible with respect to the splitting E = L_{1} ⊕(det E)⊗L_{1}^{∗} are smooth and of the expected dimension [25]. Again, the points of M_{W,E,L_{1}}^{red,0} need not be regular points of M_{W,E}^{0} since the linearization of (1.3) may not be surjective there.

We note that related transversality and compactness issues have been recently considered in approaches to defining Gromov-Witten invariants for general symplectic manifolds [59], [75], [79].

1.2. Outline. We indicate how the remainder of our article is organized. In §2.2 we prove a slice result for the configuration space C_{W,E} (Proposition 2.8) while in §2.6 we describe the elliptic deformation complex for (1.3) and compute the expected dimension of M_{W,E}. We develop the regularity theory for a generalization of the PU(2) monopole equations (1.3) (obtained by allowing additional inhomogeneous terms) in §3: the main technical result there is that an L_{1}^{2} solution to an inhomogeneous version of (1.3) and the Coulomb gauge equation is C^{∞} (Corollary 3.4). By combining this with the slice result of Proposition 2.8, we then show that any L_{k}^{2} PU(2) monopole (A, Φ) is L_{k+1}^{2} gaugeequivalent to a C^{∞} PU(2) monopole (Proposition 3.7).

We establish local estimates for L_{1}^{2} solutions to the inhomogeneous version of (1.3) in §3.3 and §3.4. We use the sharp L_{1}^{2} regularity result of Corollary 3.4 in §4 to prove the removability of point singularities for PU(2) monopoles (Theorem 4.10). In the sequels [26], [27], these regularity results and estimates are needed to prove that L_{1}^{2} gluing solutions to (1.3) are C^{∞} and to analyse the asymptotic behavior of Taubes’ gluing maps and their differentials near the lower strata of the Uhlenbeck compactification.

The proof of Theorem 1.1 relies heavily on both the regularity theory of §3 and the fact that solutions (A, Φ) to (1.3) satisfy a ‘universal energy bound’, with constant depending only on the data in the hypotheses of Theorem 1.1. The section Φ also satisfies a universal C^{0} bound: these bounds are the analogues for PU(2) monopoles of the now well-known a priori estimates for Seiberg-Witten monopoles [47], [62], [77], [98] and follow, in much the same way, from the maximum principle and the Bochner-Weitzenböck formula for D_{A} provided k ≥ 3 (see §4.1). In §4 we prove the removability of point singularities for solutions to (1.3) (Theorem 4.10) using our a priori bounds and regularity results. While the PU(2) monopole equations are not conformally invariant, they are invariant under constant rescalings of the metric (in the sense of §4.2) and, as in the case of anti-self-dual connections, this scale invariance is exploited in the proof of Theorem 1.1, whose proof is completed in §4.6.

Theorem 1.3 is initially established in §5 for C^{r} perturbations for any fixed 3 ≤ r < ∞, in order to avail of the Sard-Smale theorem for Fredholm maps of Banach manifolds [80], while in §5.1.2 we show that generic C^{∞} perturbation parameters are sufficient for transversality. (See Corollary 5.6 for the special case ℓ = 0 and §5.1.3 for its extension to the general case ℓ ≥ 0.)

As we shall explain in §5, our proof of Theorem 1.3 ultimately hinges on the fact that if (A, Φ) is a PU(2) monopole and A is reducible on a non-empty open subset containing the support of the holonomy perturbations depending on A, then A is reducible over X. The proof of this result (Theorem 5.11) occupies §5.3; the result follows from the Agmon-Nirenberg unique continuation theorem after the system (1.3) has been transformed into an ordinary differential equation for a oneparameter family of pairs over S^{3}. The unique continuation property holds for both the perturbed PU(2) monopole equations (1.3), when the initial open subset of X contains the balls in X supporting holonomy perturbations, and the unperturbed equations (1.1) for any initial open subset.

1.3. Other approaches to transversality. As we noted in Remark 1.4, transversality for the PU(2) monopole equations (1.1) has also been proved very recently using only the perturbations (τ_{0}, ϑ_{0}), together with perturbations of the Riemannian metric g on X and compatible Clifford map ρ [23], [93]. The transversality proof given in [23] is more delicate than the method we employ in the present article. When using holonomy perturbations, the main technical difficulties are due to the noncompactness of the moduli space of PU(2) monopoles, and the core transversality argument itself is more straightforward, whereas in [23] the situation is entirely reversed. To place these various transversality results in a suitable context, we briefly discuss some other approaches to transversality, both for the PU(2) monopole equations and the closely related ‘spin-ASD’ equations of [73], which Pidstrigach and Tyurin used to define spin polynomial invariants.

Suppose (A, Φ) is a solution to either equations (1.1), equations (1.1) with perturbations (τ_{0}, ϑ_{0}), or the holonomy-perturbed equations (1.3). If A is reducible, then Φ has rank less than or equal to one (see Lemma 5.22). However, as observed by Teleman [69], [92], if Φ is rank one, then A is not necessarily reducible and he describes a simple counterexample for equations (1.1) when X is a Kähler manifold with its canonical spin^{c} structure.

It is not too difficult to prove that M_{W,E}^{∗,0} is a smooth manifold of the expected dimension away from the locus of irreducible, rank-one solutions using the perturbation parameters (τ_{0}, ϑ_{0}) alone. However, as irreducible, rank-one solutions to (1.3) could be present in M_{W,E}^{∗,0}, it appears impossible to prove that the entire space M_{W,E}^{∗,0} is a smooth manifold of the expected dimension using only the parameters (τ_{0}, ϑ_{0}). A similar problem arises in the proof of transversality for the spin-ASD equations given in [73, Proposition I.3.5]; a version of these equations can be obtained from equations (1.1) by omitting the quadratic term ρ^{−1}(Φ ⊗ Φ^{∗})_{00}. In the proof of [73, Proposition I.3.5] it is claimed that if D_{A}Φ = 0 and Φ is rank one, then A is reducible [p. 277]: Teleman’s counterexample shows that this claim is incorrect, and he points out an error in their argument [69], [92]. On the other hand, the possible presence or absence of irreducible, rank-one solutions to the PU(2) monopole equations makes no difference to the transversality argument which we describe in §5 using holonomy perturbations, as these perturbations are strong enough to yield transversality without a separate analysis of the locus of irreducible, rank-one solutions.

1.4. Applications. In [25] we discuss the singularities of the moduli space M_{W,E}. We introduce cohomology classes and geometric representatives on M_{W,E}^{∗}, construct the links of the strata of antiself-dual and Seiberg-Witten moduli spaces in M_{W,E}, and compute the Chern characters of their normal bundles in suitably defined ambient manifolds. We thus obtain a relation between Donaldson and Seiberg-Witten invariants when the reducible solutions appear only in the top level of the Uhlenbeck compactification. In [26], [27], we develop the gluing theory for PU(2) monopoles: this is used to construct links of the lower-level Seiberg-Witten moduli spaces, to show that the pairing of the cohomology classes with this link is well-defined, and to eliminate the requirement of [25] that the reducible solutions appear only in the top level. A survey of some of the results contained in the present article and its sequels [25], [26], [27] is provided in [24].

Acknowledgements

The first author is especially grateful to Peter Kronheimer, Tom Mrowka, and Cliff Taubes for helpful conversations and for their enthusiasm and encouragement. He is particularly grateful to Tom Mrowka for bringing to his attention a couple of points in the manuscript that needed correcting. He would also like to thank the Harvard Mathematics Department and the National Science Foundation for their generous support. The second author is very grateful to Selman Akbulut, Ron Fintushel, and Tom Parker for helpful conversations and the Mathematics Departments at Harvard and Michigan State Universities for their hospitality and support. We warmly thank the referee for his careful reading of the manuscript and thoughtful comments.

2. The PU(2) monopole equations

We develop a framework for gauge theory for pairs in §2.1 and §2.2. In §2.1 we define the Hilbert spaces of pairs and gauge transformations, while in §2.2 we establish the main slice result which we require for the configuration space of pairs modulo gauge transformations, paying particular attention to the structure near ‘reducibles’. The relationship between the groups of SO(3) and SU(2) gauge transformations is discussed in §2.3. A few linear algebra issues which will be important for later compactness and transversality arguments are discussed in §2.4. The PU(2) monopole equations and their holonomy perturbations are introduced in §2.5. The role of the perturbations in obtaining transversality results will be explained in §5. To preserve continuity a detailed discussion of the technical points which need to be addressed when using holonomy perturbations in the present context is deferred to the Appendix. The moduli space and the elliptic deformation complex for solutions to the PU(2) monopole equations are described in §2.6.

2.1. Configuration spaces of connections and pairs. In this section we define the spaces of connections, pairs, gauge transformations, and configuration spaces which we will use throughout our work.

2.1.2. Connections on Hermitian two-plane bundles. We consider Hermitian two-plane bundles E over X whose determinant line bundles det E are isomorphic to a fixed Hermitian line bundle over X endowed with a fixed C^{∞}, unitary connection A_{e}. The Hermitian line bundle over X and its unitary connection A_{e} are fixed for the remainder of this article.

Let k ≥ 2 be an integer and let A_{E} be the space of L_{k}^{2} connections A on the U(2) bundle E all inducing the fixed determinant connection A_{e} on det E. Equivalently, following [48, §2(i)], we may view A_{E} as the space of L_{k}^{2} connections A on the PU(2) = SO(3) bundle 𝔰𝔲(E). We shall pass back and forth between these viewpoints, via the fixed connection on det E, relying on the context to make the distinction clear. Explictly, if A is a unitary connection on E and s ∈ Ω^{0}(X, E), then ∇_{A}s ∈ Ω^{1}(E). For ζ ∈ Ω^{0}(X, 𝔲(E)), then ∇_{A}ζ ∈ Ω^{1}(X, 𝔲(E)) is determined by

(∇_{A}ζ)s = ∇_{A}(ζs) − ζ(∇_{A}s),

so, if A^{τ} ∈ Ω^{1}(U, 𝔲(2)) denotes the local connection matrix defined by a choice of local frame for E over an open subset U ⊂ X, then

∇_{A}s = ds + A^{τ} s,
∇_{A}ζ = dζ + [A^{τ} , ζ] = dζ + (adA^{τ})ζ.

Similarly, the connection A on E induces connections on 𝔰𝔲(E) and det E. For example, if ξ ∈ Ω^{0}(X, 𝔰𝔲(E)), then ∇_{A}ξ ∈ Ω^{1}(X, 𝔰𝔲(E)) is also given locally by

∇_{A}ξ = dξ + [A^{τ} , ξ] = dξ + [(A^{τ})_{0}, ξ] = dζ_{0} + (adA^{τ})ζ_{0},

while if λ ∈ Ω^{0}(X, det E) then ∇_{A}λ ∈ Ω^{1}(X, det E) is given by

∇_{A}λ = dλ + (tr A^{τ})λ.

The above local connection matrices are related by

A^{τ} = (A^{τ})_{0} + \frac{1}{2}(tr A^{τ}) id_{ℂ^{2}} ∈ Ω^{1}(U, 𝔲(2)),

where (A^{τ})_{0} = A^{τ} − \frac{1}{2}(tr A^{τ}) id_{ℂ^{2}} ⊂ Ω^{1}(U, 𝔰𝔲(2)) is the traceless component of A^{τ}, while tr A^{τ} ⊂ Ω^{1}(U, 𝔲(1)) = Ω^{1}(U, iℝ) is the induced local connection form for det E given by the trace component of A^{τ}. Note that (A^{τ})_{0} ∈ Ω^{1}(U, 𝔰𝔲(2)), and that ad(A^{τ})_{0} = adA^{τ} ∈ Ω^{1}(U, 𝔰𝔬(3)) is the induced local connection matrix for the SO(3) bundle 𝔰𝔲(E), where we use the standard identification 𝔰𝔬(3) = ad(𝔰𝔲(2)).

Conversely, given a unitary connection A_{e} for det E and a Riemannian connection Â for 𝔰𝔲(E), we obtain a unitary connection for E given in terms of local connection matrices by

A^{τ} = ad^{−1}(Â^{τ}) + \frac{1}{2}A_{e}^{τ} id_{ℂ^{2}} ∈ Ω^{1}(U, 𝔲(2)).

The curvatures of these connection matrices are related by

F_{A^{τ}} = ad^{−1}(FÂ^{τ}) + \frac{1}{2}F_{A_{e}^{τ}} id_{ℂ^{2}} ∈ Ω^{2}(U, 𝔲(2)),

with FÂ^{τ} ∈ Ω^{2}(U, 𝔰𝔬(3)) and F_{A_{e}^{τ}} ∈ Ω^{2}(U, 𝔲(1)). Thus, (F_{A^{τ}})_{0} = ad^{−1}(FÂ^{τ}) and tr F_{A^{τ}} = F_{A_{e}^{τ}} = F (tr A^{τ}). Therefore, globally we have (F_{A})_{0} = ad^{−1}FÂ ∈ Ω^{2}(X, 𝔰𝔲(E)) and tr F_{A} = F_{A_{e}} ∈ Ω^{2}(X, 𝔲(1)) = Ω^{2}(X, iℝ), so that

F_{A} = ad^{−1}(FÂ) + \frac{1}{2}F_{A_{e}} id_{E} ∈ Ω^{2}(X, 𝔲(E)),

as 𝔲(E) = 𝔰𝔲(E) ⊕ (iℝ) id_{E}.

When we are not explicitly discussing connections which are reducible with respect to some splitting of E, it is generally more convenient to view our connections as being defined on 𝔰𝔲(E) rather than E as we can then avoid explicit mention of the otherwise unimportant choice of fixed connection on det E. Of course, these viewpoints are equivalent via the choice of this fixed determinant connection. The isomorphism ad : 𝔰𝔲(E) → 𝔰𝔬(𝔰𝔲(E)) will remain implicit when no confusion can arise so that, given a connection A on 𝔰𝔲(E), we write its curvature as F_{A} ∈ Ω^{2}(𝔰𝔲(E)) and associated deformation complexes on Ω^{•}(𝔰𝔲(E)) rather than F_{A} ∈ Ω^{2}(𝔰𝔬(𝔰𝔲(E))), with deformation complexes on Ω^{•}(𝔰𝔬(𝔰𝔲(E))) (see [30, Chapter 10], for comparison).

2.1.2. Spin^{c} structures. The minimal, axiomatic approach to the definition of spin^{c} structures and the Dirac operator employed by Kronheimer-Mrowka [49] and Mrowka-Ozsváth-Yu [66] is extremely useful for our purposes, so this is the method we shall follow here.

Recall that a real-linear map ρ_{+} : T^{∗}X → Hom(W^{+}, W^{−}) defines a Clifford map ρ : Λ^{•}(T^{∗}X) ⊗ ℂ → End(W), with W := W^{+} ⊕ W^{−}, if and only if [57], [77]

ρ_{+}(α)^{†}ρ_{+}(α) = g(α, α)id_{W^{+}}, α ∈ C^{∞}(T^{∗}X), (2.1)

where g denotes the Riemannian metric on T^{∗}X. The real-linear map ρ : T^{∗}X → End(W^{+} ⊕ W^{−}) is obtained by defining a real-linear map

ρ_{−} : T^{∗}X → Hom(W^{−}, W^{+}), α ↦ ρ_{−}(α) := −ρ_{+}(α)^{†},

and setting

ρ(α) := (\begin{matrix}0 & ρ_{−}(α) \\ ρ_{+}(α) & 0 \end{matrix}) . (2.2)

This extends to a linear map ρ : Λ^{•}(T^{∗}X) ⊗ ℂ → End(W^{+} ⊕ W^{−}) and satisfies

\begin{matrix}ρ(α)^{†} = −ρ(α) and ρ(α)^{†}ρ(α) = g(α, α)id_{W} , \\ α ∈ C^{∞}(T^{∗}X). \end{matrix} (2.3)

A unitary connection ∇ on W and g-compatible Clifford map ρ induce a unique SO(4) connection ∇^{g} on T^{∗}X by requiring that

[∇_{η}, ρ(α)] = ρ(∇_{η}^{g}α), (2.4)

for all η ∈ C^{∞}(T X) and α ∈ Ω^{1}(X, ℝ); the sign above is opposite to that used in [66]. The unitary connection ∇ on W uniquely determines a unitary connection on det W^{+} ≃ det W^{−} in the standard way [43]. Conversely, the preceding data uniquely determines a unitary connection ∇ on W. The connection ∇ on W is called a spin^{c} connection if the connection ∇^{g} on T^{∗}X is also torsion free, that is, if ∇^{g} is the Levi-Civita connection for the metric g.

Given a unitary connection A on an auxiliary Hermitian two-plane bundle E, we let ∇_{A} denote the induced unitary connection on W ⊗ E. The corresponding Dirac operator D_{A} is defined by

D_{A} := ∑_{μ=1}^{4} ρ(v^{μ})∇_{A,v_{μ}},

where {v_{μ}} is a local frame for T X and {v^{μ}} is the dual frame for T^{∗}X defined by v^{μ}(v_{μ}) = δ_{μν}.

2.1.3. Pre-configuration spaces and automorphism groups We denote Λ^{•} := Λ^{•}(T^{∗}X). For any fixed L_{k}^{2} connection A_{0} ∈ A_{E} we therefore write

A_{E} = A_{0} + L_{k}^{2}(Λ^{1} ⊗ 𝔰𝔲(E)),

for either fixed-determinant, unitary connections on E or determinantone, orthogonal connections on 𝔰𝔲(E). Our pre-configuration space of pairs is given by

C̃_{W,E} := A_{E} × L_{k}^{2}(W^{+} ⊗ E),

and for any fixed L_{k}^{2} pair (A_{0}, Φ_{0}) ∈ C̃_{W,E}, we have

C̃_{W,E} = (A_{0}, Φ_{0}) + L_{k}^{2}(Λ^{1} ⊗ 𝔰𝔲(E)) ⊕ L_{k}^{2}(W^{+} ⊗ E), (2.5)

for the cases of either fixed-determinant, unitary connections on E or determinant-one, orthogonal connections on 𝔰𝔲(E).

Given any C^{∞} connection A_{0} ∈ A_{E}, our Sobolev norms are defined in the usual way: for example, if a ∈ Ω^{1}(𝔰𝔲(E)), we write

∥a∥_{L_{ℓ,A_{0}}^{p}(X)} := (∑_{j=0}^{ℓ} ∥∇_{A_{0}}^{j}a∥_{L^{p}(X)}^{p})^{1/p} ,

and if (a, ϕ) ∈ Ω^{1}(𝔰𝔲(E)) ⊕ Ω^{0}(W^{+} ⊗ E), we write

∥(a, ϕ)∥_{L_{ℓ,A_{0}}^{p}(X)} := (∥a∥_{L_{ℓ,A_{0}}^{p}(X)}^{p} + ∥ϕ∥_{L_{ℓ,A_{0}}^{p}(X)}^{p})^{1/p} ,

for any 1 ≤ p ≤ ∞ and integer ℓ ≥ 0.

For convenience, we let iℝ_{Z} = (iℝ) id_{ℂ^{2}} ⊂ 𝔲(2) denote the center of the Lie algebra 𝔲(2), and let S_{Z}^{1} = exp(iℝ_{Z}) ⊂ U(2) denote the center Z(U(2)) of the Lie group U(2) given by

iℝ_{Z} = \{(\begin{matrix}iθ & 0 \\ 0 & iθ \end{matrix}) : θ ∈ ℝ\} , S_{Z}^{1} = \{(\begin{matrix}e^{iθ} & 0 \\ 0 & e^{iθ} \end{matrix}) : θ ∈ ℝ\} . (2.6)

Following [48, §2], we consider the induced action for connections on 𝔰𝔲(E) by the group G_{E} of determinant-one, unitary L_{k+1}^{2} automorphisms of E rather than the group G_{𝔰𝔲(E)} of determinant-one, orthogonal L_{k+1}^{2} automorphisms of 𝔰𝔲(E) to define quotient spaces of connections on 𝔰𝔲(E). The reasons for this choice are explained further in [25]; see also §2.3. We have

G_{E} := {u ∈ L_{k+1}^{2}(𝔤𝔩(E)) : u^{†}u = id_{E} and det u = 1 a.e.},

and

∘G_{E} := S_{Z}^{1} ×_{{±id_{E}}} G_{E}.

One can define the action of ^{∘}G_{E} on A_{E} either by push-forward or pullback, and obtain equivalent quotient spaces in either case; but the choice does affect the orientation of moduli spaces, so we specify the action here: for u ∈ ^{∘}G_{E} and (A, Φ) ∈ C̃_{W,E}, the action of ^{∘}G_{E} is defined by

u(A, Φ) := (u(A), uΦ) = (u_{∗}A, uΦ) = ((u^{−1})^{∗}A, uΦ).

The push-forward action, u(A) = u_{∗}A, agrees with the conventions of [9], [20]. For the associated covariant derivative ∇_{A} on E and u ∈ Aut E, we have

∇_{u(A)} = u ∘ ∇_{A} ∘ u^{−1},

so that u(A) = A − (∇_{A}u)u^{−1}. In terms of a local connection matrix A^{τ} ∈ Ω^{1}(U, 𝔲(2)), this gives u(A^{τ}) = uA^{τ} u^{−1} −(du)u^{−1}, where we use u to denote both the gauge transformation and the element of Ω^{0}(U, U(2)).

In order to define quotients by the action of ^{∘}G_{E}, we need to choose k ≥ 2, so gauge transformations are at least continuous. The proof of the following proposition is a standard application of the Sobolev multiplication theorem; see [30, Propositions A.2, A.3, A.9].

Proposition 2.1. Let X be a closed Riemannian four-manifold, E be a Hermitian vector bundle over X, and k ≥ 2 be an integer. Then the following hold:

(1) The space ^{∘}G_{E} is a Hilbert Lie group with Lie algebra T_{id}^{∘}G_{E} = L_{k+1}^{2}(𝔰𝔲(E)) ⊕ iℝ_{Z}.

(2) The action of ^{∘}G_{E} on C̃_{W,E} is smooth.

(3) For (A, Φ) ∈ C̃_{W,E}, the differential, at the identity id_{E} ∈ ^{∘}G_{E}, of the map ^{∘}G_{E} → C̃_{W,E} given by u ↦ u(A, Φ) = (A − (d_{A}u)u^{−1}, uΦ) is ζ ↦ −d_{A,Φ}^{0}ζ := (−d_{A}ζ, ζΦ) as a map

L_{k+1}^{2}(𝔰𝔲(E)) ⊕ iℝ_{Z} → L_{k}^{2}(Λ^{1} ⊗ 𝔰𝔲(E)) ⊕ L_{k}^{2}(W^{+} ⊗ E).

We denote B_{E} = A_{E}/G_{E} = A_{E}/^{∘}G_{E}. The configuration space of pairs is given by C_{W,E} := C̃_{W,E}/^{∘}G_{E}, that is,

C_{W,E} := (A_{E} × L_{k}^{2}(W^{+} ⊗ E)) /^{∘}G_{E} = A_{E} ×_{^{∘}G}_{E} L_{k}^{2}(W^{+} ⊗ E),

and is endowed with the quotient L_{k}^{2} topology.

If π denotes the projection C̃_{W,E} → C_{W,E}, a base for the quotient L_{k}^{2} topology of C_{W,E} is given by open subsets π^{−1}π(B_{(A,Φ)}(ε)) ⊃ B_{(A,Φ)}(ε), where B_{(A,Φ)}(ε) is the L_{k}^{2}-ball of radius ε and center (A, Φ) in C̃_{W,E} given by {(A_{1}, Φ_{1}) ∈ C̃_{W,E} : ∥(A_{1}, Φ_{1}) − (A, Φ)∥_{L_{k,A}^{2}} < ε}.

2.1.4. Stabilizers. The space C_{W,E} is not a manifold — it has singularities at points [A, Φ] with non-trivial stabilizer Stab_{A,Φ} ⊂ ^{∘}G_{E}. Recall that the stabilizer subgroup (of the group of bundle automorphisms) for a connection on a G bundle always contains the center Z(G) ⊂ G [20, §4.2.2]. We let S_{Z}^{1} ⊂ ^{∘}G_{E} denote the constant, central automorphisms of E.

Definition 2.2. Suppose (A, Φ) is a point in C̃_{W,E}. The stabilizer Stab_{A,Φ} ⊂ ^{∘}G_{E} of (A, Φ) in ^{∘}G_{E} is given by {γ ∈ ^{∘}G_{E} : γ(A, Φ) = (A, Φ)}, while the stabilizers of A and Φ are denoted by Stab_{A} and Stab_{Φ}, respectively.

(1) The point (A, Φ) is a zero-section pair if Φ ≡ 0;

(2) The point (A, Φ) is an irreducible pair if the connection A has minimal stabilizer Stab_{A} = Z(U(2)) = S_{Z}^{1} ⊂ ^{∘}G_{E} and is reducible otherwise, that is, if Stab_{A} ⊋ S_{Z}^{1}.

The point (A, Φ) is a reducible, zero-section pair if (A, Φ) is both a reducible and zero-section pair.

As usual, the stabilizer subgroup Stab_{A} ⊂ ^{∘}G_{E} may be identified with a closed subgroup of U(E_{x_{0}}) ≃ U(2) for any point x_{0} ∈ X by parallel translation with respect to the connection A [20, §4.2.2], [61]. The following lemma implies that the stabilizer in Aut E of a unitary connection on E coincides with its stabilizer on ^{∘}G_{E}.

Lemma 2.3. Let E be a Hermitian two-plane bundle over a connected four-manifold X, let A be a unitary connection on E, and let u be a unitary automorphism of E such that u ∈ Stab_{A} ⊂ Aut E. Then det u : X → S^{1} is a constant map.

Proof. The gauge transformation u may be viewed as an Ad-equivari-ant map u : P → U(2), where E = P ×_{U(2)} ℂ^{2}, so det u may be viewed as a map from P to S^{1}. Since det u is constant on the fibers of P, it descends to a map on X. Over a small enough open set U ⊂ X, we may write u = exp ζ, where ζ ∈ Ω^{0}(adP), and adP := P ×_{ad} 𝔤 ≃ 𝔲(E), with G = U(2) and 𝔤 = 𝔲(2). Differentiating the action of u on A we see that ∇_{A}ζ = 0, so that ζ is covariantly constant over U. Thus det u = exp(tr ζ) and ∇(det u) = tr(∇_{A}ζ) = 0, so det u is constant on U and therefore on X, since X is connected. q.e.d.

For a pair (A, Φ) on (E, W^{+} ⊗ E) or (𝔰𝔲(E), W^{+} ⊗ E), the Lie algebras of Stab_{A} and Stab_{A,Φ} are given by

H_{A}^{0} := Ker{d_{A} : L_{k+1}^{2}(𝔰𝔲(E)) ⊕ iℝ_{Z}
→ L_{k}^{2}(Λ^{1} ⊗ 𝔰𝔲(E))},
H_{A,Φ}^{0} := Ker{d_{A,Φ}^{0} : L_{k+1}^{2}(𝔰𝔲(E)) ⊕ iℝ_{Z}
→ L_{k}^{2}(Λ^{1} ⊗ 𝔰𝔲(E)) ⊕ L_{k}^{2}(Λ^{1} ⊗ W^{+} ⊗ E)}.

The first identification is standard (see [20, §4.2.2], [30, Chapter 3], or [61]) and the second follows by the same argument.

From [20, §4.2.2] and [30, Theorem 10.8] we have the following characterization of the stabilizer Stab_{B} ⊂ G_{V} = Aut V of a connection B on an SO(3) bundle V. Note that the four-manifold X below need not be simply-connected or closed and that H_{B}^{0} = Ker{d_{B} : L_{k+1}^{2}(𝔰𝔬(V)) → L_{k}^{2}(Λ^{1} ⊗ 𝔰𝔬(V))}.

Lemma 2.4. Let B be an orthogonal connection on an SO(3) bundle V over a connected smooth four-manifold X. The following are equivalent provided B is not flat:

(1) Stab_{B} ≃ SO(2) ≃ S^{1};

(2) H_{B}^{0} ≠ 0;

(2) B is reducible with respect to an orthogonal splitting V = N ⊕ \underline{ℝ}, where N is a complex line bundle over X and \underline{ℝ} = X × ℝ (that is, B reduces to an SO(2) connection);

(4) Stab_{B} ≠ {id_{V} }.

Finally, B is flat if Stab_{B} = SO(3).

Remark 2.5. Note that the ‘twisted reducible’ (that is, locally reducible) connections on 𝔰𝔲(E) discussed by Kronheimer and Mrowka in [48, §2(i)] are globally irreducible in the sense that they have minimal stabilizer {id_{𝔰𝔲(E)}}. The condition that B is not flat is used in the proof that (4) implies (1) [30, p. 48]: the stabilizer Stab_{B} is isomorphic to the centralizer of the holonomy group Hol_{B} so, if Stab_{B} ⊋ S^{1}, then Hol_{B} ⊊ S^{1} and hence Hol_{B} is discrete. Since the Lie algebra of Hol_{B} vanishes, the Ambrose-Singer holonomy theorem implies that the connection B is flat. In particular, we only need X to be a connected four-manifold for these arguments to hold.

As customary, we call an SO(3) connection B on V irreducible if Stab_{B} = {id_{V} }. We say that a unitary connection A on E is projectively flat if the induced connection A^{ad} on 𝔰𝔲(E) is flat. By modifying the proofs of Theorem 3.1 in [30] and Proposition II.8.10 [56], we obtain the following relation between (i) the stabilizers Stab_{A} in ^{∘}G_{E} of U(2) connections A on E or their induced SO(3) connections A^{ad} on 𝔰𝔲(E) and (ii) the stabilizers Stab_{A^{ad}} in G_{𝔰𝔲(E)} of SO(3) connections A^{ad} on 𝔰𝔲(E).

Lemma 2.6. Let A be a U(2) connection on a Hermitian two-plane bundle E over a smooth, connected four-manifold X, and let A^{ad} be the induced SO(3) connection on the bundle 𝔰𝔲(E). Then the following are equivalent:

(1) Stab_{A^{ad}} ≃ S^{1};

(2) Stab_{A} ≃ S^{1} × S_{Z}^{1};

(3) H_{A}^{0} ≃ iℝ ⊕ iℝ_{Z};

(4) H_{A,Φ}^{0} ⊋ iℝ_{Z};

(5) A is reducible with respect to an orthogonal splitting E = L_{1} ⊕ L_{2}, where L_{1} and L_{2} are complex line bundles over X (that is, A reduces to a T^{2} = S^{1} × S^{1} connection);

(6) Stab_{A} ⊋ S_{Z}^{1}.

Finally, the connection A on E is projectively flat (or the connection A^{ad} on 𝔰𝔲(E) is flat) if Stab_{A} = U(2).

Proof. (1) ⟹ (2): Let π : U(2) → SO(3) = U(2)/S_{Z}^{1} be the projection u ↦ ±(det u)^{−1/2}u. Then π^{−1}(Stab_{A^{ad}}) ≃ Stab_{A^{ad}} ×S_{Z}^{1} and so Stab_{A^{ad}} ≃ S^{1} implies that Stab_{A} ≃ S^{1} × S_{Z}^{1}.

(2) ⟹ (3): Immediate from the identification of H_{A}^{0} as the Lie algebra of Stab_{A}.

(3) ⟹ (4): Trivial.

(4) ⟹ (5): We argue as in the proof of Theorem 3.1 in [30]. Note that Ker{d_{A} : L_{k+1}^{2}(X, 𝔲(E)) → Ω^{1}(X, 𝔲(E))} is isomorphic to

Ker{d_{A} : L_{k+1}^{2}(X, 𝔰𝔲(E))} ⊕ Ker{d : L_{k+1}^{2}(X, iℝ_{Z})}
= Ker{d_{A^{ad}} : L_{k+1}^{2}(X, 𝔰𝔲(E))} ⊕ iℝ_{Z}.

Given (4), we may choose 0 ≢ ζ ∈ Ker{d_{A} : L_{k+1}^{2}Ω^{0}(X, 𝔰𝔲(E))}. The pointwise traceless, skew-Hermitian endomorphism ζ of E has eigenvalues iλ_{1}, iλ_{2} = ±iλ for some 0 ≢ λ ∈ L_{k+1}^{2}(X, ℝ). On the open subset of X where λ ≠ 0, let ξ_{j}, j = 1, 2, be L_{k+1}^{2} eigenvectors of ζ such that ζ(ξ_{j}) = iλ_{j}ξ_{j} and ⟨ξ_{j}, ξ_{k}⟩ = δ_{jk}. Just as in [30, p. 47], we find that λ is constant, the eigenvectors ξ_{j} are globally defined and d_{A}ξ_{j} = 0 for j = 1, 2. In particular, we have an orthogonal splitting E = L_{1} ⊕ L_{2}, where ξ_{j} ∈ L_{k+1}^{2}(X, E) is a section of L_{j} and d_{A} = d_{A_{1}} ⊕ d_{A_{2}} with respect to this splitting, so this gives (5).

(5) ⟹ (6): Given (5), the connection A reduces to A_{1} ⊕ A_{2} with respect to the splitting E = L_{1} ⊕ L_{2} and so has stabilizer S_{L_{1}}^{1} × S_{L_{2}}^{1}, where we identify the constant maps in Map^{k+1}(X, S^{1}) with S_{L_{i}}^{1} for i = 1, 2. Since S_{L_{1}}^{1} × S_{L_{2}}^{1} ≃ S^{1} × S_{Z}^{1}, this gives (2).

(6) ⟹ (1): Given u ∈ Stab_{A} and u ∉ S_{Z}^{1}, we obtain u_{1} = ±(det u)^{−1/2}u ∈ Stab_{A^{ad}}. (By Lemma 2.3, the determinant det u is a constant map.) If u_{1} = id_{𝔰𝔲(E)} then we would have u_{1} = ±id_{E} and u = ±(det u)^{1/2}u_{1} ∈ S_{Z}^{1}, so u_{1} ≠ id_{𝔰𝔲(E)}. Therefore, Stab_{A^{ad}} ⊋ {id_{𝔰𝔲(E)}} and Lemma 2.4 implies that Stab_{A^{ad}} ≃ S^{1}.

Lastly, A is projectively flat if Stab_{A^{ad}} = SO(3) (by Lemma 2.4) and Stab_{A^{ad}} = SO(3) ≃ U(2)/S_{Z}^{1} if and only if Stab_{A} = U(2). q.e.d.

The equivalence of (1) and (2) above can alternatively be seen by noting that H_{A^{ad}}^{0} ≃ iℝ if and only if H_{A}^{0} ≃ iℝ ⊕ iℝ_{Z}, so Stab_{A^{ad}} ≃ S^{1} if and only if Stab_{A} ≃ S^{1} × S_{Z}^{1}.

From Lemmas 2.4 and 2.6 we see that A is an irreducible U(2) connection if and only if A^{ad} is an irreducible SO(3) connection. Note that Stab_{A,Φ} = Stab_{A} ∩ Stab_{Φ}. The stabilizer Stab_{A,Φ} of a zero-section pair (A, Φ) contains S_{Z}^{1}. The stabilizer Stab_{A,Φ} of a reducible pair (A, Φ) need not contain the stabilizer Stab_{A} since Stab_{A} may not fix Φ.

We write C̃_{W,E}^{∗} (respectively, C̃_{W,E}^{0}) for the complement of the reducible pairs (respectively, zero section pairs) in C̃_{W,E}, and let C̃_{W,E}^{∗,0} = C̃_{W,E}^{∗} ∩ C̃_{W,E}^{0}. The quotients C_{W,E}^{∗}, C_{W,E}^{0}, and C_{W,E}^{∗,0} are similarly defined.

If (A, Φ) ∈ C̃_{W,E}^{∗,0}, then Stab_{A,Φ} = {id_{E}}, as the stabilizer of an irreducible connection A will be S_{Z}^{1} and if S_{Z}^{1} stabilizes the section Φ then e^{iθ}Φ = 0 for all θ ∈ ℝ and so Φ ≡ 0 ∈ L_{k}^{2}(X, W^{+} ⊗E) (note that Φ need not be continuous). Therefore, ^{∘}G_{E} acts freely on C̃_{W,E}^{∗,0} and, as we shall see in the next section, the quotient C_{W,E}^{∗,0} is a Banach manifold. Conversely, if (A, Φ) is a PU(2) monopole, A is reducible, and Φ ≢ 0, then Lemma 5.22 implies that Stab_{A,Φ} ≃ S^{1}.

2.2. A slice for the action of the group of gauge transformations. Let k ≥ 2 be an integer. The slice S_{A,Φ} ⊂ C̃_{W,E} through a pair (A, Φ) is given by S_{A,Φ} := (A, Φ) + K_{A,Φ}, where

K_{A,Φ} := Ker d_{A,Φ}^{0,∗} ⊂ L_{k}^{2}(Λ^{1} ⊗ 𝔰𝔲(E)) ⊕ L_{k}^{2}(W^{+} ⊗ E). (2.7)

If π is the projection from C̃_{W,E} onto C_{W,E} = C̃_{W,E}/^{∘}G_{E}, denoted by (A, Φ) ↦ [A, Φ], we let

B_{A,Φ}(ε) = π^{−1}B_{A,Φ}(ε) ∩ S_{A,Φ}

be the open L_{k}^{2} ball in S_{A,Φ} with center (A, Φ) and radius ε, so that

B_{A,Φ}(ε) := {(A_{1}, Φ_{1}) ∈ S_{A,Φ} : ∥(A_{1}, Φ_{1}) − (A, Φ)∥_{L_{k,A}^{2}} < ε}
= (A, Φ) + {(a, ϕ) ∈ K_{A,Φ} : ∥(a, ϕ)∥_{L_{k,A}^{2}} < ε}.

The Hilbert Lie group ^{∘}G_{E} has Lie algebra

L_{k+1}^{2}(𝔰𝔲(E)) ⊕ iℝ_{Z} ⊂ L_{k+1}^{2}(𝔰𝔲(E))

and exponential map exp : L_{k+1}^{2}(𝔰𝔲(E))⊕iℝ_{Z} → ^{∘}G_{E} given by ζ ↦ u = exp ζ. Let

Stab_{A,Φ}^{⊥} := exp((Ker d_{A,Φ}^{0})^{⊥}) ⊂ ^{∘}G_{E},

where

(Ker(d_{A,Φ}^{0}|_{L_{k+1}^{2}}))^{⊥} = Im(d_{A,Φ}^{0,∗}|_{L_{k+2}^{2}}) ⊂ L_{k+1}^{2}(𝔰𝔲(E)),

noting that

d_{A,Φ}^{0,∗} : L_{k+2}^{2}Ω^{1}(𝔰𝔲(E)) → L_{k+1}^{2}(𝔰𝔲(E)) ⊕ L_{k}^{2}(W^{+} ⊗ E)

has closed range. Recall that Stab_{A,Φ} ⊂ ^{∘}G_{E} is given by

{γ ∈ ^{∘}G_{E} : γ(A, Φ) = (A, Φ)}

and has Lie algebra

H_{A,Φ}^{0} = Ker(d_{A,Φ}^{0}|_{L_{k+1}^{2}}) ⊂ L_{k+1}^{2}(𝔰𝔲(E)) ⊕ iℝ_{Z},

so that

L_{k+1}^{2}(𝔰𝔲(E)) ⊕ iℝ_{Z} = (Ker(d_{A,Φ}^{0}|_{L_{k+1}^{2}}))^{⊥} ⊕ H_{A,Φ}^{0}. (2.8)

The subspace Stab_{A,Φ}^{⊥} ⊂ ^{∘}G_{E} is closed and is a Banach submanifold of ∘G_{E} with codimension dim H_{A,Φ}^{0}.

The map d_{A,Φ}^{0} : L_{k+1}^{2}(𝔰𝔲(E))⊕iℝ_{Z} → L_{k}^{2}(Λ^{1} ⊗𝔰𝔲(E))⊕L_{k}^{2}(W^{+} ⊗E) has closed range and so we have an L^{2}-orthogonal decomposition

T_{A,Φ}C̃_{W,E} = Im(d_{A,Φ}^{0}|_{L_{k+1}^{2}}) ⊕ K_{A,Φ} (2.9)

of the tangent space to the space of L_{k}^{2} pairs at the point (A, Φ).

The proof that the quotient space C_{W,E} is Hausdorff and our later proof of removable singularities for PU(2) monopoles make use of the following well-known technical result [20, Proposition 2.3.15], [30, Proposition A.5], [56, Theorem II.7.11]. Note that the space of L_{2}^{2} unitary automorphisms of E is neither a Hilbert Lie group nor does it act smoothly on the space of L_{2}^{2} unitary connections on E.

Lemma 2.7. Let E be a Hermitian bundle over a Riemannian manifold X and let k ≥ 2 be an integer. Suppose that {A_{α}} and {B_{α}} are sequences of L_{k}^{2} unitary connections on E, and that {u_{α}} is a sequence of unitary automorphisms of E such that u_{α}(A_{α}) = B_{α}. Then the following hold.

(1) The sequence {u_{α}} is in L_{k+1}^{2}.

(2) If {A_{α}} and {B_{α}} converge in L_{k}^{2} to limits A_{∞}, B_{∞}, then there is a subsequence {α′} ⊂ {α} such that {u_{α′}} converges in L_{k+1}^{2} to u_{∞} and B_{∞} = u_{∞}(A_{∞}).

The following slice result was established by Parker [70], but only for pairs which are neither zero-section nor reducible; see also [9]. It is, of course, the analogue of the usual result for the topology and manifold structure of the configuration space B_{E} = A_{E}/G_{E} of connections. The proof we give here is modelled on the corresponding arguments for connections given in [20, Proposition 2.3.4], [30, Theorems 3.2 & 4.4], and [56, Theorem II.10.4]. We will ultimately need a rather more involved version of this method in order to show that our gluing maps are diffeomorphisms, so we give the argument in the simpler model case below in some detail and establish some of the notation and conventions we will later require.

Proposition 2.8. Let X be a closed, oriented, Riemannian spin^{c} four-manifold, let E be a Hermitian two-plane bundle over X, and let k ≥ 2 be an integer. Then the following hold.

(1) The space C_{W,E} is Hausdorff.

(2) The subspace C_{W,E}^{∗,0} ⊂ C_{W,E} is open and is a C^{∞} Hilbert manifold with local parametrizations given by π : B_{A_{0},Φ_{0}}(ε) → C_{W,E}^{∗,0} for sufficiently small ε = ε(A_{0}, Φ_{0}, k).

(3) The projection π : C̃_{W,E}^{∗,0} → C_{W,E}^{∗,0} is a C^{∞} principal ^{∘}G_{E} bundle.

(4) For (A_{0}, Φ_{0}) ∈ C̃_{W,E}, the projection π : B_{A_{0},Φ_{0}}(ε)/ Stab_{A_{0},Φ_{0}} → C_{W,E} is a homeomorphism onto an open neighborhood of [A_{0}, Φ_{0}] ∈ C_{W,E} and a diffeomorphism on the complement of the set of points in B_{A_{0},Φ_{0}}(ε) with non-trivial stabilizer.

Proof. The stabilizer Stab_{A_{0},Φ_{0}} (which we can identify with a Lie subgroup of U(2)) acts freely on ^{∘}G_{E} and thus on the Hilbert manifold ∘G_{E} × S_{A_{0},Φ_{0}} by (u, A, Φ) ↦ γ · (u, A, Φ) = (uγ^{−1}, γ(A, Φ)), and so the quotient ^{∘}G_{E} ×_{Stab_{A_{0},Φ_{0}}} S_{A_{0},Φ_{0}} is again a Hilbert manifold. We define a smooth map

Ψ : ^{∘}G_{E} ×_{Stab_{A_{0},Φ_{0}}} S_{A_{0},Φ_{0}} → C̃_{W,E}, [u, A, Φ] ↦ u(A, Φ).

Our first task is to show that the map Ψ is a diffeomorphism onto its image upon restriction to a sufficiently small neighborhood ^{∘}G_{E} ×_{Stab_{A_{0},Φ_{0}}} B_{A_{0},Φ_{0}}(ε). Given δ > 0, let B_{id}(δ) be the ball

{u ∈ ^{∘}G_{E} : ∥u − id_{E}∥_{L_{k+1,A_{0}}^{2}} < δ},

and let B_{id}^{⊥}(δ) = B_{id}(δ) ∩ Stab_{A_{0},Φ_{0}}^{⊥}.

Claim 2.9. For small enough δ = δ(A_{0}, Φ_{0}, k), the ball B_{id}(δ) is diffeomorphic to an open neighborhood in B_{id}^{⊥}(δ)×Stab_{A_{0},Φ_{0}}, with inverse map given by (u_{0}, γ) ↦ u = u_{0}γ.

Proof. The differential of the multiplication map

Stab_{A_{0},Φ_{0}}^{⊥} × Stab_{A_{0},Φ_{0}} → ^{∘}G_{E}, (u_{0}, γ) ↦ u_{0}γ

at (id_{E}, id_{E}) is given by

Ker(d_{A_{0},Φ_{0}}^{0}|_{L_{k+1}^{2}})^{⊥} ⊕ H_{A_{0},Φ_{0}}^{0} → L_{k+1}^{2}(𝔰𝔲(E)) (ζ, χ) ↦ ζ + χ,

and is just the identity map with respect to the L^{2}-orthogonal decomposition (2.8) of the range. Hence, the Hilbert space implicit function theorem implies that there is a diffeomorphism from an open neighborhood of (id_{E}, id_{E}) onto an open neighborhood of id_{E} ∈ ^{∘}G_{E}. For small enough δ, we may suppose that if u ∈ B_{id}(δ), then u can be written uniquely as u = u_{0}γ with u_{0} ∈ B_{id}^{⊥}(δ) and γ ∈ Stab_{A_{0},Φ_{0}}. q.e.d.

Claim 2.10. For small enough ε = ε(A_{0}, Φ_{0}, k), the map Ψ is a diffeomorphism from ^{∘}G_{E} ×_{Stab_{A_{0},Φ_{0}}} B_{A_{0},Φ_{0}}(ε) onto its image in C̃_{W,E}.

Proof. We first restrict the map Ψ to a neighborhood B_{id}(δ)×_{Stab_{A_{0},Φ_{0}}} S_{A_{0},Φ_{0}}, which is diffeomorphic to the neighborhood B_{id}^{⊥}(δ) × S_{A_{0},Φ_{0}} in Stab_{A_{0},Φ_{0}}^{⊥} ×S_{A_{0},Φ_{0}} by Claim 2.9. The differential of the induced map

Ψ : Stab_{A_{0},Φ_{0}}^{⊥} ×S_{A_{0},Φ_{0}} → C̃_{W,E}, (u, A, Φ) ↦ u(A, Φ)

at (id_{E}, A_{0}, Φ_{0}) is given by

(DΨ)_{(id,A_{0},Φ_{0})} :T_{id} Stab_{A_{0},Φ_{0}}^{⊥} ⊕T_{A_{0},Φ_{0}}S_{A_{0},Φ_{0}} → T_{A_{0},Φ_{0}} C̃_{W,E},
(ζ, a, ϕ) ↦ −d_{A_{0},Φ_{0}}^{0}ζ + (a, ϕ),

where we recall that T_{A_{0},Φ_{0}}S_{A_{0},Φ_{0}} = K_{A_{0},Φ_{0}} and

T_{id} Stab_{A_{0},Φ_{0}}^{⊥} = (Ker(d_{A_{0},Φ_{0}}^{0}|_{L_{k+1}^{2}}))^{⊥} = Im(d_{A_{0},Φ_{0}}^{0,∗}|_{L_{k+2}^{2}}).

Using the L^{2}-orthogonal decomposition (2.9) of the range we see that the map

−d_{A_{0},Φ_{0}}^{0}⊕id_{E} : (Ker(d_{A_{0},Φ_{0}}^{0}|_{L_{k+1}^{2}}))^{⊥}⊕K_{A_{0},Φ_{0}} → Im(d_{A_{0},Φ_{0}}^{0}|_{L_{k+1}^{2}})⊕K_{A_{0},Φ_{0}}

given by (ζ, b, ψ) ↦ −d_{A_{0},Φ_{0}}^{0}ζ + (b, ψ) is a Hilbert space isomorphism. So, by the Hilbert space implicit function theorem, there are positive constants ε = ε(A_{0}, Φ_{0}, k) and δ = δ(A_{0}, Φ_{0}, k) and an open neighborhood U_{A_{0},Φ_{0}} ⊂ C̃_{W,E} such that the map

Ψ : B_{id}^{⊥}(δ) × B_{A_{0},Φ_{0}}(ε) → U_{A_{0},Φ_{0}}, (u, A, Φ) ↦ u(A, Φ)

gives a diffeomorphism from an open neighborhood of (id_{E}, A_{0}, Φ_{0}) onto an open neighborhood of (A_{0}, Φ_{0}). In particular, we obtain a map U_{A_{0},Φ_{0}} → Stab_{A_{0},Φ_{0}}^{⊥}, given by (A, Φ) ↦ u = u_{A,Φ}, such that

Ψ^{−1}(A, Φ) = (u, u^{−1}(A, Φ)) ∈ B_{id}^{⊥}(δ) × B_{A_{0},Φ_{0}}(ε)
⊂ Stab_{A_{0},Φ_{0}}^{⊥} ×B_{A_{0},Φ_{0}}(ε).

Hence, for any (A, Φ) ∈ U_{A_{0},Φ_{0}} there is a unique u ∈ B_{id}^{⊥}(δ) such that u^{−1}(A, Φ) − (A_{0}, Φ_{0}) ∈ K_{A_{0},Φ_{0}}:

d_{A_{0},Φ_{0}}^{0,∗}(u^{−1}(A_{1}, Φ_{1}) − (A_{0}, Φ_{0})) = 0. (2.10)

The neighborhood B_{A_{0},Φ_{0}}(ε) is Stab_{A_{0},Φ_{0}}-invariant: if (A, Φ) ∈ B_{A_{0},Φ_{0}}(ε) and γ ∈ Stab_{A_{0},Φ_{0}}, then

∥γ(A_{1}, Φ_{1}) − (A, Φ)∥_{L_{k,A_{0}}^{2}} = ∥(A_{1}, Φ_{1}) − γ^{−1}(A, Φ)∥_{L_{k,γ^{−1}(A_{0})}^{2}}
= ∥(A_{1}, Φ_{1}) − (A, Φ)∥_{L_{k,A_{0}}^{2}} < ε,

and

d_{A_{0},Φ_{0}}^{0,∗} (γ(A, Φ) − (A_{0}, Φ_{0})) = γ (d_{γ^{−1}(A_{0},Φ_{0})}^{0,∗} ((A, Φ) − (A_{0}, Φ_{0})))
= γ (d_{A_{0},Φ_{0}}^{0,∗} ((A, Φ) − (A_{0}, Φ_{0}))) = 0,

so γ(A, Φ) ∈ B_{A_{0},Φ_{0}}(ε).

The group ^{∘}G_{E} acts on ^{∘}G_{E} × S_{A_{0},Φ_{0}} by (u, A, Φ) ↦ (vu, A, Φ), and so gives a diffeomorphism

B_{id}(δ) × B_{A_{0},Φ_{0}}(ε) → B_{v}(δ) × B_{A_{0},Φ_{0}}(ε), (u, A, Φ) ↦ (vu, A, Φ),

and as this action commutes with the given action of Stab_{A_{0},Φ_{0}}, it descends to a diffeomorphism

B_{id}(δ) ×_{Stab_{A_{0},Φ_{0}}} B_{A_{0},Φ_{0}}(ε) → B_{v}(δ) ×_{Stab_{A_{0},Φ_{0}}} B_{A_{0},Φ_{0}}(ε),
[u, A, Φ] ↦ [vu, A, Φ],

for each v ∈ ^{∘}G_{E}. Consequently, the ^{∘}G_{E}-equivariant map

∘G_{E} ×_{Stab_{A_{0},Φ_{0}}} B_{A_{0},Φ_{0}}(ε) → C̃_{W,E}

is a diffeomorphism onto its image, as desired. q.e.d.

Plainly, [γ(A, Φ)] = [A, Φ] for each γ ∈ Stab_{A_{0},Φ_{0}} and (A, Φ) ∈ B_{A_{0},Φ_{0}}(ε) and hence, the projection π : B_{A_{0},Φ_{0}}(ε) → C_{W,E} factors through B_{A_{0},Φ_{0}}(ε)/ Stab_{A_{0},Φ_{0}}.

Claim 2.11. There is a positive constant δ = δ(A_{0}, Φ_{0}, k) with the following significance. If (A_{i}, Φ_{i}) ∈ B_{A_{0},Φ_{0}}(ε) for i = 1, 2 and there is a gauge transformation u ∈ B_{id}(δ) such that u(A_{1}, Φ_{1}) = (A_{2}, Φ_{2}), then u ∈ Stab_{A_{0},Φ_{0}}.

Proof. For small enough δ, Claim 2.9 implies that u ∈ B_{id}(δ) can be written uniquely as u = u_{1}γ with u_{1} ∈ B_{id}^{⊥}(δ) and γ ∈ Stab_{A_{0},Φ_{0}}. Thus, (A_{2}, Φ_{2}) = u_{1}γ(A_{1}, Φ_{1}) ∈ B_{A_{0},Φ_{0}}(ε). But the neighborhood B_{A_{0},Φ_{0}}(ε) is Stab_{A_{0},Φ_{0}}-invariant, so we also have γ(A_{1}, Φ_{1}) ∈ B_{A_{0},Φ_{0}}(ε). Therefore u_{1} = id_{E} by the uniqueness assertion of Claim 2.9 and so u = γ ∈ Stab_{A_{0},Φ_{0}}. This completes the proof of the claim. q.e.d.

Claim 2.11 shows that B_{A_{0},Φ_{0}}(ε)/ Stab_{A_{0},Φ_{0}} injects into the quotient C_{W,E} modulo the assumption that the gauge transformations are close to id_{E}. It remains to show that if (A_{i}, Φ_{i}) ∈ B_{A_{0},Φ_{0}}(ε) for i = 1, 2 and there is a gauge transformation u ∈ ^{∘}G_{E} such that u(A_{1}, Φ_{1}) = (A_{2}, Φ_{2}), then u is necessarily close to id_{E} and hence that B_{A_{0},Φ_{0}}(ε)/ Stab_{A_{0},Φ_{0}} injects into the quotient C_{W,E}.

Claim 2.12. For small enough ε(A_{0}, Φ_{0}), the projection

π : B_{A_{0},Φ_{0}}(ε)/ Stab_{A_{0},Φ_{0}} → C_{W,E}

is injective.

Proof. Suppose that (A_{i}, Φ_{i}) ∈ B_{A_{0},Φ_{0}}(ε) for i = 1, 2 and that [A_{1}, Φ_{1}] = [A_{2}, Φ_{2}] ∈ C_{W,E}, so that u(A_{1}, Φ_{1}) = (A_{2}, Φ_{2}) for some u ∈ ∘G_{E}.

Since u(A_{0}) = A_{0} − (d_{A_{0}}u)u^{−1}, we see that u ∈ Stab_{A_{0},Φ_{0}} if and only if d_{A_{0},Φ_{0}}^{0}u = (d_{A_{0}}u, −uΦ_{0}) = (0, −Φ_{0}) or, equivalently, if and only if d_{A_{0},Φ_{0}}^{0}(u − id_{E}) = 0, since id_{E} ∈ Stab_{A_{0},Φ_{0}}. Here, we view u ∈ L_{k+1}^{2}(𝔤𝔩(E)) via the isometric embedding ^{∘}G_{E} ⊂ L_{k+1}^{2}(𝔤𝔩(E)) and write

u − id_{E} = u_{0} − γ,

where u_{0} ∈ (Ker d_{A_{0},Φ_{0}}^{0})^{⊥} and γ ∈ Ker d_{A_{0},Φ_{0}}^{0}. Our first task is to estimate ∥u − id_{E}∥_{L_{3,A_{0}}^{2}} .

By assumption, d_{A_{0},Φ_{0}}^{0}γ = (d_{A_{0}}γ, γΦ) = 0, so d_{A_{0}}γ = 0 and γΦ_{0} = 0. Since u(A_{1}) ≡ A_{1} − (d_{A_{1}}u)u^{−1} = A_{2}, we have

A_{2}u = A_{1}u − d_{A_{1}}u = A_{1}u − d_{A_{0}}u − [A_{1} − A_{0}, u],

and therefore, using d_{A_{0}}id_{E} = 0 = d_{A_{0}}γ, we have

d_{A_{0}}u_{0} = d_{A_{0}}u = u(A_{1} − A_{0}) − (A_{2} − A_{0})u.

As d_{A_{0}}^{∗}(A_{i} − A_{0}) = (·Φ_{0})^{∗}(Φ_{i} − Φ_{0}), we obtain

d_{A_{0}}^{∗}d_{A_{0}}u_{0} = − ∗ (d_{A_{0}}u ∧ ∗(A_{1} − A_{0})) + ud_{A_{0}}^{∗}(A_{1} − A_{0})
− (d_{A_{0}}^{∗}(A_{2} − A_{0}))u + ∗(∗(A_{2} − A_{0}) ∧ d_{A_{0}}u)
= − ∗ (d_{A_{0}}u_{0} ∧ ∗(A_{1} − A_{0})) + u(·Φ_{0})^{∗}(Φ_{1} − Φ_{0})
− ((·Φ_{0})^{∗}(Φ_{2} − Φ_{0}))u + ∗(∗(A_{2} − A_{0}) ∧ d_{A_{0}}u_{0}).

We define the Laplacian Δ_{A_{0},Φ_{0}}^{0} by setting

Δ_{A_{0},Φ_{0}}^{0} = d_{A_{0},Φ_{0}}^{0,∗}d_{A_{0},Φ_{0}}^{0} = d_{A_{0}}^{∗}d_{A_{0}} + (·Φ_{0})^{∗}Φ_{0}.

Now u_{0} = u − id_{E} + γ and d_{A_{0}}id_{E} = 0 = d_{A_{0}}γ and γΦ_{0} = 0 so, using uΦ_{1} = Φ_{2}, we obtain

Δ_{A_{0},Φ_{0}}^{0}u_{0} = Δ_{A_{0},Φ_{0}}^{0}(u − id_{E} + γ)
= d_{A_{0}}^{∗}d_{A_{0}}(u − id_{E} + γ) + (·Φ_{0})^{∗}(u − id_{E})Φ_{0}
= d_{A_{0}}^{∗}d_{A_{0}}u + (·Φ_{0})^{∗}(uΦ_{0} − uΦ_{1} + uΦ_{1} − Φ_{0})
= d_{A_{0}}^{∗}d_{A_{0}}u_{0} + (·Φ_{0})^{∗} (u(Φ_{0} − Φ_{1}) + (Φ_{2} − Φ_{0})) .

Our assumption that (A_{i}, Φ_{i}) ∈ B_{A_{0},Φ_{0}}(ε) and the embeddings L_{k}^{2} ⊂ L_{2}^{2} imply that for i = 1, 2 we have

∥(A_{i}, Φ_{i}) − (A_{0}, Φ_{0})∥_{L_{2,A_{0}}^{2}} ≤ ∥(A_{i}, Φ_{i}) − (A_{0}, Φ_{0})∥_{L_{k,A_{0}}^{2}} < ε.

Since u_{0} ∈ (Ker d_{A_{0},Φ_{0}}^{0})^{⊥}, the standard elliptic estimate for the Laplacian Δ_{A_{0},Φ_{0}}^{0}, the fact that |u| = 1 as u ∈ ^{∘}G_{E}, the Sobolev embedding L_{2}^{2} ⊂ L^{q}, 2 ≤ q < ∞, and multiplication L_{2}^{2} × L_{2}^{2} → L_{1}^{2}, and our expression for d_{A_{0}}^{∗}d_{A_{0}}u = d_{A_{0}}^{∗}d_{A_{0}}u_{0} combine to give

∥u_{0}∥_{L_{3,A_{0}}^{2}} ≤ C∥Δ_{A_{0},Φ_{0}}^{0}u_{0}∥_{L_{1,A_{0}}^{2}}

≤ C∥d_{A_{0}}^{∗}d_{A_{0}}u_{0}∥_{L_{1,A_{0}}^{2}} + C∥(·Φ_{0})^{∗}(u(Φ_{0} − Φ_{1}) + (Φ_{2} − Φ_{0}))∥_{L_{1,A_{0}}^{2}}

≤ C∥d_{A_{0}}u_{0}∥_{L_{2,A_{0}}^{2}} (∥A_{1} − A_{0}∥_{L_{2,A_{0}}^{2}} + ∥A_{2} − A_{0}∥_{L_{2,A_{0}}^{2}} )

+ C∥u∥_{L_{2,A_{0}}^{2}} ∥Φ_{0}∥_{L_{2,A_{0}}^{2}} (∥Φ_{1} − Φ_{0}∥_{L_{2,A_{0}}^{2}} + ∥Φ_{2} − Φ_{0}∥_{L_{2,A_{0}}^{2}} )

≤ C∥u_{0}∥_{L_{3,A_{0}}^{2}} ε + C∥u∥_{L_{2,A_{0}}^{2}} ∥Φ_{0}∥_{L_{2,A_{0}}^{2}} ε
≤ C∥u_{0}∥_{L_{3,A_{0}}^{2}} ε + C∥Φ_{0}∥_{L_{2,A_{0}}^{2}} ε,

for some constant C = C(A_{0}, Φ_{0}). For small enough ε we can rearrange the preceding inequality and use the Sobolev embedding L_{3}^{2} ⊂ C^{0} to yield

∥u_{0}∥_{L_{3,A_{0}}^{2}} ≤ C∥Φ_{0}∥_{L_{2,A_{0}}^{2}} ε and ∥u_{0}∥_{C^{0}} ≤ C∥Φ_{0}∥_{L_{2,A_{0}}^{2}} ε.

Therefore, as |u| = 1, we have the pointwise bounds

1 − |u_{0}| ≤ |id_{E} − γ| ≤ 1 + |u_{0}|,

and thus, for ε small, |id_{E} − γ| > 0; the pointwise norm of z = |id_{E} − γ| is constant since d_{A_{0}}(id_{E} − γ) = 0. Consequently, γ_{0} ≡ z^{−1}(id_{E} − γ) lies in ^{∘}G_{E}: clearly, d_{A_{0}}γ_{0} = 0, so γ_{0} ∈ Stab_{A}. If Φ_{0} ≡ 0, we trivially have γ_{0} ∈ Stab_{Φ_{0}}. If Φ_{0} ≢ 0, then the equalities |γ_{0}Φ_{0}| = |Φ_{0}| (as γ_{0} is a unitary gauge transformation) and γ_{0}Φ_{0} = z^{−1}(id_{E} − γ)Φ_{0} = z^{−1}Φ_{0}, so |γ_{0}Φ_{0}| = z^{−1}|Φ_{0}| implying that z = 1 (it is enough to have equality of L^{2} norms here as z is constant). Hence, we also have γ_{0} ∈ Stab_{Φ_{0}} and in particular, γ_{0} ∈ Stab_{A_{0},Φ_{0}} = Stab_{A_{0}} ∩ Stab_{Φ_{0}}.

We now write γ_{0}^{−1}u = γ_{0}^{−1}u_{0} +γ_{0}^{−1}(id_{E} −γ) = γ_{0}^{−1}u_{0} +z id_{E}, so that

γ_{0}^{−1}u − id_{E} = (z − 1)id_{E} + γ_{0}^{−1}u_{0}.

Clearly, (z − 1)id_{E} ∈ Ker d_{A_{0},Φ_{0}}^{0} by the remarks of the last paragraph, while γ_{0}^{−1}u_{0} ∈ (Ker d_{A_{0},Φ_{0}}^{0})^{⊥} since u_{0} ∈ (Ker d_{A_{0},Φ_{0}}^{0})^{⊥} and γ_{0}^{−1} ∈ Stab_{A_{0},Φ_{0}}. Similarly, γ_{0}^{−1}u(A_{1}, Φ_{1}) = γ_{0}^{−1}(A_{2}, Φ_{2}) ∈ B_{A_{0},Φ_{0}}(ε), as the neighborhood B_{A_{0},Φ_{0}}(ε) is Stab_{A_{0},Φ_{0}}-invariant. Thus, replacing u by γ_{0}^{−1}u and (A_{2}, Φ_{2}) by (Ã_{2}, Φ̃_{2}) ≡ γ_{0}^{−1}(A_{2}, Φ_{2}) in our L_{3,A_{0}}^{2} estimate for u_{0} yields

∥γ_{0}^{−1}u_{0}∥_{L_{3,A_{0}}^{2}} ≤ C∥Φ_{0}∥_{L_{2,A_{0}}^{2}} ε,

while |z − 1| ≤ ∥u_{0}∥_{C^{0}} ≤ C∥Φ_{0}∥_{L_{2,A_{0}}^{2}} ε, so we find that

∥γ_{0}^{−1}u − id_{E}∥_{L_{3,A_{0}}^{2}} ≤ C∥Φ_{0}∥_{L_{2,A_{0}}^{2}} ε.

If ε is small enough that C∥Φ_{0}∥_{L_{2,A_{0}}^{2}} ε < δ, where δ = δ(A_{0}, Φ_{0}) is the

constant of Claim 2.11 with k = 2, then γ_{0}^{−1}u ∈ Stab_{A_{0},Φ_{0}} and so u lies in Stab_{A_{0},Φ_{0}}, as desired. q.e.d.

Claim 2.13. The quotient space C_{W,E} is Hausdorff.

Proof. Let Γ be the subspace {{(A, Φ), u(A, Φ)} : (A, Φ) ∈ C̃_{W,E} and u ∈ ^{∘}G_{E}} of C̃_{W,E} × C̃_{W,E}. If {(A_{α}, Φ_{α}), u_{α}(A_{α}, Φ_{α})} is a sequence in Γ which converges in L_{k}^{2} to a point {(A_{∞}, Φ_{∞}), (B_{∞}, Ψ_{∞})}, then Lemma 2.7 implies that there is a subsequence {α′} ⊂ {α} such that {u_{α}} converges in L_{k+1}^{2} to u_{∞} ∈ ^{∘}G_{E} and u_{∞}(A_{∞}) = B_{∞}. But then u_{α′}Φ_{α′} converges in L_{k}^{2} to u_{∞}Φ_{∞} and so Ψ_{∞} = u_{∞}Φ_{∞}. Thus, (B_{∞}, Ψ_{∞}) = u_{∞}(A_{∞}, Φ_{∞}) for some u_{∞} ∈ ^{∘}G_{E}, and therefore Γ is closed. Hence the quotient C̃_{W,E}/^{∘}G_{E} is Hausdorff. q.e.d.

Claim 2.13 gives Assertion (1) of the proposition. Thus Assertions (2), (3), and (4) now follow from the preceding arguments and Claim 2.12. This completes the proof of the proposition. q.e.d.

Remark 2.14. (1) Alternatively, one can show that the quotient L_{k}^{2} topology on C_{W,E} is metrizable via the L^{2} metric (exactly as in [20, Lemma 4.2.4]) and thus C_{W,E} is Hausdorff.

(2) As Mrowka pointed out to us, one can sharpen the assertions of Proposition 2.8, at least for the quotient space B_{E}: one finds that charts are provided by L^{4}-balls in Ker d_{A_{0}}^{∗} rather than the much smaller L_{k,A_{0}}^{2}-balls usually employed; see [22].

It is convenient to extract the following global, Coulomb gauge-fixing result (analogous to Proposition 2.3.4 in [20]) which we established in the course of proving Proposition 2.8:

Lemma 2.15. Let X be a closed, oriented, Riemannian spin^{c} fourmanifold and let E be a Hermitian vector bundle over X. Suppose that k ≥ 2 is an integer and that (A_{0}, Φ_{0}) ∈ C̃_{W,E}. Then there is a positive constant ε = ε(A_{0}, Φ_{0}, k) such that for any (A, Φ) ∈ C̃_{W,E} with

∥(A, Φ) − (A_{0}, Φ_{0})∥_{L_{k,A_{0}}^{2}(X)} < ε,

there is a gauge transformation u ∈ ^{∘}G_{E}, unique up to an element of Stab_{A_{0},Φ_{0}}, such that

d_{A_{0},Φ_{0}}^{0,∗}(u(A, Φ) − (A_{0}, Φ_{0})) = 0.

2.3. Connections on SO(3) bundles and groups of gauge transformations. For some local patching arguments over simplyconnected open regions Y ⊂ X in §3 and §4 it is very useful to be able to lift gauge transformations in G_{𝔰𝔲(E)}(Y) to gauge transformations in G_{E}(Y). The following result tells us that this is always possible when Y is simply connected; it is an extension of Theorem IV.3.1 in [61] from SU(2) to U(2) bundles.

Proposition 2.16. Let E be a Hermitian two-plane bundle over a connected manifold X. Then there is an exact sequence

1 → {±id_{E}} → G_{E} → G_{𝔰𝔲(E)} → H^{1}(X; ℤ/2ℤ).

Proof. Let P be the U(2) principal bundle underlying E so P^{ad} = P/S_{Z}^{1} is the SO(3) principal bundle underlying 𝔰𝔲(E). We can view elements of G_{E} as maps u : P → SU(2) satisfying u(pg) = g^{−1}u(p)g for p ∈ P and g ∈ U(2), and similarly for elements of G_{𝔰𝔲(E)}. The map G_{E} → G_{𝔰𝔲(E)} is then given by u → Adu. Because X is connected, the kernel of this map is {±id_{E}}, giving the exactness of all but the last terms in the sequence.

Let ν_{2} ∈ H^{1}(SO(3); ℤ/2ℤ) be the non-trivial cohomology class given by the double cover SU(2) → SO(3). Then a map u : V → SO(3) lifts to SU(2) if and only if u^{∗}ν_{2} = 0. We would like to define the map α : G_{𝔰𝔲(E)} → H^{1}(X; ℤ/2ℤ) by α(u) = u^{∗}ν_{2}. It is not immediately clear that α(u) actually lies in H^{1}(X; ℤ/2ℤ). The homotopy exact sequence

π_{1}(SO(3)) → π_{1}(P^{ad}) → π_{1}(X) → 1

gives an exact sequence in cohomology H^{1}(·; ℤ/2ℤ) = Hom(π_{1}(·), ℤ/2ℤ),

1 → H^{1}(X; ℤ/2ℤ) → H^{1}(P^{ad}; ℤ/2ℤ) → H^{1}(SO(3); ℤ/2ℤ).

Thus u^{∗}ν_{2} is pulled back from a unique element of H^{1}(X; ℤ/2ℤ) if and only if i_{x}^{∗}u^{∗}ν_{2} = 0, where i_{x} : SO(3) → P^{ad} is the inclusion of a fiber.

Because u^{∗}ν_{2} depends only on the homotopy class of u, the map α is constant on connected components. Let G_{𝔰𝔲(E)}^{x} be the subgroup of elements of G_{𝔰𝔲(E)} which are the identity over x ∈ X. The exact sequence of groups

1 → G_{𝔰𝔲(E)}^{x} → G_{𝔰𝔲(E)} → SO(3) → 1

and the corresponding exact sequence of homotopy groups (π_{0} to be specific) show that the inclusion G_{𝔰𝔲(E)}^{x} induces a surjection of connected components. Now if u ∈ G_{𝔰𝔲(E)}^{x} then i_{x}^{∗}u^{∗}ν_{2} = 0, and so i_{x}u^{∗}ν_{2} = 0 for all u ∈ G_{𝔰𝔲(E)}. This implies that the map α takes values in H^{1}(X; ℤ/2ℤ).

By the defining property of ν_{2}, we have α(u) = 0 if and only if there is a lift of u to SU(2). Because {±id_{E}} is central, any Ad-equivariant map ũ : P → SU(2) descends to P^{ad} → SU(2). Thus u ∈ G_{𝔰𝔲(E)} is induced by some ũ ∈ G_{E} if and only if α(u) = 0. q.e.d.

Remark 2.17. One can also show that the map

G_{𝔰𝔲(E)} → H^{1}(X; ℤ/2ℤ)

is surjective.

2.4. Quadratic forms for coupled spinor bundles. Give 𝔤𝔩(E) (respectively, 𝔤𝔩(W^{±})) the Hermitian fiber inner product and norm

⟨M, N⟩ := \frac{1}{2} tr(M^{†}N) and |M|^{2} = ⟨M, M⟩,

for M, N ∈ Ω^{0}(𝔤𝔩(E)) (respectively, 𝔤𝔩(W^{±})), so that |id_{E}| = 1, etc. If Φ ∈ Ω^{0}(W^{+} ⊗ E), then Φ^{∗} ∈ Ω^{0}((W^{+} ⊗ E)^{∗}) = Ω^{0}((W^{+})^{∗} ⊗ E^{∗}) is defined by Φ^{∗}(Ψ) := ⟨Ψ, Φ⟩ for Ψ ∈ Ω^{0}(W^{+} ⊗ E).

With respect to the induced Hermitian fiber inner products on 𝔤𝔩(W^{+}) and 𝔤𝔩(E), we have fiber-wise orthogonal direct sums, 𝔤𝔩(W^{+}) = ℂ id_{W^{+}} ⊕ 𝔰𝔩(W^{+}), and similarly for 𝔤𝔩(E) and thus 𝔤𝔩(W^{+}) ⊗_{ℂ} 𝔤𝔩(E). Let π_{W^{+}} (respectively, π_{E}) be the fiber-wise orthogonal projection from 𝔤𝔩(W^{+}) (respectively, 𝔤𝔩(E)) onto the subspace 𝔰𝔩(W^{+}) (respectively, 𝔰𝔩(E)), so that

π_{W^{+}}M := M − \frac{1}{2} tr M id_{W^{+}} and π_{E}N := N − \frac{1}{2} tr N id_{E}

are the projection onto the traceless components of M ∈ 𝔤𝔩(W^{+}) and N ∈ 𝔤𝔩(E), and

π_{W^{+}}^{⊥}M := \frac{1}{2} tr M id_{W^{+}} and π_{E}^{⊥}N := \frac{1}{2} tr N id_{E}

are the projections onto the trace components of M ∈ 𝔤𝔩(W^{+}) and N ∈ 𝔤𝔩(E). The orthogonal projection from 𝔤𝔩(W^{+}) ⊗_{ℂ} 𝔤𝔩(E) onto 𝔰𝔩(W^{+}) ⊗_{ℂ} 𝔰𝔩(E) is obtained by writing π_{W^{+}} = id_{W^{+}} − π_{W^{+}}^{⊥} and π_{E} = id_{E} − π_{E}^{⊥}, so that

\begin{matrix}π_{W^{+}} ⊗ π_{E} = & id_{W^{+}} ⊗ id_{E} − (π_{W^{+}} ⊗ π_{E}^{⊥} + π_{W^{+}}^{⊥} ⊗ π_{E}) \\ + π_{W^{+}}^{⊥} ⊗ π_{E}^{⊥}. \end{matrix} (2.11)

Consequently, if ϕ ∈ Ω^{0}(W^{+}) and Φ ∈ Ω^{0}(W^{+} ⊗ E), we may define

\begin{aligned}(ϕ ⊗ ϕ^{∗})_{0} & := π_{W^{+}}(ϕ ⊗ ϕ^{∗}), \\ (Φ ⊗ Φ^{∗})_{00} & := (π_{W^{+}} ⊗ π_{E})(Φ ⊗ Φ^{∗}). \end{aligned} (2.12)

Note that

𝔤𝔩(W^{+}) ⊗_{ℂ} 𝔤𝔩(E) = 𝔲(W^{+}) ⊗_{ℝ} 𝔲(E) + i𝔲(W^{+}) ⊗_{ℝ} 𝔲(E).

Plainly, (ϕ ⊗ ϕ^{∗})_{0} is a section of 𝔰𝔩(W^{+}) ∩ i𝔲(W^{+}) = i𝔰𝔲(W^{+}). The following identities are now well-known [47], [77, Chapter 8]:

Lemma 2.18. For every ϕ ∈ Ω^{0}(W^{+}) or Ω^{0}(E), T ∈ Ω^{0}(𝔰𝔩(W^{+})), and η ∈ Ω^{2}(X, ℂ), the following hold. The endomorphism (ϕ ⊗ ϕ^{∗})_{0} is a section of Ω^{0}(i𝔰𝔲(W^{+})) and satisfies:

(1)  |(ϕ ⊗ ϕ^{∗})_{0}|^{2} = \frac{1}{4}|ϕ|^{4},
(2)  |ρ(η)|^{2} = 2|η^{+}|^{2} and |ρ^{−1}T |^{2} = \frac{1}{2}|T |^{2}.

Similarly, Φ⊗Φ^{∗} is a section of i𝔲(W^{+} ⊗_{ℂ}E) = 𝔲(W^{+})⊗_{ℝ}𝔲(E) and (Φ ⊗ Φ^{∗})_{00} is a section of 𝔰𝔲(W^{+}) ⊗_{ℝ}𝔰𝔲(E). We then have the following analogue of Lemma 2.18:

Lemma 2.19. For every Φ ∈ Ω^{0}(W^{+} ⊗ E) and

M ∈ Ω^{0}(𝔤𝔩(W^{+} ⊗ E)),

the following identities hold:

(1)  ⟨Φ ⊗ Φ^{∗}, M⟩ = \frac{1}{2}⟨MΦ, Φ⟩,
(2)  (\frac{5}{4} − \frac{1}{\sqrt{2}})|Φ|^{4} ≤ ⟨(Φ ⊗ Φ^{∗})_{00}Φ, Φ⟩ ≤ \frac{1}{\sqrt{2}}|Φ|^{4},
(3)  (\frac{5}{8} − \frac{1}{2\sqrt{2}})|Φ|^{4} ≤ |(Φ ⊗ Φ^{∗})_{00}|^{2} ≤ \frac{1}{2\sqrt{2}}|Φ|^{4}.

Proof. Using Φ = c^{a}Φ^{a} (we employ the summation convention), so MΦ = c^{a}MΦ^{a} and Φ^{∗}Φ^{a} = ⟨Φ^{a}, Φ⟩ = c̄^{a}, we have

⟨Φ ⊗ Φ^{∗}, M⟩ = \frac{1}{2} tr((Φ ⊗ Φ^{∗})^{†}M) = \frac{1}{2}⟨(Φ ⊗ Φ^{∗})MΦ^{a}, Φ^{a}⟩
= \frac{1}{2}⟨MΦ^{a}, (Φ ⊗ Φ^{∗})Φ^{a}⟩ = \frac{1}{2}⟨MΦ^{a}, Φ(Φ^{∗}Φ^{a})⟩
= \frac{1}{2}⟨MΦ^{a}, c̄^{a}Φ⟩ = \frac{1}{2}⟨c^{a}MΦ^{a}, Φ⟩ = \frac{1}{2}⟨MΦ, Φ⟩,

which gives (1). Next, we see that

|Φ ⊗ Φ^{∗}|^{2} = ⟨Φ ⊗ Φ^{∗}, Φ ⊗ Φ^{∗}⟩ = \frac{1}{2}⟨(Φ ⊗ Φ^{∗})Φ, Φ⟩
= \frac{1}{2}|Φ|^{2}⟨Φ, Φ⟩ = \frac{1}{2}|Φ|^{4},

and so the upper bound in (2) is obtained by

|⟨(Φ ⊗ Φ^{∗})_{00}Φ, Φ⟩| ≤ |(Φ ⊗ Φ^{∗})_{00}||Φ|^{2} ≤ |Φ ⊗ Φ^{∗}||Φ|^{2}
≤ (1/\sqrt{2})|Φ|^{4} < \frac{3}{4}|Φ|^{4},

where we use the fact that ( ·)_{00} = π_{W^{+}} ⊗π_{E} is an orthogonal projection to obtain the second inequality.

To obtain the lower bound in (2), suppose that Φ = ϕ^{a} ⊗ξ^{a} (implied summation), where {ϕ^{a}} ⊂ Ω^{0}(W^{+}) and {ξ^{a}} ⊂ Ω^{0}(E), then Φ ⊗ Φ^{∗} = (ϕ^{a} ⊗ ϕ^{b,∗}) ⊗ (ξ^{a} ⊗ ξ^{b,∗}) and

(π_{W^{+}}^{⊥} ⊗ π_{E}^{⊥})(Φ ⊗ Φ^{∗}) = π_{W^{+}}^{⊥}(ϕ^{a} ⊗ ϕ^{b,∗}) ⊗ π_{E}^{⊥}(ξ^{a} ⊗ ξ^{b,∗})
= \frac{1}{4} tr(ϕ^{a} ⊗ ϕ^{b,∗}) tr(ξ^{a} ⊗ ξ^{b,∗}) id_{W^{+}⊗E}
= \frac{1}{4}⟨ϕ^{a}, ϕ^{b}⟩⟨ξ^{a}, ξ^{b}⟩ id_{W^{+}⊗E}
= \frac{1}{4}⟨ϕ^{a} ⊗ ξ^{a}, ϕ^{b} ⊗ ξ^{b}⟩ id_{W^{+}⊗E}
= \frac{1}{4}⟨Φ, Φ⟩ id_{W^{+}⊗E}.

Using the last identity together with (Φ ⊗ Φ^{∗})Φ = Φ⟨Φ, Φ⟩ = Φ|Φ|^{2} and (2.11) gives

⟨(Φ ⊗ Φ^{∗})_{00}Φ, Φ⟩ = ⟨(Φ ⊗ Φ^{∗})Φ, Φ⟩
− ⟨(π_{W^{+}}^{⊥} ⊗ π_{E} + π_{W^{+}} ⊗ π_{E}^{⊥})(Φ ⊗ Φ^{∗})Φ, Φ⟩
+ ⟨\frac{1}{4}|Φ|^{2}Φ, Φ⟩
= \frac{5}{4}|Φ|^{4} − ⟨(π_{W^{+}}^{⊥} ⊗ π_{E} + π_{W^{+}} ⊗ π_{E}^{⊥})(Φ ⊗ Φ^{∗})Φ, Φ⟩.

But π_{W^{+}}^{⊥} ⊗ π_{E} + π_{W^{+}} ⊗ π_{E}^{⊥} is just the orthogonal projection onto the middle two factors of the orthogonal decomposition of 𝔤𝔩(W^{+})⊗_{ℂ}𝔤𝔩(E), so

|⟨(π_{W^{+}}^{⊥} ⊗ π_{E} + π_{W^{+}} ⊗ π_{E}^{⊥})(Φ ⊗ Φ^{∗})Φ, Φ⟩|
≤ |(π_{W^{+}}^{⊥} ⊗ π_{E} + π_{W^{+}} ⊗ π_{E}^{⊥})(Φ ⊗ Φ^{∗})| |Φ|^{2}
≤ |Φ ⊗ Φ^{∗}||Φ|^{2} ≤ (1/\sqrt{2})|Φ|^{4} < \frac{3}{4}|Φ|^{4},

and therefore

|⟨(Φ ⊗ Φ^{∗})_{00}Φ, Φ⟩| ≥ \frac{5}{4}|Φ|^{4} − (1/\sqrt{2})|Φ|^{4} > \frac{1}{2}|Φ|^{4}.

Consequently, using (Φ ⊗ Φ^{∗})_{00}^{†} = (Φ ⊗ Φ^{∗})_{00}, we have

\frac{1}{2}⟨(Φ ⊗ Φ^{∗})_{00}Φ, Φ⟩ = ⟨(Φ ⊗ Φ^{∗})_{00}, Φ ⊗ Φ^{∗}⟩ = ⟨(Φ ⊗ Φ^{∗})_{00}, (Φ ⊗ Φ^{∗})_{00}⟩
= |(Φ ⊗ Φ^{∗})_{00}|^{2},

which gives (3). q.e.d.

The fact that |(Φ⊗Φ^{∗})_{0}|^{2} ≥ \frac{1}{4}|Φ|^{4}, for Φ ∈ Ω^{0}(W^{+}⊗E), is used in §4 to show that the moduli space of PU(2) monopoles has an Uhlenbeck compactification. (The analogous equality, |(ϕ ⊗ ϕ^{∗})_{0}|^{2} = \frac{1}{4}|ϕ|^{4} for ϕ ∈ Ω^{0}(W^{+}), is used in [47], [98] to show that the moduli space of Seiberg-Witten monopoles is compact.)

We adapt some terminology from the proofs of Proposition 3.4 in [30] and Lemma 4.3.25 in [20] which we will need for our proof of transversality for the PU(2) monopole moduli space in §5:

Definition 2.20. The rank of a section Φ ∈ Ω^{0}(W^{±} ⊗ E) at a point x ∈ X is the rank of Φ|_{x} considered as a complex linear map (W^{±})^{∗}|_{x} → E|_{x} (that is, its rank as a complex two-by-two matrix). We say that Φ is rank one on a subset U ⊂ X if it has rank less than or equal to one at all points in U. Similarly, for v ∈ Ω^{0}(Λ^{+}(T^{∗}X) ⊗ 𝔰𝔲(E)), the rank of v at a point x ∈ X is the rank of v considered as a real linear map from Λ^{+}(T X)|_{x} → 𝔰𝔲(E)|_{x} (that is, its rank as a real three-by-three matrix). The rank of v on a set U ⊂ X is the maximum rank of v|_{x} over all points x ∈ U. Finally, for M ∈ Ω^{0}(𝔰𝔲(W^{+}) ⊗ 𝔰𝔲(E)), the rank of M at a point x ∈ X is the rank of M considered as a real linear map from 𝔰𝔲(W^{+})^{∗}|_{x} → 𝔰𝔲(E)|_{x} (again, its rank as a real three-by-three matrix). The rank of M on a set U ⊂ X is the maximum rank of M|_{x} over all points x ∈ U.

For the proof of Lemma 2.21 below we shall need a simple linear algebra identity. Suppose that Φ = ϕ ⊗ ξ, for some ϕ ∈ Ω^{0}(W^{+}) and ξ ∈ Ω^{0}(E), and that Ψ = ψ ⊗ ζ, with ϕ ∈ Ω^{0}(W^{+}) and ξ ∈ Ω^{0}(E). Then

Φ ⊗ Ψ^{∗} = (ϕ ⊗ ξ) ⊗ (ψ ⊗ ζ)^{∗} = (ϕ ⊗ ψ^{∗}) ⊗ (ξ ⊗ ζ^{∗}) ∈ 𝔤𝔩(W^{+}) ⊗_{ℂ} 𝔤𝔩(E).

Writing B ∈ 𝔤𝔩(W^{+}) (or 𝔤𝔩(E)) as B = B_{h} +B_{s}, where B_{h} = \frac{1}{2}(B +B^{†}) is Hermitian and B_{s} = \frac{1}{2}(B − B^{†}) is skew-Hermitian, we have

Φ ⊗ Ψ^{∗} = \frac{1}{4}(ϕ ⊗ ψ^{∗} + ψ ⊗ ϕ^{∗}) ⊗ (ξ ⊗ ζ^{∗} + ζ ⊗ ξ^{∗})
+ \frac{1}{4}(ϕ ⊗ ψ^{∗} − ψ ⊗ ϕ^{∗}) ⊗ (ξ ⊗ ζ^{∗} − ζ ⊗ ξ^{∗})
+ \frac{1}{4}(ϕ ⊗ ψ^{∗} + ψ ⊗ ϕ^{∗}) ⊗ (ξ ⊗ ζ^{∗} − ζ ⊗ ξ^{∗})
+ \frac{1}{4}(ϕ ⊗ ψ^{∗} − ψ ⊗ ϕ^{∗}) ⊗ (ξ ⊗ ζ^{∗} + ζ ⊗ ξ^{∗}),

and similarly for Ψ ⊗ Φ^{∗}. Therefore,

Φ ⊗ Ψ^{∗} + Ψ ⊗ Φ^{∗} = \frac{1}{2}(ϕ ⊗ ψ^{∗} + ψ ⊗ ϕ^{∗}) ⊗ (ξ ⊗ ζ^{∗} + ζ ⊗ ξ^{∗})
+ \frac{1}{2}(ϕ ⊗ ψ^{∗} − ψ ⊗ ϕ^{∗}) ⊗ (ξ ⊗ ζ^{∗} − ζ ⊗ ξ^{∗})

= −\frac{1}{2}i(ϕ ⊗ ψ^{∗} + ψ ⊗ ϕ^{∗}) ⊗ i(ξ ⊗ ζ^{∗} + ζ ⊗ ξ^{∗}) (2.13)

+ \frac{1}{2}(ϕ ⊗ ψ^{∗} − ψ ⊗ ϕ^{∗}) ⊗ (ξ ⊗ ζ^{∗} − ζ ⊗ ξ^{∗}).

We shall need the following elementary observation in our proof of transversality for the moduli space of PU(2) monopoles in §5. Note that as ρ : Λ^{+} → 𝔰𝔲(W^{+}) is an isomorphism, the rank of a section v of Λ^{+} ⊗ 𝔰𝔲(E) is equal to the rank of the section ρ(v) of 𝔰𝔲(W^{+}) ⊗ 𝔰𝔲(E).

Lemma 2.21. If Φ ∈ C^{0}(W^{+} ⊗ E) and x ∈ X, then the following hold:

(1) Rank_{ℝ}(Φ(x) ⊗ Φ^{∗}(x))_{00} = 1 if and only if Rank_{ℂ} Φ(x) = 1,

(2) Rank_{ℝ}(Φ(x) ⊗ Φ^{∗}(x))_{00} = 3 if and only if Rank_{ℂ} Φ(x) = 2.

Proof. For convenience we write Φ, W^{+}, and E for Φ|_{x}, W^{+}|_{x}, and E|_{x}. If Φ ∈ Hom(W^{+}, E) has complex rank one, we can write Φ = ϕ⊗ξ, for ϕ ∈ W^{+} and ξ ∈ E. Then (Φ ⊗ Φ^{∗})_{00} = −i(ϕ ⊗ ϕ^{∗})_{0} ⊗ i(ξ ⊗ ξ^{∗})_{0} and so (Φ ⊗ Φ^{∗})_{00} ∈ Hom(𝔰𝔲(W^{+}), 𝔰𝔲(E)) has real rank one.

Conversely, suppose (Φ ⊗ Φ^{∗})_{00} has real rank one. Let {ξ_{1}, ξ_{2}} be an orthonormal basis for E and write Φ = ϕ_{1} ⊗ ξ_{1} + ϕ_{2} ⊗ ξ_{2}: if ϕ_{1}, ϕ_{2} ∈ E are linearly dependent, then Φ has complex rank one and we are done, so suppose that ϕ_{1}, ϕ_{2} are linearly independent. Using the standard basis {σ_{1}, σ_{2}, σ_{3}} of Pauli matrices (2.14) for 𝔰𝔲(2), namely

σ_{1} = (\begin{matrix}i & 0 \\ 0 & −i \end{matrix}) , σ_{2} = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) , σ_{3} = (\begin{matrix}0 & i \\ i & 0 \end{matrix}) , (2.14)

we obtain bases for the real vector spaces 𝔰𝔲(W^{+}) and 𝔰𝔲(E):

σ_{1}^{e} = i(ξ_{1} ⊗ ξ_{1}^{∗} − ξ_{2} ⊗ ξ_{2}^{∗}), σ_{2}^{e} = ξ_{1} ⊗ ξ_{2}^{∗} − ξ_{2} ⊗ ξ_{1}^{∗},
σ_{3}^{e} = i(ξ_{1} ⊗ ξ_{2}^{∗} + ξ_{2} ⊗ ξ_{1}^{∗}),
σ_{1}^{w} = i(ϕ_{1} ⊗ ϕ_{1}^{∗} − ϕ_{2} ⊗ ϕ_{2}^{∗}), σ_{2}^{w} = ϕ_{1} ⊗ ϕ_{2}^{∗} − ϕ_{2} ⊗ ϕ_{1}^{∗},
σ_{3}^{w} = i(ϕ_{1} ⊗ ϕ_{2}^{∗} + ϕ_{2} ⊗ ϕ_{1}^{∗}).

Writing Φ = Φ_{1} + Φ_{2}, we have

(Φ_{a} ⊗ Φ_{a}^{∗})_{00} = −i(ϕ_{a} ⊗ ϕ_{a}^{∗})_{0} ⊗ i(ξ_{a} ⊗ ξ_{a}^{∗})_{0}

for a = 1, 2 and

(Φ ⊗ Φ^{∗})_{00} = (Φ_{1} ⊗ Φ_{1}^{∗})_{00} + (Φ_{1} ⊗ Φ_{2}^{∗} + Φ_{2} ⊗ Φ_{1}^{∗})_{00} + (Φ_{2} ⊗ Φ_{2}^{∗})_{00}.

So, by the identity (2.13) and noting that

i(ξ_{1} ⊗ ξ_{1}^{∗})_{0} = \frac{1}{2}i(ξ_{1} ⊗ ξ_{1}^{∗} − ξ_{2} ⊗ ξ_{2}^{∗}) = \frac{1}{2}σ_{1}^{e} = −i(ξ_{2} ⊗ ξ_{2}^{∗})_{0},

and

i(ϕ_{1} ⊗ ϕ_{1}^{∗})_{0} = \frac{1}{2}i(ϕ_{1} ⊗ ϕ_{1}^{∗} − ϕ_{2} ⊗ ϕ_{2}^{∗}) = \frac{1}{2}σ_{1}^{w} = −i(ϕ_{2} ⊗ ϕ_{2}^{∗})_{0}

we see that

(Φ ⊗ Φ^{∗})_{00} = −i(ϕ_{1} ⊗ ϕ_{1}^{∗})_{0} ⊗ i(ξ_{1} ⊗ ξ_{1}^{∗})_{0} − i(ϕ_{2} ⊗ ϕ_{2}^{∗})_{0} ⊗ i(ξ_{2} ⊗ ξ_{2}^{∗})_{0}
− \frac{1}{2}i(ϕ_{1} ⊗ ϕ_{2}^{∗} + ϕ_{2} ⊗ ϕ_{1}^{∗}) ⊗ i(ξ_{1} ⊗ ξ_{2}^{∗} + ξ_{2} ⊗ ξ_{1}^{∗})
+ \frac{1}{2}(ϕ_{1} ⊗ ϕ_{2}^{∗} − ϕ_{2} ⊗ ϕ_{1}^{∗}) ⊗ (ξ_{1} ⊗ ξ_{2}^{∗} − ξ_{2} ⊗ ξ_{1}^{∗})
= −\frac{1}{4}σ_{1}^{w} ⊗ σ_{1}^{e} − \frac{1}{4}σ_{1}^{w} ⊗ σ_{1}^{e} − \frac{1}{2}σ_{3}^{w} ⊗ σ_{3}^{e} + \frac{1}{2}σ_{2}^{w} ⊗ σ_{2}^{e}
= −\frac{1}{2}σ_{1}^{w} ⊗ σ_{1}^{e} + \frac{1}{2}σ_{2}^{w} ⊗ σ_{2}^{e} − \frac{1}{2}σ_{3}^{w} ⊗ σ_{3}^{e}.

Thus, (Φ ⊗ Φ^{∗})_{00} would have real rank three, a contradiction, and so Φ must have complex rank one, which proves Assertion 1.

The preceding argument also shows that if Φ has complex rank two, then (Φ ⊗ Φ^{∗})_{00} has real rank three. Conversely, if (Φ ⊗ Φ^{∗})_{00} has real rank three, then Φ cannot have complex rank one by Assertion 1, so Φ has complex rank two. q.e.d.

2.5. The PU(2) monopole equations and holonomy perturbations. In §2.5.1 we describe the PU(2) monopole equations in their unperturbed form, following [71], [74]. In §2.5.2 we introduce the holonomy perturbations and the perturbed PU(2) monopole equations, deferring a detailed discussion of most of the technical regularity issues concerning holonomy perturbations to the Appendix. In Donaldson’s application to the extended anti-self-dual equation, some important features ensure that the requisite analysis is relatively tractable:

(i) reducible connections can be excluded from the compactification of the extended moduli spaces [18, p. 283], (ii) the cohomology groups for the elliptic complex of his extended equations have simple weak semicontinuity properties with respect to Uhlenbeck limits [18, Proposition 4.33], and (iii) the zero locus being perturbed is cut out of a finitedimensional manifold [18, p. 281, Lemma 4.35, & Corollary 4.38]. For the development of Donaldson’s method for PU(2) monopoles described here, none of these simplifying features hold and so the corresponding transversality argument is rather complicated. Indeed, from Proposition 7.1.32 in [20] one can see that because of the Dirac operator, the behavior of the cokernels of the linearization of the PU(2) monopole equations can be quite involved under Uhlenbeck limits. The holonomy perturbations considered by Donaldson in [18] are inhomogeneous, as he uses the perturbations to kill the cokernels Coker d_{A}^{+} directly. In contrast, the perturbations which we consider in (1.3) are homogeneous and we shall argue indirectly in §5 that the cokernels of the linearization vanish away from the reducible and zero-section solutions.

2.5.1. The unperturbed PU(2) monopole equations. Recall that we consider Hermitian two-plane bundles E over X whose determinant line bundles det E are isomorphic to a fixed Hermitian line bundle over X endowed with a fixed C^{∞}, unitary connection A_{e}. Let (ρ, W^{+}, W^{−}) be a spin^{c} structure on X, where ρ : T^{∗}X → End W is the Clifford map, and the Hermitian four-plane bundle W = W^{+} ⊕ W^{−} is endowed with a C^{∞} spin^{c} connection. Given a connection A on E with curvature F_{A} ∈ L_{k−1}^{2}(Λ^{2} ⊗ 𝔲(E)), then (F_{A}^{+})_{0} ∈ L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) denotes the traceless part of its self-dual component. Equivalently, if A is a connection on 𝔰𝔲(E) with curvature F_{A} ∈ L_{k−1}^{2}(Λ^{2} ⊗ 𝔰𝔬(𝔰𝔲(E))), then F_{A}^{+} ∈ L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) is its self-dual component, viewed as a section of Λ^{+} ⊗ 𝔰𝔲(E) via the implicit isomorphism ad : 𝔰𝔲(E) → 𝔰𝔬(𝔰𝔲(E)). Let D_{A} : L_{k}^{2}(W^{+} ⊗ E) → L_{k−1}^{2}(W^{−} ⊗ E) be the corresponding Dirac operator.

For an L_{k}^{2} section Φ of W^{+}⊗E, let Φ^{∗} be its pointwise Hermitian dual and let (Φ ⊗ Φ^{∗})_{00} be the component of the Hermitian endomorphism Φ ⊗ Φ^{∗} of W^{+} ⊗ E which lies in 𝔰𝔲(W^{+}) ⊗ 𝔰𝔲(E). The Clifford map ρ defines an isomorphism ρ : Λ^{+} → 𝔰𝔲(W^{+}) and thus an isomorphism ρ = ρ ⊗ id_{𝔰𝔲(E)} of Λ^{+} ⊗ 𝔰𝔲(E) with 𝔰𝔲(W^{+}) ⊗ 𝔰𝔲(E). Then

\begin{aligned}F_{A}^{+} − ρ^{−1}(Φ ⊗ Φ^{∗})_{00} & = 0, \\ D_{A}Φ & = 0, \end{aligned} (2.15)

are the unperturbed equations considered in [67], [68], [71], [74] (with slightly differing trace conditions) for a pair (A, Φ) consisting of a connection A on 𝔰𝔲(E) and a section Φ of W^{+} ⊗ E. Equivalently, given a pair (A, Φ) with A a fixed-determinant connection on E, equations (2.15) take the same form except that F_{A}^{+} is replaced by (F_{A}^{+})_{0}.

2.5.2. The perturbed PU(2) monopole equations. We next introduce perturbations of the PU(2) monopole equations (2.15) which will enable us to prove the transversality result described in Theorem 1.3. The perturbations in question make use of holonomy and their construction is partly modelled on related perturbations introduced by Donaldson, Floer and Taubes [8], [17], [29], [18], [86]. All of these constructions require that a unitary connection A on a Hermitian two-plane bundle E over X be regular enough that parallel translation along a C^{∞} path in X is well-defined. For example, Floer employs a configuration space of L_{1}^{4} connections modulo L_{2}^{4} gauge-transformations over a threemanifold, so these connections are at least continuous (as L_{1}^{4} ⊂ C^{0} in dimension three) and the standard theory of ordinary differential equations applies to define parallel translation without further qualification [29]. In dimension four, parallel translation can be defined without difficulty using L_{k}^{2} connections with k ≥ 3. With a little more care one can see that parallel translation can be defined by L_{2}^{2} connections, even though L_{2}^{2} ⊄ C^{0} [61, §3.2]. These regularity issues become rather more intractable, as one can see from the Sobolev restriction theorem, for connections which are only L_{1}^{2} [2, Theorem V.5.4]. As explained in the Appendix — to which we refer the reader for a discussion of the more technical points — we shall ultimately restrict our attention to configuration spaces of L_{k}^{2} connections with k ≥ 3. The perturbations will, by definition, be zero on a neighborhood of a point in X where the curvature is large and so our regularity theory for L_{1}^{2} monopoles will apply near points where curvature has bubbled off in order to prove removability of singularities (see §4.5.2).

We follow standard convention by saying that a connection A on a G bundle E over a connected manifold Y is ‘irreducible’ if its stabilizer Stab_{A} is trivial, that is, the center of the Lie group G [20, p. 133], rather than saying (more correctly) that its holonomy group Hol_{A}(y_{0}) is not a proper Lie subgroup of Aut(E_{y_{0}}) ≃ G, where y_{0} ∈ Y is any basepoint. However, the holonomy will be our primary concern in this section, so some care is required as the two notions do not coincide in general. Recall from [20, Lemma 4.2.8] that Stab_{A} is isomorphic to the centralizer of Hol_{A}(y_{0}) in G. If Y is simply connected, then Hol_{A}(y_{0}) is a connected Lie subgroup of G [44, Theorem II.4.2]. Thus, if G = SU(2) or SO(3) and Y is simply connected, then A has trivial stabilizer in G if and only if Hol_{A}(y_{0}) ≃ G [20, p. 133]. Indeed, if G = SO(3) and Stab_{A} = {id}, then H := Hol_{A}(y_{0}) = SO(3); otherwise, we would have:

• H = SO(2), with centralizer Z(H) = SO(2) (by [11, Theorem IV.2.3(ii)]), contradicting Stab_{A} = {id}, or

• H = {id}, with centralizer Z(H) = SO(3), again contradicting Stab_{A} = {id}.

The same argument holds for G = SU(2), but not for higher-dimensional Lie groups. For example, if G = U(2) = SU(2) ×_{{±id}} S^{1}, we cannot exclude the possibility that a U(2) connection A with Stab_{A} = S_{Z}^{1} reduces to an SU(2) connection, that is, A has holonomy Hol_{A}(y_{0}) = SU(2).

With the preceding comments in mind, the first ingredient in our construction of the holonomy perturbations is a local section of 𝔰𝔲(E) given by the holonomy of an SO(3) connection A on 𝔰𝔲(E). Let γ ⊂ X be a C^{∞} loop based at a point x_{0} ∈ X and let

h_{γ,x_{0}}(A) ∈ SO(𝔰𝔲(E))|_{x_{0}}

be the holonomy of the connection A around the loop γ. The exponential map exp : 𝔰𝔬(3) → SO(3) gives a diffeomorphism from a ball 2B around the origin in 𝔰𝔬(3) to a ball around the identity in SO(3). Let ψ : ℝ → [0, 1] be a C^{∞} cutoff function such that ψ(|ζ|) = 1 for ζ ∈ \frac{1}{2}B, ψ(|ζ|) > 0 for ζ ∈ B, and ψ(|ζ|) = 0 for ζ ∈ 𝔰𝔬(3) − B. Then

(2.16) 𝔥_{γ,x_{0}}(A) := ψ (| exp^{−1}(h_{γ,x_{0}}(A))|) · ad^{−1}(exp^{−1}(h_{γ,x_{0}}(A)))

defines a gauge-equivariant map A_{E}(X) → 𝔰𝔲(E)|_{x_{0}}, where ad : 𝔰𝔲(E) → 𝔰𝔬(𝔰𝔲(E)) is the standard isomorphism.

Lemma 2.22. Let U ⊂ X be a simply connected open subset and let x_{0} be a point in U. If A|_{U} is an irreducible SO(3) connection on 𝔰𝔲(E)|_{U}, then there are loops {γ_{l}}_{l=1}^{3} ⊂ U, depending on A, such that the set {h_{γ_{l},x_{0}}(A)}_{l=1}^{3} ⊂ SO(𝔰𝔲(E))|_{x_{0}} lies in the open subset of SO(𝔰𝔲(E))|_{x_{0}} given by the image under exp of the ball B ⊂ 𝔰𝔬(𝔰𝔲(E))|_{x_{0}} around the identity. The set {𝔥_{γ_{l},x_{0}}(A)}_{l=1}^{3} is then a basis for 𝔰𝔲(E)|_{x_{0}}.

Proof. Since A|_{U} is an irreducible SO(3) connection over a simply connected manifold U, the holonomy group Hol_{A|U}(x_{0}) is equal to SO(𝔰𝔲(E))|_{x_{0}} by the remarks preceding the statement of the lemma. Hence, there are three loops γ_{1}, γ_{2}, γ_{3} such that the holonomies

h_{γ_{l},x_{0}}(A) ∈ B ⊂ SO(𝔰𝔲(E))|_{x_{0}}

give a basis {𝔥_{γ_{l},x_{0}}(A)}_{l=1}^{3} for 𝔰𝔲(E)|_{x_{0}}. q.e.d.

For a C^{∞} connection A we may extend 𝔥_{γ,x_{0}}(A) to a C^{∞} section 𝔥̂_{γ}(A) of 𝔰𝔲(E) by radial parallel translation, with respect to A over a small ball B(x_{0}, 2R_{0}) and then multiplying by a C^{∞} cutoff function φ on X which is positive on B(x_{0}, R_{0}) and identically zero on X − B(x_{0}, R_{0}). Thus if the set {𝔥_{γ_{l},x_{0}}(A)}_{l=1}^{3} spans 𝔰𝔲(E)|_{x_{0}}, then the set {φ𝔥̂_{γ_{l}}(A)|_{y}}_{l=1}^{4} will span 𝔰𝔲(E)|_{y} for y ∈ B(x_{0}, R_{0}). The constant R_{0} is chosen so that 4R_{0} is smaller than the injectivity radius of (X, g).

In general, for an L_{k}^{2} connection A with k ≥ 2, the section 𝔥̂_{γ}(A) will not be in L_{k+1}^{2} and so we use the Neumann heat operator, for fixed small t > 0,

K_{t}(A|_{B(x_{0},2R_{0})}) : L^{2}(B(x_{0}, 2R_{0}), 𝔰𝔲(E)) → L_{k+1}^{2}(B(x_{0}, 2R_{0}), 𝔰𝔲(E)),

as discussed in Appendix A.1 to construct an L_{k+1}^{2} section

𝔥_{γ}(A) := K_{t}(A|_{B(x_{0},2R_{0})})𝔥̂_{γ}(A) (2.17)

of 𝔰𝔲(E) over B(x_{0}, 2R_{0}) which converges to 𝔥̂_{γ}(A) in C^{0}(B(x_{0}, R_{0})) as t → 0. Therefore, for small enough t = t(A), the set {φ𝔥_{γ_{l}}(A)|_{y}}_{l=1}^{3}, will span 𝔰𝔲(E)|_{y} for all y ∈ B(x_{0}, R_{0}) just as before.

Lemma 2.23. Let k ≥ 2 be an integer and let A_{0} be an L_{k}^{2} unitary connection on E|_{B(x_{0},2R_{0})}. Let γ_{1}, γ_{2}, γ_{3} be loops in B(x_{0}, 2R_{0}) based at x_{0} such that {𝔥_{γ_{l},x_{0}}(A_{0})}_{l=1}^{3} spans 𝔰𝔲(E)|_{x_{0}}. Then there is a positive constant ε(A_{0}, {γ_{l}}) such that if A is an L_{k}^{2} unitary connection on E|_{B(x_{0},2R_{0})} satisfying

∥A − A_{0}∥_{L_{k,A_{0}}^{2}(B(x_{0},2R_{0}))} < ε,

then the set {𝔥_{γ_{l},x_{0}}(A)}_{l=1}^{3} spans 𝔰𝔲(E)|_{x_{0}}.

Proof. According to Lemma A.3 the holonomy maps

h_{γ,x_{0}} : A_{E}(X) → U(E)|_{x_{0}}

are continuous and so the maps 𝔥_{γ_{l},x_{0}} : A_{E}(X) → 𝔰𝔲(E)|_{x_{0}} are continuous on small open neighborhoods of A_{0} ∈ A_{E}(X). Hence, for small enough ε, the set {𝔥_{γ_{l},x_{0}}(A)}_{l=1}^{3} spans 𝔰𝔲(E)|_{x_{0}}. q.e.d.

We now specify the loops to be used in this construction, essentially following the argument used in the proof of Lemma 2.5 in [17]. Let {B(x_{j}, 4R_{0})}_{j=1}^{N_{b}}, be a disjoint collection of open balls, where N_{b} is a fixed integer to be specified later (see §4.5.2). According to the slice results of [20, p. 192], [87, Proposition 2.1] for manifolds with boundary, the quotient space B_{E}^{∗}(B(x_{j}, 2R_{0})) of irreducible L_{k}^{2} connections on E|_{B(x_{j},2R_{0})} is a C^{∞} manifold modelled on a separable Hilbert space. The quotient L_{k}^{2} topology on B_{E}^{∗}(B(x_{j}, 2R_{0})) is clearly metrizable and so Stone’s Theorem implies that B_{E}^{∗}(B(x_{j}, 2R_{0})) is paracompact and thus admits partitions of unity [55, Corollary II.3.8].

For each j = 1, …, N_{b} and each point [A_{0}] in B_{E}^{∗}(B(x_{j}, 2R_{0})), we can find loops {γ_{j,l,A_{0}}}_{l=1}^{3}, contained in B(x_{j}, 2R_{0}) and based at x_{j} such that {𝔥_{γ_{j,l,A_{0}}} (A_{0})}_{l=1}^{3}, spans 𝔰𝔲(E)|_{x_{j}}. For each such point [A_{0}], Lemma 2.23 implies that there is an L_{k,A_{0}}^{2} ball

B_{[A_{0}]}(ε_{A_{0}}) := {[A] ∈ B_{E}^{∗}(B(x_{j}, 2R_{0})) : dist_{L_{k,A_{0}}^{2}} ([A], [A_{0}]) < ε_{A_{0}}}

such that for all [A] ∈ B_{[A_{0}]}(ε_{A_{0}}), the set {𝔥_{γ_{j,l,A_{0}}} (A)}_{l=1}^{3} spans 𝔰𝔲(E)|_{x_{j}}. These balls give an open cover of B_{E}^{∗}(B(x_{j}, 2R_{0})), and hence there is a locally finite refinement of this open cover, {U_{j,α}}_{α=1}^{∞}, and a positive partition χ_{j,α} (see Proposition A.12) subordinate to {U_{j,α}}_{α=1}^{∞} in the sense that

∑_{α} χ_{j,α}[A] > 0, [A] ∈ B_{E}^{∗}(B(x_{j}, 2R_{0})).

Hence, for each U_{j,α}, we obtain loops {γ_{j,l,α}}_{l=1}^{3} ⊂ B(x_{j}, 2R_{0}) such that for all [A] ∈ U_{j,α}, the sections 𝔥_{γ_{j,l,α}}(A) span 𝔰𝔲(E)|_{x_{j}}.

Let β be a smooth cutoff function on ℝ such that β(t) = 1 for t ≤ \frac{1}{2} and β(t) = 0 for t ≥ 1, with β(t) > 0 for t < 1. Then the C^{∞} gauge-invariant maps A_{E}(X) → ℝ, A ↦ β_{j}[A] given by

β_{j}[A] := β (\frac{1}{ε_{0}^{2}} ∫_{B(x_{j},4R_{0})} β (\frac{dist_{g}(·, x_{j})}{4R_{0}}) |F_{A}|^{2} dV ) (2.18)

are zero when the energy of the connection A is greater than or equal to \frac{1}{2}ε_{0}^{2} over a ball B(x_{j}, 2R_{0}). Here, ε_{0} is the constant of Corollary 3.16. Finally, we define C^{∞} cutoff functions on X by setting

φ_{j}(x) := β (\frac{dist_{g}(x, x_{j})}{R_{0}}) , x ∈ X, (2.19)

so that φ_{j} is positive on the ball B(x_{j}, R_{0}) and zero on its complement in X.

We can now define a gauge equivariant C^{∞} map

A_{E}(X) → L_{k+1}^{2}(X, 𝔰𝔲(E)), A ↦ 𝔪_{j,l,α}(A)

by setting

𝔪_{j,l,α}(A) := β_{j}[A]χ_{j,α}[A|_{B(x_{j},2R_{0})}]φ_{j}𝔥_{γ_{j,l,α}}(A). (2.20)

Thus at each point A ∈ A_{E}(X) only a finite number of the 𝔪_{j,l,α}(A) are non-zero and each map 𝔪_{j,l,α} is C^{∞} with uniformly bounded derivatives of all orders on A_{E}(X) (see Appendix A.4).

To define the perturbation of the Dirac operator in (2.15), we need elements of Ω^{0}(Hom_{ℂ}(W^{+}, W^{−})) and these are conveniently provided by complex one-forms using the Clifford isomorphism

ρ : Λ_{ℂ}^{1} → Hom_{ℂ}(W^{+}, W^{−}),

where Λ_{ℂ}^{1} = Λ^{1}⊗_{ℝ}ℂ. These one-forms are parametrized by the following Banach space: Let 𝔸 be the index set

{(j, l, α) : 1 ≤ j ≤ N_{b}, 1 ≤ l ≤ 3, α ∈ ℕ},

let δ = (δ_{α})_{α=1}^{∞} ∈ ℓ^{1}(ℝ) be the sequence of positive weights described in Appendix A.4, let r ≥ k + 1, and let

P_{ϑ}^{r} := ℓ_{δ}^{1}(𝔸, C^{r}(X, Λ_{ℂ}^{1})) (2.21)

be the set of sequences ϑ⃗ := (ϑ_{j,l,α}) in C^{r}(X, Λ_{ℂ}^{1}) such that

∥ϑ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))} := ∑_{j,l,α} δ_{α}^{−1}∥ϑ_{j,l,α}∥_{C^{r}(X)} < ∞.

Then P_{ϑ}^{r} is a Banach space with respect to the above norm [40, §1.7].

Remark 2.24.

1. Note that we measure reducibility of connections and allow the loops γ_{j,l,α} to be contained in the larger balls B(x_{j}, 2R_{0}) while the perturbations are supported on the smaller balls B(x_{j}, R_{0}). Thus, if A ∈ A_{E}(X) is a connection such that A|_{B(x_{j},2R_{0})} is reducible for all j ∈ {1, …, N_{b}} such that β_{j}[A] > 0 and Φ ≢ 0, then [A, Φ] cannot be a point in M_{W,E}^{∗,0}: our unique continuation result for PU(2) monopoles (see Theorem 5.11) which are reducible on an open subset of X containing all balls B̄(x_{j}, R_{0}) with β_{j}[A] > 0 would imply that A is reducible on all of X and so (A, Φ) would be a reducible PU(2) monopole. Consequently, if [A, Φ] ∈ M_{W,E}^{∗,0} then A must be irreducible on some ball B(x_{j}, 2R_{0}); otherwise, A would be reducible on all balls B(x_{j}, 2R_{0}) with β_{j}[A] > 0 and so reducible on X.

2. Note that the sections 𝔪_{j,l,α} are defined on the entire space A_{E}(X): They are zero for connections A ∈ A_{E}(X) which are reducible when restricted to B(x_{j}, 2R_{0}).

3. Our energy bound (4.2) for solutions (A, Φ) to the perturbed PU(2) monopole equations ensures that there is always at least one ball B(x_{j}, R_{0}) where the sections {𝔪_{j,l,α}(A)}_{l=1}^{3} span 𝔰𝔲(E)|_{B(x_{j},R_{0})}; see §4.5.2.

Because only a finite number of the sections 𝔪_{j,l,α}(A) are non-zero at each A ∈ A_{E}(X), the sum

ϑ⃗ · 𝔪⃗(A) := ∑_{j,l,α} ρ(ϑ_{j,l,α}) ⊗_{ℂ} 𝔪_{j,l,α}(A) (2.22)

gives a well-defined L_{k+1}^{2} section of

Hom_{ℂ}(W^{+}, W^{−}) ⊗_{ℂ} 𝔰𝔩(E) ≃ Λ_{ℂ}^{1} ⊗_{ℂ} 𝔰𝔩(E),

noting that although each 𝔪_{j,l,α}(A) is a section of 𝔰𝔲(E), we have 𝔰𝔩(E) = 𝔰𝔲(E) ⊗_{ℝ} ℂ. The map

A_{E}(X) → L_{k+1}^{2}(X, Hom_{ℂ}(W^{+}, W^{−}) ⊗_{ℂ} 𝔰𝔩(E)), A ↦ ϑ⃗ · 𝔪⃗(A)

is C^{∞} and gauge equivariant and so defines a C^{∞} section of the vector bundle

A_{E}^{∗}(X) ×_{G_{E}} L_{k+1}^{2}(X, Hom_{ℂ}(W^{+}, W^{−}) ⊗_{ℂ} 𝔰𝔩(E)) → B_{E}^{∗}(X).

By construction (see Appendix A.4) the sum ϑ⃗ · 𝔪⃗(A) satisfies a C^{0} estimate of the form

\begin{aligned}sup_{A∈A_{E}(X)} ∥ϑ⃗ · 𝔪⃗(A)∥_{L_{k+1,A}^{2}(X)} & ≤ C∥ϑ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))} \\ and ∥ϑ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))} & ≤ ε_{ϑ}, \end{aligned} (2.23)

where C = C(g, k), and ε_{ϑ} is a positive constant which we are free to specify.

We shall also need to construct a gauge equivariant C^{∞} map from A_{E}^{∗}(X) to L_{k+1}^{2}(𝔤𝔩(Λ^{+}) ⊗_{ℝ}𝔰𝔬(𝔰𝔲(E))). This map will define a perturbation of the quadratic form in (2.15) using the representation ad : 𝔰𝔲(E) → 𝔰𝔬(𝔰𝔲(E)) and is parametrized by the Banach space

P_{τ}^{r} := ℓ_{δ}^{1}(𝔸, C^{r}(X, 𝔤𝔩(Λ^{+}))) (2.24)

of sequences τ⃗ := (τ_{j,l,α}) in C^{r}(X, 𝔤𝔩(Λ^{+})) such that

∥τ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))} := ∑_{j,l,α} δ_{α}^{−1}∥τ_{j,l,α}∥_{C^{r}(X)} < ∞.

Then the sum

τ⃗ · 𝔪⃗(A) := ∑_{j,l,α} τ_{j,l,α} ⊗_{ℝ} ad(𝔪_{j,l,α}(A)) (2.25)

is pointwise finite and gives a well-defined L_{k+1}^{2} section of

𝔤𝔩(Λ^{+}) ⊗_{ℝ} 𝔰𝔬(𝔰𝔲(E)).

The map

A_{E}(X) → L_{k+1}^{2}(X, 𝔤𝔩(Λ^{+}) ⊗_{ℝ} 𝔰𝔬(𝔰𝔲(E))), A ↦ τ⃗ · 𝔪⃗(A)

is C^{∞} and gauge equivariant and so defines a C^{∞} section of the vector bundle

A_{E}^{∗}(X) ×_{G_{E}} L_{k+1}^{2}(X, 𝔤𝔩(Λ^{+}) ⊗_{ℝ} 𝔰𝔬(𝔰𝔲(E))) → B_{E}^{∗}(X).

By construction (see Appendix A.4) the sum τ⃗ · 𝔪⃗(A) satisfies C^{0} estimates

\begin{aligned}sup_{A∈A_{E}(X)} ∥τ⃗ · 𝔪⃗(A)∥_{L_{k+1,A}^{2}(X)} & ≤ C∥τ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))} \\ and ∥τ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))} & ≤ ε_{τ} , \end{aligned} (2.26)

where C = C(g, k), and ε_{τ} is a positive constant which we are free to specify.

Our perturbed PU(2) monopole equations for a pair (A, Φ) on (𝔰𝔲(E), W^{+} ⊗ E) then take the form

𝔖_{1}(A, Φ) :=F_{A}^{+} − (id + τ_{0} ⊗ id_{𝔰𝔲(E)}

+τ⃗ · 𝔪⃗(A)) ρ^{−1}(Φ ⊗ Φ^{∗})_{00} = 0, (2.27)

𝔖_{2}(A, Φ) :=D_{A}Φ + ρ(ϑ_{0})Φ + ϑ⃗ · 𝔪⃗(A)Φ = 0,

where τ_{0} ∈ C^{r}(X, 𝔤𝔩(Λ^{+})) and ϑ_{0} ∈ C^{r}(X, Λ_{ℂ}^{1}) are additional perturbation parameters. For brevity, we shall often denote τ⃗_{A} := τ⃗ · 𝔪⃗(A) and ϑ⃗_{A} := ϑ⃗ · 𝔪⃗(A).

Remark 2.25.

1. In [25] we consider the question of transversality of the spaces of reducible solutions or U(1) monopoles, which are identified with moduli spaces of Seiberg-Witten monopoles. In particular, we show that the moduli spaces of U(1) monopoles are cut out transversely for generic τ_{0} ∈ Ω^{0}(𝔤𝔩(Λ^{+})).

2. If the section Φ is identically zero, then the PU(2) monopole equations reduce to those for a projectively anti-self-dual unitary connection A on E. If the connection A is reducible, then all of the sections 𝔪_{j,l,α}(A) are identically zero on X; the perturbations τ⃗ · 𝔪⃗(A) and ϑ⃗ · 𝔪⃗(A) in (2.27) are then zero.

3. We emphasize that the holonomy sections in the sequence 𝔪⃗(A) are always L_{k+1}^{2} by construction for L_{k}^{2} connections A (with k ≥ 2) and that the sums defining τ⃗ · 𝔪⃗(A) and ϑ⃗ · 𝔪⃗(A) are always finite for each A, although the number of terms may tend to infinity as A approaches a reducible connection.

The proof of the existence of an Uhlenbeck compactification for M_{W,E} in §4 requires the perturbations τ⃗ · 𝔪⃗(A) and ϑ⃗ · 𝔪⃗(A) to satisfy the following estimates in order to obtain universal a priori L_{1,A}^{2} bounds for Φ and L^{2} bounds for F_{A} (see Lemmas 4.2 and 4.3):

\begin{aligned}sup_{A∈A_{E}} ∥τ⃗ · 𝔪⃗(A)∥_{L^{∞}(X)} & ≤ 1, \\ sup_{A∈A_{E}} ∥ϑ⃗ · 𝔪⃗(A)∥_{L^{4}(X)} & ≤ 1. \end{aligned} (2.28)

To obtain the more delicate universal a priori L^{∞} bounds for Φ and F_{A}^{+} (see Lemmas 2.26 and 4.4) required by our Uhlenbeck compactness argument in §4, the perturbations τ_{0}, τ⃗ ·𝔪⃗(A), and ϑ⃗·𝔪⃗(A) must satisfy the following stronger estimates:

\begin{aligned}∥τ_{0}∥_{L^{∞}(X)} + sup_{A∈A_{E}} ∥τ⃗ · 𝔪⃗(A)∥_{L^{∞}(X)} & ≤ \frac{1}{64}, \\ ∥ϑ_{0}∥_{L_{1}^{∞}(X)} + sup_{A∈A_{E}} ∥ϑ⃗ · 𝔪⃗(A)∥_{L_{1,A}^{∞}(X)} & ≤ 1. \end{aligned} (2.29)

We take a constant 1 on the right-hand sides of (2.28) and the second inequality in (2.29) for notational convenience only: these bounds do not need to be ‘small’. There are continuous Sobolev embeddings L_{4}^{2}(X, ℝ) ⊂ L^{∞}(X, ℝ) and L_{4}^{2}(X, ℝ) ⊂ L_{1}^{∞}(X, ℝ) over a four-manifold X. The estimates (2.29) then follow from the estimates (2.23) and (2.26) when k ≥ 3. Therefore, to obtain the inequalities (2.29), we require that k ≥ 3 in our configuration spaces of L_{k}^{2} connections B_{E} and pairs C_{W,E}; see §A.4. The bounds in (2.29) then follow for small enough choices of ε_{τ} and ε_{ϑ} in the inequalities (2.23) and (2.26).

We shall need a slight generalization of Lemma 2.19 which applies to the perturbed quadratic form τρ^{−1}(Φ ⊗ Φ^{∗})_{00} when τ ≠ id_{Λ^{+}}. Without loss of generality, we may restrict our attention to automorphisms τ_{ρ} := ρτρ^{−1} of 𝔰𝔲(W^{+}) such that |τ_{ρ} − id_{𝔰𝔲(W^{+})}| = |ρτρ^{−1} − id| < ε_{τ} and thus

⟨Φ_{1}, Φ_{2}⟩ − ε_{τ} |Φ_{1}||Φ_{2}| ≤ ⟨τ(Φ_{1}), Φ_{2}⟩ ≤ ⟨Φ_{1}, Φ_{2}⟩ + ε_{τ} |Φ_{1}||Φ_{2}|

for some universal positive constant ε_{τ} ≪ 1 and all

Φ_{1}, Φ_{2} ∈ Ω^{0}(W^{+} ⊗ E).

Hence, Lemma 2.19 yields the following bounds for the perturbed quadratic form:

Lemma 2.26. There is a universal positive constant ε_{τ} ≪ 1 such that for all ∥τ − id_{Λ^{+}}∥_{C^{0}(X)} < ε_{τ} and Φ ∈ Ω^{0}(W^{+} ⊗ E) the following inequalities hold:

(1)  \frac{1}{2}|Φ|^{4} ≤ ⟨τ_{ρ}(Φ ⊗ Φ^{∗})_{00}Φ, Φ⟩ ≤ \frac{3}{4}|Φ|^{4},
(2)  \frac{1}{4}|Φ|^{4} ≤ |τ_{ρ}(Φ ⊗ Φ^{∗})_{00}|^{2} ≤ \frac{3}{8}|Φ|^{4}.

Remark 2.27. The constant ε_{τ} = 1/64 will suffice. The constraint ∥τ − id∥_{C^{0}(X)} < ε_{τ} is only used in §4.1 and consequently in §4.6, where we establish the ‘universal’ a priori bounds for PU(2) monopoles and prove the existence of an Uhlenbeck compactification, respectively.

2.6. The moduli space and the elliptic deformation complex. We define the moduli space of PU(2) monopoles and compute the index of its elliptic deformation complex.

For any integer k ≥ 2, the PU(2) monopole equations (2.27) define a C^{∞} map 𝔖 := (𝔖_{1}, 𝔖_{2}) of Hilbert manifolds,

𝔖 : C̃_{W,E} → L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) ⊕ L_{k−1}^{2}(W^{−} ⊗ E), (2.30)

given by

(A, Φ) ↦ (\begin{matrix}𝔖_{1}(A, Φ) \\ 𝔖_{2}(A, Φ) \end{matrix})

:= (\begin{matrix}F_{A}^{+} − (id + τ_{0} ⊗ id_{𝔰𝔲(E)} + τ⃗ · 𝔪⃗(A))ρ^{−1}(Φ ⊗ Φ^{∗})_{00} \\ D_{A}Φ + +ρ(ϑ_{0})Φ + ϑ⃗ · 𝔪⃗(A)Φ \end{matrix}) .

We can then define the moduli space of PU(2) monopoles by setting

M_{W,E} := {(A, Φ) ∈ C̃_{W,E} : 𝔖(A, Φ) = 0}/^{∘}G_{E} (2.31) and denote the moduli space of PU(2) monopoles which are neither reducible nor zero-section pairs by

M_{W,E}^{∗,0} := M_{W,E} ∩ C_{W,E}^{∗,0}.

We let M_{E}^{asd} := {[A] : F_{A}^{+} = 0} ⊂ B_{E} be the moduli space of anti-self-dual connections on 𝔰𝔲(E).

The SO(3) bundle 𝔰𝔲(E) has first Pontrjagin number and second Stiefel-Whitney class given by

\begin{matrix}κ := −\frac{1}{4}p_{1}(𝔰𝔲(E)) = c_{2}(E) − \frac{1}{4}c_{1}(E)^{2} \\ w_{2}(𝔰𝔲(E)) = c_{1}(E) (mod 2), \end{matrix} (2.32)

and, for a connection A on 𝔰𝔲(E), we have the following Chern-Weil integral identity,

−\frac{1}{4}p_{1}(𝔰𝔲(E)) = \frac{1}{8π^{2}} ∫_{X} (|F_{A}^{−}|^{2} − |F_{A}^{+}|^{2}) dV, (2.33)

where we view F_{A} as a section of Λ^{2} ⊗ 𝔰𝔲(E) via the isomorphism ad : 𝔰𝔲(E) ≃ 𝔰𝔬(𝔰𝔲(E)) (see [20, §2.1.4]).

Since the map 𝔖 is^{∘}G_{E}-equivariant, it defines a section of the Hilbert vector bundle 𝔙 over C_{W,E}^{∗,0} with total space

𝔙 := C̃_{W,E}^{∗,0} ×_{^{∘}G}_{E} (L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) ⊕ L_{k−1}^{2}(W^{−} ⊗ E)) . (2.34)

While the equation 𝔖[A, Φ] = 0 is, of course, defined on C_{W,E}, 𝔙 does not extend from C_{W,E}^{∗,0} to a vector bundle over C_{W,E}. The moduli space M_{W,E}^{∗,0} ⊂ C_{W,E}^{∗,0} is then the zero locus of the section 𝔖 of Hilbert vector bundle 𝔙 over the Hilbert manifold C_{W,E}^{∗,0}: it will be a regular submanifold if 𝔖 vanishes transversely, that is, if the differential

(D𝔖)_{A,Φ} : T_{[A,Φ]}C_{W,E}^{∗,0} → L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) ⊕ L_{k−1}^{2}(W^{−} ⊗ E) (2.35)

is surjective at all points [A, Φ] in 𝔖^{−1}(0)∩C_{W,E}^{∗,0}; recall from §2 that the tangent space T_{[A,Φ]}C_{W,E}^{∗,0} is canonically identified with Ker d_{A,Φ}^{0,∗} ⊂ C̃_{W,E}^{∗,0}.

Suppose (A, Φ) is a pair in A_{E} × Ω^{0}(W^{+} ⊗ E). Recall from Proposition 2.1 that the differential at the identity id_{E} ∈ ^{∘}G_{E}, of the map ∘G_{E} → A_{E} × Ω^{0}(W^{+} ⊗ E) given by u ↦ u(A, Φ) is

Ω^{0}(𝔰𝔲(E)) ⊕ iℝ_{Z} → Ω^{1}(𝔰𝔲(E)) ⊕ Ω^{0}(W^{+} ⊗ E),
ζ ↦ −d_{A,Φ}^{0}ζ = (−d_{A}ζ, ζΦ).

Similarly, the differential (a, ϕ) ↦ d_{A,Φ}^{1}(a, ϕ) := (D𝔖)_{A,Φ}(a, ϕ),

Ω^{1}(𝔰𝔲(E)) ⊕ Ω^{0}(W^{+} ⊗ E) → Ω^{+}(𝔰𝔲(E)) ⊕ Ω^{0}(W^{−} ⊗ E),

of the map 𝔖 at the point (A, Φ) is given by

d_{A,Φ}^{1}(a, ϕ)

:= (d_{A}^{+}a − τ⃗ · (δ𝔪⃗)(a)(Φ ⊗ Φ^{∗})_{00}−(id + τ_{0} ⊗ id_{𝔰𝔲(E)} +τ⃗ · 𝔪⃗(A))ρ^{−1}(Φ ⊗ ϕ^{∗} + ϕ ⊗ Φ^{∗})_{00} D_{A}ϕ + ρ(ϑ_{0})ϕ + ϑ⃗ · 𝔪⃗(A)ϕ + ρ(a)Φ + ϑ⃗ · (δ𝔪⃗)(a)Φ) , (2.36)

where δ𝔪⃗ = δ𝔪⃗/δA. (In the sequels we also find it convenient to use L_{A,Φ} to denote the linearization of the map 𝔖 at the point (A, Φ).) The differential of the composition ^{∘}G_{E} ↦ Ω^{+}(𝔰𝔲(E)) ⊕ Ω^{0}(W^{−} ⊗ E) given by

u ↦ u(A, Φ) ↦ 𝔖(u(A, Φ)) = u(𝔖(A, Φ)) = (u𝔖_{1}(A, Φ)u^{−1}, u𝔖_{2}(A, Φ))

is then

Ω^{0}(𝔰𝔲(E)) ⊕ iℝ_{Z} → \begin{matrix}Ω^{+}(𝔰𝔲(E)) \\ ⊕ \\ Ω^{0}(W^{−} ⊗ E) \end{matrix},
ζ ↦ d_{A,Φ}^{1} ∘ d_{A,Φ}^{0}ζ = ([ζ, 𝔖_{1}(A, Φ)], ζ𝔖_{2}(A, Φ)) .

Therefore, d_{A,Φ}^{1} ∘ d_{A,Φ}^{0} = 0 if and only if 𝔖(A, Φ) = 0, that is, if and only if (A, Φ) is a PU(2) monopole. Consequently, the sequence

Ω^{0}(𝔰𝔲(E)) ⊕ iℝ_{Z} ⟶^{d_{A,Φ}^{0}} \begin{matrix}Ω^{1}(𝔰𝔲(E)) \\ ⊕ \\ Ω^{0}(W^{+} ⊗ E) \end{matrix} ⟶^{d_{A,Φ}^{1}} \begin{matrix}Ω^{+}(𝔰𝔲(E)) \\ ⊕ \\ Ω^{0}(W^{−} ⊗ E) \end{matrix} (2.37)

is a complex if and only if (A, Φ) is a PU(2) monopole. The L^{2} adjoint of −d_{A,Φ}^{0} is given by

−d_{A,Φ}^{0,∗}(a, ϕ) = −d_{A}^{∗}a + (·Φ)^{∗}ϕ. (2.38)

The operator

(2.39) D_{A,Φ} := d_{A,Φ}^{0,∗} + d_{A,Φ}^{1} : \begin{matrix}Ω^{1}(𝔰𝔲(E)) \\ ⊕ \\ Ω^{0}(W^{+} ⊗ E) \end{matrix} ⟶ \begin{matrix}Ω^{0}(𝔰𝔲(E)) ⊕ iℝ_{Z} \\ ⊕ \\ Ω^{+}(𝔰𝔲(E)) \\ ⊕ \\ Ω^{0}(W^{−} ⊗ E) \end{matrix}

is elliptic (thus Fredholm) and so (2.37) is an elliptic deformation complex for the PU(2) monopole equations (2.27), with cohomology groups

H_{A,Φ}^{0} := Ker d_{A,Φ}^{0}, H_{A,Φ}^{1} := Ker d_{A,Φ}^{1}/ Im d_{A,Φ}^{0}, H_{A,Φ}^{2} := Coker d_{A,Φ}^{1},

analogous to the usual elliptic deformation complex [20, Eq. (4.2.26)] for the anti-self-dual equation, F_{A}^{+} = 0. (An elliptic deformation complex for stable pairs for holomorphic bundles is given by Bradlow and Daskalopoulos in [9, §2].)

Recall that H_{A,Φ}^{0} = Ker d_{A,Φ}^{0} is just the Lie algebra of the stabilizer Stab_{A,Φ} of the point [A, Φ] ∈ M_{W,E} and H_{A,Φ}^{1} is the Zariski or formal tangent space. Thus, for any point [A, Φ] ∈ M_{W,E}^{∗,0} we have H_{A,Φ}^{0} = 0. If H_{A,Φ}^{2} = 0, then Coker(D𝔖)_{A,Φ} = 0 and so [A, Φ] is a regular point of the zero locus of the section 𝔖 of 𝔙 over C_{W,E}^{∗,0}. Thus, if H_{A,Φ}^{0} = 0 and H_{A,Φ}^{2} = 0, then [A, Φ] is a smooth point of M_{W,E} with tangent space Ker D_{A,Φ} = Ker(d_{A,Φ}^{0,∗} + d_{A,Φ}^{1}) = H_{A,Φ}^{1}, as we see from (2.35). Provided the zero set 𝔖^{−1}(0) is regular, then M_{W,E}^{∗,0} will be a smooth manifold of dimension − Ind D_{A,Φ}.

The perturbation terms in (2.27) define gauge equivariant maps

C̃_{W,E}^{∗,0} → L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) ⊕ L_{k−1}^{2}(W^{−} ⊗ E),

given by

(A, Φ) ↦ (\begin{matrix}τ⃗ · 𝔪⃗(A)ρ^{−1}(Φ ⊗ Φ^{∗})_{00} \\ ϑ⃗ · 𝔪⃗(A)Φ \end{matrix}) .

For k ≥ 2, the Sobolev multiplication theorem implies that ϑ⃗ · 𝔪⃗(A)Φ is in L_{k}^{2}(W^{−} ⊗ E), while τ⃗ · 𝔪⃗(A)ρ^{−1}(Φ ⊗ Φ^{∗})_{00} is in L_{k}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) when k ≥ 3 and in L_{2}^{p}(Λ^{+} ⊗ 𝔰𝔲(E)), 1 ≤ p < 2 when k = 2. By the Rellich embedding theorem, the inclusions L_{k}^{2} ⊂ L_{k−1}^{2} and L_{2}^{p} ⊂ L_{1}^{2}, p > 1, are compact. In particular, it follows that the linearization of the perturbed PU(2) monopole equations (2.36) differs from the linearization of the unperturbed equations (2.15) by a compact operator [2, Theorem VI.2].

Proposition 2.28. If the map 𝔖 of (2.30) vanishes transversely for the parameters (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗), then M_{W,E}^{∗,0}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) is a smooth manifold of dimension

dim M_{W,E}^{∗,0} = −2p_{1}(𝔰𝔲(E)) − 3(1 − b^{1}(X) + b^{+}(X))
+ \frac{1}{2}p_{1}(𝔰𝔲(E)) + \frac{1}{2}(c_{1}(W^{+}) + c_{1}(E))^{2} − \frac{1}{2}σ(X) − 1
= dim M_{E}^{asd} + 2 Ind_{ℂ} D_{A} − 1,

where σ(X) is the signature of X.

Proof. Since H_{A,Φ}^{0} = 0 and H_{A,Φ}^{2} = 0 (by hypothesis) at any point [A, Φ] in M_{W,E}^{∗,0}, we have dim M_{W,E}^{∗,0} = − Ind D_{A,Φ}. By our regularity result, Proposition 3.7, any point [A, Φ] in M_{W,E}^{∗,0} has a smooth representative (A, Φ). Therefore, from the expressions for d_{A,Φ}^{0,∗} in (2.38) and for d_{A,Φ}^{1} in (2.36) and the Sobolev multiplication and embedding theorems, we find that the operator D_{A,Φ} = d_{A,Φ}^{0,∗} + d_{A,Φ}^{1} : L_{k}^{2} → L_{k−1}^{2} differs from

(\begin{matrix}d_{A}^{∗} + d_{A}^{+} & 0 \\ 0 & D_{A} \end{matrix}) : \begin{matrix}L_{k}^{2}(Λ^{1} ⊗ 𝔰𝔲(E)) \\ ⊕ \\ L_{k}^{2}(W^{+} ⊗ E) \end{matrix} → \begin{matrix}L_{k−1}^{2}(𝔰𝔲(E)) ⊕ iℝ_{Z} \\ ⊕ \\ L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) \\ ⊕ \\ L_{k−1}^{2}(W^{−} ⊗ E) \end{matrix}

by a compact operator and so has the same (real) index:

Ind D_{A,Φ} = Ind(d_{A}^{∗} + d_{A}^{+}) + Ind_{ℝ} D_{A} − 1.

We recall from [20, Eq. (4.2.22)] that the operator

d_{A}^{∗} + d_{A}^{+} : L_{k}^{2}(Λ^{1} ⊗ 𝔰𝔲(E)) → L_{k−1}^{2}(𝔰𝔲(E)) ⊕ L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E))

has index

Ind(d_{A}^{∗} + d_{A}^{+}) = −2p_{1}(𝔰𝔲(E)) − 3(1 − b^{1}(X) + b^{+}(X)).

The complex index of the Dirac operator D_{A} is given by

Ind_{ℂ} D_{A} = ⟨(Â(X)) ch(E)e^{\frac{1}{2} c_{1}(W^{+})}, [X]⟩
= ⟨(1 − \frac{1}{24}p_{1}(X))(2 + c_{1}(E)
+\frac{1}{2}c_{1}(E)^{2} − c_{2}(E))(1 + \frac{1}{2}c_{1}(W^{+}) + \frac{1}{8}c_{1}(W^{+})^{2}), [X]⟩
= ⟨−\frac{1}{12}p_{1}(X) + \frac{1}{4}c_{1}(W^{+})^{2}
+ \frac{1}{2}c_{1}(E)^{2} − c_{2}(E) + \frac{1}{2}c_{1}(E)c_{1}(W^{+}), [X]⟩

= −\frac{1}{4}σ(X) + \frac{1}{4}c_{1}(W^{+})^{2} + \frac{1}{2}c_{1}(E)^{2} − c_{2}(E) + \frac{1}{2}c_{1}(E)c_{1}(W^{+})

= (\frac{1}{4}c_{1}(E)^{2} − c_{2}(E)) + \frac{1}{4}((c_{1}(W^{+}) + c_{1}(E))^{2} − σ(X))
= \frac{1}{4}p_{1}(𝔰𝔲(E)) + \frac{1}{4}((c_{1}(W^{+}) + c_{1}(E))^{2} − σ(X)).

We obtain our dimension formula by adding these indices and noting that Ind_{ℝ} D_{A} = 2 Ind_{ℂ} D_{A}. q.e.d.

3. Regularity

Our goal in this section is to establish the basic regularity results for solutions to the PU(2) monopole equations (2.27). In §3.1 we show that global L_{1}^{2} solutions to the first-order elliptic system comprising (2.27) and the Coulomb gauge equation (see §2) are necessarily C^{∞}, while in §3.2 we show that global L_{k}^{2} solutions to (2.27) are equivalent via an L_{k+1}^{2} determinant-one, unitary automorphism of E — provided k ≥ 2 — to C^{∞} solutions to (2.27) in Coulomb gauge relative to some C^{∞} reference pair. In §3.3 and §3.4 we establish local versions of the results of §3.1 and §3.2. The regularity results and estimates of this section will be needed repeatedly throughout our work, so we state them here in sufficient generality to cover all of these applications. In the present article, we require the regularity results for our proof of transversality for the moduli space of PU(2) monopoles (see §5) and they form the cornerstone of our proofs of removable singularities and existence of an Uhlenbeck compactification (see §4). Furthermore, in sequels to this article [26], [27], the regularity results of this section are used to show that L_{1}^{2} gluing solutions to (2.27) are necessarily C^{∞} and to analyse the Uhlenbeck-boundary behavior of the gluing and obstruction maps parametrizing the moduli space ends.

In order to simultaneously address all of the intended applications, the equations we find it convenient to consider here are a quasi-linear, inhomogeneous elliptic system consisting of a generalization of the equations (2.27) and Coulomb gauge equation for a pair (A, Φ) ∈ C̃_{W,E}(X). Specifically, we allow inhomogeneous, right-hand terms: the need for this generalization arises in our proofs of removable singularities, of regularity for gluing solutions to (2.27), and in analysing the Uhlenbeckboundary behavior of gluing maps. Suppose that (A_{0}, Φ_{0}) is a fixed C^{∞} reference pair in C̃_{W,E}(X): writing (A, Φ) = (A_{0}, Φ_{0})+(a, ϕ), combining (2.27) with the Coulomb gauge equation, and allowing inhomogenous terms, we obtain an elliptic system of equations for a pair (a, ϕ) in Ω^{1}(X, 𝔰𝔲(E)) ⊕ Ω^{0}(X, W^{+} ⊗ E),

(3.1)  d_{A_{0}}^{∗}a − (·Φ_{0})^{∗}ϕ = ζ,
𝔖(A_{0} + a, Φ_{0} + ϕ) = (v_{0}, ψ_{0}),

Considering A to be a connection on 𝔰𝔲(E) and using the isomorphism ad : 𝔰𝔲(E) → 𝔰𝔬(𝔰𝔲(E)) to view F_{A} as a section of Λ^{2} ⊗ 𝔰𝔲(E), we write (3.1) as

d_{A_{0}}^{∗}a − (·Φ_{0})^{∗}ϕ = ζ,
F_{A_{0}+a}^{+} − (id + τ_{0} ⊗ id_{𝔰𝔲(E)} + τ⃗ · 𝔪⃗(A)) ρ^{−1}((Φ_{0} + ϕ) ⊗ (Φ_{0} + ϕ)^{∗})_{00} = v_{0},
(D_{A_{0}+a} + ρ(ϑ_{0}) + ϑ⃗ · 𝔪⃗(A))(Φ_{0} + ϕ) = ψ_{0},

where (v_{0}, ψ_{0}) ∈ Ω^{+}(X, 𝔰𝔲(E)) ⊕ Ω^{0}(X, W^{−} ⊗ E). Recalling that d_{A_{0},Φ_{0}}^{0} and d_{A_{0},Φ_{0}}^{1} are the differential operators in the elliptic deformation complex (2.37) for the PU(2) monopole equations (2.27), the above system may be rewritten in the form

d_{A_{0},Φ_{0}}^{0,∗}(a, ϕ) = ζ,
d_{A_{0},Φ_{0}}^{1}(a, ϕ) + {(a, ϕ), (a, ϕ)} = −𝔖(A_{0}, Φ_{0}) + (v_{0}, ψ_{0}) =: (v, ψ),

where (v, ψ) ∈ Ω^{+}(X, 𝔰𝔲(E)) ⊕ Ω^{0}(X, W^{−} ⊗ E), and the differentials d_{A_{0},Φ_{0}}^{0,∗} and d_{A_{0},Φ_{0}}^{1} are given by (2.38) and (2.36). It will be convenient to view the quadratic term {(a, ϕ), (a, ϕ)} as being defined via the following bilinear form,

{(a, ϕ), (b, φ)}

:= (\begin{matrix}(a ∧ b)^{+} − (id + τ_{0} ⊗ id_{𝔰𝔲(E)} + τ⃗ · 𝔪⃗(A)) ρ^{−1}(ϕ ⊗ φ^{∗})_{00} \\ ρ(a)φ \end{matrix}) ,

where (b, φ) ∈ Ω^{1}(X, 𝔰𝔲(E))⊕Ω^{0}(X, W^{+}⊗E); we will further abbreviate {(a, ϕ), (a, ϕ)} by q(a, ϕ) when convenient. Our elliptic system (3.1) then takes the simple shape

D_{A_{0},Φ_{0}}(a, ϕ) + {(a, ϕ), (a, ϕ)} = (ζ, v, ψ), (3.2)

recalling from (2.39) that D_{A_{0},Φ_{0}} = d_{A_{0},Φ_{0}}^{0,∗} + d_{A_{0},Φ_{0}}^{1}. This is the form of the (inhomogeneous) Coulomb gauge and PU(2) monopole equations we will use for the majority of the basic regularity arguments.

Some of the regularity results and estimates of this section generalize corresponding results for the first-order anti-self-dual equation [20],

[30] and, to a certain extent, those of Uhlenbeck [94] and Parker [70] for the second-order (coupled) Yang-Mills equations. As usual for a quasilinear, first-order elliptic system with a quadratic non-linearity, over an n-dimensional manifold, the main difficulty is to prove regularity for L_{1}^{n/2} solutions (that is, at the critical Sobolev exponent). Once the solutions are known to be in L^{∞}, then standard linear elliptic regularity theory can be applied [37], [64]. It is worth noting at the outset, though, that despite an extensive literature on quasi-linear elliptic systems due to Ladyzhenskaya-Ural’tseva, Morrey and others [54], [64], these classical results do not meet the usual demands of gauge theory and this explains why we develop the precise results we require here. The constants appearing in our estimates generally depend on the underlying Riemannian metric on X, the fixed spin^{c} connection on W, the fixed unitary connection on det E, and the perturbations (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗): this dependence is not explicitly noted, as these ingredients in the PU(2) monopole equations (2.27) are fixed once and for all.

3.1. Regularity for L_{1}^{2} solutions to the inhomogeneous Coulomb gauge and PU(2) monopole equations. We show in this subsection that an L_{1}^{2} solution (a, ϕ) to the PU(2) monopole and Coulomb-gauge equations (3.2), with an L_{k}^{2} inhomogeneous term (with k ≥ 1) is in L_{k+1}^{2}. Thus, if the inhomogeneous term is in C^{∞}, then (a, ϕ) is in C^{∞}. In passing from an L_{1}^{2} to an L_{2}^{2} solution we need only consider the case where (τ⃗, ϑ⃗) = 0, while no restriction is placed on the perturbation (τ⃗, ϑ⃗) ∈ P given an L_{2}^{2} solution (a, ϕ).

Our regularity result contains, as special cases, Proposition 4.4.13 in [20] and Theorem 8.8 in [30], for anti-self-dual connections, and Theorem 8.11 in [77] for Seiberg-Witten monopoles. The proof we give below for PU(2) monopoles is rather different. We provide a fairly detailed argument here since regularity of L_{2}^{2} solutions to an elliptic equation with a quadratic non-linear term does not quite follow directly from the standard theory for non-linear elliptic systems (for example, [64, Theorem 6.8.1]).

The two principal steps are, first, to get L_{1}^{p} regularity of an L_{1}^{2} solution (a, ϕ) when 2 < p < 4 and (a, ϕ) is sufficiently L^{4}-small and, second, to apply elliptic boostrapping and the Sobolev multiplication and embedding theorems to get C^{∞} regularity of an L_{1}^{p} solution (a, ϕ) when 2 < p < ∞. We will use these sharp regularity results and estimates repeatedly throughout our work, so we give the argument in some detail here. The main ingredient in the first step is supplied by Proposition 3.2 and uses a Fredholm alternative argument to pass from L_{1}^{2} to the slightly stronger L_{1}^{p} regularity [33, Theorem 5.3]. Although we will often be able to simply assume that the inhomogenous term is in C^{∞}, rather than just in L^{p}, we will later need these intermediate regularity results in our development of the gluing theory for PU(2) monopoles [26], [27] to show that L_{1}^{2} solutions (a, ϕ) to the system (3.2) with a certain inhomogeneous term (ζ, v, ψ), where the latter is initially only known to be in L_{1}^{2} or L_{2}^{2}, are actually in C^{∞} (together, of course, with (ζ, v, ψ)).

Lemma 3.1. Let 𝔅_{1}, 𝔅_{2} be Banach spaces and let

T ∈ Hom(𝔅_{1}, 𝔅_{2})

have a right (left) inverse P. If T′ ∈ Hom(𝔅_{1}, 𝔅_{2}) satisfies

∥T′ − T ∥ < ∥P ∥^{−1},

then T′ also has a right (left) inverse.

Proof. If P ∈ Hom(𝔅_{2}, 𝔅_{1}) is a right inverse for T, so T P = id_{1}, then

∥(T′ − T)P ∥ ≤ ∥T′ − T ∥∥P ∥ < 1,

and id_{1} + (T′ − T)P is an invertible element of the Banach algebra End(𝔅_{1}). Define P′ = P (1 + (T′ − T)P)^{−1}, so T′P′ = id_{1} and P′ is a right inverse for T′. Similarly for left inverses. q.e.d.

The preceding elementary consequence of the usual characterization of invertible elements of a Banach algebra will be frequently invoked in this section and in particular, in the proof of the proposition below.

Proposition 3.2. Let X be a closed, oriented, Riemannian fourmanifold with metric g and spin^{c} structure (ρ, W), and let E be a Hermitian two-plane bundle over X. Let (A_{0}, Φ_{0}) be a C^{∞} pair on the C^{∞} bundles (𝔰𝔲(E), W^{+} ⊗E) over X and let 2 ≤ p < 4. Then there are positive constants ε = ε(A_{0}, Φ_{0}, p) and C = C(A_{0}, Φ_{0}, p) with the following significance. Suppose that (a, ϕ) ∈ L_{1}^{2}(X, Λ^{1} ⊗ 𝔰𝔲(E)) ⊕ L_{1}^{2}(X, W^{+} ⊗ E) is an L_{1}^{2} solution on (𝔰𝔲(E), W^{+} ⊗ E) to the elliptic system (3.2) over X with (τ⃗, ϑ⃗) = 0, where (ζ, v, ψ) is in L^{p}. If ∥(a, ϕ)∥_{L^{4}(X)} < ε, then (a, ϕ) is in L_{1}^{p} and

∥(a, ϕ)∥_{L_{1,A_{0}}^{p}(X)} ≤ C (∥(ζ, v, ψ)∥_{L^{p}(X)} + ∥(a, ϕ)∥_{L^{2}(X)}) .

Proof. The operator D_{A_{0},Φ_{0}} is Fredholm (since it is elliptic with C^{∞} coefficients and X is closed), and so has a finite-dimensional kernel and cokernel. In particular, Ker D_{A_{0},Φ_{0}}|_{L_{1}^{2}} ⊂ C^{∞}. Let Π_{1} be the L^{2}-orthogonal projection onto Ker D_{A_{0},Φ_{0}}|_{L_{1}^{2}} and let Π_{2} be the L^{2}- orthogonal projection onto Ker D_{A_{0},Φ_{0}}^{∗}|_{L^{2}} = (Im D_{A_{0},Φ_{0}}|_{L_{1}^{2}})^{⊥}, the L^{2}- orthogonal complement of the image of Im D_{A_{0},Φ_{0}}|_{L_{1}^{2}}. The L^{2}-adjoint D_{A_{0},Φ_{0}}^{∗} is again a first-order linear elliptic operator with C^{∞} coefficients and so Ker D_{A_{0},Φ_{0}}^{∗}|_{L^{2}} ⊂ C^{∞}. We may then rewrite our quasi-linear elliptic equation (3.2) in the form

Π_{2}^{⊥}D_{A_{0},Φ_{0}}Π_{1}^{⊥}(a, ϕ) + Π_{2}^{⊥}{(a, ϕ), Π_{1}^{⊥}(a, ϕ)}
= −Π_{2} (D_{A_{0},Φ_{0}}(a, ϕ) + {(a, ϕ), Π_{1}^{⊥}(a, ϕ)})
− {(a, ϕ), Π_{1}(a, ϕ)} + (ζ, v, ψ) =: Υ,

where Π_{i}^{⊥} = id−Π_{i} for i = 1, 2. Thus, we need to consider the existence and uniqueness problem for solutions (b, φ) to

Π_{2}^{⊥}D_{A_{0},Φ_{0}}(b, φ) + Π_{2}^{⊥}{(a, ϕ), (b, φ)} = Υ, (3.3)

where (b, φ) ∈ L_{1}^{p} ∩ (Ker D_{A_{0},Φ_{0}})^{⊥}. Plainly, the operator

Π_{2}^{⊥}D_{A_{0},Φ_{0}} : L_{1}^{p} ∩ (Ker D_{A_{0},Φ_{0}})^{⊥} → L_{1}^{p} ∩ (Ker D_{A_{0},Φ_{0}}^{∗})^{⊥}

is (left and right) invertible and hence this will also be true for any operator from L_{1}^{p}∩(Ker D_{A_{0},Φ_{0}})^{⊥} to L^{p}∩(Ker D_{A_{0},Φ_{0}}^{∗})^{⊥}, such as Π_{2}^{⊥}D_{A_{0},Φ_{0}} + Π_{2}^{⊥}{(a, ϕ), · }, which is sufficiently close in the Hom(L_{1,A_{0}}^{p}, L^{p}) operator norm by Lemma 3.1. Since D_{A_{0},Φ_{0}} and D_{A_{0},Φ_{0}}^{∗} are first-order linear elliptic operators with C^{∞} coefficients and D_{A_{0},Φ_{0}}Π_{1} = 0 and D_{A_{0},Φ_{0}}^{∗}Π_{2} = 0, standard elliptic theory implies that

\begin{aligned}∥Π_{2}(ξ, w, ϱ)∥_{L_{k,A_{0}}^{r}} & ≤ C∥(ξ, w, ϱ)∥_{2},  (ξ, w, ϱ) ∈ L_{k}^{r}, \\ ∥Π_{1}(b, φ)∥_{L_{k,A_{0}}^{r}} & ≤ C∥(b, φ)∥_{2},  (b, φ) ∈ L_{k}^{r}, \end{aligned} (3.4)

for some constant C = C(A_{0}, Φ_{0}, k, r), whenever k ≥ 1 and rk ≥ 4/3 or k = 0 and r ≥ 2.

Since (a, ϕ) is in L_{1}^{2}, the terms D_{A_{0},Φ_{0}}(a, ϕ) and {(a, ϕ), (a, ϕ)} and {(a, ϕ), Π_{1}^{⊥}(a, ϕ)} are in L^{2}, while the term {(a, ϕ), Π_{1}(a, ϕ)} is in L_{1}^{2}. The terms Π_{2}D_{A_{0},Φ_{0}}(a, ϕ) and Π_{2}{(a, ϕ), Π_{1}^{⊥}(a, ϕ)} are each in C^{∞}, while (ζ, v, ψ) is in L^{p}, and so the right-hand side Υ of (3.3) is in L^{p} ∩ (Ker D_{A_{0},Φ_{0}}^{∗})^{⊥}.

Let q = 4p/(4 − p). Then 4 < q < ∞; there is a continuous multiplication map L^{4} × L^{q} → L^{p} and an embedding L_{1}^{p} ⊂ L^{q}. So, as Π_{2}^{⊥} = id − Π_{2},

∥Π_{2}^{⊥}{(a, ϕ), (b, φ)}∥_{L^{p}} ≤ ∥{(a, ϕ), (b, φ)}∥_{L^{p}} + C∥{(a, ϕ), (b, φ)}∥_{L^{2}}
≤ C∥(a, ϕ)∥_{L^{4}}∥(b, φ)∥_{L_{1,A_{0}}^{p}} ,

for some positive constant C = C(A_{0}, Φ_{0}, p). By Lemma 3.1 there is a positive constant ε = ε(A_{0}, Φ_{0}, p) such that if ∥(a, ϕ)∥_{L^{4}} < ε, the linear operator Π_{2}^{⊥}D_{A_{0},Φ_{0}} + Π_{2}^{⊥}{(a, ϕ), · } is (left and right) invertible as a map L_{1}^{p} ∩ (Ker D_{A_{0},Φ_{0}})^{⊥} → L^{p} ∩ (Ker D_{A_{0},Φ_{0}}^{∗})^{⊥}. Since Π_{1}^{⊥}(a, ϕ) is a solution to (3.3) when p = 2, it is the unique solution for p = 2.

Let (a_{p}, ϕ_{p}) ∈ L_{1}^{p} ∩ (Ker D_{A_{0},Φ_{0}})^{⊥} be the L_{1}^{p} solution to (3.3). Then (a_{p}, ϕ_{p}) is also an L_{1}^{2} solution and by the uniqueness assertion we must have that (a_{p}, ϕ_{p}) = Π_{1}^{⊥}(a, ϕ). Thus, Π_{1}^{⊥}(a, ϕ) is in L_{1}^{p} and so (a, ϕ) is in L_{1}^{p}, since Π_{1}(a, ϕ) is in C^{∞}. Finally, the standard estimate for D_{A_{0},Φ_{0}}, the estimate (3.4), and equation (3.2) yield

∥(a, ϕ)∥_{L_{1,A_{0}}^{p}} ≤ C (∥D_{A_{0},Φ_{0}}(a, ϕ)∥_{L^{p}} + ∥Π_{1}(a, ϕ)∥_{L^{2}})

≤ C (∥{(a, ϕ), (a, ϕ)}∥_{L^{p}} + ∥(ζ, v, ψ)∥_{L^{p}} + ∥Π_{1}(a, ϕ)∥_{L^{2}}) ,

and so the desired bound for (a, ϕ) follows by the Sobolev multiplication L^{4} × L^{q} → L^{p}, the embedding L_{1}^{p} ⊂ L^{q}, and rearrangement. q.e.d.

Proposition 3.2 will have two main applications: the first is in our proof of removable singularities for PU(2) monopoles, and the second is in our proof of C^{∞} regularity for L_{1}^{2} solutions to the PU(2) monopole equation obtained by gluing [26], [27].

As is well-known from standard gauge theory, it is not possible to construct a well-defined quotient space using L_{1}^{2} pairs modulo L_{2}^{2} gauge transformations. We construct a quotient using L_{k}^{2} pairs modulo L_{k+1}^{2} gauge transformations, with k ≥ 2. We first establish a regularity result for the inhomogeneous PU(2) monopole and Coulomb gauge equations, while in §3.2 we show that any PU(2) monopole in L_{k}^{2} is L_{k+1}^{2}-gauge equivalent to a PU(2) monopole in C^{∞}.

Proposition 3.3. Continue the notation of Proposition 3.2. Let k ≥ 1 be an integer and let 2 < p < ∞. Let (A_{0}, Φ_{0}) be a C^{∞} pair on the C^{∞} bundles (𝔰𝔲(E), W^{+} ⊗ E) over X. Suppose that either

• (a, ϕ) ∈ L_{1}^{p}(X, Λ^{1} ⊗ 𝔰𝔲(E)) ⊕ L_{1}^{p}(X, W^{+} ⊗ E), with (τ⃗, ϑ⃗) = 0, or

• (a, ϕ) ∈ L_{2}^{2}(X, Λ^{1} ⊗ 𝔰𝔲(E)) ⊕ L_{2}^{2}(X, W^{+} ⊗ E)

is a solution on (𝔰𝔲(E), W^{+} ⊗ E) to the elliptic system (3.2) over X, where (ζ, v, ψ) is in L_{k}^{2}. Then (a, ϕ) is in L_{k+1}^{2} and there is a universal polynomial Q_{k}(x, y), with positive real coefficients depending at most on (A_{0}, Φ_{0}), k, such that Q_{k}(0, 0) = 0 and

∥(a, ϕ)∥_{L_{k+1,A_{0}}^{2}(X)} ≤ Q_{k} (∥(ζ, v, ψ)∥_{L_{k,A_{0}}^{2}(X)}, ∥(a, ϕ)∥_{L_{1,A_{0}}^{p}(X)}) .

In particular, if (ζ, v, ψ) is in C^{∞}, then (a, ϕ) is in C^{∞}, and if (ζ, v, ψ) = 0, then

∥(a, ϕ)∥_{L_{k+1,A_{0}}^{2}(X)} ≤ C∥(a, ϕ)∥_{L_{1,A_{0}}^{p}(X)}.

Proof. We consider the cases k = 1, k = 2, and k ≥ 3 separately. We may further assume without loss of generality that 2 < p < 4.

Case (k = 1) . Let p_{0} = p and q_{0} = q = 4p/(4 − p). Then 1/p = 1/4 + 1/q and 4 < q < ∞. We have a continuous Sobolev multiplication map L^{4} × L^{q} → L^{p} and an embedding L_{1}^{p} ⊂ L^{q}. Since

D_{A_{0},Φ_{0}}(a, ϕ) = −{(a, ϕ), (a, ϕ)} + (ζ, v, ψ), (3.5)

elliptic regularity for D_{A_{0},Φ_{0}} implies that (a, ϕ) ∈ L_{1}^{p_{1}}, where p_{1} = 2p/(4 − p) and q_{1} = 4p_{1}/(4 − p_{1}), so 2 < p_{1} < 4 and 4 < q_{1} < ∞. Here, we have used the Sobolev multiplication L^{4} × L^{q_{1}} → L^{p_{1}} to get {(a, ϕ), (a, ϕ)} in L^{p_{1}} and the embedding L_{1}^{2} ⊂ L^{p_{1}} to get (ζ, v, ψ) in L^{p_{1}}. Let p = 2 + ε and δ = ε/(2 − ε), and note that

p_{1} = \frac{2p}{4 − p} = \frac{2p}{2 − ε} = (1 + δ)p > 2 + ε.

If p_{j} < 4, we inductively define p_{j+1} = 2p_{j}/(4 − p_{j}) and q_{j+1} = 4p_{j}/(4 − p_{j}) for j ≥ 0. Therefore, we have p_{j} > 2 + ε. Thus

p_{j+1} = \frac{2p_{j}}{4 − p_{j}} > \frac{2p_{j}}{2 − ε} = (1 + δ)p_{j} > p_{j} > 2 + ε,

and so p_{j+1} > (1 + δ)^{j+1}p for j ≥ 0.

By repeating the above regularity argument when 2 < p_{j+1} < 4, using (3.5) at each stage, we see that (a, ϕ) ∈ L_{1}^{p_{j}} for j ≥ 0. We continue the induction until for large enough j ≥ 0, we find that p′ := p_{j+1} ≥ 8/3 and q′ := q_{j+1} ≥ 8, so (a, ϕ) ∈ L_{1}^{8/3} ⊂ L^{8}. Therefore, with (ζ, v, ψ) ∈ L_{1}^{2} ⊂ L^{4} and using the Sobolev multiplication L^{8} × L^{8} → L^{4} to get {(a, ϕ), (a, ϕ)} in L^{4}, equation (3.5) gives (a, ϕ) ∈ L_{1}^{4}. The Sobolev multiplication L_{1}^{4} × L_{1}^{4} → L_{1}^{2} implies that the quadratic term {(a, ϕ), (a, ϕ)} is in L_{1}^{2} and so (3.5) yields (a, ϕ) ∈ L_{2}^{2}, as required.

Case (k = 2) . From the case k = 1 we continue the induction until p′ > 8/3 and q′ > 4. Hence, with (ζ, v, ψ) now in L_{2}^{2} ⊂ L_{1}^{p′}, equation (3.5) gives (a, ϕ) ∈ L_{2}^{p′} ⊂ C^{0}. The Sobolev multiplication L_{2}^{p′} × L_{2}^{2} → L_{2}^{2} implies that the quadratic term {(a, ϕ), (a, ϕ)} is then in L_{2}^{2} and so (3.5) gives (a, ϕ) ∈ L_{4}^{2}, as required.

Case (k ≥ 3) . There is continuous Sobolev multiplication map L_{k}^{2} × L_{k}^{2} → L_{k}^{2} and so the quadratic term {(a, ϕ), (a, ϕ)} is in L_{k}^{2}, since (a, ϕ) is in L_{k}^{2}. Therefore, (3.5) gives (a, ϕ) ∈ L_{k+1}^{2}.

This completes the proof of the proposition. q.e.d.

By combining Propositions 3.2 and 3.3 we obtain the desired regularity result for L_{1}^{2} solutions to the inhomogeneous Coulomb gauge and PU(2) monopole equations:

Corollary 3.4. Continue the notation of Proposition 3.2. Let (A_{0}, Φ_{0}) be a C^{∞} pair on the C^{∞} bundles (𝔰𝔲(E), W^{+} ⊗ E) over X. Then there is a positive constant ε = ε(A_{0}, Φ_{0}) such that the following hold. Suppose that either

• (a, ϕ) ∈ L_{1}^{2}(X, Λ^{1} ⊗ 𝔰𝔲(E)) ⊕ L_{1}^{2}(X, W^{+} ⊗ E), with (τ⃗, ϑ⃗) = 0, or

• (a, ϕ) ∈ L_{2}^{2}(X, Λ^{1} ⊗ 𝔰𝔲(E)) ⊕ L_{2}^{2}(X, W^{+} ⊗ E)

is a solution on (𝔰𝔲(E), W^{+} ⊗ E) to the elliptic system (3.2) over X, where (ζ, v, ψ) is in L_{k}^{2} and ∥(a, ϕ)∥_{L^{4}(X)} < ε and k ≥ 0 is an integer. Then (a, ϕ) is in L_{k+1}^{2} and there is a universal polynomial Q_{k}(x, y), with positive real coefficients depending at most on (A_{0}, Φ_{0}), k, such that Q_{k}(0, 0) = 0 and

∥(a, ϕ)∥_{L_{k+1,A_{0}}^{2}(X)} ≤ Q_{k} (∥(ζ, v, ψ)∥_{L_{k,A_{0}}^{2}(X)}, ∥(a, ϕ)∥_{L^{2}(X)}) .

In particular, if (ζ, v, ψ) is in C^{∞} then (a, ϕ) is in C^{∞}, and if (ζ, v, ψ) = 0, then

∥(a, ϕ)∥_{L_{k+1,A_{0}}^{2}(X)} ≤ C∥(a, ϕ)∥_{L^{2}(X)}.

Remark 3.5. A similar sharp regularity result for solutions to the Coulomb and anti-self-dual equations (that is, (d_{Γ}^{∗}+d_{Γ}^{+})a+(a∧a)^{+} = v) on the product bundle over S^{4} is given by Proposition 4.4.13 in [20]. The reader is forwarned that Corollary 3.4 does not apply directly to show that L_{1}^{2} gluing solutions to the PU(2) monopole equation in [26] are C^{∞} due to the unfavorable dependence of the constant ε(A′, Φ′, p) on the approximate PU(2) monopole (A′, Φ′) when p > 2. However, a local version of this result, namely Corollary 3.11 below, is applicable in this situation. The point is explained further in [26].

3.2. Regularity of L_{k}^{2} solutions to the PU(2) monopole equations. We explain in this subsection why an L_{k}^{2} monopole (A, Φ) on the C^{∞} bundles (𝔰𝔲(E), W^{+} ⊗ E) over X is gauge equivalent, via an L_{k+1}^{2} determinant-one, unitary automorphism u of the bundle E, to a C^{∞} solution u(A, Φ) on (𝔰𝔲(E), W^{+} ⊗ E) over X when k ≥ 2. This regularity result contains, as special cases, Theorem 8.8 in [30], for anti-self-dual connections, and Theorem 8.11 in [77] for Seiberg-Witten monopoles.

We will need the following observation concerning the symmetry of the Coulomb gauge equation for pairs; the corresponding fact for connections is explained in [20, p. 56].

Lemma 3.6. Let (A, Φ), (A_{0}, Φ_{0}) be L_{k}^{2} pairs on the bundles (𝔰𝔲(E), W^{+} ⊗ E) over X. If (A_{0}, Φ_{0}) is in Coulomb gauge relative to (A, Φ), so d_{A,Φ}^{0,∗}((A_{0}, Φ_{0}) − (A, Φ)) = 0, then (A, Φ) is in Coulomb gauge relative to (A_{0}, Φ_{0}), so d_{A_{0},Φ_{0}}^{0,∗}((A, Φ) − (A_{0}, Φ_{0})) = 0.

Proof. The equation d_{A,Φ}^{0,∗}((A_{0}, Φ_{0}) − (A, Φ)) = 0 is the Euler-Lagrange equation for the functional

G_{E}^{2,k+1} ∋ u ↦ ∥u(A_{0}, Φ_{0}) − (A, Φ)∥_{L^{2}}^{2},

while the equation d_{A_{0},Φ_{0}}^{0,∗}((A, Φ) − (A_{0}, Φ_{0})) = 0 is the Euler-Lagrange equation for the functional G_{E}^{2,k+1} ∋ v ↦ ∥v(A, Φ) − (A_{0}, Φ_{0})∥_{L^{2}}^{2}.

But for any u ∈ G_{E}^{2,k+1} we have

∥u(A_{0}, Φ_{0}) − (A, Φ)∥_{L^{2}} = ∥u^{−1}(A, Φ) − (A_{0}, Φ_{0})∥_{L^{2}},

and so if the functional u ↦ ∥u(A_{0}, Φ_{0}) − (A, Φ)∥_{L^{2}}^{2} has a critical point at u = id_{E}, then the functional u^{−1} ↦ ∥u^{−1}(A, Φ) − (A_{0}, Φ_{0})∥_{L^{2}}^{2} will also have a critical point at u = id_{E}. q.e.d.

Proposition 3.7. Let X be a closed, oriented, Riemannian fourmanifold with spin^{c} bundle W and let E be a Hermitian two-plane bundle over X. Let k ≥ 2 be an integer and suppose that (A, Φ) is an L_{k}^{2} solution to (2.27) on the C^{∞} bundles (𝔰𝔲(E), W^{+} ⊗ E) over X. Then there is a L_{k+1}^{2} determinant-one, unitary automorphism u of the bundle E over X such that u(A, Φ) is C^{∞} over X.

Proof. It suffices, of course, to show that there is an L_{k+1}^{2} gauge transformation u of E over such that u(A, Φ) is in L_{k+1}^{2}. The C^{∞} pairs (A_{0}, Φ_{0}) on (𝔰𝔲(E), W^{+} ⊗ E) form a dense subspace of the space of L_{k}^{2} pairs and so, given ε = ε(A, Φ) > 0, there is a C^{∞} pair (A_{0}, Φ_{0}) such that

∥(A, Φ) − (A_{0}, Φ_{0})∥_{L_{k,A_{0}}^{2}} < ε.

For small enough ε, Proposition 2.8 gives an L_{k+1}^{2} determinant-one, unitary automorphism u of the bundle E such that u^{−1}(A_{0}, Φ_{0}) is in Coulomb gauge relative to (A, Φ):

d_{A,Φ}^{0,∗}(u^{−1}(A_{0}, Φ_{0}) − (A, Φ)) = 0.

Now u(d_{A,Φ}^{0,∗}(u^{−1}(A_{0}, Φ_{0}) − (A, Φ))) = d_{u(A,Φ)}^{0,∗}((A_{0}, Φ_{0}) − u(A, Φ)) and so

d_{u(A,Φ)}^{0,∗}((A_{0}, Φ_{0}) − u(A, Φ)) = 0.

Therefore, (A_{0}, Φ_{0}) is in Coulomb gauge relative to u(A, Φ) and Lemma 3.6 implies that u(A, Φ) is in Coulomb gauge relative to (A_{0}, Φ_{0}),

d_{A_{0},Φ_{0}}^{0,∗}(u(A, Φ) − (A_{0}, Φ_{0})) = 0.

Let (a, ϕ) = u(A, Φ) − (A_{0}, Φ_{0}), so that

(a, ϕ) ∈ L_{k}^{2}(X, Λ^{1} ⊗ 𝔰𝔲(E)) ⊕ L_{k}^{2}(X, W^{+} ⊗ E);

the Coulomb gauge condition then takes the simpler form d_{A_{0},Φ_{0}}^{0,∗}(a, ϕ) = 0. Since u(A, Φ) = (A_{0}, Φ_{0}) + (a, ϕ) is an L_{k}^{2} monopole, 𝔖((A_{0}, Φ_{0}) + (a, ϕ)) = 0 and so (a, ϕ) is an L_{k}^{2} solution to the quasi-linear elliptic equation

D_{A_{0},Φ_{0}}(a, ϕ) + {(a, ϕ), (a, ϕ)} = −𝔖(A_{0}, Φ_{0}),

with C^{∞} data −𝔖(A_{0}, Φ_{0}). The conclusion now follows from Proposition 3.3 with ζ = 0 and (v, ψ) = −𝔖(A_{0}, Φ_{0}). q.e.d.

One of the convenient practical consequences of Proposition 3.7 is that we can always work, modulo global L_{k+1}^{2} gauge transformations, with C^{∞} rather than L_{k}^{2} monopoles. For the remainder of this article, therefore, the term ‘PU(2) monopole’ is generally reserved for C^{∞} solutions to the perturbed PU(2) monopole equations (2.27). In a similar vein, we generally reserve the terms ‘gauge transformation’ or ‘bundle map’ for gauge transformations or bundle maps which are in C^{∞}.

Given Proposition 3.7, we then have the following analogue of Proposition 4.2.16 in [20], the corresponding result for the moduli space of anti-self-dual connections. The proof is standard and so is left to the reader.

Corollary 3.8. Continue the hypotheses of Proposition 3.7. Then for any k ≥ 2 the natural inclusion of topological spaces M_{W,E}^{k} ↪ M_{W,E}^{k+1} is a homeomorphism.

Thus, the topology of the moduli space M_{W,E}^{k} of L_{k}^{2} monopoles is independent of the Sobolev spaces used in its construction for k ≥ 2 and so we simply denote the moduli space by M_{W,E}.

3.3. Local regularity and interior estimates for L_{1}^{2} solutions to the inhomogeneous Coulomb gauge and PU(2) monopole equations. In this section we specialize the results of §3.1 to the case where the reference pair is a trivial PU(2) monopole, so (A_{0}, Φ_{0}) = (Γ, 0) on the bundles (𝔰𝔲(E), W^{+} ⊗ E), over an open subset Ω ⊂ X, where Γ is a flat connection.

We continue to assume that X is a closed, oriented four-manifold with metric g, spin^{c} bundle W, and Hermitian two-plane bundle E extending those on Ω ⊂ X. We use the inhomogenous estimates and regularity results in our proof of removable singularities for PU(2) monopoles in §4.3 and in our development of the gluing theory for PU(2) monopoles in sequels [26], [27] to the present article — especially to show that a global L_{1}^{2} gluing solution (a, ϕ) is actually C^{∞}. We use the homogeneous estimates and regularity results in §4 for our proof of the existence of an Uhlenbeck compactification for the moduli space of PU(2) monopoles.

We have the following local versions of Propositions 3.2 and 3.3 and Corollary 3.4:

Proposition 3.9. Continue the notation of the preceding paragraph. Let Ω′ ⋐ Ω be a precompact open subset and let 2 ≤ p < 4. Then there are positive constants ε = ε(Ω, p) and C = C(Ω′, Ω, p) with the following significance. Suppose that (a, ϕ) is an L_{1}^{2}(Ω) solution to the elliptic system (3.2) over Ω for (τ⃗, ϑ⃗) = 0, with (A_{0}, Φ_{0}) = (Γ, 0) and where (ζ, v, ψ) is in L^{p}(Ω). If ∥(a, ϕ)∥_{L^{4}(Ω)} < ε, then (a, ϕ) is in L_{1}^{p}(Ω′) and

∥(a, ϕ)∥_{L_{1,Γ}^{p}(Ω′)} ≤ C (∥(ζ, v, ψ)∥_{L^{p}(Ω)} + ∥(a, ϕ)∥_{L^{2}(Ω)}) .

Proof. Choose an open subset Ω′′ such that Ω′ ⋐ Ω′′ ⋐ Ω. Let χ be C^{∞} cutoff function such that supp χ ⊂ Ω′′ and χ = 1 on Ω′. Let β be a cutoff function such that β = 1 on supp χ and supp β ⊂ Ω′′. Since (a, ϕ) is a solution to (3.2) with right-hand side (ζ, v, ψ) over Ω, then χ(a, ϕ) solves

D_{Γ,0}χ(a, ϕ) + {(a, ϕ), χ(a, ϕ)}
= χD_{Γ,0}(a, ϕ) + dχ ⊗ (a, ϕ) + χ{(a, ϕ), (a, ϕ)}
= χ(ζ, v, ψ) + dχ ⊗ (a, ϕ) =: (ζ′, v′, ψ′).

Thus, χ(a, ϕ) is an L_{1}^{2} solution over X to the linear elliptic system over X,

D_{Γ,0}(b, φ) + {β(a, ϕ), (b, φ)} = (ζ′, v′, ψ′),

with L_{1}^{2} coefficient β(a, ϕ) and L_{1}^{2} right-hand side (ζ′, v′, ψ′). Since ∥β(a, ϕ)∥_{L^{4}(X)} ≤ ∥(a, ϕ)∥_{L^{4}(Ω)}, the proof of Proposition 3.2 implies that χ(a, ϕ) is in L_{1}^{p} over X if ε = ε(g, Ω, p) is sufficiently small. Thus, (a, ϕ) is in L_{1}^{p} over Ω′ with

∥χ(a, ϕ)∥_{L_{1,Γ}^{p}(X)} ≤ C (∥(ζ′, v′, ψ′)∥_{L^{p}(X)} + ∥χ(a, ϕ)∥_{L^{2}(X)}) ,

and therefore, for 2 ≤ p < 4, we have

∥(a, ϕ)∥_{L_{1,Γ}^{p}(Ω′)} ≤ C (∥(ζ, v, ψ)∥_{L^{p}(Ω′′)} + ∥(a, ϕ)∥_{L^{p}(Ω′′)}) .

The preceding bound and the Sobolev embedding L_{1}^{2} ⊂ L^{p} give the estimate

∥(a, ϕ)∥_{L^{p}(Ω′′)} ≤ c∥(a, ϕ)∥_{L_{1,Γ}^{2}(Ω′′)} ≤ C (∥(ζ, v, ψ)∥_{L^{2}(Ω)} + ∥(a, ϕ)∥_{L^{2}(Ω)}) .

Combining these inequalities then yields the required L_{1}^{p} estimate for (a, ϕ) over Ω′. q.e.d.

Proposition 3.10. Continue the notation of Proposition 3.9. Let k ≥ 1 be an integer, and let 2 < p < ∞. Suppose that either

• (a, ϕ) is an L_{1}^{p}(Ω), when (τ⃗, ϑ⃗) = 0, or

• (a, ϕ) is an L_{2}^{2}(Ω)

solution to the elliptic system (3.2) over Ω with (A_{0}, Φ_{0}) = (Γ, 0), where (ζ, v, ψ) is in L_{k}^{2}(Ω). Then (a, ϕ) is in L_{k+1}^{2}(Ω′) and there is a universal polynomial Q_{k}(x, y), with positive real coefficients depending at most on k, Ω′, Ω, such that Q_{k}(0, 0) = 0 and

∥(a, ϕ)∥_{L_{k+1,Γ}^{2}(Ω′)} ≤ Q_{k} (∥(ζ, v, ψ)∥_{L_{k,Γ}^{2}(Ω)}, ∥(a, ϕ)∥_{L_{1,Γ}^{p}(Ω)}) .

If (ζ, v, ψ) is in C^{∞}(Ω), then (a, ϕ) is in C^{∞}(Ω′) and if (ζ, v, ψ) = 0, then

∥(a, ϕ)∥_{L_{k+1,Γ}^{2}(Ω′)} ≤ C∥(a, ϕ)∥_{L_{1,Γ}^{p}(Ω)}.

Proof. Again, choose an open subset Ω′′ such that Ω′ ⋐ Ω′′ ⋐ Ω, let χ be C^{∞} cutoff function such that supp χ ⊂ Ω, and χ = 1 on Ω′′ and let β be a cutoff function such that β = 1 on supp χ and supp β ⊂ Ω. Since (a, ϕ) is a solution to (3.2) with right-hand side (ζ, v, ψ) over Ω, χ(a, ϕ) solves

D_{Γ,0}χ(a, ϕ) + {χ(a, ϕ), χ(a, ϕ)}
= χD_{Γ,0}(a, ϕ) + dχ ⊗ (a, ϕ) + χ{(a, ϕ), (a, ϕ)}
+ χ(χ − 1){(a, ϕ), (a, ϕ)}
= χ(ζ, v, ψ) + dχ ⊗ (a, ϕ) + χ(χ − 1){(a, ϕ), (a, ϕ)}
=: (ζ′, v′, ψ′).

Note that χ(ζ, v, ψ) is in L_{k}^{2}(X), while dχ ⊗ (a, ϕ) is in L_{1}^{p}(X), and the Sobolev multiplication L^{q}×L^{q} → L^{p_{1}} implies that χ(χ−1){(a, ϕ), (a, ϕ)} is in L^{p_{1}}(X), where p_{1} = q/2 = 2p/(4 − p) > p. Thus, (ζ′, v′, ψ′) is in L^{p_{1}}(X).

The proof of the case k = 1 in Proposition 3.3 then implies that χ(a, ϕ) is an L_{1}^{p_{1}}(X), so (a, ϕ) is in L_{1}^{p_{1}}(Ω′′). We now repeat this process for each of the remaining steps in the proof of Proposition 3.3, at each stage on successively smaller open subsets Ω′′′ such that Ω′ ⋐ Ω′′′ ⋐ Ω′′, until we obtain (a, ϕ) in L_{k+1}^{2}(Ω′) and the desired L_{k+1}^{2}(Ω′) estimate. q.e.d.

Corollary 3.11. Continue the notation of Proposition 3.10. Then there is a positive constant ε = ε(Ω) with the following significance. Suppose that either

• (a, ϕ) is an L_{1}^{2}(Ω), when (τ⃗, ϑ⃗) = 0, or

• (a, ϕ) is an L_{2}^{2}(Ω)

solution to the elliptic system (3.2) over Ω with (A_{0}, Φ_{0}) = (Γ, 0), where (ζ, v, ψ) is in L_{k}^{2}(Ω) and ∥(a, ϕ)∥_{L^{4}(Ω)} < ε. Then (a, ϕ) is in L_{k+1}^{2}(Ω′) and there is a universal polynomial Q_{k}(x, y), with positive real coefficients, depending at most on k, Ω′, Ω, such that Q_{k}(0, 0) = 0 and

∥(a, ϕ)∥_{L_{k+1,Γ}^{2}(Ω′)} ≤ Q_{k} (∥(ζ, v, ψ)∥_{L_{k,Γ}^{2}(Ω)}, ∥(a, ϕ)∥_{L^{2}(Ω)}) .

If (ζ, v, ψ) is in C^{∞}(Ω) then (a, ϕ) is in C^{∞}(Ω′), and if (ζ, v, ψ) = 0, then

∥(a, ϕ)∥_{L_{k+1,Γ}^{2}(Ω′)} ≤ C∥(a, ϕ)∥_{L^{2}(Ω)}.

Corollary 3.11 thus yields a sharp local elliptic regularity result for PU(2) monopoles (A, Φ) in L_{1}^{2} which are given to us in Coulomb gauge relative to (Γ, 0): this regularity result is the key ingredient in our proof (given in §4.3) of removable point singularities for PU(2) monopoles.

Proposition 3.12. Continue the notation of Corollary 3.11. Then there is a positive constant ε = ε(Ω) and, if k ≥ 1 is an integer, there is a positive constant C = C(Ω′, Ω, k) with the following significance. Suppose that either

• (A, Φ) is an L_{1}^{2}, when (τ⃗, ϑ⃗) = 0, or

• (A, Φ) is an L_{2}^{2}

solution to the PU(2) monopole equations (2.27) over Ω, which is in Coulomb gauge over Ω relative to (Γ, 0), so d_{Γ}^{∗}(A − Γ) = 0, and obeys ∥(A−Γ, Φ)∥_{L^{4}(Ω)} < ε. Then (A−Γ, Φ) is in C^{∞}(Ω′) and for any k ≥ 1,

∥(A − Γ, Φ)∥_{L_{k,Γ}^{2}(Ω′)} ≤ C∥(A − Γ, Φ)∥_{L^{2}(Ω)}.

Proof. Corollary 3.11 applies to the L_{1}^{2}(Ω) pair (a, ϕ) = (A − Γ, Φ) and yields the required regularity and estimates for (A − Γ, Φ) with (ζ, v, ψ) = 0 in (3.2). q.e.d.

3.4. Estimates for PU(2) monopoles in a good local gauge. It remains to combine the local regularity results and estimates of §3.3, for PU(2) monopoles (A, Φ) where the connection A is assumed to be in Coulomb gauge relative to the product SO(3) connection Γ, with Uhlenbeck’s local, Coulomb gauge-fixing theorem. We then obtain regularity results and estimates for PU(2) monopoles (A, Φ) with small curvature F_{A}, parallel to those of Theorem 2.3.8 and Proposition 4.4.10 in [20] for anti-self-dual connections.

In order to apply Corollary 3.11 we need Uhlenbeck’s Coulomb gauge-fixing result [95, Theorem 2.1 & Corollary 2.2]). Let B (respectively, B̄) be the open (respectively, closed) unit ball centered at the origin in ℝ^{4} and let G be a compact Lie group. In order to provide universal constants we assume ℝ^{4} has its standard metric, though the results of this subsection naturally hold for any C^{∞} Riemannian metric, with comparable constants for metrics which are suitably close.

Theorem 3.13. There are positive constants c and ε with the following significance. If 2 ≤ p < 4 is a constant and

A ∈ L_{1}^{p}(B, Λ^{1} ⊗ 𝔤) ∩ L_{1}^{p}(∂B, Λ^{1} ⊗ 𝔤)

is a connection matrix whose curvature satisfies ∥F_{A}∥_{L^{p}(B)} < ε, then there is an gauge transformation u ∈ L_{2}^{p}(B, G) ∩ L_{2}^{p}(∂B, G) such that u(A) := uAu^{−1} − (du)u^{−1} satisfies

(1)  d^{∗}u(A) = 0 on B,
(2)  \frac{∂}{∂r} ⌟u(A) = 0 on ∂B,
(3)  ∥u(A)∥_{L_{1}^{p}(B)} ≤ c∥F_{A}∥_{L^{p}(B)}.

If A is in L_{k}^{p}(B), for k ≥ 2, then u is in L_{k+1}^{p}(B), and the gauge transformation u is unique up to multiplication by a constant element of G.

Remark 3.14. If G is abelian, then the requirement that

∥F_{A}∥_{L^{p}(B)} < ε

can be omitted.

It is often useful to rephrase Theorem 3.13 in two other slightly different ways. Suppose A is an L_{k}^{2} connection on a C^{∞} principal G bundle P over B with k ≥ 2 and ∥F_{A}∥_{L^{2}(B)} < ε. Then the assertions of Theorem 3.13 are equivalent to each of the following:

• There is an L_{k+1}^{2} trivialization τ : P → B × G such that

(i) d_{Γ}^{∗}(τ(A) − Γ) = 0, where Γ is the product connection on B × G,

(ii) \frac{∂}{∂r} ⌟(τ(A) − Γ) = 0, and

(iii) ∥(τ(A) − Γ)∥_{L_{1}^{2}(B)} ≤ c∥F_{A}∥_{L^{2}(B)}.

• There is an L_{k+1}^{2} flat connection Γ on P such that

(i) d_{Γ}^{∗}(A − Γ) = 0,

(ii) \frac{∂}{∂r} ⌟(A − Γ) = 0, and

(iii) ∥(A − Γ)∥_{L_{1}^{2}(B)} ≤ c∥F_{A}∥_{L^{2}(B)}, and an L_{k+1}^{2} trivialization P |_{B} ≃ B × G taking Γ to the product connection.

We can now combine Theorem 3.13 with Proposition 3.12 to give the following analogue of Theorem 2.3.8 in [20] — the interior estimate for anti-self-dual connections with L^{2}-small curvature.

Corollary 3.15. Let B ⊂ ℝ^{4} be the open unit ball with center at the origin with spin^{c} structure (ρ, W), let U ⋐ B be an open subset, and let Γ be the product connection on B × SO(3). Then there is a positive constant ε and if ℓ ≥ 1 is an integer, there is a positive constant C(ℓ, U) with the following significance. Suppose that either

• (A, Φ) is an L_{1}^{2}, when (τ⃗, ϑ⃗) = 0, or

• (A, Φ) is an L_{k}^{2}, k ≥ max{2, ℓ},

solution to the PU(2) monopole equations (2.27) over B and that the curvature of the SO(3) connection matrix A obeys ∥F_{A}∥_{L^{2}(B)} < ε. Then there is an L_{k+1}^{2} gauge transformation u : B → SU(2) such that (u(A)− Γ, uΦ) is in C^{∞}(B) with d^{∗}(u(A) − Γ) = 0 over B and

∥(u(A) − Γ, uΦ)∥_{L_{ℓ,Γ}^{2}(U)} ≤ C∥F_{A}∥_{L^{2}(B)}^{1/2}.

Proof. Let ε_{1} be the constant in Theorem 3.13 and note that for ε ≤ ε_{1}, Theorem 3.13 (taking G = SO(3)) and the Sobolev embedding L_{1}^{2}(B) ⊂ L^{4}(B) imply that there is an L_{k+1}^{2} gauge transformation u : B → SU(2) (by lifting the SO(3) gauge transformation) such that

d_{Γ}^{∗}(u(A) − Γ) = 0,
∥u(A) − Γ∥_{L^{4}(B)} ≤ c_{1}∥F_{A}∥_{L^{2}(B)} < c_{1}ε.

On the other hand, the quadratic equation for Φ in (2.27) and Lemma 2.26 give the L^{4} and L^{2} estimates,

∥uΦ∥_{L^{4}(B)} = ∥Φ∥_{L^{4}(B)} ≤ 2∥F_{A}^{+}∥_{L^{2}(B)}^{1/2},
∥uΦ∥_{L^{2}(B)} ≤ 2∥F_{A}∥_{L^{1}(B)}^{1/2}.

Let ε_{2} be the constant in Proposition 3.12. Hence, if c_{1}ε ≤ ε_{2} and 4\sqrt{ε} ≤ ε_{2}, then Proposition 3.12 implies that (u(A) − Γ, uΦ) obeys

∥(u(A) − Γ, uΦ)∥_{L_{ℓ,Γ}^{2}(U)} ≤ C∥(u(A) − Γ, uΦ)∥_{L^{2}(B)},

since d_{Γ}^{∗}(u(A) − Γ) = 0 and ∥(u(A) − Γ, uΦ)∥_{L^{4}(B)} < ε_{2}. The desired estimate follows by combining these inequalities for small enough ε ≤ 1.

q.e.d.

Again, it is often useful to rephrase Corollary 3.15 in the two other slightly different ways. Suppose that ℓ ≥ 1 and that (A, Φ) is a PU(2) monopole in L_{k}^{2} on (𝔰𝔲(E), W^{+} ⊗ E) over the unit ball B ⊂ ℝ^{4} with k ≥ max{2, ℓ}, ∥F_{A}∥_{L^{2}(B)} < ε and U ⋐ B. Then the assertions of Corollary 3.15 are equivalent to each of the following:

• There is a C^{∞} trivialization τ : E|_{B} → B × ℂ^{2} and an L_{k+1}^{2} determinant-one, unitary bundle automorphism u of E|_{B} such that, with respect to the product connection Γ on B × 𝔰𝔲(2), we have

(i) d_{Γ}^{∗}(τu(A) − Γ) = 0, and

(ii) ∥(τu(A) − Γ, τuΦ)∥_{L_{ℓ,Γ}^{2}(U)} ≤ C∥F_{A}∥_{L^{2}(B)}^{1/2}.

• There is an L_{k+1}^{2} flat connection Γ on 𝔰𝔲(E)|_{B} such that

(i) d_{Γ}^{∗}(A − Γ) = 0, and

(ii) ∥(A − Γ, Φ)∥_{L_{ℓ,Γ}^{2}(U)} ≤ c∥F_{A}∥_{L^{2}(B)}^{1/2}, and an L_{k+1}^{2} trivialization 𝔰𝔲(E)|_{B} ≃ B × 𝔰𝔲(2) taking Γ to the product connection.

Corollary 3.15 immediately yields the following local compactness result for PU(2) monopoles analogous to the local compactness result for anti-self-dual connections in [20, Corollary 2.3.9].

Corollary 3.16. Let B ⊂ ℝ^{4} be the open unit ball and spin^{c} structure (ρ, W). Then there is a positive constant ε_{0}(g, A_{det W^{+}}, A_{det E}) with the following significance. Let U ⋐ B be an open subset and let k ≥ 2 be an integer. If (A_{α}, Φ_{α}) is a sequence of PU(2) monopoles in L_{k}^{2} on (𝔰𝔲(E), W^{+} ⊗ E) over B such that

∥F_{A_{α}}∥_{L^{2}(B)} < ε_{0},

then there is a subsequence {α′} ⊂ {α}, a sequence of determinantone, unitary L_{k+1}^{2} automorphisms {u_{α′}} of E|_{B} and a sequence of gauge equivalent pairs (Ã_{α′}, Φ̃_{α′}) := u_{α′}(A_{α′}, Φ_{α′}) which converge in L_{k,loc}^{2} on U to a PU(2) monopole (Ã, Φ̃) over U.

We will also need interior estimates for PU(2) monopoles in a good local gauge over more general simply-connected regions than the open balls considered in Corollary 3.15. Specifically, recall that a domain Ω ⊂ X is strongly simply-connected if it has an open covering by balls D_{1}, …, D_{m} (not necessarily geodesic) such that for 1 ≤ r ≤ m the intersection D_{r} ∩ (D_{1} ∪ ⋯ ∪ D_{r−1}) is connected. We recall (see [20, Proposition 2.2.3] or [43, Proposition I.2.6]):

Proposition 3.17. If Γ is a C^{∞} flat connection on a principal G bundle P over a simply-connected manifold Ω, then there is a C^{∞} isomorphism P ≃ Ω × G taking Γ to the product connection on Ω × G.

More generally, if A is a C^{∞} connection on a G bundle P over a simply-connected manifold-with-boundary Ω̄ = Ω ∪ ∂Ω with L^{p}-small curvature (with p > 2), then Uhlenbeck’s theorem implies that A is L_{2}^{p}-gauge equivalent to a connection which is L_{1}^{p}-close to an L_{1}^{p} flat connection on P (see [96, Corollary 4.3] or [20, p. 163]). The following a priori interior estimate is a straightforward generalization of [20, Proposition 4.4.10]: the method of proof is identical to that described in [20, pp. 161–162] and uses a patching argument for gauge transformations employed by Uhlenbeck in the proof of Theorem 3.6 in [95]. The required bound for the connection in terms of its curvature is obtained by covering the given open region with balls and applying the estimate of Corollary 3.15 in place of Theorem 2.3.8 in [20].

We recall from Proposition 2.16 that any automorphism of the SO(3) bundle 𝔰𝔲(E)|_{Ω} over a simply-connected open subset Ω ⊂ X lifts to a determinant-one, unitary bundle automorphism of E|_{Ω}. The method of [20, pp. 161–162] then yields:

Proposition 3.18. Let X be a closed, oriented, Riemannian fourmanifold with spin^{c} structure (ρ, W) and let Ω ⊂ X be a strongly simplyconnected open subset. Then there is a positive constant ε(Ω) with the following significance. For Ω′ ⋐ Ω a precompact open subset and an integer ℓ ≥ 1, there is a constant C(ℓ, Ω′, Ω) such that the following holds. Suppose (A, Φ) is a PU(2) monopole in L_{k}^{2} on (𝔰𝔲(E), W^{+} ⊗ E) over Ω with k ≥ max{2, ℓ} such that

∥F_{A}∥_{L^{2}(Ω)} < ε.

Then there is an L_{k}^{2} flat connection Γ on 𝔰𝔲(E)|_{Ω′} such that

∥(A − Γ, Φ)∥_{L_{ℓ,Γ}^{2}(Ω′)} ≤ C∥F_{A}∥_{L^{2}(Ω)}^{1/2},

and an L_{k+1}^{2} trivialization 𝔰𝔲(E)|_{Ω′} ≃ Ω′ × 𝔰𝔲(2) taking Γ to the product connection.

4. Uhlenbeck compactification for the moduli space of PU(2) monopoles

Our goal in this section is to prove the existence of an Uhlenbecktype compactification of the moduli space of PU(2) monopoles analogous to the Uhlenbeck compactification of the moduli space of anti-self-dual connections [20]. In §4.1 we establish the Bochner-Weitzenböck formulas for the coupled Dirac operators D_{A} and D_{A}^{∗} and the a priori bounds satisfied by the section Φ and the curvature F_{A}, if (A, Φ) is a PU(2) monopole on (𝔰𝔲(E), W^{+} ⊗E); these generalize the well-known a priori bounds for Seiberg-Witten monopoles [47], [62], [77]. In our proof of removable singularities we need to take into account the variation of spin^{c} structures with the Riemannian metric and the rescaling behavior of the PU(2) equations; this variation is discussed in §4.2.

In §4.3 we prove the removability of point singularities for PU(2) monopoles. T. Parker established the removability of point singularities for solutions to the second-order coupled Yang-Mills equations, namely, d_{A}^{∗}F_{A} = q(Φ) and D_{A}Φ = 0, for a unitary connection A on E and a section Φ of W ⊗ E, where q is a certain quadratic form [70]. This generalizes the corresponding results of Uhlenbeck in the case Φ = 0, where the above system then reduces to the second-order Yang-Mills equation [30], [94]; proofs of removable singularities for anti-self-dual connections are given in [20] (see also [96]). D. Salamon has given a proof of removable singularities for Seiberg-Witten monopoles [77, Chapter 9]. The arguments of Uhlenbeck and Parker rely on pointwise curvature and energy decay estimates; Salamon’s method relies on energy decay estimates and elliptic regularity results for Seiberg-Witten monopoles. The proof we give for PU(2) monopoles is rather different and instead relies heavily on our C^{∞} regularity result for Coulomb-gauge PU(2) monopoles in L_{1}^{2} (Proposition 3.12); this is similar to the strategy used by Donaldson and Kronheimer in [20].

The technical ingredients we need for patching sequences of local gauge transformations are described in §4.4, and the Uhlenbeck closure M̄_{W,E} is defined in §4.5, by analogy with the corresponding definition for the moduli space of anti-self-dual connections in [20, §4.4]. In §4.5.2 we describe how the holonomy perturbations extend continously with respect to Uhlenbeck limits and induce holonomy perturbations on all lower-level moduli spaces of PU(2) monopoles. We then define the Uhlenbeck closure for the moduli space of perturbed PU(2) monopoles.

In §4.6 we prove Theorem 1.1, which asserts that the Uhlenbeck closure M̄_{W,E} is compact. The main analytical ingredients in the proof comprise the regularity results and estimates of §3; the a priori bounds of §4.1 provide a ‘universal energy bound’ for a PU(2) monopole (A, Φ) and this bound plays the same role here in establishing the existence of an Uhlenbeck compactification as the usual topological bound for the energy of an anti-self-dual connection in [20]. The scale invariance of the PU(2) monopole equation described in §4.2 is used here in much the same way that the conformal invariance of the anti-self-dual equation is exploited in [20].

The parameters τ_{0}, ϑ_{0}, τ⃗, ϑ⃗ are chosen so that the perturbation estimates (2.29) are satisfied. These bounds follow from Proposition A.13, with k ≥ 2 for the first inequality in (2.29) and k ≥ 3 for the second, if

∥ϑ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))} < ε_{ϑ} and ∥τ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))} < ε_{τ} , (4.1)

with suitable positive constants ε_{τ} and ε_{ϑ}. These constraints are needed in the proofs of the a priori estimates in Lemmas 4.2, 4.3, and 4.4 of §4.1 and hence in the proof of Theorem 4.20 in §4.6. For the remainder of the article we therefore require that k ≥ 3.

4.1. Bochner formulas and a priori estimates for PU(2) monopoles. In this section we apply the Bochner-Weitzenböck formula for D_{A}^{∗}D_{A} together with Kato’s inequality and the maximum principle to derive a priori estimates for solutions (A, Φ) to the PU(2) monopole equations (2.27). These estimates then lead to a vanishing result generalizing that of [98, pp. 781–782].

We first have a generalization of the usual Bochner-Weitzenböck identity for Seiberg-Witten monopoles [47], [62], [77], [89], [98].

Lemma 4.1. Let X be an oriented, Riemannian, four-manifold with spin^{c} structure (ρ, W), and let E be a Hermitian bundle over X. If A denotes a U(2) connection on E,

∇_{A} : Ω^{0}(W^{±} ⊗ E) → Ω^{1}(W^{±} ⊗ E)

are the covariant derivatives defined by A, and

D_{A} : Ω^{0}(W^{+} ⊗ E) → Ω^{0}(W^{−} ⊗ E)

the Dirac operator, then

(1a)  D_{A}^{∗}D_{A} = ∇_{A}^{∗}∇_{A} + \frac{1}{4}R + ρ_{+}(F_{A}^{+}) + \frac{1}{2}ρ_{+}(F_{A_{L}}^{+}),
(2a)  D_{A}D_{A}^{∗} = ∇_{A}^{∗}∇_{A} + \frac{1}{4}R + ρ_{−}(F_{A}^{−}) + \frac{1}{2}ρ_{−}(F_{A_{L}}^{+}),

where R is the scalar curvature of the Riemannian metric and A_{L} = A_{det W^{+}} is the induced connection on L = det W^{+} ≃ det W^{−}. If A denotes an SO(3) connection on 𝔰𝔲(E), then

(1b)  D_{A}^{∗}D_{A} = ∇_{A}^{∗}∇_{A} + \frac{1}{4}R + ρ_{+}(F_{A}^{+}) + \frac{1}{2}ρ_{+}(F_{A_{L}}^{+} + F_{A_{e}}^{+}),
(2b)  D_{A}D_{A}^{∗} = ∇_{A}^{∗}∇_{A} + \frac{1}{4}R + ρ_{−}(F_{A}^{−}) + \frac{1}{2}ρ_{−}(F_{A_{L}}^{−} + F_{A_{e}}^{+}),

where A_{e} = A_{det E} is the fixed connection on det E, and the identification ad : 𝔰𝔲(E) ≃ 𝔰𝔬(𝔰𝔲(E)) is implicit.

Proof. We just consider the first pair of identities, as the second follow immediately from the fact that

F (A_{E}) = F (A_{𝔰𝔲(E)}) + \frac{1}{2}F (A_{det E}) ⊗ id_{E}.

Over any sufficiently small local coordinate neighborhood U in X we have a local spin structure and a Hermitian spin bundle S such that W |_{U} = S ⊗ L^{1/2}, where L^{1/2} is a Hermitian line bundle over U such that (L^{1/2})^{⊗2} = L|_{U} and having an induced unitary connection \frac{1}{2}A_{L}. Applying the Bochner-Weitzenböck identity of [57, Theorem II.8.17] to the Hermitian bundle S ⊗ L^{1/2} ⊗ E over U with unitary connection ∇_{A}, given by the tensor product of ∇^{S}, ∇_{2^{−1}A_{L}}^{L^{1/2}}, and ∇_{A}^{E}, yields

D_{A}^{2} = ∇_{A}^{∗}∇_{A} + \frac{1}{4}R + ρ(F_{L^{1/2}⊗E}),

where F_{L^{1/2}⊗E} is the curvature of the tensor product connection ∇_{2^{−1}A_{L}}^{L^{1/2}} ⊗ id_{E} + id_{L^{1/2}} ⊗ ∇_{A}^{E} on L^{1/2} ⊗ E. Since

F_{L^{1/2}⊗E} = F_{2^{−1}A_{L}} ⊗ id_{E} + id_{L^{1/2}} ⊗ F_{A} = \frac{1}{2}F_{A_{L}} ⊗ id_{E} + id_{L^{1/2}} ⊗ F_{A},

we have

ρ(F_{L^{1/2}⊗E}) = \frac{1}{2}ρ(F_{A_{L}}) ⊗ id_{E} + id_{L^{1/2}} ⊗ ρ(F_{A}),

and hence on Ω^{0}(U, W ⊗ E),

D_{A}^{2} = ∇_{A}^{∗}∇_{A} + \frac{1}{4}R + \frac{1}{2}ρ(F_{A_{L}}) + ρ(F_{A}),

which is plainly independent of the local splitting W = S ⊗ L^{1/2} and so gives an identity on Ω^{0}(X, W ⊗E). From the decomposition ρ = ρ_{+}⊕ρ_{−} we see that

ρ^{±}(F_{A})Φ^{±} = ρ^{±}(F_{A}^{±})Φ^{±} and ρ^{±}(F_{A_{L}})Φ^{±} = ρ^{±}(F_{A_{L}}^{±})Φ^{±},

for any Φ^{±} ∈ Ω^{0}(X, W^{±} ⊗ E), and so the result follows. q.e.d.

As in the case of the abelian monopole equations, the equations (2.27) and Lemma 4.1 combine to give a priori estimates for solutions (A, Φ) which play essential role in the proof of existence of the Uhlenbeck compactification.

Lemma 4.2. Let X be a closed, oriented, Riemannian four-manifold with spin^{c} structure (ρ, W), and Hermitian line bundle det E with fixed unitary connection. Then there is a positive constant K_{1}, depending only on the L^{2} norms of the scalar curvature R and the curvatures F (A_{det E}) and F (A_{det W^{+}}), such that the following holds. If (A, Φ) is an L_{1}^{2} solution on (𝔰𝔲(E), W^{+} ⊗ E) to the PU(2) monopole equations (2.27) over X, then

∥Φ∥_{L^{4}(X)} ≤ K_{1} and ∥∇_{A}Φ∥_{L^{2}(X)} ≤ K_{1}.

Proof. We may assume that (A, Φ) is a C^{∞} pair. Then Lemma 4.1 gives

(D_{A}^{∗}D_{A}Φ, Φ) =(∇_{A}^{∗}∇_{A}Φ, Φ) + \frac{1}{4}(RΦ, Φ) + (ρ(F_{A}^{+})Φ, Φ)
+ \frac{1}{2}(ρ(F_{A_{L}}^{+} + F_{A_{e}}^{+})Φ, Φ),

while the first equation in (2.27) implies that

ρ(F_{A}^{+}) = ρτ⃗_{A}ρ^{−1}(Φ ⊗ Φ^{∗})_{00}.

So, using the second equation in (2.27) and integration by parts, we obtain

∥ϑ⃗_{A}Φ∥_{L^{2}}^{2} = ∥D_{A}Φ∥_{L^{2}}^{2}
= ∥∇_{A}Φ∥_{L^{2}}^{2} + \frac{1}{4}(RΦ, Φ) + (ρτ⃗_{A}ρ^{−1}(Φ ⊗ Φ^{∗})_{00}Φ, Φ)
+ \frac{1}{2}(ρ(F_{A_{L}}^{+} + F_{A_{e}}^{+})Φ, Φ).

Consequently, Lemmas 2.18 and 2.26, Hölder’s inequality, and the estimate for τ⃗_{A} in (2.29) leads to the bound

\frac{1}{2}∥Φ∥_{L^{4}}^{4} + ∥∇_{A}Φ∥_{L^{2}}^{2}
≤ (\frac{1}{4}∥R∥_{L^{2}} + \frac{1}{2}∥F_{A_{L}}^{+}∥_{L^{2}} + \frac{1}{2}∥F_{A_{e}}^{+}∥_{L^{2}} + ∥ϑ⃗_{A}∥_{L^{4}}^{2}) ∥Φ∥_{L^{4}}^{2}.

Thus, if Φ ≢ 0, the above inequality and the estimate for ϑ⃗_{A} in (2.29) give the required L^{4} estimate for Φ. Then the above inequality and the L^{4} estimate for Φ yield the L^{2} bound for ∇_{A}Φ. The bounds hold trivially if Φ ≡ 0. q.e.d.

The preceding a priori L^{4} estimate on the section Φ yields a priori L^{2} bounds on the components of the curvature, F_{A}^{+} and F_{A}^{−}, if (A, Φ) is a PU(2) monopole:

Lemma 4.3. Continue the notation and hypotheses of Lemma 4.2. Then there is a positive constant K_{2}^{+}, depending only on the L^{2} norms of the scalar curvature R, the curvatures F (A_{det E}) and F (A_{det W^{+}}), and a positive constant K_{2}^{−}, depending only on K_{2}^{+} and p_{1}(𝔰𝔲(E)), such that if (A, Φ) is an L_{1}^{2} solution on (𝔰𝔲(E), W^{+} ⊗ E) to the PU(2) monopole equations (2.27), then

∥F_{A}^{+}∥_{L^{2}} ≤ K_{2}^{+} and ∥F_{A}^{−}∥_{L^{2}} ≤ K_{2}^{−}.

Proof. Lemma 2.26 and the first PU(2) monopole equation in (2.27) imply that

∥F_{A}^{+}∥_{L^{2}} ≤ \frac{3}{4}∥Φ∥_{L^{4}}^{2},

and so the L^{4} bound for Φ in Lemma 4.2 gives the first estimate. The Chern-Weil integral identity (2.33) yields

∫_{X} |F_{A}^{−}|^{2}dV ≤ ∫_{X} |F_{A}^{+}|^{2}dV + 8π^{2} (c_{2}(E) − \frac{1}{4}c_{1}(E)^{2}) ,

and so the second estimate follows from the first. q.e.d.

Lemmas 4.2 and 4.3 imply an a priori bound K on the ‘energy’ of a PU(2) monopole (A, Φ) in terms of the scalar curvature R, the connections on det W^{+} and det E and p_{1}(𝔰𝔲(E)):

∫_{X} (|F_{A}|^{2} + |Φ|^{4} + |∇_{A}Φ|^{2}) dV ≤ K. (4.2)

We use Lemmas 4.2 and 4.3 to provide the ‘energy bound’ assumed in our proof of removable singularities (Theorem 4.10) and we use Lemma 4.3 to give a lower bound on the second Chern class of an ideal monopole appearing in the Uhlenbeck compactification of the moduli space of PU(2) monopoles (see §4.5 and the conclusion of the proof of Theorem 4.20 in §4.6).

As in the case of the U(1) monopole equations [47], [98], the Bochner-Weitzenböck identity and the maximum principle yield a priori C^{0} estimates for Φ and F_{A}^{+} when (A, Φ) is a PU(2) monopole.

Lemma 4.4. Continue the notation and hypotheses of Lemma 4.2. Then there is a non-negative constant K_{3}, depending only on the C^{0} norms of the scalar curvature R and the curvatures F (A_{det E}) and F (A_{det W^{+}}), such that the following holds. If (A, Φ) is a C^{1} solution on (𝔰𝔲(E), W^{+} ⊗ E) to the PU(2) monopole equations (2.27), then

∥Φ∥_{C^{0}(X)}^{2} ≤ K_{3} and ∥F_{A}^{+}∥_{C^{0}(X)} ≤ K_{3}.

Proof. We may assume that the pair (A, Φ) is C^{∞}. The analogue of equation (6.18) of [30] for Hermitian bundles (see [77, §8.3]) reads

\frac{1}{2}Δ|Φ|^{2} + |∇_{A}Φ|^{2} = Re⟨∇_{A}^{∗}∇_{A}Φ, Φ⟩,

where Δ = d^{∗}d on Ω^{0}(X, ℝ). Since Φ is continuous, |Φ| achieves its maximum at some point x_{0} ∈ X, so Δ|Φ|^{2}(x_{0}) = −∇_{e_{μ}}∇_{e_{μ}}|Φ|^{2}(x_{0}) ≥ 0 and thus at x_{0}:

Re⟨∇_{A}^{∗}∇_{A}Φ, Φ⟩ ≥ |∇_{A}Φ|^{2}.

Let {e_{μ}} be a local oriented, orthonormal frame for the tangent bundle T X such that (∇_{e_{μ}}e^{ν})_{x_{0}} = 0, where {e^{μ}} is the dual coframe for the cotangent bundle T^{∗}X. Since D_{A}Φ = −ϑ⃗_{A}Φ by the second equation in (2.27), at x_{0} we have

⟨D_{A}^{∗}D_{A}Φ, Φ⟩ = −⟨D_{A}^{∗}(ϑ⃗_{A}Φ), Φ⟩
= −⟨ρ(e^{μ})(∇_{e_{μ}} ϑ⃗_{A})Φ, Φ⟩ − ⟨ρ(e^{μ})ϑ⃗_{A}∇_{e_{μ}}^{A}Φ, Φ⟩,

and so (using the inequality ab ≤ \frac{1}{4}a^{2} + b^{2}) we get the following bound at x_{0}:

|⟨D_{A}^{∗}D_{A}Φ, Φ⟩| ≤ 4|∇_{A}ϑ⃗_{A}||Φ|^{2} + 4|ϑ⃗_{A}||∇_{A}Φ||Φ|
≤ 4|∇_{A}ϑ⃗_{A}||Φ|^{2} + 4 (|ϑ⃗_{A}|^{2}|Φ|^{2} + \frac{1}{4}|∇_{A}Φ|^{2})
= 4 (|∇_{A}ϑ⃗_{A}| + |ϑ⃗_{A}|^{2}) |Φ|^{2} + |∇_{A}Φ|^{2}.

Recall that the first PU(2) monopole equation in (2.27) gives

ρ(F_{A}^{+}) = ρτ⃗_{A}ρ^{−1}(Φ ⊗ Φ^{∗})_{00}.

The endomorphism ρτ⃗_{A}ρ^{−1}(Φ⊗Φ^{∗})_{00} of W^{+} ⊗E lies in 𝔰𝔲(W^{+})⊗𝔰𝔲(E) and in particular is Hermitian, so

Re⟨ρτ⃗_{A}ρ^{−1}(Φ ⊗ Φ^{∗})_{00}Φ, Φ⟩ = ⟨ρτ⃗_{A}ρ^{−1}(Φ ⊗ Φ^{∗})_{00}Φ, Φ⟩,

and similarly for the endomorphism ρ(F_{A}^{+}). We now combine Lemmas 2.26 and 4.1 and the first equation in (2.27) to get an estimate for Φ at the point x_{0}:

\frac{1}{2}|Φ|^{4} ≤ ⟨ρτ⃗_{A}ρ^{−1}(Φ ⊗ Φ^{∗})_{00}Φ, Φ⟩
= ⟨ρ(F_{A}^{+})Φ, Φ⟩
= Re⟨D_{A}^{∗}D_{A}Φ, Φ⟩ − Re⟨∇_{A}^{∗}∇_{A}Φ, Φ⟩ − \frac{1}{4}⟨RΦ, Φ⟩
− \frac{1}{2}⟨ρ(F_{A_{L}}^{+} + F_{A_{e}}^{+})Φ, Φ⟩.

We now combine the last inequality with our estimates for ⟨D_{A}^{∗}D_{A}Φ, Φ⟩ and ⟨∇_{A}^{∗}∇_{A}Φ, Φ⟩ to obtain the following bound for Φ at x_{0}:

\frac{1}{2}|Φ|^{4} ≤ 4 (|∇_{A}ϑ⃗_{A}| + |ϑ⃗_{A}|^{2}) |Φ|^{2} + |∇_{A}Φ|^{2}
− |∇_{A}Φ|^{2} − \frac{1}{4}R|Φ|^{2} + \frac{1}{2} (|F_{A_{L}}^{+}| + |F_{A_{e}}^{+}|) |Φ|^{2}

≤ 4 (|∇_{A}ϑ⃗_{A}| + |ϑ⃗_{A}|^{2}) |Φ|^{2} − \frac{1}{4}R|Φ|^{2} + \frac{1}{2} (|F_{A_{L}}^{+}| + |F_{A_{e}}^{+}|) |Φ|^{2}.

Either Φ(x_{0}) = 0, and so Φ is identically zero, or at the point x_{0} the preceding inequality implies that

|Φ|^{2} ≤ 8 (|∇_{A}ϑ⃗_{A}| + |ϑ⃗_{A}|^{2}) − \frac{1}{2} inf_{X} R + |F_{A_{L}}^{+}| + |F_{A_{e}}^{+}|,

which gives the first desired estimate. Then the second desired estimate follows from Lemma 2.26, the first equation in (2.27), and the estimate for ϑ⃗_{A} in (2.29). q.e.d.

We use Lemma 4.4 in §4.6 to show that the curvature of a PU(2) monopole connection A concentrates at points with integer multiplicities given by the second Chern classes of limiting (ideal) anti-self-dual connections over S^{4} (see Lemma 4.21). These C^{0} estimates yield the following analogue of Witten’s vanishing theorem [98, §3] for U(1) monopoles over four-manifolds with non-negative scalar curvature.

Corollary 4.5. Continue the notation and hypotheses of Lemma 4.4 and suppose K_{3} ≤ 0. If (A, Φ) is a C^{1} solution on (𝔰𝔲(E), W^{+} ⊗ E) to the PU(2) monopole equations (2.27), then Φ ≡ 0 and F_{A}^{+} ≡ 0.

Remark 4.6. The proof of Lemma 4.4 shows that

K_{3} = max \{0, −\frac{1}{2} inf_{X} R + 8∥ϑ⃗_{A}∥_{L_{1,A}^{∞}(X)} + ∥F_{A_{L}}^{+}∥_{C^{0}(X)} + ∥F_{A_{e}}^{+}∥_{C^{0}(X)}\} .

In particular, we see that if X = S^{4} has its round metric of scalar curvature R = 1, standard spin^{c} structure with c_{1}(W^{+}) = 0 and F_{A_{L}} = 0, the Hermitian bundle E has c_{1}(E) = 0 and F_{A_{e}} = 0, and we have ϑ⃗ = 0, then Φ ≡ 0 and A is an anti-self-dual SO(3) connection.

4.2. Scale invariance of the PU(2) monopole equations. In this section we describe the behavior of the PU(2) monopole under rescaling of the Riemannian metric. As is well-known, the anti-self-dual equation is conformally invariant. Although the PU(2) monopole equations are not conformally invariant they are, like the Seiberg-Witten equations, invariant under constant rescalings of the metric in a sense we describe below. The proof of the existence of an Uhlenbeck compactification (Theorem 1.1) in §4 relies on local regularity and removable singularity results for solutions to (2.27) over the unit ball B in ℝ^{4}. The requirements that the L^{2} norm of the curvature F_{A} and the L_{1}^{2} norm of section Φ be sufficiently small are met via a rescaling argument.

Suppose that λ > 0 is a constant and that the Riemannian metric g on T^{∗}X is replaced by λ^{2}g. Since the Clifford map ρ is compatible with g, the Clifford map

ρ_{λ^{2}g} ≡ λρ_{g} : T^{∗}X → End(W)

is compatible with λ^{2}g, as for any α ∈ Ω^{1}(X, ℝ) it satisfies

ρ_{λ^{2}g}(α)^{†}ρ_{λ^{2}g}(α) = λ^{2}ρ_{g}(α)^{†}ρ_{g}(α) = λ^{2}g(α, α)id_{W} .

It extends in the usual way to a linear map

ρ_{λ^{2}g} : Λ^{•}(T^{∗}X) ⊗ ℂ → End(W).

The Levi-Civita connection on T^{∗}X for the metric g coincides with the Levi-Civita connection on T^{∗}X for the rescaled metric λ^{2}g, so the SO(4) connection on T^{∗}X induced by the unitary connection on W and the Clifford map ρ_{λ^{2}g} : T^{∗}X → End(W) is still torsion free.

Lemma 4.7. If (A, Φ) is a solution to the PU(2) monopole equations (2.27) for the metric g on T^{∗}X, then (A, λΦ) is a solution to the PU(2) monopole equations for the rescaled metric λ^{2}g on T^{∗}X, where λ is a positive constant.

Proof. The projection P^{+}(λ^{2}g) = \frac{1}{2}(1 + ∗_{λ^{2}g}) from Λ^{2}(T^{∗}X) to Λ^{+}(T^{∗}X) is given by P^{+}(λ^{2}g) = P^{+}(g), while the induced map ρ_{λ^{2}g} : Λ^{2}(T^{∗}X) → End(W) is given by ρ_{λ^{2}g} = λ^{2}ρ_{g}. Therefore,

ρ_{λ^{2}g}(ϑ) = λρ_{g}(ϑ), ϑ ∈ Ω^{1}(X, ℂ),
D_{A}^{λ^{2}g} = λD_{A}^{g},
ρ_{λ^{2}g}(P^{+}(λ^{2}g)F_{A}) = λ^{2}ρ_{g}(P^{+}(g)F_{A}).

Consequently, we see from (2.27) that if (A, Φ) is a solution for the metric g, then (A, λΦ) is a solution for the metric λ^{2}g. q.e.d.

Remark 4.8. By adapting the proof of Theorem II.5.24 in [57] we see that if g is replaced by the conformally equivalent metric h^{−2}g, then D_{A}^{h^{−2}g} = h^{5/2}D_{A}^{g}h^{−3/2}. Thus, while the proof of Lemma 4.7 adapts without change to show that the first equation in (2.27) is invariant under the transformation (A, Φ) ↦ (A, hΦ) when g ↦ h^{−2}g, the second equation (when ϑ = 0) is invariant under the transformation (A, Φ) ↦ (A, h^{3/2}Φ). It is this incompatibility which prevents the PU(2) monopole equations from being conformally invariant, although they are scale invariant in the sense described above.

4.3. Removable singularities. Given the sharp local elliptic regularity result of Proposition 3.12 for PU(2) monopoles in Coulomb gauge in L_{1}^{2}, we can now establish a removable singularities theorem for PU(2) monopoles analogous to Theorem 4.1 in [94] in the case of the Yang-Mills equations, and Theorem 8.1 in [70] in the case of the coupled Yang-Mills equations. Our method is modelled on the proof of Theorem 4.4.12 in [20] — the removable singularities result for the anti-self-dual equation — which uses a local elliptic regularity result for L_{1}^{2} solutions to the Coulomb gauge and inhomogeneous anti-self-dual equation (namely, Proposition 4.4.13 in [20]) and which in turn has its antecedent in the proof of Theorem 4.5 in [96]. This, of course, is not the only possible approach: Uhlenbeck’s original argument [94] employed a differential inequality to obtain a pointwise decay estimate for solutions near the singular point, and this was the method generalized by Parker to the case of certain coupled Yang-Mills equations; see [77, §9.2] for a proof of removable singularities for Seiberg-Witten monopoles which also uses differential inequalities.

It will be convenient to define the following annuli in X, given a point x_{0} ∈ X and a positive constant r:

Ω(x_{0}; r) := \{x ∈ X : \frac{1}{8}r < dist_{g}(x, x_{0}) < r\},
Ω′(x_{0}; r) := \{x ∈ X : \frac{1}{4}r < dist_{g}(x, x_{0}) < \frac{1}{2}r\} ⋐ Ω(x_{0}, r).

If X = ℝ^{4} and x_{0} = 0 and r = 1, we denote Ω = Ω(1) and Ω′ = Ω′(1). We will need the following special case of Proposition 3.18.

Lemma 4.9. Let ℝ^{4} have a C^{∞} Riemannian metric g, let Ω ⊂ ℝ^{4} be an open subset with spin^{c} structure (ρ, W), let E be a Hermitian twoplane product bundle over Ω, and let Ω′ ⋐ Ω be an open subset. Then there are positive constants C, ε such that if (A, Φ) is a PU(2) monopole in C^{∞} on (𝔰𝔲(E), W^{+} ⊗ E) over Ω with

∥F_{A}∥_{L^{2}(Ω)} < ε,

then there is a C^{∞} flat connection Γ on 𝔰𝔲(E)|_{Ω′} such that

∥A − Γ∥_{L^{4}(Ω′)} + ∥∇_{Γ}(A − Γ)∥_{L^{2}(Ω′)} ≤ c∥F_{A}∥_{L^{2}(Ω)},

and a C^{∞} trivialization 𝔰𝔲(E)|_{Ω′} ≃ Ω′ × 𝔰𝔲(2) taking Γ to the product connection.

The proof of Theorem 4.10 relies on a cutting off procedure to ‘smooth out’ the PU(2) monopole near the singular point, using a family of cutoff functions which we now define. Let χ : ℝ → [0, 1] be a bump function such that χ(t) = 1 if t ≥ \frac{1}{2} and χ(t) = 0 if t ≤ \frac{1}{4}. For any r ∈ (0, ϱ), where ϱ is the injectivity radius of (X, g), define a C^{∞} cutoff function on X by setting χ_{r}(·) = χ(dist_{g}(·, x_{0})/r). Thus, we have χ_{r} = 0 on the ball B(x_{0}, \frac{1}{4}r), while χ_{r} = 1 on X − B(x_{0}, \frac{1}{2}r) and so dχ_{r} is supported in Ω′(x_{0}, r).

Theorem 4.10. Let B ⊂ ℝ^{4} be a geodesic ball with C^{∞} metric g and center at the origin, spin^{c} structure (ρ, W) over B, and Hermitian two-plane bundle E over B\{0}. Suppose (A, Φ) is a C^{∞} solution to the PU(2) monopole equations (2.27) on (𝔰𝔲(E), W^{+} ⊗ E) with (τ⃗, ϑ⃗) = 0 over the punctured ball B \ {0} and finite energy,

∫_{B\{0}} (|F_{A}|^{2} + |Φ|^{4} + |∇_{A}Φ|^{2}) dV < ∞.

Then there are a Hermitian two-plane bundle Ẽ over B with det Ẽ = det E, a C^{∞} PU(2) monopole (Ã, Φ̃) on (𝔰𝔲(Ẽ), W^{+}⊗Ẽ) over the ball B, and a C^{∞}, determinant-one unitary bundle isomorphism u : E|_{B\{0}} → Ẽ|_{B\{0}} such that

u(A, Φ) = (Ã, Φ̃) over B \ {0}.

Remark 4.11. We restrict our attention to the case of (τ⃗, ϑ⃗) = 0 in the PU(2) monopole equations (2.27) since the holonomy perturbations are undefined for the L_{1}^{2} connections which arise in the proof of Theorem 4.10. However, there is no loss of generality in making this restriction as the holonomy perturbations vanish near points where curvature has bubbled off.

Proof. We may suppose without loss of generality that the ball B has radius less than or equal to one. Since ∥F_{A}∥_{L^{2}(B)} < ∞, then ∥F_{A}∥_{L^{2}(B_{r_{0}})} tends to zero as r_{0} tends to zero. Hence, for small enough r_{0} ∈ (0, 1], we may suppose that ∥F_{A}∥_{L^{2}(B_{r_{0}})} < ε, where ε is the constant of Lemma 4.9. Thus, for any r ∈ (0, r_{0}), Lemma 4.9 provides a C^{∞} flat connection Γ′_{r} on 𝔰𝔲(E)|_{Ω′(r)} such that

∥A − Γ′_{r}∥_{L^{4}(Ω′(r))} ≤ c∥F_{A}∥_{L^{2}(Ω(r))}, (4.3)

where the constant c is independent of r ∈ (0, r_{0}). To see that the constant c is indeed scale invariant, note that by Lemma 4.7 the pair (A, r^{−1}Φ) is a PU(2) monopole over B with respect to the rescaled metric g_{r} := r^{−2}g, so Ω′_{g}(r) = Ω′_{g_{r}}(1) and Ω_{g}(r) = Ω_{g_{r}}(1). We then apply Lemma 4.9 to the annuli Ω′_{g_{r}}(1) ⋐ Ω_{g_{r}}(1) and observe that the L^{4} norm on one-forms and the L^{2} norm on two-forms are scale invariant.

Lemma 4.9 also provides a C^{∞} trivialization 𝔰𝔲(E)|_{Ω′(r)} ≃ Ω′(r) × 𝔰𝔲(2) taking Γ′_{r} to the product connection on Ω′(r) × 𝔰𝔲(2). We can then define a smooth SO(3) bundle V_{r} over B by setting

V_{r} := \begin{cases}B(\frac{1}{2}r) × 𝔰𝔲(2) & over B(\frac{1}{2}r), \\ 𝔰𝔲(E) & over B − B(\frac{1}{4}r), \end{cases}

recalling that Ω′(r) = B(\frac{1}{2}r) − B̄(\frac{1}{4}r). Let Γ′_{r} denote the C^{∞} flat connection on V_{r} over the ball B(\frac{1}{2}r), extending Γ′_{r} on 𝔰𝔲(E)|_{Ω′(r)} via the product connection on B(\frac{1}{2}r) × 𝔰𝔲(2), and let E_{r} be the smooth U(2) bundle over B with det E_{r} = det E and 𝔰𝔲(E_{r}) = V_{r} over B.

Now let (A_{r}, Φ_{r}) be the C^{∞} pair on B defined by

(A_{r}, Φ_{r}) := \begin{cases}(Γ′_{r} + χ_{r}(A − Γ′_{r}), χ_{r}Φ) & over B(\frac{1}{2}r), \\ (A, Φ) & over B \ B(\frac{1}{4}r), \end{cases}

where we note that χ_{r} = 0 on B(\frac{1}{4}r) and χ_{r} = 1 on B − B(\frac{1}{2}r). To estimate the L^{2} norm of F_{A_{r}}, note that over B −B(\frac{1}{2}r) we have A_{r} = A and F_{A_{r}} = F_{A}, while over B(\frac{1}{2}r),

F_{A_{r}} = χ_{r}F_{A} + dχ_{r} ∧ (A − Γ′_{r}) + (χ_{r}^{2} − χ_{r})(A − Γ′_{r}) ∧ (A − Γ′_{r}).

Hence, by (4.3), we have

∥F_{A_{r}}∥_{L^{2}(Ω′(r))} ≤∥F_{A}∥_{L^{2}(Ω′(r))} + ∥dχ_{r}∥_{L^{4}(Ω′(r))}∥A − Γ′_{r}∥_{L^{4}(Ω′(r))}
+ ∥A − Γ′_{r}∥_{L^{4}(Ω′(r))}^{2}.

Therefore, since ∥dχ_{r}∥_{L^{4}(Ω′(r))} ≤ c_{0}, for a constant c_{0} independent of r ∈ (0, ∞), there is a positive constant c, independent of r ∈ (0, r_{0}), such that

∥F_{A_{r}}∥_{L^{2}(B_{r_{0}})} ≤ c (∥F_{A}∥_{L^{2}(B_{r_{0}})} + ∥F_{A}∥_{L^{2}(B_{r_{0}})}^{2}) ≤ c∥F_{A}∥_{L^{2}(B_{r_{0}})}. (4.4)

Thus, ∥F_{A_{r}}∥_{L^{2}(B_{r_{0}})} tends to zero as r_{0} → 0, uniformly with respect to r ∈ (0, r_{0}). Fix r_{0} small enough so that ∥F_{A_{r}}∥_{L^{2}(B_{r_{0}})} < ε_{1} for all r ∈ (0, r_{0}), where ε_{1} is the constant of Theorem 3.13. Hence, there is a family of C^{∞} flat connections Γ_{r}, r ∈ (0, r_{0}), on the SO(3) bundles 𝔰𝔲(E_{r}) over B_{r_{0}} such that

d_{Γ_{r}}^{∗}(A_{r} − Γ_{r}) = 0 and ∥A_{r} − Γ_{r}∥_{L_{1,Γr}^{2} (B_{r_{0}})} ≤ c∥F_{A_{r}}∥_{L^{2}(B_{r_{0}})}, (4.5)

for a positive constant c independent of r ∈ (0, r_{0}), and a family of C^{∞} bundle isomorphisms τ_{r} : 𝔰𝔲(E_{r})|_{B_{r_{0}}} ≃ B_{r_{0}} × 𝔰𝔲(2) inducing C^{∞} bundle isomorphisms τ_{r} : E_{r} ≃ B_{r_{0}} × ℂ^{2}, via a choice of fixed C^{∞} trivialization det E_{r}|_{B_{r_{0}}} = det E|_{B_{r_{0}}} ≃ B_{r_{0}} × ℂ.

Since ∥Φ∥_{L_{1,A}^{2}(B)} < ∞, then ∥Φ∥_{L_{1,A}^{2}(B_{r_{0}})} and ∥Φ∥_{L^{4}(B_{r_{0}})} tend to zero as r_{0} as tends to zero. As Φ_{r} = χ_{r}Φ over B(\frac{1}{2}r) and using (4.3), we have

∥Φ_{r}∥_{L_{1,Γr}^{2} (B_{r_{0}})} ≤ ∥Φ_{r}∥_{L^{2}(B_{r_{0}})} + ∥∇_{Γ_{r}}Φ_{r}∥_{L^{2}(B_{r_{0}})}
≤ ∥Φ_{r}∥_{L^{2}(B_{r_{0}})} + ∥(A_{r} − Γ_{r}) · Φ_{r}∥_{L^{2}(B_{r_{0}})}
+ ∥∇_{A_{r}}Φ_{r}∥_{L^{2}(B_{r_{0}})}
≤ c (1 + ∥(A_{r} − Γ_{r})∥_{L^{4}(B_{r_{0}})}) ∥Φ_{r}∥_{L^{4}(B_{r_{0}})}
+ ∥∇_{A_{r}}Φ_{r}∥_{L^{2}(B_{r_{0}})}
≤ c (1 + ∥F_{A_{r}}∥_{L^{2}(B_{r_{0}})}) ∥Φ∥_{L^{4}(B_{r_{0}})} + ∥∇_{A_{r}}Φ_{r}∥_{L^{2}(B_{r_{0}})}
≤ c∥Φ∥_{L^{4}(B_{r_{0}})} + ∥∇_{A_{r}}(χ_{r}Φ)∥_{L^{2}(Ω( \frac{1}{4} r, \frac{1}{2} r))}
+ ∥∇_{A}Φ∥_{L^{2}(Ω( \frac{1}{2} r,r_{0}))}
≤ c∥Φ∥_{L_{1,A}^{2}(B_{r_{0}})} + ∥dχ_{r} · Φ∥_{L^{2}(Ω′(r))}
+ ∥χ_{r}∇_{A_{r}}Φ∥_{L^{2}(Ω′(r))}

≤ c∥Φ∥_{L_{1,A}^{2}(B_{r_{0}})} + ∥dχ_{r}∥_{L^{4}(B_{r_{0}})}∥Φ∥_{L^{4}(B_{r_{0}})}
+ ∥∇_{A}Φ∥_{L^{2}(Ω′(r))}
+ ∥(χ_{r} − 1)(A − Γ′_{r}) · Φ∥_{L^{2}(Ω′(r))}
≤ c∥Φ∥_{L_{1,A}^{2}(B_{r_{0}})} + ∥A − Γ′_{r}∥_{L^{4}(Ω′(r))}∥Φ∥_{L^{4}(Ω′(r))},

using A_{r} = Γ′_{r} + χ_{r}(A − Γ′_{r}) = A + (χ_{r} − 1)(A − Γ′_{r}) over Ω′(r), and so

∥Φ_{r}∥_{L_{1,Γr}^{2} (B_{r_{0}})} ≤ c∥Φ∥_{L_{1,A}^{2}(B_{r_{0}})}, (4.6)

for some constant c independent of r ∈ (0, r_{0}). Therefore, ∥Φ_{r}∥_{L_{1,Γr}^{2} (B_{r_{0}})} tends to zero as r_{0} → 0, uniformly with respect to r ∈ (0, r_{0}).

Hence, the estimates (4.4), (4.5), and (4.6) combine to give a uniform bound,

∥(A_{r} − Γ_{r}, Φ_{r})∥_{L_{1,Γr}^{2} (B_{r_{0}})} ≤ c (∥F_{A}∥_{L^{2}(B_{r_{0}})} + ∥Φ∥_{L_{1,A}^{2}(B_{r_{0}})}) , (4.7)

for some constant c independent of r ∈ (0, r_{0}). Note that

L_{1,Γ_{r}}^{2}(B_{r_{0}}, Λ^{1} ⊗ 𝔰𝔲(E_{r})) ⊕ L_{1,Γ_{r}}^{2}(B_{r_{0}}, W^{+} ⊗ E_{r})
= L_{1,Γ}^{2}(B_{r_{0}}, Λ^{1} ⊗ 𝔰𝔲(2)) ⊕ L_{1,Γ}^{2}(B_{r_{0}}, W^{+} ⊗ ℂ^{2}),

via the C^{∞} isomorphisms τ_{r} : E_{r}|_{B_{r_{0}}} → B_{r_{0}} × ℂ^{2}, with

∥(A_{r} − Γ_{r}, Φ_{r})∥_{L_{1,Γr}^{2} (B_{r_{0}})} = ∥(τ_{r}(A_{r}) − Γ, τ_{r}(Φ_{r}))∥_{L_{1,Γ}^{2}(B_{r_{0}})},
d_{Γ}^{∗}(τ_{r}(A_{r}) − Γ) = 0.

By the weak compactness of the unit ball in the Hilbert space L_{1}^{2}(B_{r_{0}}), there is a sequence r_{α} → 0 such that the pairs (A_{r_{α}} −Γ_{r_{α}}, Φ_{r_{α}}) converge weakly in L_{1}^{2}(B_{r_{0}}) to a limit (Ã, Φ̃) in L_{1}^{2}(B_{r_{0}}). For brevity, we denote (A_{r}^{τ}, Φ_{r}^{τ}) := (τ_{r}(A_{r}), τ_{r}(Φ_{r})).

Claim 4.12. Continue the above notation. Then the following hold:

(1) After passing to a subsequence, the pairs (A_{r_{α}}^{τ}, Φ_{r_{α}}^{τ}) converge in C^{∞} over compact subsets of B_{r_{0}} \ {0} to (Ã^{τ} , Φ̃^{τ}) and so (Ã^{τ} , Φ̃^{τ}) is in C^{∞} over B_{r_{0}} \ {0};

(2) The pair (Ã^{τ} , Φ̃^{τ}) is an L_{1}^{2} solution over B_{r_{0}} to the elliptic system

𝔖(Ã^{τ} , Φ̃^{τ}) = 0 and d_{Γ}^{∗}(Ã^{τ} − Γ) = 0.

Proof. For any open subset U ⋐ B_{r_{0}} \ \{0\}, the ball B(\frac{1}{2}r) containing the support of the cutoff function χ_{r} does not meet U when r is sufficiently small and so (A_{r}^{τ}, Φ_{r}^{τ}) is a PU(2) monopole in C^{∞} over U in Coulomb gauge with respect to (Γ, 0) with uniformly L^{2}-small curvature. Hence, choosing U′ ⋐ B_{r_{0}} \ {0} so that U′ ⋑ U, Proposition 3.12 implies that for any integer k ≥ 1 and r small enough that B(\frac{1}{2}r) ∩ U′ = ∅, we have uniform bounds

∥(A_{r}^{τ} − Γ, Φ_{r}^{τ})∥_{L_{k,Γ}^{2}(U)} ≤ C (∥(A_{r}^{τ} − Γ, Φ_{r}^{τ})∥_{L^{2}(U′)} + ∥F (A_{r}^{τ})∥_{L^{2}(U′)})
≤ C (∥Φ∥_{L^{4}(B)} + ∥F_{A}∥_{L^{2}(B)}) < ∞,

for some constant C(g, k, U) independent of r ∈ (0, r_{0}). Therefore, by passing to a subsequence, the pairs (A_{r_{α}}^{τ}, Φ_{r_{α}}^{τ}) converge in C^{∞} over compact subsets of B_{r_{0}} \ {0} to a C^{∞} pair (Ã^{τ} , Φ̃^{τ}) over B_{r_{0}} \ {0} as r_{α} → 0. But for any U ⋐ B_{r_{0}} \ {0} and small enough r ∈ (0, r_{0}), we have 𝔖(A_{r}^{τ}, Φ_{r}^{τ}) = 𝔖(A_{r}, Φ_{r}) = 0 over U, and so

𝔖(Ã^{τ} , Φ̃^{τ}) = lim_{α→∞} 𝔖(A_{r_{α}}^{τ}, Φ_{r_{α}}^{τ}) = 0 over U.

Hence, 𝔖(Ã^{τ} , Φ̃^{τ}) = 0 over B_{r_{0}}\{0} and so (Ã^{τ} , Φ̃^{τ}) is a PU(2) monopole in C^{∞} over B_{r_{0}} \ {0}. This proves Assertion (1) of the claim.

Since (Ã^{τ} , Φ̃^{τ}) is a C^{∞} monopole over B_{r_{0}} \ {0}, then 𝔖(Ã^{τ} , Φ̃^{τ}) = 0 a.e. over B_{r_{0}} and so (Ã^{τ} , Φ̃^{τ}) is an L_{1}^{2} monopole over B_{r_{0}}. Let W_{0}^{2,1}(B_{r_{0}}) ⊂ L_{1}^{2}(B_{r_{0}}) be the closure in L_{1}^{2}(B_{r_{0}}) of the pairs

C_{0}^{∞}(B_{r_{0}}, Λ^{1} ⊗ 𝔰𝔲(E)) ⊕ C_{0}^{∞}(B_{r_{0}}, W^{+} ⊗ E)

with compact support in the open ball B_{r_{0}}. Then, for any (b, φ) ∈ W_{0}^{2,1}(B_{r_{0}}) we have

(b, d_{Γ}^{∗}(Ã^{τ} − Γ))_{L^{2}(B_{r_{0}})} = (d_{Γ}b, Ã^{τ} − Γ)_{L^{2}(B_{r_{0}})}

= lim_{α→∞}(d_{Γ}b, (A_{r_{α}}^{τ} − Γ))_{L^{2}(B_{r_{0}})}

= lim_{α→∞}(b, d_{Γ}^{∗}(A_{r_{α}}^{τ} − Γ))_{L^{2}(B_{r_{0}})} = 0,

and so d_{Γ}^{∗}(Ã^{τ} − Γ) = 0, as required. This completes the proof of the claim. q.e.d.

By Claim 4.12, the pair (Ã^{τ} , Φ̃^{τ}) is an L_{1}^{2} monopole over B_{r_{0}} in Coulomb gauge relative to (Γ, 0). From the estimate (4.7) and the Sobolev embedding L_{1}^{2}(B_{r_{0}}) ⊂ L^{4}(B_{r_{0}}), we can ensure that for sufficiently small r_{0},

∥(A_{r}^{τ} − Γ, Φ_{r}^{τ})∥_{L^{4}(B_{r_{0}})} < ε_{2},

for all r ∈ (0, r_{0}) and so ∥(Ã^{τ} , Φ̃^{τ})∥_{L^{4}(B_{r_{0}})} < ε_{2}, where ε_{2} is the constant of Proposition 3.12 and therefore, (Ã^{τ} , Φ̃^{τ}) is a C^{∞} monopole over B_{r_{0}}. (As usual this means, more precisely, that there is a C^{∞} monopole over B_{r_{0}} which coincides with (Ã^{τ} , Φ̃^{τ}) over B_{r_{0}} except over a subset of measure zero; since (Ã^{τ} , Φ̃^{τ}) is already C^{∞} on the punctured ball B_{r_{0}} \ {0}, this C^{∞} monopole is equal to (Ã^{τ} , Φ̃^{τ}) except possibly at the origin.)

Finally, the C^{∞} bundle isomorphisms τ_{r} : 𝔰𝔲(E_{r})|_{B_{r_{0}}} ≃ B_{r_{0}} × 𝔰𝔲(2) may be viewed as SU(2) automorphisms acting on the SO(3) bundle B_{r_{0}} × 𝔰𝔲(2) by initially choosing a fixed C^{∞} trivialization

E|_{B\{0}} ≃ B \ {0} × ℂ^{2}, with 𝔰𝔲(E)|_{B\{0}} ≃ B \ {0} × 𝔰𝔲(2).

Then Lemma 2.7 implies, after passing to a subsequence, that the sequence of SU(2) automorphisms τ_{r_{α}} converges in C^{∞} over compact subsets of B_{r_{0}}\{0} to a C^{∞} limit σ over B_{r_{0}}\{0}. Then (Ã^{τ} , Φ̃^{τ}) = (A^{σ}, Φ^{σ}) over the punctured ball B_{r_{0}} \ {0}, while (Ã^{τ} , Φ̃^{τ}) is smooth over B_{r_{0}}. Thus, the finite-energy C^{∞} monopole (A, Φ) over the punctured ball B_{r_{0}} \ {0} is equivalent via a C^{∞}, determinant-one, unitary bundle isomorphism to (Ã^{τ} , Φ̃^{τ}) over B_{r_{0}} \ {0}, where (Ã^{τ} , Φ̃^{τ}) is a C^{∞} monopole over B_{r_{0}}. This completes the proof of the theorem. q.e.d.

Remark 4.13. The proof of Theorem 4.10 does not imply, of course, that the section Φ̃^{τ} is zero at the center of the ball B. Even though the C^{∞} sections Φ_{r_{α}}^{τ} are zero at the center, the subsequence only converges in L_{1}^{2}(B_{r_{0}}) over B_{r_{0}} to a limit Φ̃^{τ}. Similarly, while Lemma 4.4 provides a uniform C^{0} bound for the sections Φ_{r_{α}}^{τ} over B_{r_{0}}, we would need a uniform, C^{0,ν} Hölder bound, for some ν ∈ (0, 1), in order to extract a convergent subsequence.

4.4. Patching arguments. The standard proof of the compactness theorem for the moduli space of anti-self-dual connections employs a patching argument for gauge transformations to obtain C^{∞} convergence (modulo gauge transformations) on compact subsets of X \ {x_{1}, …, x_{m}} for a sequence of anti-self-dual connections A_{α} on a Hermitian bundle E over X. The gauge transformations that require patching are obtained by repeated application of Corollary 2.3.9 in [20] to geodesic balls where the L^{2} norm of the curvature F_{A_{α}} is less than ε: since the L^{2} norm of the curvature is scale invariant, these possibly small balls may be rescaled to standard size with metrics which are approximately Euclidean as in [20, Corollary 2.3.9].

Throughout this subsection, (A_{α}, Φ_{α}) will denote a sequence of C^{∞} pairs (not necessarily PU(2) monopoles) on (𝔰𝔲(E), W^{+} ⊗ E) over Ω and u_{α} will denote a sequence of C^{∞} determinant-one, unitary automorphisms of a Hermitian bundle E (that is, gauge transformations in G_{E} and not ^{∘}G_{E}), where Ω is an oriented, Riemannian four-manifold with spin^{c} structure (ρ, W). Convergence will mean convergence in C^{∞} on compact subsets which, as usual, can be replaced by L_{k+1,loc}^{2} convergence of L_{k}^{2} pairs (A_{α}, Φ_{α}) provided k ≥ 2.

The following four patching results follow almost immediately from the proofs of Lemmas 4.4.5–4.4.7 and Corollary 4.4.8 in [20] (where the sequence of connections A_{α} is not assumed to be anti-self-dual). Their proofs are omitted and instead we refer the reader to [20] or [95] for a detailed account; patching arguments of this type are used by Uhlenbeck in her proof of Theorem 3.6 [95], where the connections (not necessarily anti-self-dual or Yang-Mills) are just assumed to be in L_{1}^{p} and the gauge transformations are in L_{2}^{p} with p > 2.

Lemma 4.14. Suppose that (A_{α}, Φ_{α}) is a sequence of pairs on (𝔰𝔲(E), W^{+} ⊗ E) over a base manifold Ω (possibly non-compact), and let Ω̃ ⋐ Ω be an interior domain. Suppose that there are gauge transformations u_{α} ∈ G_{E} and ũ_{α} ∈ G_{E|Ω̃} such that u_{α}(A_{α}, Φ_{α}) converges

over Ω and ũ_{α}(A_{α}, Φ_{α}) converges over Ω̃. Then for any compact set K ⋐ Ω̃ there are a subsequence {α′} ⊂ {α} and gauge transformations w_{α′} ∈ G_{E} such that w_{α′} = ũ_{α′} on a neighborhood of K and the pairs w_{α′}(A_{α′}, Φ_{α′}) converge over Ω.

We have the following two extensions of this result.

Lemma 4.15. Let Ω be exhausted by an increasing sequence of precompact open subsets U_{1} ⋐ U_{2} ⋐ ⋯ ⋐ Ω with ∪_{n}U_{n} = Ω. Suppose (A_{α}, Φ_{α}) is a sequence of pairs on (𝔰𝔲(E), W^{+} ⊗ E) over Ω and that for each n there are a subsequence {α′} ⊂ {α} and gauge transformations u_{α′} ∈ G_{E|U_{n}} such that u_{α′}(A_{α′}, Φ_{α′}) converges over U_{n}. Then there are a subsequence {α′′} ⊂ {α} and gauge transformations u_{α′′} ∈ G_{E} such that u_{α′′}(A_{α′′}, Φ_{α′′}) converges over Ω.

Lemma 4.16. Suppose that Ω = Ω_{1} ∪ Ω_{2} and that (A_{α}, Φ_{α}) is a sequence of pairs on (𝔰𝔲(E), W^{+} ⊗ E) over Ω. If there are sequences of gauge transformations v_{α} ∈ G_{E|Ω_{1}} and w_{α} ∈ G_{E|Ω_{2}} such that v_{α}(A_{α}, Φ_{α}) converges over Ω_{1} and w_{α}(A_{α}, Φ_{α}) converges over Ω_{2}, then there are a subsequence {α′} ⊂ {α} and gauge transformations u_{α′} ∈ G_{E} such that the pairs u_{α′}(A_{α′}, Φ_{α′}) converge over Ω.

Lemmas 4.14, 4.15, and 4.16 combine to yield the following analogue of Corollary 4.4.8 in [20].

Corollary 4.17. Suppose that (A_{α}, Φ_{α}) is a sequence of pairs on (𝔰𝔲(E), W^{+} ⊗ E) over Ω such that the following holds. For each point x ∈ Ω there are a neighborhood D of x, a subsequence {α′} ⊂ {α}, and gauge transformations v_{α′} ∈ G_{E|_{D}} such that the pairs v_{α′}(A_{α′}, Φ_{α′}) converge over D. Then there are a single subsequence {α′′} ⊂ {α}, and gauge transformations u_{α′′} ∈ G_{E} such that the pairs u_{α′′}(A_{α′′}, Φ_{α′′}) converge over Ω.

We now assume that (A_{α}, Φ_{α}) is a sequence of PU(2) monopoles on (𝔰𝔲(E), W^{+} ⊗ E) over Ω and obtain the required convergence from our local elliptic estimates for PU(2) monopoles and Uhlenbeck’s gaugefixing theorem. The following result is the analogue of Proposition 4.4.9 in [20], which applies to a sequence of anti-self-dual connections.

Proposition 4.18. Let Y be an oriented four-manifold with Riemannian metric g and spin^{c} structure (ρ, W). Suppose that (A_{α}, Φ_{α}) is a sequence of PU(2) monopoles, on the bundles (𝔰𝔲(E), W^{+} ⊗ E) over Y, with the following property. For each point y ∈ Y there is a geodesic ball B_{g}(y, r_{y}) with center y, radius r_{y}, and index α_{y} such that

∥F_{A_{α}}∥_{L^{2}(B_{g}(y,r_{y}))} < ε_{0}, α ≥ α_{y},

where ε_{0}(g, A_{det W^{+}}, A_{det E}) is the constant of Corollary 3.16. Then there are a subsequence {α′′} ⊂ {α} and a sequence of C^{∞} gauge transformations u_{α′′} ∈ G_{E} such that u_{α′′}(A_{α′′}, Φ_{α′′}) converges in C^{∞} on compact subsets over Y.

Proof. Fix a point y ∈ Y. If B_{g}(y, r_{y}) is a geodesic ball with center y and g-radius r_{y}, then B_{g_{r}}(y, 1) is a geodesic ball with center y and g_{r}-radius one, where g_{r} = r_{y}^{−2}g. Thus, (A_{α}, r_{y}^{−1}Φ_{α}) is sequence of PU(2) monopoles over B_{g_{r}}(y, 1) such that

∥F_{A_{α}}∥_{L^{2}(B_{gr} (y,1))} < ε_{0}, α ≥ α_{y}.

Corollary 3.16 implies that there are a subsequence {α′} ⊂ {α} and a sequence of gauge transformations {u_{α′}} over B_{g_{r}}(y, 1) such that the sequence u_{α′}(A_{α′}, r_{y}^{−1}Φ_{α′}) converges over B_{g_{r}}(y, \frac{1}{2}).

Therefore, for each point y ∈ Y, we have a sequence of gauge transformations {u_{α′}} over B_{g}(y, r_{y}) such that the sequence u_{α′}(A_{α′}, Φ_{α′}) converges over B_{g}(y, \frac{1}{2}r_{y}). The conclusion now follows from Corollary 4.17. q.e.d.

4.5. Definition of the Uhlenbeck closure. The definition of the Uhlenbeck closure of the moduli space of solutions to the perturbed PU(2) monopole equations (2.27) is slightly more involved than that of the unperturbed PU(2) monopole equations (2.15). For this reason it is convenient to define the Uhlenbeck closure in the unperturbed case before considering the general case.

4.5.1. Definition of the Uhlenbeck closure for the moduli space of unperturbed PU(2) monopoles. Let M_{W,E} (temporarily) denote the moduli space of gauge-equivalence classes of solutions (A, Φ) on (𝔰𝔲(E), W^{+} ⊗ E) to the unperturbed PU(2) monopole equations (2.15). We define the Uhlenbeck closure M̄_{W,E} of the moduli space M_{W,E} by analogy with the definition of the Uhlenbeck closure of the moduli space of anti-self-dual connections [20, §4.4]. The moduli set IM_{W,E} of unperturbed ideal PU(2) monopoles on (𝔰𝔲(E), W^{+} ⊗ E) is given by

IM_{W,E} := ⋃_{ℓ=0}^{N} M_{W,E_{−ℓ}} × Sym^{ℓ}(X),

where N ≥ N_{p}, N_{p} is the constant defined in equation (4.15), E_{−ℓ} denotes a Hermitian two-plane bundle with det E_{−ℓ} = det E, and c_{2}(E_{−ℓ}) = c_{2}(E) − ℓ, for ℓ ≥ 0.

Definition 4.19. Suppose that [A_{α}, Φ_{α}, y_{α}] is a sequence of points in IM_{W,E} and that [A_{0}, Φ_{0}, x] is a point in IM_{W,E}, where (A_{α}, Φ_{α}) and (A_{0}, Φ_{0}) are monopoles on (𝔰𝔲(E_{α}), W^{+} ⊗ E_{α}) and (𝔰𝔲(E_{0}), W^{+} ⊗ E_{0}) over X, respectively, with det E_{α} = det E_{0} = det E and c_{2}(E_{α}), c_{2}(E_{0}) ≤ c_{2}(E). Then the sequence of points [A_{α}, Φ_{α}, y_{α}] converges to [A_{0}, Φ_{0}, x] (or, the sequence of triples (A_{α}, Φ_{α}, y_{α}) converges weakly to (A_{0}, Φ_{0}, x)) if the following hold:

• There is a sequence of L_{k+1,loc}^{2} determinant-one, unitary bundle isomorphisms u_{α} : E_{α}|_{X\x} → E_{0}|_{X\x} such that the sequence of PU(2) monopoles u_{α}(A_{α}, Φ_{α}) converges to (A_{0}, Φ_{0}) in L_{k,loc}^{2} over X \ x.

• The sequence |F_{A_{α}}|^{2} + 8π^{2} ∑_{y∈y_{α}} δ(y) converges in the weak-* topology on measures to |F_{A_{0}}|^{2} + 8π^{2} ∑_{x∈x} δ(x).

• We have c_{2}(E) = c_{2}(E_{0}) + |x|.

Give IM_{W,E} the Uhlenbeck topology specified by Definition 4.19 and let M̄_{W,E} ⊂ IM_{W,E} be the closure of M_{W,E} in IM_{W,E}. The topological space IM_{W,E} is second-countable and Hausdorff.

4.5.2. Definition of the Uhlenbeck closure for the moduli space of perturbed PU(2) monopoles. The basic idea underlying the choice of holonomy perturbations described in §2.5.2 is a generalization of an earlier construction due to S. K. Donaldson for the moduli spaces of solutions to the ‘extended anti-self-dual equations’, to which the Freed-Uhlenbeck generic metrics theorem does not apply [18, §IV(v), pp. 282–286]. As in the case of the moduli space of anti-self-dual connections, we shall see in §4.6 that there is an upper bound M (which is determined by g, A_{det W^{+}}, A_{det E}, and p_{1}(𝔰𝔲(E))) on the total energy ∥F_{A}∥_{L^{2}(X)}^{2} for any solution (A, Φ) to the perturbed PU(2) monopole equations (2.27) and so an upper bound 2M/ε_{0}^{2} on the number of disjoint balls B(x_{j}, 4R_{0}) with energy greater than or equal to \frac{1}{2}ε_{0}^{2}. Hence, if N_{b} ≥ 2M/ε_{0}^{2} + 1, at least one ball B(x_{j}, 4R_{0}) in the collection \{B(x_{j}, 4R_{0})\}_{j=1}^{N_{b}} has energy less than \frac{1}{2}ε_{0}^{2}.

Suppose (A_{α}, Φ_{α}) is a sequence of PU(2) monopoles which converges to an ideal PU(2) monopole (A_{0}, Φ_{0}, x) in C̃_{W,E_{−ℓ}} × Sym^{ℓ}(X). If a point x ∈ x lies in a ball B(x_{j}, 2R_{0}), the corresponding sections 𝔪_{j,l,β}(A_{α}) supported on B̄(x_{j}, R_{0}) converge to zero (by construction) for all l, β. Thus, the solution (A_{0}, Φ_{0}) will satisfy a version of the PU(2) monopole equations (2.27) with the perturbations supported on B̄(x_{j}, R_{0}) omitted. (In the situation considered by Donaldson, ideal extended antiself-dual connections also satisfy a family of equations [18, Eq. (4.37)] which depend on the bubble points in X.) Therefore, the ideal limit [A_{0}, Φ_{0}, x] is a point in the fiber M_{W,E_{−ℓ}}|_{x} over a point x in the base Sym^{ℓ}(X). Here, M_{W,E_{−ℓ}}|_{x} is simply the moduli space of solutions to the perturbed monopole equations (2.27) with the connection energy cutoff functions β_{j}[A] of (2.18) (used in the definition of the perturbing sections 𝔪_{j,l,β}(A) of (2.20)) replaced by cutoff functions

\begin{matrix}β_{j}[A_{0}, x] \\ := β (\frac{1}{ε_{0}^{2}} ∫_{B(x_{j},4R_{0})} β (\frac{dist_{g}(·, x_{j})}{4R_{0}}) (|F_{A_{0}}|^{2} + 8π^{2} ∑_{x∈x} δ(x)) dV ) \end{matrix}, (4.8)

where ε_{0} is the constant of Corollary 3.16. Then M_{W,E_{−ℓ}} is the moduli space of triples (A_{0}, Φ_{0}, x) solving the ‘lower-level’ PU(2) monopole equations

F_{A}^{+} − (id + τ_{0} ⊗ id_{𝔰𝔲(E)} + τ⃗ · 𝔪⃗(A, x)) ρ^{−1}(Φ ⊗ Φ^{∗})_{00} = 0,
D_{A}Φ + ρ(ϑ_{0})Φ + ϑ⃗ · 𝔪⃗(A, x)Φ = 0.

(4.9)

In the more familiar case of the Uhlenbeck compactification of the moduli space of solutions to the unperturbed PU(2) monopole equations (2.15), the spaces M_{W,E_{−ℓ}} above would be replaced by the products M_{W,E_{−ℓ}} × Sym^{ℓ}(X).

The definition of the Uhlenbeck topology for the moduli space of solutions to the perturbed PU(2) monopole equations (2.27) is almost identical to that of the unperturbed case. The only difference is in the definition of the set of ideal solutions to (2.27). In the presence of holonomy perturbations, the Uhlenbeck closure M̄_{W,E} is therefore defined to be the closure of M_{W,E} in

IM_{W,E} := ⋃_{ℓ=0}^{N} M_{W,E_{−ℓ}} ⊂ ⋃_{ℓ=0}^{N} C_{W,E_{−ℓ}} × Sym^{ℓ}(X),

where M_{W,E_{−0}} := M_{W,E}, while N ≥ N_{p} and N_{p} is the constant defined in equation (4.15).

4.6. Sequential compactness. In this section we apply our elliptic regularity and removable singularity results to prove our main compactness result, namely Theorem 1.1, which asserts the existence of an Uhlenbeck compactification for the moduli space of PU(2) monopoles, analogous to that given by Theorem 4.4.3 [20] in the case of the moduli space of anti-self-dual connections.

As in [20], the proof of Theorem 1.1 follows by an entirely routine argument (which we leave to the reader) from the special case below which is an analogue of similar compactness results for anti-self-dual connections; see, for example, [20, Theorem 4.4.4], [31, Theorem 3.2], [30, Chapter 8], [78, Theorem 3.1], [83, Proposition 4.4], and [84, Proposition 5.1].

Theorem 4.20. Let X be a closed, oriented, smooth four-manifold with C^{∞} Riemannian metric, spin^{c} structure (ρ, W) with spin^{c} connection, and a Hermitian two-plane bundle E with unitary connection on det E. Then there is a positive integer N_{p}, depending at most on the curvatures of the fixed connections on W and det E together with c_{2}(E), such that for all N ≥ N_{p}, any infinite sequence in M_{W,E} has a weakly convergent subsequence, with limit point in ∪_{ℓ=0}^{N}M_{W,E_{−ℓ}}.

Proof. The basic argument follows that of [20, pp. 163–165] and [83, Proposition 4.4] for the moduli space of anti-self-dual connections. Let [A_{α}, Φ_{α}] be a sequence of points in M_{W,E} and let (A_{α}, Φ_{α}) be a corresponding sequence of PU(2) monopoles in C^{∞} on (E, W^{+}⊗E). By passing to a subsequence we can assume that the sequence of positive measures μ_{α} := |F_{A_{α}}|^{2} on X converges to a measure μ_{∞} on X in the weak-* topology on measures, so

lim_{α→∞} ∫_{X} |F_{A_{α}}|^{2} dV = ∫_{X} μ_{∞} =: M_{∞} ≤ K,

where K < ∞ is the constant in our universal energy bound (4.2) for a PU(2) monopole over X. Hence there are at most M_{∞}/ε_{0}^{2} distinct points in X, labelled {x_{1}, …, x_{m}}, which do not lie in a geodesic ball B(x, r) of μ_{∞}-measure less than ε_{0}^{2}, where ε_{0} is the constant of Proposition 4.18 and which appears in (2.18) and (4.8). Thus, for any r > 0, we have

lim_{α→∞} ∫_{B(x_{i},r)} |F_{A_{α}}|^{2} dV = ∫_{B(x_{i},r)} μ_{∞} ≥ ε_{0}^{2}, i = 1, …, m,

and so we may define real numbers κ_{i} ≥ ε_{0}^{2}/8π^{2} by setting

κ_{i} := lim_{r→0} lim_{α→∞} \frac{1}{8π^{2}} ∫_{B(x_{i},r)} |F_{A_{α}}|^{2} dV = lim_{r→0} \frac{1}{8π^{2}} ∫_{B(x_{i},r)} μ_{∞}.

We may suppose, without loss of generality, that m ≥ 1. If a point x_{i} lies in a ball B̄(x_{j}, 2R_{0}) then the holonomy perturbation sections are zero over B(x_{j}, R_{0}) since 8π^{2}κ_{i} ≥ \frac{1}{2}ε_{0}^{2}. Hence, the points \{x_{1}, …, x_{m}\} are contained in a large open subset of X where (τ⃗, ϑ⃗) = 0.

By passing to a subsequence, Proposition 4.18 supplies determinantone, unitary gauge transformations u_{α} over X \ {x_{1}, …, x_{m}} such that the sequence u_{α}(A_{α}, Φ_{α}) converges over X \ {x_{1}, …, x_{m}} to a pair (A_{0}, Φ_{0}) on (𝔰𝔲(E), W^{+}⊗E)|_{X\{x_{1},...,x_{m}}}, such that the triple (A_{0}, Φ_{0}, x) solves the lower-level PU(2) monopole equations (4.9). Plainly,

∫_{X\{x_{1},...,x_{m}}} (|F_{A_{0}}|^{2} + |Φ_{0}|^{4} + |∇_{A_{0}}Φ_{0}|^{2}) dV ≤ K < ∞.

By the removability of point singularities for finite-energy PU(2) monopoles (Theorem 4.10), there are a Hermitian two-plane bundle E_{0} with det E_{0} = det E over X, a PU(2) monopole (Ã_{0}, Φ̃_{0}) on (𝔰𝔲(E_{0}), W^{+}⊗E_{0}), and a determinant-one, unitary bundle isomorphism u_{0} from E|_{X\{x_{1},...,x_{m}}} to E_{0}|_{X\{x_{1},...,x_{m}}} such that u_{0}(A_{0}, Φ_{0}) = (Ã_{0}, Φ̃_{0}) over X \ {x_{1}, …, x_{m}}.

The limiting measure μ_{∞} has the form

μ_{∞} = |F_{A_{0}}|^{2} + 8π^{2} ∑_{i=1}^{m} κ_{i}δ_{x_{i}},

where the δ_{x_{i}} have unit mass concentrated at the points x_{i}. It remains to show that the κ_{i} are positive integers. For this purpose we use an argument similar to that used to prove Lemma 3.8 in [31] (due to Taubes) which fits better with our later development of the gluing theory for PU(2) monopoles, though one could also use the Chern-Simons functional for this purpose as in [20, p. 164]. The proof of Theorem 3.2 in [31] is a modification of an earlier compactness result, Proposition 4.4 in [83], for connections with L^{2} bounded curvature but which are only approximately anti-self-dual in a suitable sense. Theorem 4.20 follows easily from the next lemma:

Lemma 4.21. The bundle E_{0} has Chern classes c_{1}(E_{0}) = c_{1}(E) and c_{2}(E_{0}) = c_{2}(E) − ℓ, where ℓ = ∑_{i=1}^{m} κ_{i} and the constants κ_{i} are positive integers for i = 1, …, m.

Proof. The equality of the first Chern classes follows from the remarks in the preceding paragraph. The proof that each κ_{i} is an integer requires a brief digression in order to discuss the limiting behavior of the connections A_{α} near the points x_{i} ∈ X.

Fix an index i ∈ {1, …, m}, let ϱ be the injectivity radius of (X, g), and fix a constant δ ∈ (0, \frac{1}{4}ϱ). Choose an orthogonal frame for 𝔰𝔲(E_{0})|_{x_{i}}, use parallel translation via the connection A_{0} along radial geodesics from x_{i} ∈ X to trivialize 𝔰𝔲(E_{0}) over the ball B(x_{i}, ϱ), and let w_{0,i} : 𝔰𝔲(E_{0})|_{B(x_{i},ϱ)} → B(x_{i}, ϱ) × 𝔰𝔲(2) be the resulting smooth bundle map. We have ∥F_{A_{0}}∥_{L^{∞}(X)} ≤ C, for some positive constant C, and so

∥F_{A_{0}}∥_{L^{2}(B(x_{i},δ))} ≤ Cδ^{2}.

Thus, we may suppose that δ is fixed small enough so that Theorem 3.13 provides an SU(2) gauge transformation v_{0,i} of B(x_{i}, δ)×𝔰𝔲(2) such that

∥a_{0,i}∥_{L^{4}(B(x_{i},δ))} + ∥∇_{Γ}a_{0,i}∥_{L^{2}(B(x_{i},δ))} ≤ c∥F_{A_{0}}∥_{L^{2}(B(x_{i},2δ))},

where a_{0,i} := u_{0,i}(A_{0}) − Γ ∈ Ω^{1}(B(x_{i}, δ), 𝔰𝔲(2)) and u_{0,i} := v_{0,i} ∘ w_{0,i}, and Γ is the product connection on B(x_{i}, δ) × 𝔰𝔲(2). The sequence of connections u_{α}(A_{α}) converges in C^{∞} on compact subsets of the punctured balls B(x_{i}, δ)\{x_{i}} to the C^{∞} connection A_{0} on 𝔰𝔲(E_{0})|_{B(x_{i},δ)\{x_{i}}}; therefore, the sequence of connections u_{0,i}u_{α}(A_{α}) converges in C^{∞} on compact subsets of the punctured balls B(x_{i}, δ) \ {x_{i}} to the C^{∞} connection u_{0,i}(A_{0}) on B(x_{i}, δ) \ {x_{i}} × 𝔰𝔲(2).

Write u_{0,i}u_{α}(A_{α}) = Γ + a_{i,α} over B(x_{i}, δ) \ {x_{i}}, where a_{i,α} ∈ Ω^{1}(B(x_{i}, δ) \ \{x_{i}\}, 𝔰𝔲(2)). Let Ω(x_{i}; \frac{1}{2}r, 2r) denote the open annulus \{x ∈ X : \frac{1}{2}r < dist_{g}(x, x_{i}) < 2r\} in X. Then, for any r ∈ (0, \frac{1}{2}δ), there is an index α_{0}(r) such that

∫_{Ω(x_{i}, \frac{1}{2} r,2r)} (|a_{i,α} − a_{0,i}|^{4} + |∇_{Γ}(a_{i,α} − a_{0,i})|^{2}) dV < r^{4}, α ≥ α_{0}.

Since ∥F_{A_{0}}∥_{L^{∞}(X)} ≤ C, we have

∫_{B(x_{i},2r)} (|a_{0,i}|^{4} + |∇_{Γ}a_{0,i}|^{2}) dV ≤ c ∫_{B(x_{i},2r)} |F_{A_{0}}|^{2} dV ≤ Cr^{4}, (4.10)

and therefore

∫_{Ω(x_{i}, \frac{1}{2} r,2r)} (|a_{i,α}|^{4} + |∇_{Γ}a_{i,α}|^{2}) dV ≤ Cr^{4}, α ≥ α_{0}. (4.11)

Let χ : ℝ → [0, 1] be a bump function such that χ(t) = 0 for t ≤ \frac{1}{2} and χ(t) = 1 for t ≥ 2. Define a cutoff function χ_{i,r} : X → [0, 1] by setting χ_{i,r}(x) = 1 − χ(dist_{g}(x, x_{i})/r) so that χ_{i,r} = 1 on B(x_{i}, \frac{1}{2}r) and χ_{i,r} = 0 on X − B(x_{i}, 2r). Fix a Riemannian metric g_{i} on S^{4} which coincides with g on B(x_{i}, δ) = B(n, δ) (after identifying the point x_{i} ∈ X with the north pole n ∈ S^{4}) and extends g outside B(x_{i}, 2δ) = B(n, 2δ) to a smooth metric on S^{4}. Define a sequence of SO(3) bundles V_{i,r,α} over S^{4} by setting

V_{i,r,α} := \begin{cases}𝔰𝔲(E) & over B(n, 2r), \\ S^{4} \ \{n\} × 𝔰𝔲(2) & over S^{4} \ \{n\}, \end{cases}

where the identification of the SO(3) bundles 𝔰𝔲(E) and S^{4} \{n}×𝔰𝔲(2) over the annulus B(n, 2r) \ {n} = B(x_{i}, 2r) \ {x_{i}} is induced from the the SO(3) bundle isomorphism

u_{0,i} ∘ u_{α} : 𝔰𝔲(E)|_{B(n,2r)\{n}} → B(n, 2r) \ {n} × 𝔰𝔲(2).

We cut off the sequence of connections A_{α} on 𝔰𝔲(E) over the annulus Ω(x_{i}; \frac{1}{2}r, 2r) and thus obtain a sequence of C^{∞} connections A_{i,r,α} on the sequence of SO(3) bundles V_{i,r,α} over S^{4} by setting

A_{i,r,α} := \begin{cases}A_{α} & on 𝔰𝔲(E)|_{B(n, \frac{1}{2} r)}, \\ Γ + χ_{i,r}a_{i,α} & on S^{4} \ \{n\} × 𝔰𝔲(2). \end{cases}

Recall from Lemma 4.4 that there is a constant C independent of α such that ∥F_{A_{α}}^{+}∥_{L^{∞}(X)} ≤ C and so

∥F_{A_{α}}^{+}∥_{L^{2}(B(x_{i},2r))} ≤ Cr^{2} for all α. (4.12)

Since

F_{A_{i,r,α}}^{+} = χ_{i,r}F_{A_{α}}^{+} + (dχ_{i,r} ∧ a_{i,α})^{+} + χ_{i,r}(χ_{i,r} − 1)(a_{i,α} ∧ a_{i,α})^{+},

the estimates (4.10), (4.11), and (4.12) imply that

∥F_{A_{i,r,α}}^{+}∥_{L^{2}(S^{4})} ≤ ∥F_{A_{α}}^{+}∥_{L^{2}(B(x_{i},2r))}
+ \sqrt{2}∥dχ_{i,r}∥_{L^{4}(X)}∥a_{i,α}∥_{L^{4}(Ω(x_{i}; \frac{1}{2} r,2r))}
+ \sqrt{2}∥a_{i,α}∥_{L^{4}(Ω(x_{i}; \frac{1}{2} r,2r))}^{2}
≤ C(r + r^{2}), α ≥ α_{0}.

Therefore,

lim_{r→0} lim_{α→∞} ∥F_{A_{i,r,α}}^{+}∥_{L^{2}(S^{4})} = 0. (4.13)

Similarly, as

F_{A_{i,r,α}} = χ_{i,r}F_{A_{α}} + dχ_{i,r} ∧ a_{i,α} + χ_{i,r}(χ_{i,r} − 1)a_{i,α} ∧ a_{i,α},

the estimates (4.10) and (4.11) yield

∥F_{A_{i,r,α}} − F_{A_{α}}∥_{L^{2}(Ω(x_{i}; \frac{1}{2} r,2r))}
≤ ∥∇_{Γ}a_{i,α}∥_{L^{2}(Ω(x_{i}; \frac{1}{2} r,2r))} + c∥a_{i,α}∥_{L^{4}(Ω(x_{i}; \frac{1}{2} r,2r))}
+ ∥a_{i,α}∥_{L^{4}(Ω(x_{i}; \frac{1}{2} r,2r))}^{2}
< C(r + r^{2}), α ≥ α_{0}.

Therefore,

lim_{r→0} lim_{α→∞} ∥F_{A_{i,r,α}} − F_{A_{α}}∥_{L^{2}(Ω(x_{i}; \frac{1}{2} r,2r))} = 0. (4.14)

We can now complete the argument that the κ_{i} are integers:

Claim 4.22. There is an SU(2) bundle E_{i} over S^{4} such c_{2}(E_{i}) = κ_{i} for each i = 1, …, m.

Proof. Fix an index i ∈ {1, …, m}. Over S^{4}, the SO(3) bundles V_{i,r,α} lift to SU(2) bundles E_{i,r,α} with V_{i,r,α} = 𝔰𝔲(E_{i,r,α}) and p_{1}(V_{i,r,α}) = −4c_{2}(E_{i,r,α}). The second Chern classes of the SU(2) bundles E_{i,r,α} are given by

c_{2}(E_{i,r,α}) = \frac{1}{8π^{2}} ∫_{S^{4}} (|F_{A_{i,r,α}}^{−}|^{2} − |F_{A_{i,r,α}}^{+}|^{2}) dV,

recalling that the isomorphisms ad : 𝔰𝔲(E_{i,r,α}) → 𝔰𝔬(𝔰𝔲(E_{i,r,α})) are implicit and that we view F_{A_{i,r,α}} as sections of Λ^{2} ⊗ 𝔰𝔲(E_{i,r,α}). Therefore, by (4.14) and (4.13) and the fact that F_{A_{i,r,α}} = F_{A_{α}} on B(x_{i}, \frac{1}{2}r), we have

lim_{r→0} lim_{α→∞} c_{2}(E_{i,r,α}) = lim_{r→0} lim_{α→∞} \frac{1}{8π^{2}} ∫_{S^{4}} |F_{A_{i,r,α}}^{−}|^{2} dV

= lim_{r→0} lim_{α→∞} \frac{1}{8π^{2}} ∫_{S^{4}} (|F_{A_{i,r,α}}^{−}|^{2} + |F_{A_{i,r,α}}^{+}|^{2}) dV

= lim_{r→0} lim_{α→∞} \frac{1}{8π^{2}} ∫_{B(x_{i},2r)} |F_{A_{i,r,α}}|^{2} dV

= lim_{r→0} lim_{α→∞} \frac{1}{8π^{2}} ∫_{B(x_{i},2r)} |F_{A_{α}}|^{2} dV = κ_{i},

where the final equality follows by definition of κ_{i}. Thus, for small enough r and large enough α, we have c_{2}(E_{i,r,α}) = c_{2}(E_{i}) for some fixed SU(2) bundle E_{i} over S^{4} and so κ_{i} = c_{2}(E_{i}), completing the proof of the claim. q.e.d.

By Claim 4.22 the κ_{i} are positive integers for i = 1, …, m. We can now compute the second Chern class of the limit bundle E_{0}. The Chern-Weil identity (2.33) implies that, for all α,

−\frac{1}{4}p_{1}(𝔰𝔲(E)) = c_{2}(E) − \frac{1}{4}c_{1}(E)^{2} = \frac{1}{8π^{2}} ∫_{X} (|F_{A_{α}}^{−}|^{2} − |F_{A_{α}}^{+}|^{2}) dV.

Therefore, by (4.12) we have

c_{2}(E) − \frac{1}{4}c_{1}(E)^{2} = lim_{r→0} lim_{α→∞} \frac{1}{8π^{2}} ∫_{X} (|F_{A_{α}}^{−}|^{2} − |F_{A_{α}}^{+}|^{2}) dV

= lim_{r→0} lim_{α→∞} \frac{1}{8π^{2}} ∫_{X\∪_{i=1}^{m}B(x_{i},2r)} (|F_{A_{α}}^{−}|^{2} − |F_{A_{α}}^{+}|^{2}) dV

+ ∑_{i=1}^{m} lim_{r→0} lim_{α→∞} \frac{1}{8π^{2}} ∫_{B(x_{i},2r)} (|F_{A_{α}}^{−}|^{2} − |F_{A_{α}}^{+}|^{2}) dV

= \frac{1}{8π^{2}} ∫_{X} (|F_{A_{0}}^{−}|^{2} − |F_{A_{0}}^{+}|^{2}) dV + ∑_{i=1}^{m} κ_{i}

= c_{2}(E_{0}) − \frac{1}{4}c_{1}(E_{0})^{2} + ∑_{i=1}^{m} κ_{i}.

Now c_{1}(E) = c_{1}(E_{0}) and thus c_{2}(E_{0}) = c_{2}(E)−∑_{i=1}^{m} κ_{i}. This completes the proof of Lemma 4.21. q.e.d.

Therefore, after passing to a subsequence, the sequence of points [A_{α}, Φ_{α}] in M_{W,E} converges to an ideal monopole [A_{0}, Φ_{0}, x] in M_{L,E_{0}} × Sym^{ℓ}(X), for some integer ℓ ≥ 0, and Lemma 4.21 implies the Chern classes of the limit bundle E_{0} are given by c_{1}(E_{0}) = c_{1}(E) and c_{2}(E_{0}) = c_{2}(E) − ℓ.

It remains to give an upper bound for the integer ℓ = c_{2}(E)−c_{2}(E_{0}). The Chern-Weil identity (2.33) implies that

ℓ = c_{2}(E) − c_{2}(E_{0}) =\frac{1}{8π^{2}} ∫_{X} (|F_{A}^{−}|^{2} − |F_{A}^{+}|^{2}) dV

− \frac{1}{8π^{2}} ∫_{X} (|F_{A_{0}}^{−}|^{2} − |F_{A_{0}}^{+}|^{2}) dV,

and thus, by Lemma 4.3, we have

ℓ ≤ \frac{1}{8π^{2}} ∫_{X} |F_{A}^{−}|^{2} dV + \frac{1}{8π^{2}} ∫_{X} |F_{A_{0}}^{+}|^{2} dV ≤ N_{p}, (4.15)

for some positive integer N_{p} = N_{p}(c_{1}(E), c_{2}(E), g, F (A_{det W}), F (A_{det E})). This completes the proof of Theorem 4.20. q.e.d.

Remark 4.23. The compactness result in this section for the moduli space of PU(2) monopoles has an antecedent in [31, Theorem 3.2] (due to Taubes) in the following sense. Taubes’ theorem provides a weak compactness result for connections A satisfying the ‘infinite-dimensional part’ of the anti-self-dual equation, namely Π_{A;μ}F_{A}^{+} = 0, where μ ∉ Spec d_{A}^{+}d_{A}^{∗}, and Π_{A;μ} is the L^{2}-orthogonal projection onto the eigenvectors of d_{A}^{+}d_{A}^{∗} with eigenvalue less than μ, together with the curvature bounds ∥F_{A}∥_{L^{2}(X)} + ∥d_{A}F_{A}^{+}∥_{L^{2}(X)} ≤ C for some constant C independent of A. The analogous point here is that although ∫_{X} |F_{A}|^{2} dV is not a topological invariant unless F_{A}^{+} = 0 or F_{A}^{−} = 0, just as in the case of the PU(2) monopoles, it is enough for the purposes of obtaining a weak compactness result to have uniform bounds on the L^{2} norm of F_{A} together with an L^{p} bound on F_{A}^{+} for some p > 2.

There is one further compactness result we will need, analogous to Uhlenbeck’s original compactness theorem for connections (not necessarily satisfying any elliptic equation) with L^{p} bounds on curvature with p > 2 [95, Theorem 1.5].

Proposition 4.24. Let p > 2 and K > 0 be constants. If [A_{α}, Φ_{α}] is an infinite sequence in M_{W,E} satisfying

∥F_{A_{α}}∥_{L^{p}(X)} ≤ K,

then there is a subsequence {α′} ⊂ {α} such that the sequence [A_{α′}, Φ_{α′}] converges to a point [A_{∞}, Φ_{∞}] ∈ M_{W,E}.

Proof. Let (A_{α}, Φ_{α}) be a corresponding sequence of C^{∞} pairs. Since p > 2, Hölder’s inequality implies that for any geodesic ball B(x, r) ⊂ X we have

∥F_{A_{α}}∥_{L^{2}(B(x,r)} ≤ cr^{2−(4/p)}∥F_{A_{α}}∥_{L^{p}(B(x,r))} ≤ cKr^{2−(4/p)}.

Hence, for small enough r, Proposition 4.18 applies and there are a subsequence {α′} ⊂ {α} and a sequence of C^{∞} gauge transformations u_{α′} such that the sequence u_{α′}(A_{α′}, Φ_{α′}) converges in C^{∞} to a limit (A_{∞}, Φ_{∞}) over all of X, with no exceptional points. q.e.d.

As we shall see in §5.1.2, Proposition 4.24 allows us to work with perturbation parameters (τ_{0}, τ⃗, ϑ⃗) and a metric g which are C^{∞} rather than just C^{r}, as required by the application of the Sard-Smale theorem in our proof of transversality in §5.

5. Transversality

In this section we show that for generic perturbation parameters (ϑ_{0}, τ_{0}, τ⃗, ϑ⃗) the moduli space of solutions to the perturbed PU(2) monopole equations (2.27) is a smooth manifold away from the zero-section and reducible pairs.

The outline of the proof is of the now standard form introduced in [14] and [30]. In §5.1 we define a parametrized moduli space and explain why transversality for the moduli space (Corollary 5.3) follows from transversality for the parametrized moduli space (Theorem 5.2) via the Sard-Smale theorem. In §5.2 we show that the parametrized moduli space is a smooth Banach manifold (Theorem 5.2). The proof of Theorem 5.2 relies on the fact that a PU(2) monopole, which is reducible on an admissible open subset of the manifold X, is reducible on the entire manifold (Theorem 5.11) and this is proved in §5.3.

The proof of Theorem 1.3 does not apply to PU(2) monopoles which are zero-sections or which are reducible. We describe the cokernels of D𝔖 evaluated at these pairs in the sequel [25] to the present article.

5.1. The parametrized moduli space. It is convenient to first consider the question of transversality for the top stratum M_{W,E}^{∗,0} of the Uhlenbeck compactification M̄_{W,E} and then consider the very slight modification required to obtain simultaneous transversality for all the lower-level moduli spaces M_{W,E_{−ℓ}}^{∗,0}|_{Σ} ⊂ C_{W,E_{−ℓ}}^{∗,0} × Σ, for smooth strata Σ ⊂ Sym^{ℓ}(X).

5.1.1. Transversality for the top-level moduli space. The condition in Proposition 2.28 that the section 𝔖, vanish transversely is, of course, not necessarily true for all the parameters (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗), on which 𝔖 depends. As in the cases of the moduli spaces of anti-self-dual connections [14], [20], [30] and Seiberg-Witten monopoles [47], [98], we first show that the family of moduli spaces parametrized by the perturbations (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) is smooth and then apply the Sard-Smale theorem [80] to conclude that for generic perturbations (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) (that is, a subset of perturbations which is the complement of some first-category subset), the moduli space 𝔖^{−1}(0) = M_{W,E}^{∗,0}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) is smooth.

Set P_{0}^{r} := C^{r}(X, 𝔤𝔩(Λ^{+})) ⊕ C^{r}(X, Λ^{1} ⊗ ℂ), let P^{r} := P_{0}^{r} ⊕ P_{τ}^{r} ⊕ P_{ϑ}^{r} denote our Banach space of C^{r} perturbation parameters, and define a ∘G_{E}-equivariant map

\underline{𝔖} := (\underline{𝔖}_{1}, \underline{𝔖}_{2}) : P^{r} × C̃_{W,E} → L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) ⊕ L_{k−1}^{2}(W^{−} ⊗ E)) by setting

\underline{𝔖}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗, A, Φ)

:= (\begin{matrix}\underline{𝔖}_{1}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗, A, Φ) \\ \underline{𝔖}_{2}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗, A, Φ) \end{matrix}) (5.1)

= (\begin{matrix}F_{A}^{+} − (id + τ_{0} ⊗ id_{𝔰𝔲(E)} + τ⃗ · 𝔪⃗(A)) ρ^{−1}(Φ ⊗ Φ^{∗})_{00} \\ D_{A}Φ + ρ(ϑ_{0})Φ + ϑ⃗ · 𝔪⃗(A)Φ \end{matrix}) ,

where (A, Φ) is a pair on (𝔰𝔲(E), W^{+} ⊗ E) and the isomorphism ad : 𝔰𝔲(E) ≃ 𝔰𝔬(𝔰𝔲(E)) is implicit, ^{∘}G_{E} acts trivially on the space of perturbations P^{r}, and so \underline{𝔖}^{−1}(0)/^{∘}G_{E} is a subset of P^{r} × C_{W,E}. We let 𝔐_{W,E} denote the parametrized moduli space \underline{𝔖}^{−1}(0)/^{∘}G_{E} and let 𝔐_{W,E}^{∗,0} = 𝔐_{W,E} ∩ (P^{r} × C_{W,E}^{∗,0}).

Remark 5.1. While we assumed for convenience in §3 and §4 that the parameters g, τ_{0}, ϑ_{0}, τ⃗, ϑ⃗ were C^{∞}, the only difference, if the parameters are only assumed to be C^{r} for some finite r, is the slight increase in bookkeeping required to keep track of the regularity of solutions to (2.27) and other associated elliptic systems.

Just as in §2.6, the ^{∘}G_{E}-equivariant map \underline{𝔖} defines a section of a Banach vector bundle \underline{𝔙} over P^{r} × C_{W,E}^{∗,0} with total space

\underline{𝔙} := P^{r} × C̃_{W,E}^{∗,0} ×_{^{∘}G}_{E} (L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) ⊕ L_{k−1}^{2}(W^{−} ⊗ E)) ,

so 𝔖 := \underline{𝔖}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗, ·) is a section over C̃_{W,E}^{∗,0} of the Banach vector bundle 𝔙 := \underline{𝔙}|_{(τ_{0},ϑ_{0},τ⃗,ϑ⃗)} in (2.34). In particular, the parametrized

moduli space 𝔐_{W,E}^{∗,0} is the zero set of the section \underline{𝔖} of the vector bundle \underline{𝔙} over P^{r} × C_{W,E}^{∗,0}.

Theorem 5.2. The zero set in P^{r} × C_{W,E}^{∗,0} of the section \underline{𝔖} is regular and, in particular, the moduli space 𝔐_{W,E}^{∗,0} is a smooth Banach submanifold of P^{r} × C_{W,E}^{∗,0}.

To preserve continuity, we defer the proof of Theorem 5.2 to §5.2. The differential D\underline{𝔖} := (D\underline{𝔖})_{[τ_{0},ϑ_{0},τ⃗,ϑ⃗,A,Φ]} of the section \underline{𝔖} at a point

[τ_{0}, ϑ_{0}, τ⃗, ϑ⃗, A, Φ] in P^{r} × C_{W,E}^{∗,0} is given by

D\underline{𝔖}(δτ_{0}, δϑ_{0}, δτ⃗, δϑ⃗, a, ϕ) = (\begin{matrix}D\underline{𝔖}_{1}(δτ_{0}, δϑ_{0}, δτ⃗, δϑ⃗, a, ϕ) \\ D\underline{𝔖}_{2}(δτ_{0}, δϑ_{0}, δτ⃗, δϑ⃗, a, ϕ) \end{matrix}) , (5.2)

where (a, ϕ) ∈ K_{A,Φ} ⊂ L_{k}^{2}(Λ^{1} ⊗ 𝔰𝔲(E)) ⊕ L_{k}^{2}(W^{+} ⊗ E) represents a vector in the tangent space (T C_{W,E}^{∗,0})_{[A,Φ]} and (δτ_{0}, δϑ_{0}, δτ⃗, δϑ⃗) ∈ P^{r}. The differential of the first component in (5.2) is given explicitly by

D\underline{𝔖}_{1} (δτ_{0}, δϑ_{0}, δτ⃗, δϑ⃗, a, ϕ)
= d_{A}^{+}a − δτ_{0}ρ^{−1}(Φ ⊗ Φ^{∗})_{00}
− (id + τ_{0} ⊗ id_{𝔰𝔲(E)} + τ⃗ · 𝔪⃗(A)) ρ^{−1}(ϕ ⊗ Φ^{∗} + Φ ⊗ ϕ^{∗})_{00}

− ∑_{j,l,α}(δτ_{j,l,α} ⊗ ad(𝔪_{j,l,α}(A))ρ^{−1}(Φ ⊗ Φ^{∗})_{00} (5.3)

− ∑_{j,l,α} τ_{j,l,α} ⊗ ad (\frac{δ𝔪_{j,l,α}}{δA} a) ρ^{−1}(Φ ⊗ Φ^{∗})_{00},

and the second component by

D\underline{𝔖}_{2} (δτ_{0}, δϑ_{0}, δτ⃗, δϑ⃗, a, ϕ)
= D_{A}ϕ + ϑ⃗ · 𝔪⃗(A)ϕ + ρ(a)Φ

+ ρ(δϑ_{0})Φ + ∑_{j,l,α} ρ(δϑ_{j,l,α}) ⊗ 𝔪_{j,l,α}(A)Φ (5.4)

+ ∑_{j,l,α} ρ(ϑ_{j,l,α}) ⊗ (\frac{δ𝔪_{j,l,α}}{δA} a) Φ.

We note that from their definitions in §2.5.2 the perturbations (and their variations) are zeroth order, unlike the first order perturbations considered in [92].

Recall from the arguments of §2.2 that D\underline{𝔖}(·, d_{A,Φ}^{0}ζ) = 0 for all ζ ∈ L_{k+1}^{2}(𝔰𝔲(E)) ⊕ iℝ_{Z} since \underline{𝔖} is ^{∘}G_{E}-equivariant. By Proposition 3.7 we may assume, without loss of generality, that the pair (A, Φ) in C̃_{W,E}^{∗,0} is a C^{r} representative for the point [A, Φ] in the zero set 𝔖^{−1}(0) ⊂ C_{W,E}^{∗,0}. Since the tangent space (T C_{W,E}^{0,∗})_{[A,Φ]} may be identified with K_{A,Φ} := Ker d_{A,Φ}^{0,∗} (see §2.2), we have

D\underline{𝔖}(0, 0, 0, 0, a, ϕ) = d_{A,Φ}^{1}(a, ϕ) = (d_{A,Φ}^{0,∗} + d_{A,Φ}^{1})(a, ϕ),

for (a, ϕ) ∈ K_{A,Φ}, so the differential D\underline{𝔖}|_{\{0\}×T C_{W,E}^{∗,0}} is Fredholm, where

\{0\} × T C_{W,E}^{0,∗} = T (\{τ_{0}, ϑ_{0}, τ⃗, ϑ⃗\} × C_{W,E}^{∗,0}). Thus, \underline{𝔖} is a Fredholm section when restricted to the fixed parameter fibers

{τ_{0}, ϑ_{0}, τ⃗, ϑ⃗} × C_{W,E}^{∗,0} ⊂ P^{r} × C_{W,E}^{∗,0}.

The Sard-Smale theorem (in the form of Proposition 4.3.11 in [20]) then implies that there is a first-category subset of the space P^{r} such that the zero sets in C_{W,E}^{∗,0} of the sections 𝔖 = \underline{𝔖}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗, ·) are regular for all C^{r} perturbations (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) in the complement of this subset. Now

M_{W,E}^{∗,0}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) = 𝔖^{−1}(0) ∩ C_{W,E}^{∗,0},

and so for generic parameters (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗), the moduli space M_{W,E}^{∗,0}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) is a smooth manifold of the expected dimension. In summary, we have:

Corollary 5.3. There is a first-category subset of the space P^{r}, such that for all C^{r} perturbations (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) in the complement of this subset, the zero locus of the section 𝔖 is regular and so the moduli space M_{W,E}^{∗,0}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) = 𝔖^{−1}(0) ∩ C_{W,E}^{∗,0} is a smooth submanifold of C_{W,E}^{∗,0} with the expected dimension.

We recall that a subset S of a topological space P is a set of the first category if its complement P − S is a countable intersection of dense open sets or, equivalently, if S is a countable union of closed subsets of P with empty interior; if P is a complete metric space, then Baire’s theorem implies that P − S is dense in P [76]. In our applications, P will either be a Banach or Fréchet space (with a complete metric), so P − S will always be dense if S is a first-category subset.

5.1.2. Reduction to the case of C^{∞} parameters. The restriction to C^{r} parameters (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗), necessary to apply the Sard-Smale theorem in §5.1.1, proves inconvenient in practice. We shall see that these restrictions can now be removed, so we need only use C^{∞} parameters (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗). Although we did not need the metric g to be generic in order for our transversality proof to work, we will nonetheless require the metric g to be generic in the sequels to the present article and so, a priori, g would also be restricted to a certain Banach manifold of metrics on X. An argument almost identical to the one which we describe here can be used to show that one need only consider generic C^{∞} metrics in those applications.

There is an argument due to Taubes — for the moduli space of Seiberg-Witten monopoles — which reduces the case of transversality for C^{∞} parameters to the case of Hölder or Sobolev parameters [77, §9.4]. (A related result for generic metrics due to Freed and Uhlenbeck appears as Proposition 3.20 in [30], although it is only stated for the moduli space of anti-self-dual SU(2) connections with second Chern class one over a simply-connected, negative definite four-manifold.)

We adapt Taubes argument here to the case of the moduli spaces of PU(2) monopoles. We define

P : = P_{0} ⊕ P_{τ} ⊕ P_{ϑ},
P_{0} : = Ω^{0}(X, 𝔤𝔩(Λ^{+})) ⊕ Ω^{1}(X, ℂ),
P_{τ} = ℓ_{δ}^{1}(𝔸, Ω^{0}(X, 𝔤𝔩(Λ^{+}))) := ⋂_{r=0}^{∞} ℓ_{δ}^{1}(𝔸, C^{r}(X, 𝔤𝔩(Λ^{+}))),

P_{ϑ} = ℓ_{δ}^{1}(𝔸, Ω^{1}(X, ℂ)) := ⋂_{r=0}^{∞} ℓ_{δ}^{1}(𝔸, C^{r}(X, Λ^{1} ⊗ ℂ)),

and let P^{r} be the Banach space

P^{r} : = P_{0}^{r} ⊕ P_{τ}^{r} ⊕ P_{ϑ}^{r}
= C^{r}(X, 𝔤𝔩(Λ^{+})) ⊕ C^{r}(X, Λ^{1} ⊗ ℂ)
⊕ ℓ_{δ}^{1}(𝔸, C^{r}(X, 𝔤𝔩(Λ^{+}))) ⊕ ℓ_{δ}^{1}(𝔸, C^{r}(X, Λ^{1} ⊗ ℂ)).

Define metrics d_{0}, d_{τ} and d_{ϑ} on Met(X), P_{0}, P_{τ} and P_{ϑ} by setting

d_{0}(τ_{01}, ϑ_{01}; τ_{02}, ϑ_{02}) := ∑_{r=0}^{∞} \frac{2^{−r}(∥τ_{01} − τ_{02}∥_{C^{r}} + ∥ϑ_{01} − ϑ_{02}∥_{C^{r}})}{1 + ∥τ_{01} − τ_{02}∥_{C^{r}} + ∥ϑ_{01} − ϑ_{02}∥_{C^{r}}},

d_{τ} (τ⃗_{1}, τ⃗_{2}) := ∑_{r=0}^{∞} \frac{2^{−r}∥τ⃗_{1} − τ⃗_{2}∥_{ℓ_{δ}^{1}(C^{r})}}{1 + ∥τ⃗_{1} − τ⃗_{2}∥_{ℓ_{δ}^{1}(C^{r})}} ,

d_{ϑ}(ϑ⃗_{1}, ϑ⃗_{2}) := ∑_{r=0}^{∞} \frac{2^{−r}∥ϑ⃗_{1} − ϑ⃗_{2}∥_{ℓ_{δ}^{1}(C^{r})}}{1 + ∥ϑ⃗_{1} − ϑ⃗_{2}∥_{ℓ_{δ}^{1}(C^{r})}} ,

and observe that P_{0}, P_{τ} and P_{ϑ} are complete metric spaces, with the above metrics inducing the C^{∞} topologies. Thus, P is a complete metric space with respect to the product metric d = d_{0} ×d_{τ} ×d_{ϑ}, which induces the C^{∞} topology on P. The C^{r} topology of P^{r} is induced by the product metric d^{r} := d_{0}^{r} × d_{τ}^{r} × d_{ϑ}^{r}.

Let P_{reg} ⊂ P be the subspace of parameters for which the zero set of 𝔖_{p} is regular, and note that

P_{reg} = ⋂_{n≥1} P_{n,reg},

where P_{n,reg} ⊂ P_{reg} is the subspace of C^{∞} parameters p such that the differential D𝔖_{p} is surjective for all pairs (A, Φ) in 𝔖_{p}^{−1}(0) ∩ C̃_{W,E}^{∗,0}, satisfying

∥F_{A}∥_{L^{p}} ≤ n and ν_{0}(A, Φ; p) ≥ 1/n, (5.5)

for some p > 2, where ν_{0}(A, Φ; p) is the least eigenvalue of the Laplacian Δ_{A,Φ;p}^{0} := d_{A,Φ}^{0,∗}d_{A,Φ}^{0} computed with respect to the metric g. Define, in the analogous way, the subspaces P_{reg}^{r} and P_{n,reg}^{r} of P^{r}.

The uniform upper L^{p} bound on F_{A} precludes bubbling, while the uniform lower bound on ν_{0}(A, Φ) keeps (A, Φ) bounded away from the reducible or zero-section pairs. Let ν_{2}(A, Φ; p) be the least eigenvalue of the Laplacian Δ_{A,Φ;p}^{2} := d_{A,Φ}^{1}d_{A,Φ}^{1,∗} computed with respect to the parameters p = (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗). Then (D𝔖_{p})_{A,Φ} is surjective if and only if ν_{2}(A, Φ; p) > 0.

Claim 5.4. The subspace P_{n,reg} ⊂ P is open in the C^{∞} topology and P_{n,reg}^{r} ⊂ P^{r} is open in the C^{r} topology.

Proof. Let {p_{α}}_{α=1}^{∞} ⊂ P \ P_{n,reg} be a sequence of parameters and suppose that p_{α} converges to p ∈ P in the C^{∞} topology. Then there is a sequence of solutions (A_{α}, Φ_{α}) to (2.27) in C̃_{W,E}^{∗,0}, with parameters p_{α}, which satisfy the bounds in (5.5) and for which ν_{2}(A_{α}, Φ_{α}; p_{α}) = 0. Proposition 4.24 and the L^{p} bounds in (5.5) imply that, after passing to a subsequence, there is a sequence of gauge transformations {u_{α}} ⊂ G_{E} such that u_{α}(A_{α}, Φ_{α}) converges (strongly) in L_{k}^{2} to a solution (A, Φ) in C̃_{W,E}^{∗,0} to (2.27) which satisfies the curvature bound in (5.5). Standard perturbation theory implies that the eigenvalues ν_{i}(u_{α}(A_{α}, Φ_{α}); p_{α}) converge to ν_{i}(A, Φ; p) for i = 0, 2 [41], so that the triple (A, Φ; p) satisfies the eigenvalue bound in (5.5). The eigenvalue ν_{2}(A, Φ; p) must be zero — otherwise the eigenvalues ν_{2}(u_{α}(A_{α}, Φ_{α}); p_{α}) would be positive for large enough α. Hence, p ∉ P_{n,reg} and so P_{n,reg} is open. The proof that P_{n,reg}^{r} is an open subset of P^{r} is identical. q.e.d.

Claim 5.5. The subspace P_{n,reg} ⊂ P is dense in the C^{∞} topology.

Proof. By Corollary 5.3, the space P_{reg}^{r} is the complement in P^{r} of a first-category subset and so is dense by Baire’s theorem; clearly, P_{n,reg}^{r} is also dense in P^{r}, since P_{reg}^{r} ⊂ P_{n,reg}^{r}. Let p ∈ P be a C^{∞} parameter, and let {p_{α}} ⊂ P_{n,reg}^{r} be a sequence of C^{r} parameters such that d^{r}(p, p_{α}) < 2^{−α−1}, so p_{α} converges in C^{r} to p. Since P_{n,reg}^{r} ⊂ P^{r} is open by Claim 5.4 and the C^{∞} parameters P are dense in P_{n,reg}^{r}, we may choose, for each α, a C^{∞} parameter p′_{α} ∈ P_{n,reg}^{r} such that d^{r}(p_{α}, p′_{α}) < 2^{−α−1}. Since p′_{α} is C^{∞}, then {p′_{α}} ⊂ P_{n,reg} and by construction we have d^{r}(p, p′_{α}) < 2^{−α} and so the sequence {p′_{α}} converges in C^{r} to p ∈ P.

Therefore, for each r, we obtain a sequence {p′_{α}(r)} ⊂ P_{n,reg} which converges in C^{r} to p ∈ P_{reg}. But then the diagonal sequence {p′_{α}(α)} ⊂ P_{n,reg} converges to p in C^{r} for each r (it satisfies d^{r}(p, p′_{α}(α)) < 2^{−α} for all α ≥ r) and so the sequence converges in C^{∞} to p ∈ P, as required. q.e.d.

From Claims 5.4 and 5.5 we conclude that P_{reg} is a countable intersection of dense, open subsets of P and hence is the complement of a first-category subset (in particular, the subset P_{reg} ⊂ P is dense by Baire’s theorem). Hence, the space P_{reg} of C^{∞} parameters (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) such that the moduli spaces M_{W,E}^{∗,0}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) are regular is the complement of a first-category subset of P. From this and Corollary 5.3 we conclude:

Corollary 5.6. Let X be a closed, oriented, smooth four-manifold with C^{∞} metric g. There is a first-category subset of the space P, such that for all C^{∞} perturbations (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) in the complement of this subset, the zero locus of the section 𝔖 is regular and so the moduli space M_{W,E}^{∗,0}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗) = 𝔖^{−1}(0) is a smooth submanifold of C_{W,E}^{∗,0} with the expected dimension.

Remark 5.7. The same argument shows that the standard Freed-Uhlenbeck generic metrics theorems (specifically, Corollaries 4.3.15, 4.3.18, and 4.3.19 in [20] and the refinement Lemma 2.4 in [48] for the non-simply connected case) for the moduli spaces of anti-self-dual connections on an SU(2) or SO(3) bundle over X continue to hold for the complement of a first-category subset of C^{∞} metrics, rather than just C^{r} metrics.

5.1.3. Simultaneous transversality for the top and lowerlevel moduli spaces. Let Σ be a smooth stratum of Sym^{ℓ}(X). Recall from §4.5.2 that a universal choice of a sufficiently large constant N_{b} guarantees that if [A, Φ, x] is any point in M̄_{W,E} and A is irreducible, then A has at least one ball B(x_{j}, R_{0}) which supports holonomy perturbations.

The PU(2) monopole equations cutting out the locus

M_{W,E_{−ℓ}}^{∗,0}|_{Σ} ⊂ C_{W,E_{−ℓ}}^{∗,0} × Σ

from the Uhlenbeck compactification M̄_{W,E} are equations for triples (A, Φ, x) ∈ C̃_{W,E_{−ℓ}}^{∗,0} ×Σ. We can again define a ^{∘}G_{E}-equivariant C^{∞} map

\underline{𝔖} : P^{r} × C̃_{W,E_{−ℓ}}^{∗,0} × Σ → L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) ⊕ L_{k−1}^{2}(W^{−} ⊗ E)

by setting

\underline{𝔖}(τ_{0}, ϑ_{0}, τ⃗, ϑ⃗, A, Φ, x)

:= (\begin{matrix}F_{A}^{+} − (id + τ_{0} ⊗ id_{𝔰𝔲(E)} + τ⃗ · 𝔪⃗(A, x)) ρ^{−1}(Φ ⊗ Φ^{∗})_{00} \\ D_{A}Φ + ρ(ϑ_{0})Φ + ϑ⃗ · 𝔪⃗(A, x)Φ \end{matrix}) .

The proof of Corollary 5.6 now shows that M_{W,E_{−ℓ}}^{∗,0}|_{Σ} ⊂ C_{W,E_{−ℓ}}^{∗,0} × Σ is a smooth submanifold of the expected dimension for generic parameters (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗),

(5.6) dim M_{W,E_{−ℓ}}^{∗,0}|_{Σ} = dim M_{W,E_{−ℓ}}^{∗,0} + dim Σ.

Furthermore, by considering regular values of the projection maps onto the second factors Σ, Sard’s Theorem also shows that the fibers M_{W,E_{−ℓ}}^{∗,0}|_{x} are smooth manifolds of the expected dimension for generic points x ∈ Sym^{ℓ}(X). Indeed, the only tangent vectors in each stratum Σ, which might not appear in the image of the projection, are those arising from the radial vector on the annuli B(x_{j}, 4R_{0})\B̄(x_{j}, 2R_{0}). This observation shows that the projection from M_{W,E_{−ℓ}}^{∗,0} to Σ is transverse to certain submanifolds of Σ which would allow dimension-counting arguments similar to those in [25]. Issues related to dimension-counting in the presence of holonomy perturbations are also discussed by Donaldson in [18, pp. 282–287]. We can now conclude the proof of our main transversality result:

Proof of Theorem 1.3, given Corollary 5.6. For the case ℓ = 0, the transversality assertion is given by Corollary 5.6 and the dimension formula is provided by Proposition 2.28. The case ℓ > 0 then follows from the discussion in the preceding paragraphs. q.e.d.

5.2. Smoothness of the parametrized moduli space. We prove Theorem 5.2 by showing that the ^{∘}G_{E}-equivariant map

\underline{𝔖} : P^{r} × C̃_{W,E}^{∗,0} → L_{k−1}^{2}(Λ^{+} ⊗ 𝔰𝔲(E)) ⊕ L_{k−1}^{2}(W^{+} ⊗ E)

vanishes transversely, and so the parametrized moduli space 𝔐_{W,E}^{∗,0} = \underline{𝔖}^{−1}(0)/^{∘}G_{E} is a smooth Banach manifold. The broad strategy is reminiscent of that of [30, Chapter 3] and [20, §4.3.5], where the analogous result is established for the moduli space of anti-self-dual connections parametrized by the Banach space of C^{r} (conformal classes of) metrics: our proof of surjectivity of the differential D\underline{𝔖} at a point (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗, A, Φ) in the zero set \underline{𝔖}^{−1}(0) ultimately relies on the fact that, for Φ ≢ 0, a monopole (A, Φ) which is reducible on an admissible open subset of X is necessarily reducible over all of X (Theorem 5.11). If there are sections 𝔪_{j,l,α}(A) which are non-zero on B(x_{j}, R_{0}), we say that the connection A has holonomy perturbations supported on B(x_{j}, R_{0}); the set {𝔪_{j,l,α}(A)}_{l=1}^{3} then spans 𝔰𝔲(E)|_{B(x_{j},R_{0})} for at least one index α. Then an admissible open set for the pair (A, Φ) is one containing the B̄(x_{j}, R_{0}) for all j such that β_{j}[A] > 0. (The supports of all the sections 𝔪_{j,l,α}(A) are contained in ∪_{j=1}^{N_{b}}B̄(x_{j}, R_{0}), and so any open subset of X containing ∪_{j=1}^{N_{b}}B̄(x_{j}, R_{0}) is admissible.) Recall from §4.5.2 that because of our choice of constant N_{b}, if [A, Φ] is any point in M_{W,E}^{∗}, then at least one ball B(x_{j}, R_{0}) supports holonomy perturbations for A.

Theorem 5.2 is an almost immediate consequence of

Proposition 5.8. Suppose (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗, A, Φ) is a point in \underline{𝔖}^{−1}(0) with A irreducible and Φ ≢ 0. If (v, ψ) is in the cokernel of the differential D\underline{𝔖} at the point (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗, A, Φ), then (v, ψ)|_{B(x_{j},R_{0})} ≡ 0 for each ball B(x_{j}, R_{0}) supporting holonomy perturbations for A.

We first observe that the elements of the cokernel of

D\underline{𝔖} := D\underline{𝔖}_{(τ_{0},ϑ_{0},τ⃗,ϑ⃗,A,Φ)}

have a restricted form of the unique continuation property (sufficient for our purposes) by Aronszajn’s theorem [5]:

Lemma 5.9. If (v, ψ) ∈ Ker D\underline{𝔖}(D\underline{𝔖})^{∗} and (v, ψ)|_{U} ≡ 0 on some non-empty open subset U ⊂ X containing all balls B(x_{j}, R_{0}) supporting holonomy perturbations for A, then (v, ψ) ≡ 0 on X.

Proof. By hypothesis, the pair (v, ψ) solves the second-order elliptic equation

(D\underline{𝔖})(D\underline{𝔖})^{∗}(v, ψ) = 0 on X,

where the Laplacian D\underline{𝔖} = (D\underline{𝔖}_{1}, D\underline{𝔖}_{2}), given by equations (4.3) and (4.4), has C^{r−1} coefficients and (v, ψ) is at least C^{r+1}. Also, (v, ψ) ≡ 0 on the set of closed balls supporting holonomy perturbations,

B̄_{I}(A) := ⋃_{j∈I(A)} B̄(x_{j}, R_{0}),

where I(A) := {j : 1 ≤ j ≤ N_{b}, β_{j}[A] > 0, and A|_{B(x_{j},2R_{0})} is irreducible}. Now, on the subset X − B_{I(A)} where all of the holonomy perturbations and especially their derivatives with respect to A vanish (see their definition in §2.5.2), the Laplacian (D\underline{𝔖})(D\underline{𝔖})^{∗} is a purely differential operator. In particular, it extends to a differential operator with C^{r−1} coefficients over X, say (D\underline{𝔖}^{0})(D\underline{𝔖}^{0})^{∗}, given by the linearized PU(2) monopole equations (5.3) and (5.4) with all terms involving holonomy perturbations and their derivatives set equal to zero. On the other hand, the pair (v, ψ) also solves the resulting second-order elliptic differential equation

(D\underline{𝔖}^{0})(D\underline{𝔖}^{0})^{∗}(v, ψ) = 0 on X,

since (v, ψ) = 0 on B̄_{I}(A) and D\underline{𝔖}^{0} ≠ D\underline{𝔖} only on B_{I}(A), while D\underline{𝔖}^{0} = D\underline{𝔖} on X − B_{I(A)}.

Without loss of generality, we may scale the Dirac equation in (2.21) by 1/\sqrt{2}. We then have

D\underline{𝔖}^{0}(D\underline{𝔖}^{0})^{∗} = (\begin{matrix}d_{A}^{+}d_{A}^{+,∗} & 0 \\ 0 & \frac{1}{2}D_{A}D_{A}^{∗} \end{matrix}) + First-order differential terms,

and so by the Bochner formulas of Lemma 4.1 and [30, Eq. (6.26)], the Laplacian D\underline{𝔖}^{0}(D\underline{𝔖}^{0})^{∗} is a second order elliptic differential operator with scalar principal symbol (given by the metric \frac{1}{2}g on T^{∗}X). The desired conclusion then follows from Aronszajn’s unique continuation theorem [5]. q.e.d.

Remark 5.10.

1. Aronszajn’s theorem does not apply without the given restriction on the open set U in the statement of Lemma 5.9, as the Laplacian (D\underline{𝔖})(D\underline{𝔖})^{∗} is not a purely differential operator over all of X. One can see from equations (5.3) and (5.4) that the problem terms are those appearing in the last line of each displayed equation: the operator δ𝔪_{j,ℓ,α}/δA acting on a ∈ Ω^{1}(𝔰𝔲(E)) is an integral operator, as is clear from the formula (A.7) for the differential of the holonomy with respect to the connection.

2. Lemma 5.9 can also be proved without using Aronszajn’s theorem explicitly and instead applying the Agmon-Nirenberg unique continuation theorem (Theorem 5.25) to the equation (D\underline{𝔖})^{∗}(v, ψ) = 0, and mimicking the existing application in the proof of Theorem 5.11. Indeed, this second proof of Lemma 5.9 is virtually identical to the proof of Theorem 5.11. We leave the details to the interested reader, as the preceding use of Aronszajn’s theorem appears easier to us. We note that Aronszajn’s theorem can be derived from that of Agmon-Nirenberg (see [3]).

If (v, ψ) is an L_{k−1}^{2} element of the cokernel of D\underline{𝔖}, then elliptic regularity for the Laplacian D\underline{𝔖}(D\underline{𝔖})^{∗}, with C^{r−1} coefficients, implies that (v, ψ) is in C^{r+1}. Our proof of Theorem 5.2 also relies on the following ‘unique continuation’ result for reducible monopoles:

Theorem 5.11. If (A, Φ) is a C^{r} solution to the perturbed PU(2) monopole equations (2.27) with Φ ≢ 0 over a connected, oriented, smooth four-manifold X with C^{r} Riemannian metric, and (A, Φ) is reducible on a non-empty open subset U ⊂ X with B̄(x_{j}, R_{0}) ⊂ U for all j such that β_{j}[A] > 0, then (A, Φ) is reducible on X.

The proof of Theorem 5.11 is lengthy, so we defer it to §5.3.

Proof of Theorem 5.2, given Proposition 5.8 and Theorem 5.11. Let (τ_{0}, ϑ_{0}, τ⃗, ϑ⃗, A, Φ) be a C^{r} representative for a point in 𝔐_{W,E}^{∗,0}, so that A is irreducible and Φ ≢ 0, and suppose (v, ψ) is in the cokernel of D\underline{𝔖}. By definition of N_{b} in §2.5.2, the set

J(A) := {j : 1 ≤ j ≤ N_{b} and β_{j}[A] > 0}

is non-empty. Since A is irreducible on X, Theorem 5.11 implies that A|_{B(x_{j′},2R_{0})} must be irreducible for some j′ ∈ {1, …, N_{b}} such that β_{j′}[A] > 0; otherwise, A|_{B(x_{j′},2R_{0})} would be reducible for all j such that β_{j}[A] > 0, and Theorem 5.11 would imply that A would be reducible over all of X, contradicting our assumption that A is irreducible. By Proposition 5.8 we see that (v, ψ)|_{B(x_{j},R_{0})} ≡ 0 for all j such that β_{j}[A] > 0 and A|_{B(x_{j},2R_{0})} is irreducible, so (v, ψ) ≡ 0 on X by Lemma 5.9.

q.e.d.

The proof of Proposition 5.8 occupies the remainder of this subsection. We first note that since Φ is in the kernel D_{A} + ϑ⃗ · 𝔪⃗(A), it has the unique continuation property by Aronszajn’s Theorem [5]:

Lemma 5.12. If (D_{A} + ρ(ϑ_{0}) + ϑ⃗ · 𝔪⃗(A))Φ = 0 and Φ|_{U} ≡ 0 for some non-empty open subset U ⊂ X, then Φ ≡ 0.

Proof. The perturbed Dirac operator D_{A} + ρ(ϑ_{0}) + ϑ⃗ · 𝔪⃗(A) differs from D_{A} by a zeroth order term and so

(D_{A} + ρ(ϑ_{0}) + ϑ⃗ · 𝔪⃗(A))^{∗}(D_{A} + ρ(ϑ_{0}) + ϑ⃗ · 𝔪⃗(A))
= D_{A}^{∗}D_{A} + First order terms.

The Bochner formula of Lemma 4.1 then implies that the above Laplacian is a second order elliptic differential operator with scalar principal symbol (given by the metric g on T^{∗}X). The conclusion now follows from Aronszajn’s unique continuation theorem [5]. q.e.d.

We shall use the following linear algebra result to show that v ∈ C^{r+1}(X, Λ^{+} ⊗ 𝔰𝔲(E)) vanishes on a ball:

Lemma 5.13. Let M, N be elements of (Λ^{+} ⊗ 𝔰𝔲(E))|_{x}. Suppose 𝔪_{1}, 𝔪_{2}, 𝔪_{3} span 𝔰𝔲(E)|_{x}. If

⟨τ_{0}M, N⟩ + ∑_{l=1}^{3}⟨(τ_{l} ⊗ ad(𝔪_{l}))M, N⟩ = 0 (5.7)

for all τ_{0}, τ_{1}, τ_{2}, τ_{3} ∈ 𝔤𝔩(Λ^{+})|_{x}, then either M = 0 or N = 0.

Proof. If ⟨τ_{0}M, N⟩ = 0 for all τ_{0} ∈ 𝔤𝔩(Λ^{+})|_{x}, then by the proof of [30, Lemma 3.7], the images in 𝔰𝔲(E)|_{x} of M, N ∈ Hom(Λ^{+}|_{x}, 𝔰𝔲(E)|_{x}) are orthogonal. (Although their lemma refers to an element of Hom(Λ^{+}|_{x}, 𝔰𝔲(E)|_{x}) and an element of Hom(Λ^{−}|_{x}, 𝔰𝔲(E)|_{x}), we can choose any isomorphism between Λ^{+}|_{x} and Λ^{−}|_{x} to translate the result.) We can therefore assume that M has rank one and N has rank less than or equal to two. (If M is rank two and N is rank one, we can reverse their roles by using adjoints.) If both M ≠ 0 and N ≠ 0 let

M = u ⊗ m and N = v_{1} ⊗ n_{1} + v_{2} ⊗ n_{2},

where u, v_{1}, v_{2} ∈ Λ^{+}|_{x} and m, n_{1}, n_{2} ∈ 𝔰𝔲(E)|_{x}. Since the images of M and N in 𝔰𝔲(E)|_{x} are orthogonal, we have ⟨m, n_{1}⟩ = 0 = ⟨m, n_{2}⟩; without loss of generality, we can assume that ⟨n_{1}, n_{2}⟩ = 0. (If N is rank one, n_{2} can be any element of 𝔰𝔲(E)|_{x} completing m, n_{1} to an orthogonal basis of 𝔰𝔲(E)|_{x}.) Under the isomorphism 𝔰𝔲(E)|_{x} ≃ ℝ^{3}, the adjoint representation is given by the cross-product. We can find f_{l} ∈ ℝ such that n_{2} = ∑_{l} f_{l}𝔪_{l}, so

(5.8)  ∑_{l=1}^{3}⟨ad(f_{l}𝔪_{l})m, n_{1}⟩ = ⟨[n_{2}, m], n_{1}⟩ ≠ 0,

(5.9)  ∑_{l=1}^{3}⟨ad(f_{l}𝔪_{l})m, n_{2}⟩ = ⟨[n_{2}, m], n_{2}⟩ = 0.

By assumption, M ≠ 0 and N ≠ 0, so u ≠ 0 and either v_{1} ≠ 0 or v_{2} ≠ 0; we may suppose without loss of generality that v_{1} ≠ 0. Thus, we can find τ ∈ 𝔤𝔩(Λ^{+})|_{x} such that τu = v_{1}, and so choosing τ_{l} = f_{l}τ for l = 1, 2, 3 and δτ_{0} = 0, we have

∑_{l=1}^{3}⟨(τ_{l} ⊗ ad(𝔪_{l})) M, N⟩ = ∑_{l=1}^{3}⟨τ ⊗ ad(f_{l}𝔪_{l})M, N⟩
= ⟨(τ ⊗ ad(n_{2}))M, N⟩

= ⟨(τ ⊗ ad(n_{2}))(u ⊗ m), v_{1} ⊗ n_{1} + v_{2} ⊗ n_{2}⟩

= ⟨τu ⊗ [n_{2}, m], v_{1} ⊗ n_{1} + v_{2} ⊗ n_{2}⟩

= ⟨τu, v_{1}⟩⟨[n_{2}, m], n_{1}⟩ + ⟨τu, v_{2}⟩⟨[n_{2}, m], n_{2}⟩

= |v_{1}|^{2}⟨[n_{2}, m], n_{1}⟩ ≠ 0 by (5.8) and (5.9),

contradicting our hypothesis in (5.7). Hence, either M = 0 or N = 0, as desired. q.e.d.

Remark 5.14. Lemma 5.13 does not hold if the rank of E is greater than two. If a, b ∈ 𝔰𝔲(E), the above arguments would only allow one to conclude that

0 = ⟨[𝔪, a], b⟩ = ⟨𝔪, [a, b]⟩

for all 𝔪 ∈ 𝔰𝔲(E), so [a, b] = 0 and a, b are simultaneously diagonalizable. The subspace of diagonal elements of 𝔰𝔲(n) has dimension n − 1, so this would not contradict the orthogonality of a, b if n > 2.

Lemma 5.15. Continue the hypotheses of Proposition 5.8 and suppose B(x_{j}, R_{0}) is a ball supporting holonomy perturbations for A. Then v ≡ 0 on B(x_{j}, R_{0}).

Proof. By hypothesis, there are holonomy sections 𝔪_{l} ≡ 𝔪_{j,l,α}(A), l = 1, 2, 3, which span 𝔰𝔲(E)|_{y}, for any point y ∈ B(x_{j}, R_{0}). Let δτ_{l} := δτ_{j,l,α} ∈ Ω^{0}(𝔤𝔩(Λ^{+})), l = 1, 2, 3, denote the corresponding coefficients, and let δτ⃗ be a sequence with all other coefficients equal to zero.

By the hypothesis of Proposition 5.8, we have

(D\underline{𝔖}(δτ_{0}, δϑ_{0}, δτ⃗, δϑ⃗, a, ϕ), (v, ψ))_{L^{2}} = 0

for all (δτ_{0}, δϑ_{0}, δτ⃗, δϑ⃗, a, ϕ), and so

0 = (D\underline{𝔖}(δτ_{0}, 0, δτ⃗, 0, 0, 0), (v, ψ))_{L^{2}}

= (D\underline{𝔖}_{1}(δτ_{0}, 0, δτ⃗, 0, 0, 0), v)_{L^{2}}

= ((δτ_{0} ⊗ id_{𝔰𝔲(E)} + δτ⃗ · 𝔪⃗(A))ρ^{−1}(Φ ⊗ Φ^{∗})_{00}), v)_{L^{2}}

= (δτ_{0}ρ^{−1}(Φ ⊗ Φ^{∗})_{00}, v)_{L^{2}} + ∑_{l=1}^{3}(δτ_{l} ⊗ ad(𝔪_{l})ρ^{−1}(Φ ⊗ Φ^{∗})_{00}, v)_{L^{2}}.

Taking a sequence of δτ_{l}’s which approximate δτ_{l,y}δ(·, y), where δ(·, y) is the Dirac delta distribution supported at y and δτ_{l,y} ∈ 𝔤𝔩(Λ^{+})|_{y}, we obtain the pointwise identity

⟨δτ_{0,y}ρ^{−1}(Φ ⊗ Φ^{∗})_{00}|_{y}, v|_{y}⟩ + ∑_{l=1}^{3}⟨δτ_{l,y} ⊗ ad(𝔪_{l})ρ^{−1}(Φ ⊗ Φ^{∗})_{00}|_{y}, v|_{y}⟩ = 0,

for all δτ_{l,y} ∈ 𝔤𝔩(Λ^{+})|_{y}, l = 0, 1, 2, 3. Lemma 5.13 then implies that either ρ^{−1}(Φ ⊗ Φ^{∗})_{00}|_{y} = 0, and thus Φ|_{y} = 0 by Lemma 2.19, or else v|_{y} = 0. If v|_{y} ≠ 0, we see that Φ would be zero on the nonempty open subset {v ≠ 0} ∩ B(x_{j}, R_{0}). But then Lemma 5.12 would imply that Φ ≡ 0 on X, contradicting our assumption that Φ ≢ 0. Thus, v ≡ 0 on B(x_{j}, R_{0}), as desired. q.e.d.

The following similar argument shows that ψ ≡ 0 on the ball B(x_{j}, R_{0}). Note that having only v ≡ 0 or ψ ≡ 0 on an open set does not suffice to contradict Lemma 5.9, as the non-vanishing result of Lemma 5.9 applies to the pair (v, ψ). We again begin with a linear algebra lemma:

Lemma 5.16. Let S^{+} ∈ (W^{+} ⊗ E)|_{x} and S^{−} ∈ (W^{−} ⊗ E)|_{x}. If 𝔪_{0}, …, 𝔪_{3} span 𝔲(E)|_{x} and

∑_{l=0}^{3}⟨(ϑ_{l} ⊗ 𝔪_{l})S^{+}, S^{−}⟩ = 0

for all ϑ_{0}, …, ϑ_{3} ∈ Hom_{ℂ}(W^{+}, W^{−})|_{x}, then S^{+} = 0 or S^{−} = 0.

Proof. Because {𝔪_{l}}_{l=0}^{3} spans 𝔲(E)|_{x} and 𝔤𝔩(E)|_{x} = 𝔲(E)|_{x}⊕ i𝔲(E)|_{x}, we have 𝔤𝔩(E)|_{x} = 𝔲(E)|_{x} ⊗_{ℝ} ℂ, and the set {𝔪_{l}}_{l=0}^{3} is a complex basis for 𝔤𝔩(E)|_{x}. Thus, any element of

Hom_{ℂ}(W^{+} ⊗_{ℂ} E, W^{−} ⊗_{ℂ} E)|_{x} ≃ Hom_{ℂ}(W^{+}, W^{−})|_{x} ⊗_{ℂ} 𝔤𝔩(E)|_{x}

can be written as ∑_{l=0}^{3} ϑ_{l} ⊗ 𝔪_{l}, for some ϑ_{l} ∈ Hom_{ℂ}(W^{+}, W^{−})|_{x}, l = 0, …, 3. Thus, if S^{+} ≠ 0, the hypothesis implies that S^{−} = 0 and conversely, if S^{−} ≠ 0, then S^{+} = 0. q.e.d.

Lemma 5.17. Continue the hypotheses of Proposition 5.8 and suppose B(x_{j}, R_{0}) is a ball supporting holonomy perturbations for A. Then ψ ≡ 0 on B(x_{j}, R_{0}).

Proof. By hypothesis, there are holonomy sections 𝔪_{l} := 𝔪_{j,l,α}(A), l = 1, 2, 3, which span 𝔰𝔲(E)|_{y}, for any point y ∈ B(x_{j}, R_{0}), and so {𝔪_{l}}_{l=0}^{3} spans 𝔲(E)|_{y}, where 𝔪_{0} := i·id_{E}. Let δϑ_{l} := δϑ_{j,l,α} ∈ Ω^{1}(X, ℂ), l = 1, 2, 3, denote the corresponding coefficients, and let δϑ⃗ be a perturbation sequence with all other coefficients equal to zero.

By the hypothesis of Proposition 5.8, we have

(D\underline{𝔖}(δτ_{0}, δϑ_{0}, δτ⃗, δϑ⃗, a, ϕ), (v, ψ))_{L^{2}} = 0

for all (δτ_{0}, δϑ_{0}, δτ⃗, δϑ⃗, a, ϕ), and so

0 = (D\underline{𝔖}(0, δϑ_{0}, 0, δϑ⃗, 0, 0), (v, ψ))_{L^{2}}

= (D\underline{𝔖}_{2}(0, δϑ_{0}, 0, δϑ⃗, 0, 0), ψ)_{L^{2}}

= (ρ(δϑ_{0})Φ + δϑ⃗ · 𝔪⃗(A)Φ, ψ)_{L^{2}}

= ∑_{l=0}^{3}(ρ(δϑ_{l}) ⊗ 𝔪_{l}Φ, ψ)_{L^{2}}.

Taking a sequence of δϑ_{l}’s which approximate δϑ_{l,y}δ(·, y), where δ(·, y) is the Dirac delta distribution supported at y and δϑ_{l,y} ∈ T^{∗}X|_{y} ⊗ ℂ, we obtain the pointwise identity

∑_{l=0}^{3}⟨ρ(δϑ_{l,y}) ⊗ 𝔪_{l}Φ|_{y}, ψ|_{y}⟩ = 0,

for all δϑ_{l,y} ∈ T^{∗}X|_{y} ⊗ ℂ, l = 0, 1, 2, 3. Lemma 5.16 then implies that either Φ|_{y} = 0 or ψ|_{y} = 0. If ψ|_{y} ≠ 0, then Φ would be zero on the nonempty open subset {ψ ≠ 0}∩B(x_{j}, R_{0}). Consequently, Lemma 5.12 would imply that Φ ≡ 0 on X, again contradicting our assumption that Φ ≢ 0. Thus, ψ ≡ 0 on B(x_{j}, R_{0}), as desired. q.e.d.

We can now conclude the proof of Proposition 5.8:

Proof of Proposition 5.8. If (v, ψ) is in the cokernel of D\underline{𝔖} and B(x_{j}, R_{0}) is a ball supporting holonomy perturbations for A, then (v, ψ) ≡ 0 on B(x_{j}, R_{0}) by Lemmas 5.15 and 5.17. q.e.d.

5.3. Local reducibility implies global reducibility. The goal of this section is to prove Theorem 5.11. The argument has two main ingredients: a local extension result for stabilizers of pairs which are reducible on a ball and a description of how these local stabilizers fit together to give a stabilizer and thus a reducible pair on the whole manifold.

Remark 5.18. The fact that an anti-self-dual connection which is reducible on an open subset is necessarily reducible on all of X is an essential part of Donaldson and Kronheimer’s proof of transversality for the moduli space of anti-self-dual connections in [20, §4.3]. The original argument of Freed and Uhlenbeck [30, pp. 57-58] constructs a parallel section ζ of 𝔰𝔲(E) on the set {F_{A} ≠ 0}. Because the connection A is anti-self-dual and therefore Yang-Mills, so d_{A}^{∗}F_{A} = 0 = d_{A}F_{A}, the set {F_{A} ≠ 0} is open, dense and connected. The section ζ cannot develop any holonomy on {F_{A} = 0}, so it extends across all of X, showing that A is globally reducible. This argument does not work in the case of PU(2) monopoles because the connection A is not necessarily Yang-Mills and our argument does not show that the existence of a nonzero element (v, ψ) in Coker D\underline{𝔖} implies that the connection A is reducible on a dense open subset of X.

We first state the local extension result for pair stabilizers and defer its lengthy proof until after that of Theorem 5.11. Generalizations due to Taubes of the analogous result for anti-self-dual connections, namely Lemma 4.3.21 in [20], appear as Theorems 4 and 5 in [88]. As Taubes points out in [88, p. 35], unique continuation theorems for solutions to the anti-self-dual equation do not seem to follow from standard results for elliptic partial differential equations (such as those of Aronszajn [5]) since the anti-self-dual equation does not linearize as an elliptic equation for the connection. Because the PU(2) monopole equations do not linearize as an elliptic system for pairs, the same remarks apply here as well. Rather than rely on the Agmon-Nirenberg theorem for the unique continuation property for a general class of ordinary differential equations (Theorem 5.25), Taubes proves the required unique continuation property directly for the ordinary differential equation induced by the anti-self-dual equation on a cylinder. (As Mrowka pointed out to us, it should also be possible to deduce the unique continuation results of [88] by studying the anti-self-dual equation on a ball and applying the Fredholm theory of [6].) Recall that B(x_{0}, r_{0}) ⊂ X denotes an open geodesic ball with center at the point x_{0} and radius r_{0}. Also, recall that if [A, Φ] is a point in M_{W,E}, then Proposition 3.7 implies that it has a ‘smooth’ (that is, C^{r}) representative (A, Φ) solving (2.27).

Proposition 5.19. Let X be an oriented, smooth four-manifold with C^{r} Riemannian metric g and injectivity radius ϱ = ϱ(x_{0}) at a point x_{0}. Suppose that 0 < r_{0} < r_{1} ≤ \frac{1}{2}ϱ. Let (A, Φ) be a C^{r} pair solving the PU(2) monopole equations (2.27) on X. If u is a C^{r+1} gauge transformation of E|_{B(x_{0},r_{0})} satisfying u(A, Φ) = (A, Φ) on B(x_{0}, r_{0}), and if either B(x_{j}, R_{0}) ∩ B(x_{0}, r_{1}) = ∅ or B(x_{j}, R_{0}) ⊂ B(x_{0}, r_{0}), for all balls B(x_{j}, R_{0}) for which β_{j}[A] > 0, then there is an extension of u to a C^{r+1} gauge transformation û of E|_{B(x_{0},r_{1})} with û(A, Φ) = (A, Φ) on B(x_{0}, r_{1}).

Remark 5.20. This extension result only holds on domains B(x_{0}, r_{1})\B̄(x_{0}, r_{0}) where the perturbations vanish because the perturbation terms 𝔪_{j,l,α}(A) depend not just on the connection A and its derivatives at a point, but rather on the connection A over open neighborhoods in X. Although the unique continuation theorem of [4] does allow certain integral terms, it still does not cover the perturbations we consider here because of their non-local dependence on A.

We digress briefly to introduce some useful facts about stabilizer subgroups of ^{∘}G_{E}.

Lemma 5.21. If u ∈ Stab_{Φ} for Φ ∈ C^{0}(X, W^{+} ⊗ E) and u ≠ id_{E}, then Φ is rank one. If u ∈ S_{Z}^{1} and u ≠ id_{E}, then Φ ≡ 0 on X.

Proof. Because u and Φ are continuous, the equality uΦ = Φ holds at each point x ∈ X and so u|_{x} ∈ 𝔤𝔩(E)|_{x} must be the identity on the image of Φ in E|_{x}. If Φ|_{x} is rank two, then u|_{x} must be the identity on E|_{x}, while if u|_{x} ≠ id_{E_{x}}, then Φ|_{x} can be at most rank one. If u ∈ S_{Z}^{1}, then uΦ = e^{iθ}Φ = Φ and so Φ = 0. q.e.d.

Next we consider the stabilizers of reducible pairs. Recall from [44, Chapter II], [61, §III.3.3] that the stabilizer Stab_{A} ⊂ ^{∘}G_{E} may be identified with a subgroup of Aut(E|_{x}), for any point x ∈ X, by parallel translation with respect to the connection A and hence identified with a subgroup of U(2) by choosing an orthonormal frame for E|_{x}; these subgroups are again denoted by Stab_{A}. If E = L_{1} ⊕ L_{2}, let S_{L_{1}}^{1} denote the group of gauge transformations given by e^{iθ}id_{L_{1}} ⊕ id_{L_{2}}.

Lemma 5.22. Let (A, Φ) be a PU(2) monopole in C^{r} on (E, W^{+} ⊗ E). Let U ⊂ X be a connected open set, with U ∩ B(x_{j}, R_{0}) empty or B(x_{j}, 2R_{0}) ⊂ U for all balls B(x_{j}, R_{0}) such that β_{j}[A] > 0. If Φ ≢ 0 and A|_{U} is reducible, then A|_{U} is reducible with respect to a splitting E|_{U} = L_{1} ⊕L_{2} where Φ is rank one on U, with image contained either in L_{1} or in L_{2}. In the first case, Stab_{(A,Φ)|_{U}} = S_{L_{2}}^{1}, while in the second case, Stab_{(A,Φ)|_{U}} = S_{L_{1}}^{1}.

Proof. Because A|_{U} is reducible, all the holonomy sections 𝔪_{j,l,α}(A) vanish on U if B(x_{j}, 2R_{0}) ⊂ U. If B(x_{j}, R_{0})∩U is empty, the holonomy sections also vanish on U, as they are supported on B(x_{j}, R_{0}). Since Φ ≢ 0, we have (F_{A}^{+})_{0} ≢ 0 on U by Lemma 2.19 and the equation (F_{A}^{+})_{0} = (id + τ_{0})ρ^{−1}(Φ ⊗ Φ^{∗})_{00} of (2.27), so A|_{U} is not projectively flat. Therefore, Stab_{A|_{U}} ≃ T^{2} and A|_{U} is reducible with respect to a splitting E|_{U} = L_{1} ⊕ L_{2} by Lemma 2.6.

Because the connection A|_{U} is reducible with respect to the splitting E|_{U} = L_{1} ⊕ L_{2}, we can write A|_{U} = A_{1} ⊕ A_{2}, where A_{1}, A_{2} are unitary connections on L_{1}, L_{2}, so F_{A}|_{U} = F_{A_{1}} ⊕ F_{A_{2}} and

(F_{A}^{+})_{0} = (\begin{matrix}\frac{1}{2}(F_{A_{1}}^{+} − F_{A_{2}}^{+}) & 0 \\ 0 & −\frac{1}{2}(F_{A_{1}}^{+} − F_{A_{2}}^{+}) \end{matrix}) = ϖ ⊗ σ_{1},

where ϖ = −\frac{i}{2}(F_{A_{1}}^{+} − F_{A_{2}}^{+}) ∈ Ω^{+}(U, ℝ) and σ_{1} ∈ 𝔰𝔲(2) is one of the Pauli matrices (2.14). Hence, (F_{A}^{+})_{0} is rank one on U, and the equation (F_{A}^{+})_{0} = (id + τ_{0})ρ^{−1}(Φ ⊗ Φ^{∗})_{00} implies that (Φ ⊗ Φ^{∗})_{00} is rank one on U. Lemma 2.21 then implies that Φ is also rank one on U and so Φ|_{U} = ϕ ⊗ ξ for some ϕ ∈ C^{r}Ω^{0}(U, W^{+}) and ξ ∈ C^{r}Ω^{0}(U, E), and

(Φ ⊗ Φ^{∗})_{00} = −i(ϕ ⊗ ϕ^{∗})_{0} ⊗ i(ξ ⊗ ξ^{∗})_{0}.

Writing ξ = ξ_{1} + ξ_{2} for ξ_{j} ∈ C^{r}Ω^{0}(U, L_{j}), we see that

(ξ ⊗ ξ^{∗})_{0} = (\begin{matrix}\frac{1}{2}(|ξ_{1}|^{2} − |ξ_{2}|^{2}) & ξ_{1} ⊗ ξ_{2}^{∗} \\ ξ_{2} ⊗ ξ_{1}^{∗} & −\frac{1}{2}(|ξ_{1}|^{2} − |ξ_{2}|^{2}) \end{matrix}) .

Since (id + τ_{0})ρ^{−1}(Φ ⊗ Φ^{∗})_{00} = −(id + τ_{0})ρ^{−1}(i(ϕ ⊗ ϕ^{∗})_{0}) ⊗ i(ξ ⊗ ξ^{∗})_{0}, we have

(id + τ_{0})ρ^{−1}(Φ ⊗ Φ^{∗})_{00}

= −(id + τ_{0})ρ^{−1}(i(ϕ ⊗ ϕ^{∗})_{0}) ⊗ (\begin{matrix}\frac{i}{2}(|ξ_{1}|^{2} − |ξ_{2}|^{2}) & i(ξ_{1} ⊗ ξ_{2}^{∗}) \\ i(ξ_{2} ⊗ ξ_{1}^{∗}) & −\frac{i}{2}(|ξ_{1}|^{2} − |ξ_{2}|^{2}) \end{matrix}) ,

and by comparison with our matrix expression for (F_{A}^{+})_{0}, we see that ξ_{1} ⊗ ξ_{2}^{∗} = 0 on U and thus at each point of U, either ξ_{1} = 0 or ξ_{2} = 0. Since ∇_{A}ξ = ∇_{A_{1}}ξ_{1}⊕∇_{A_{2}}ξ_{2} and because the perturbations vanish on U, the equation (D_{A} +ϑ⃗·𝔪⃗(A))Φ = 0 reduces to (D_{A_{1}} +ρ(ϑ_{0}))(ϕ⊗ξ_{1}) = 0 and (D_{A_{2}} + ρ(ϑ_{0}))(ϕ ⊗ ξ_{2}) = 0. The unique continuation result for the perturbed Dirac operator, Lemma 5.12, implies that if ϕ ⊗ ξ_{1} vanishes on an open subset of U, then ϕ ⊗ ξ_{1} ≡ 0 on U, and similarly for ϕ ⊗ ξ_{2}. If ξ_{1} is non-zero at a point and thus non-zero on an open neighborhood, ξ_{2} vanishes on this open set and by unique continuation, ξ_{2} ≡ 0 on all of the connected set U. Symmetrically, if ξ_{2} is non-zero at a point, then ξ_{1} ≡ 0 on U. Thus, Φ = ϕ ⊗ ξ_{1} or Φ = ϕ ⊗ ξ_{2}.

The stabilizer of A|_{U} is S_{L_{1}}^{1} × S_{L_{2}}^{1}. If ξ_{1} = 0, then Stab_{Φ|_{U}} = Map(U, S_{L_{1}}^{1}) while if ξ_{2} = 0, then Stab_{Φ|_{U}} = Map(U, S_{L_{2}}^{1}). q.e.d.

We see that elements of the stabilizer of a pair cannot exhibit holonomy, in the sense of the following lemma.

Lemma 5.23. Suppose (A, Φ) is a PU(2) monopole with Φ ≢ 0 and that U_{1}, U_{2} are connected open subsets of X. If there are gauge transformations u_{i} ∈ ^{∘}G_{E|U_{i}}, i = 1, 2, such that u_{i} ∈ Stab_{A|_{U_{i}}}, u_{i} ∈ Stab_{(A,Φ)|_{U_{1}∩U_{2}}} , u_{i} ≠ id, and there is a point x ∈ U_{1} ∩ U_{2} such that u_{1} = u_{2} on E|_{x}, then u_{1} = u_{2} on E|_{U_{1}∩U_{2}}.

Proof. Let V ⊂ U_{1} ∩ U_{2} be the dense open subset of points {Φ ≠ 0} ∩ U_{1} ∩ U_{2}. Because there is a gauge transformation u_{i} ∈ ^{∘}G_{E|U_{i}} with u_{i}Φ = Φ over U_{i}, then Φ|_{U_{1}∩U_{2}} must be rank one by Lemma 5.21 and there is an orthogonal decomposition E|_{V} = Im Φ ⊕ (Im Φ)^{⊥}. Since u_{i}Φ = Φ on V, both u_{i} respect this decomposition and must be the identity on Im Φ|_{V}. Thus, on V we can write

u_{i} = (\begin{matrix}1 & 0 \\ 0 & e^{iθ_{i}} \end{matrix})

with respect to this decomposition. Now det u_{i} = e^{iθ_{i}} and because u_{i} ∈ ^{∘}G_{E|U_{i}}, the function det u_{i} is constant on U_{i} and e^{iθ_{i}} ∈ S^{1}. Hence, if u_{1} = u_{2} on E|_{x}, then u_{1} = u_{2} on all points in V which can be connected to x by a path in U_{1} and a path in U_{2} (note that these need not be the same paths). Because U_{1} and U_{2} are connected, u_{1} = u_{2} over all of V. But V is dense in U_{1} ∩ U_{2} and the u_{i} are continuous, so u_{1} = u_{2} over all of U_{1} ∩ U_{2}. q.e.d.

Theorem 5.11 now follows from Proposition 5.19:

Proof of Theorem 5.11, given Proposition 5.19. Let (A, Φ) be a C^{r} solution to the PU(2) monopole equations (2.27) with Φ ≢ 0 and A reducible on a non-empty open set U ⊂ X. Let J(A) = {j : 1 ≤ j ≤ N_{G} and β_{j}[A] > 0} and let B̄_{J}(A) = ∪_{j∈J(A)}B̄(x_{j}, R_{0}). Let U^{∘} be a connected component of U − B̄_{J}(A). Lemma 5.22 then implies that Stab_{(A,Φ)|_{U^{∘}}} ≃ S^{1} ≠ S_{Z}^{1}, and so there is a C^{r+1} gauge transformation u of E|_{U^{∘}} such that u(A, Φ) = (A, Φ) on U^{∘}. Proposition 5.19 allows the extension of u to open subsets of X\B̄_{J}(A) containing U^{∘}. To be explicit, let x ∈ U^{∘} and r_{0} = dist_{g}(x, ∂U^{∘}) and choose r_{1} > r_{0} such that r_{1} ≤ \frac{1}{2}ϱ and r_{1} ≤ min_{j∈J(A)} dist_{g}(x, B(x_{j}, R_{0})). By Proposition 5.19 there is an extension of u ∈ Stab_{(A,Φ)|_{B(x,r_{0})}} to an element û ∈ Stab_{(A,Φ)|_{B(x,r_{1})}}. By Lemma 5.23 we have û = u on U^{∘} ∩ B(x_{0}, r_{1}) and not just B(x_{0}, r_{0}). This gives an extension of the stabilizer u for (A, Φ) over U^{∘} to a stabilizer û over the slightly larger open set U^{∘} ∪ B(x, r_{1}). Since we do not assume that X is simply connected, we must check that the extension obtained by repeating this process yields a single-valued gauge transformation over X − B̄_{J}(A).

The consistency of two extensions follows from Lemma 5.23. Let u_{i}, i = 1, 2, be two extensions of u to connected open sets U_{i}^{∘} containing U^{∘}, so we have u_{i} ∈ Stab_{(A,Φ)|_{U^{∘}_{i}}} with u_{i} = u on E|_{U^{∘}}. Because u_{1} = u = u_{2} on E|_{U^{∘}}, Lemma 5.23 implies that u_{1} = u_{2} on E|_{U_{1}^{∘}∩U_{2}^{∘}}. Therefore, the extensions of u ∈ Stab_{(A,Φ)|_{U^{∘}}} fit together to form a global gauge transformation û ∈ Stab_{A,Φ} on X − B̄_{J}(A) such that û = u on E|_{U^{∘}}.

Thus, given that A is reducible on an open set U containing B̄_{J}(A), the above argument produces an element u ∈ Stab_{(A,Φ)|_{X−B̄_{J} (A)}} with u ∉ S_{Z}^{1}. Consequently, A is reducible on X −B̄_{J}(A) by Lemma 2.6. For j = 1, …, N_{b}, let V_{j} be an open, connected subset of U ∩ B(x_{j}, 2R_{0}) containing B̄(x_{j}, R_{0}), such that V_{j} ∩ (X − B_{J}(A)) is connected, and set

X_{1} = X − B̄_{J}(A),

X_{j} = (X − B̄_{J}(A)) ∪ ⋃_{1≤k≤j−1} V_{k}, 2 ≤ j ≤ N_{b} + 1.

Each subset X_{j} and V_{j} ∩ X_{j} is connected, and X_{1} ⊂ X_{2} ⊂ ⋯ ⊂ X_{N_{b}+1} = X. We extend u ∈ Stab_{(A,Φ)|_{X−B̄_{J} (A)}} inductively over each X_{j}. Plainly, we have u ∈ Stab_{A|_{V_{j} ∩X_{j}}} and Stab_{A|_{V_{j}}} ⊂ Stab_{A|_{V_{j} ∩X_{j}}}, so we first check that Stab_{A|_{V_{j}}} = Stab_{A|_{V_{j} ∩X_{j}}}. By hypothesis, A|_{U} is reducible and so for each subset V_{j} ⊂ U, we have that A|_{V_{j}} is reducible. Because Φ|_{V_{j}∩X_{j}} ≢ 0 and thus F_{A} ≢ 0 on V_{j} ∩ X_{j} by (2.27) — so A|_{V_{j}∩X_{j}} is not flat — then Lemma 2.6 implies that Stab_{A|_{V_{j} ∩X_{j}}} ≃ T^{2}. The same argument yields Stab_{A|_{V_{j}}} ≃ T^{2}. (We use the assumptions on the connectedness of V_{j} and V_{j} ∩ X_{j} here: if the sets were not connected, the stabilizers would be ⊕T^{2}, a direct sum over connected components.) Thus, Stab_{A|_{V_{j}}} = Stab_{A|_{V_{j} ∩X_{j}}} ≃ T^{2}.

Hence, there is an element u′′ ∈ Stab_{A|_{V_{j}}} such that u′′ = u′ on V_{j} ∩ X_{j}, where u′ ∈ Stab_{(A,Φ)|_{X_{j}}} with u′ ≠ id. Together, u′ and u′′ give an element u ∈ Stab_{A|_{X_{j+1}}} which is not in S_{Z}^{1}. The connection A is then reducible on X_{j+1}, which implies that all the holonomy perturbations vanish on X_{j+1}. Lemma 5.22 therefore shows that the pair (A, Φ) is reducible on X_{j+1} and we obtain a stabilizer u ∈ Stab_{(A,Φ)|_{X_{j+1}}} , with u ≠ id. The construction of u ∈ Stab_{A,Φ}, u ≠ id, is thus completed by induction on j. q.e.d.

Remark 5.24. The analogue of Theorem 5.11 (namely, that local reducibility implies global reducibility) does not hold for anti-self-dual connections without further restrictions on the topology of X. For example, in [30] it is assumed that the four-manifold X is simply connected (see Lemma 4.3.21). As described by Kronheimer and Mrowka in [47] one can have locally reducible anti-self-dual connections (called ‘twisted reducibles’ in [47]); see their Lemma 2.4 for a sharp version of the Freed-Uhlenbeck generic metrics theorem (Corollaries 4.3.15, 4.3.18, and 4.3.19 in [20]) which holds when the requirement that X be simply connected is dropped. In our case, we see from the proof of Theorem 5.11 that globally irreducible, locally reducible solutions (A, Φ) to (2.27) do not exist (at least when Φ ≢ 0) because the stabilizer u ∈ Stab_{A,Φ} must stabilize the section Φ and not just the connection A.

The proof of Proposition 5.19 takes up the remainder of this section.

5.3.1. The Agmon-Nirenberg unique continuation theorem. As in the case of the anti-self-dual equation [20, Lemma 4.3.21], our proof of the unique continuation property for PU(2) monopoles in radial gauge relies on the following special case of a more general result due to Agmon and Nirenberg for an ordinary differential equation on a Hilbert space [4]:

Theorem 5.25 [4, Theorem 2 (ii)]. Let ℌ be a Hilbert space and let P : Dom(P(r)) ⊂ ℌ → ℌ be a family of symmetric linear operators for r ∈ [r_{0}, R). Suppose that η ∈ C^{1}([r_{0}, R), ℌ), with η(r) ∈ Dom(P(r)) and Pη ∈ C^{0}([r_{0}, R), ℌ) such that

∥\frac{dη}{dr} − P(r)η(r)∥ ≤ c_{1} ∥η(r)∥ , (5.10)

for some positive constant c_{1} and all r ∈ [r_{0}, R). If the function r ↦ (η(r), P(r)η(r)) is differentiable for r ∈ [r_{0}, R) and satisfies

\frac{d}{dr} (η, Pη) − 2 Re (\frac{dη}{dr} , Pη) ≥ −c_{2} ∥Pη∥ ∥η∥ − c_{3}∥η∥^{2}, (5.11)

for positive constants c_{2}, c_{3} and every r ∈ [r_{0}, R), then the following holds: If η(r_{0}) = 0, then η(r) = 0 for all r ∈ [r_{0}, R).

Applications of [4, Theorem 2] to the proof of unique continuation results for first-order elliptic and parabolic partial differential equations were considered by Agmon in [3, Chapter II]. In our application, P(r) will be a family of first-order partial differential operators which are selfadjoint over the closed manifold X. Theorem 5.25 has also been applied by D. Salamon to prove the unique continuation property for harmonic spinors [77, Appendix E]. One of the difficulties in applying the Agmon-Nirenberg theorem to the PU(2) monopole equations (in Gaussian polar coordinates, (r, θ)) is the requirement that the one-parameter family of operators be self-adjoint with respect to a fixed inner product on a fixed vector space. The additional complication, not present in [20, §4.3.4], is that the induced spin^{c} structures on geodesic spheres in X vary with the induced family of r-dependent metrics.

Remark 5.26 [4, Remark, p. 209]. If Dom(P(r)) = D is independent of r, and P(r), r ∈ [r_{0}, R), is a differentiable family of self-adjoint operators, then the left-hand side of (5.11) simplifies, of course, to give the condition

(η, \frac{dP}{dr} η) ≥ −c_{2} ∥Pη∥ ∥η∥ − c_{3}∥η∥^{2}, (5.12)

since, noting that P is self-adjoint,

\frac{d}{dr} (η, Pη) − 2 Re (\frac{dη}{dr} , Pη) = (\frac{dη}{dr} , Pη) + (η, \frac{dP}{dr} η) + (η, P \frac{dη}{dr} )

− (\frac{dη}{dr} , Pη) − (Pη, \frac{dη}{dr} )

= (η, \frac{dP}{dr} η) .

If P(r), r ∈ [r_{0}, R), is a differentiable family of self-adjoint operators, then it is easy to see that (5.12) follows from the simpler condition

∥\frac{dP}{dr} η∥ ≤ c_{2}∥Pη∥ + c_{3}∥η∥, (5.13)

provided Pη ∈ C^{0}([r_{0}, R), ℌ).

5.3.2. The PU(2) monopole equations in Gaussian polar coordinates. Our first task is to write the pair of PU(2) monopole equations (2.27) as an ordinary differential equation with respect to Gaussian polar coordinates (r, θ) centered at a point x_{0} ∈ X. For the analogous ordinary differential equation in the case of the anti-self-dual equation, see [20], [63], [88], and for the Seiberg-Witten equations; see [47], [77].

Recall that ϱ is the injectivity radius of (X, g) at the point x_{0}, so exp_{x_{0}} : B(0, ϱ) ⊂ (T X)_{x_{0}} → B(x_{0}, ϱ) ⊂ X is a diffeomorphism. For each ξ in the unit sphere S^{3} ⊂ (T X)_{x_{0}}, let {e_{i}(ξ)}_{i=1}^{3} be an oriented, orthonormal basis for (ℝξ)^{⊥} = (T S^{3})_{ξ} ⊂ (T X)_{x_{0}}; that is, let {e_{i}} be an orthonormal frame for T S^{3}. Let γ_{ξ}(r) be the geodesic exp_{x_{0}}(rξ), r ∈ [0, ϱ), so that γ_{ξ}(0) = x_{0} and |γ′_{ξ}(r)| = |ξ|, and let τ_{ξ}(r) : (T X)_{x_{0}} → (T X)_{γ_{ξ}(r)} denote parallel translation with respect to the metric’s Levi-Civita connection along γ_{ξ}(r). Let e_{i}(r, ξ) := τ_{ξ}(r)e_{i}(ξ) for r ≥ 0, so that {γ′_{ξ}(r), e_{i}(r, ξ)} is an orthonormal frame for (T X)_{γ_{ξ}(r)} which is parallel along the radial geodesics γ_{ξ}(r) and satisfies γ′_{ξ}(0) = ξ and e_{i}(0, ξ) = e_{i}(ξ). Denote the radial vector γ′_{ξ}(r) ∈ (T X)_{γ_{ξ}(r)} by \frac{∂}{∂r} := \frac{∂}{∂r} |_{γ_{ξ}(r)} when no confusion can arise. Thus, \{\frac{∂}{∂r} , e_{i}\} is an oriented, orthonormal frame for T X over B(x_{0}, ϱ) \ {0}, which is parallel along radial geodesics; let {dr, e^{i}} be the corresponding dual frame for T^{∗}X over B(x_{0}, ϱ) \ {0}.

With respect to the parametrization S^{3} × (0, ϱ) ≃ B(x_{0}, ϱ) − {x_{0}}, given by (r, ξ) ↦ exp_{x_{0}}(rξ), the metric g on B(x_{0}, ϱ) − {x_{0}} pulls back to

g = (dr)^{2} + g_{r},

where g_{r} is the metric on S^{3} pulled back from the restriction g|_{S^{3}(x_{0},r)} to the geodesic sphere S^{3}(x_{0}, r) := {x ∈ X : dist_{g}(x, x_{0}) = r}. Let ∗_{g_{r}} denote the Hodge star operator for the metric g_{r} on S^{3} and, for emphasis, we write ∗_{g} for the Hodge star operator for the metric g on X.

Suppose that a pair (A, Φ) on (𝔰𝔲(E), W^{+} ⊗ E) is a C^{r} solution to the PU(2) monopole equations (2.27) over X,

ρ(F_{A}^{+}) = ρτρ^{−1}(Φ ⊗ Φ^{∗})_{00},
D_{A,ϑ_{0}}Φ = 0,

where τ := id_{Λ^{+}} + τ_{0} is an automorphism of Λ^{+} and

D_{A,ϑ_{0}}Φ := D_{A}Φ + ρ(ϑ_{0})Φ.

We have not included the holonomy perturbations τ⃗ · 𝔪⃗(A) and ϑ⃗ · 𝔪⃗(A) because they vanish near x_{0} by hypothesis.

We obtain an isomorphism E|_{B(x_{0},ϱ)} ≃ E_{0} × B(x_{0}, ϱ) of complex two-plane bundles by choosing a unitary frame for E_{0} := E|_{x_{0}} and using parallel translation via the U(2) connection on E defined by A and A_{det E} along radial geodesics emanating from x_{0}. Let A = B + Cdr denote the induced SO(3) connection on the bundle 𝔰𝔲(E_{0}) × S^{3} × (0, ϱ) over S^{3} × (0, ϱ) and note that A is in radial gauge with respect to the point x_{0}, so C := A(\frac{∂}{∂r}) = 0. We let B = B(r), r ∈ (0, ϱ), denote the resulting one-parameter family of SO(3) connections on the bundle 𝔰𝔲(E_{0}) × S^{3} over S^{3}. In exactly the same way, we obtain an induced one-parameter family of U(2) connections on the bundle W_{0}^{+} × S^{3} over S^{3}, r ∈ (0, ϱ), induced by the isomorphism W^{+}|_{B(x_{0},ϱ)−{x_{0}}} ≃ W_{0}^{+} × S^{3} × (0, ϱ).

A section Φ of the bundle W^{+} ⊗ E over B(x_{0}, ϱ) pulls back, via the isomorphism (W^{+} ⊗ E)|_{B(x_{0},ϱ)−{x_{0}}} ≃ (W_{0}^{+} ⊗ E_{0}) × S^{3} × (0, ϱ), to a one-parameter family of sections Ψ(r) of the bundle W_{0}^{+} ⊗ E_{0} × S^{3} over S^{3}. The automorphism τ of Λ^{+,g} pulls back to a one-parameter family of automorphisms σ(r) of T^{∗}S^{3}, for r ∈ (0, ϱ), using the isomorphism

(0, ϱ) × T^{∗}S^{3} → Λ^{+,g}(T^{∗}X), (r, α) ↦ ∗_{g_{r}}α + dr ∧ α.

The g-compatible Clifford map ρ : T^{∗}X → Hom(W^{+}, W^{−}) and the isomorphism W^{+}|_{B(x_{0},ϱ)−{x_{0}}} ≃ W_{0}^{+} × S^{3} × (0, ϱ) define a family of g_{r}-compatible Clifford maps γ(r) : T^{∗}S^{3} → End(W_{0}^{+}) by setting

γ(r) := ρ(dr)ρ( · ).

Indeed, to see this, observe that g(dr, dr) = 1 and so for a family of one-forms α(r) on S^{3}, defined by the isomorphism

B(x_{0}, ϱ) − {x_{0}} ≃ S^{3} × (0, ϱ)

and a one-form α on B(x_{0}, ϱ), we have

γ_{r}(α)^{†}γ_{r}(α) = ρ(α)^{†}ρ(dr)^{†}ρ(dr)ρ(α) = ρ(α)^{†}ρ(α)
= g_{r}(α, α) id_{W_{0}^{+}},

as required. The map γ extends to a one-parameter family of Clifford maps γ(r) : Λ^{•}(T^{∗}S^{3}) ⊗ ℂ → End(W_{0}^{+}) in the usual way. For example, γ_{r}(α ∧ β) := γ_{r}(α)γ_{r}(β), for α, β ∈ Ω^{1}(S^{3}), in which case we see that γ_{r}(α ∧ β) = ρ(α)ρ(β).

With the above understood, we can proceed to rewrite the PU(2) monopole equations (2.27) over the ball B(x_{0}, ϱ) as an ordinary differential equation for a one-parameter family of pairs (B(r), Ψ(r)) on (𝔰𝔲(E_{0}), W_{0}^{+} ⊗ E_{0}) over S^{3}. The curvature F_{A} of the connection A over B(x_{0}, ϱ) is given by

F_{A} = F_{B} − \frac{dB}{dr} ∧ dr.

For any ω ∈ Ω^{2}(X, ℝ), we have ρ_{+}(ω) = ρ_{+}(∗_{g}ω), since ρ_{+}|_{Λ^{−}} = 0. If the radial component of ω vanishes and we consider ω|_{B(x_{0},ϱ)} as a one-parameter family of two-forms ω(r) on S^{3}, then we see that ∗_{g}ω = −(∗_{g_{r}}ω) ∧ dr. Combining these observations yields

ρ(F_{A}^{+}) = ρ_{+}(F_{A}) = ρ_{+} (F_{B} − \frac{dB}{dr} ∧ dr)

= ρ_{+} (−(∗_{g_{r}}F_{B}) ∧ dr − \frac{dB}{dr} ∧ dr)

= ρ_{−}(dr)ρ_{+} (∗_{g_{r}}F_{B} + \frac{dB}{dr} ) ,

and therefore,

ρ(F_{A}^{+}) = γ (∗_{g_{r}}F_{B} + \frac{dB}{dr} ) .

The section ρτρ^{−1}(Φ⊗Φ^{∗})_{00} of 𝔰𝔲(W^{+})⊗𝔰𝔲(E) over B(x_{0}, ϱ) pulls back to the one-parameter family of sections γσγ^{−1}(Ψ ⊗ Ψ^{∗})_{00} of

𝔰𝔲(W_{0}^{+}) ⊗ 𝔰𝔲(E_{0}) × S^{3}

over S^{3} via the isomorphism

W^{+}|_{B(x_{0},ϱ)−{x_{0}}} ≃ W_{0}^{+} × S^{3} × (0, ϱ),

and similarly for E.

Let θ(r) be the induced one-parameter family of complex one-forms on S^{3} defined by ϑ_{0} on B(x_{0}, ϱ) and the isomorphism

B(x_{0}, ϱ) − {x_{0}} ≃ S^{3} × (0, ϱ),

so ϑ_{0} = −fdr + θ on S^{3} × (0, ϑ), where f(r) ∈ Ω^{0}(S^{3}, ℂ). Given the preceding identifications, the Dirac operator term in (2.27) can then be written over S^{3} × {r} as

D_{A,ϑ_{0}}Φ = ρ(dr)∇A_{\frac{∂}{∂r}} Ψ + ∑_{i=1}^{3} ρ(e^{i})∇_{e_{i}}^{A}Ψ + ρ(ϑ_{0})Ψ

= ρ(dr)\frac{dΨ}{dr} + ∑_{i=1}^{3} ρ(e^{i})∇_{e_{i}}^{B}Ψ + ρ(θ)Ψ − fρ(dr)Ψ

= ρ(dr) (\frac{dΨ}{dr} − ∑_{i=1}^{3} ρ(dr)ρ(e^{i})∇_{e_{i}}^{B}Ψ − ρ(dr)ρ(θ)Ψ − fΨ)

= ρ(dr) (\frac{dΨ}{dr} − ∑_{i=1}^{3} γ(e^{i})∇_{e_{i}}^{B}Ψ − γ(θ)Ψ − fΨ) ,

and therefore,

D_{A,ϑ_{0}}Φ = ρ(dr) (\frac{dΨ}{dr} − D_{B,θ,f}Ψ) .

Hence, the PU(2) monopole equations (2.27) can be written as

γ (\frac{dB}{dr} + ∗_{g_{r}}F_{B}) = γσγ^{−1}(Ψ ⊗ Ψ^{∗})_{00},
ρ(dr)\frac{dΨ}{dr} = ρ(dr)D_{B,θ,f}Ψ.

We use D_{B} : Ω^{0}(S^{3}, W_{0}^{+} ⊗ E_{0}) → Ω^{0}(S^{3}, W_{0}^{+} ⊗ E_{0}) to denote the one-parameter family of Dirac operators defined by the family of g(r)- compatible Clifford maps γ(r) : T^{∗}S^{3} → End W_{0}^{+}, the family of U(2) connections on W_{0}^{+} × S^{3}, the family of SO(3) connections B(r) on 𝔰𝔲(E_{0}) × S^{3} over S^{3}, and the family of determinant connections on det E_{0} × S^{3} over S^{3}. Since the Clifford map gives an isomorphism γ : T^{∗}S^{3} ≃ 𝔰𝔲(W_{0}^{+}) × S^{3}, the PU(2) monopole equations then take the shape

\begin{aligned}\frac{dB}{dr} + ∗_{g_{r}}F_{B} & = σγ^{−1}(Ψ ⊗ Ψ)_{00}, \\ \frac{dΨ}{dr} & = D_{B,θ,f}Ψ, \end{aligned} (5.14)

for a one-parameter family (B(r), Ψ(r)), r ∈ (0, ϱ), on (E_{0}, W_{0}^{+} ⊗ E_{0}) over S^{3}.

5.3.3. The ordinary differential equation for the difference pair. Let (A, Φ) be a C^{r} solution to the PU(2) monopole equations (2.27) over X. Suppose, as in the hypothesis of Proposition 5.19, that there is a C^{r+1} gauge transformation u of E|_{B(x_{0},r_{0})} such that u(A, Φ) = (A, Φ) on B(x_{0}, r_{0}). Let (A, Φ) again denote the induced pair defined by the isomorphism E|_{B(x_{0},ϱ)} ≃ E_{0} × S^{3} × (0, ϱ) (given by a choice of unitary frame for E_{0} = E_{x_{0}} and parallel, radial translation via A) and let u be the induced gauge transformation on E_{0} × S^{3} × (0, r_{0}). Then u(A) = uAu^{−1} − (d_{A}u)u^{−1} and A = B + C dr, where C = A(\frac{∂}{∂r}) = 0, and so du/dr = 0. We now extend u by parallel translation via A along radial geodesics emanating from x_{0} to a gauge transformation û on E_{0} × S^{3} × (0, ϱ).

Let (Â, Φ̂) = û(A, Φ) be the gauge-equivalent pair on S^{3} × (0, ϱ), so (Â, Φ̂) is a C^{r} solution to (2.27) over S^{3} ×(0, ϱ), with Â = û(A) in radial gauge. In particular, (Â, Φ̂) = (A, Φ) over S^{3} × (0, r_{0}): we need to show that (Â, Φ̂) = (A, Φ) over S^{3} ×(0, r_{1}) in order to prove Proposition 5.19.

The one-parameter family of pairs (B̂(r), Ψ̂(r)) on (𝔰𝔲(E_{0}), W_{0}^{+} ⊗ E_{0}) over S^{3} also satisfies the ordinary differential equation in (5.14) so, subtracting these two pairs of ordinary differential equations, we obtain for r ≥ r_{0},

\frac{d(B̂ − B)}{dr} + ∗_{g_{r}}(FB̂ − F_{B}) = σγ^{−1}(Ψ̂ ⊗ Ψ̂^{∗} − Ψ ⊗ Ψ^{∗})_{00},
\frac{d(Ψ̂ − Ψ)}{dr} = D_{B,θ,f}Ψ − D_{B̂,θ,f} Ψ̂.

Since

F_{B} = d(B − Γ) + (B − Γ) ∧ (B − Γ)

and

D_{B,θ,f} = D + γ(B − Γ) + γ(θ) + fid_{W_{0}^{+}⊗E_{0}},

we obtain an ordinary differential equation for the difference pair

(b, ψ) := (B̂ − B, Ψ̂ − Ψ) ∈ Ω^{1}(S^{3}, 𝔰𝔲(E_{0})) ⊕ Ω^{0}(S^{3}, W_{0}^{+} ⊗ E_{0}),

so that,

\frac{db}{dr} = − ∗_{g_{r}}db + ∗_{g_{r}}(B̂ ∧ b + b ∧ B)

+ σγ^{−1}(Ψ̂ ⊗ ψ^{∗} + ψ ⊗ Ψ^{∗})_{00}, (5.15)

\frac{dψ}{dr} =Dψ + γ(B̂ − Γ)ψ + γ(b)Ψ + γ(θ)ψ + fψ,

where r ∈ (r_{0}, ϱ), and Γ is the product connection on 𝔰𝔲(E_{0}) × S^{3} over S^{3} defined by our trivialization. The above system has the schematic form

\frac{d(b, ψ)}{dr} = (\begin{matrix}− ∗_{g_{r}} d & 0 \\ 0 & D \end{matrix}) (b, ψ) + Z_{r}(b, ψ), (5.16)

where Z_{r} is a one-parameter family of zeroth-order operators with coefficients depending on g_{r}, B, σ, and Ψ.

5.3.4. Reduction to the Agmon-Nirenberg theorem. We first observe that the operator − ∗_{g_{r}} d on Ω^{1}(S^{3}, 𝔰𝔲(E_{0})) is self-adjoint with respect to the L^{2} inner product induced by the metric g_{r} on S^{3}. Indeed, as ∗_{g_{r}}^{2} = 1 on Ω^{1}(S^{3}) and d tr(b ∧ b′) = tr(db ∧ b′) − tr(b ∧ db′), we have

∫_{S^{3}}⟨b, − ∗_{g_{r}} db′⟩_{r} d vol_{r} = − ∫_{S^{3}} tr(b ∧ ∗_{g_{r}}(− ∗_{g_{r}} db′))
= ∫_{S^{3}} tr(b ∧ db′) = ∫_{S^{3}} tr(db ∧ b′)
= ∫_{S^{3}} tr(b ∧ db′) = ∫_{S^{3}} tr(b′ ∧ ∗_{g_{r}}^{2}db)
= − ∫_{S^{3}}⟨b′, ∗_{g_{r}}db⟩_{r} d vol_{r},

and therefore,

∫_{S^{3}}⟨b, − ∗_{g_{r}} db′⟩_{r} d vol_{r} = ∫_{S^{3}}⟨− ∗_{g_{r}} db, b′⟩_{r} d vol_{r} .

The Dirac operators D : Ω^{1}(S^{3}, W_{0}^{+} ⊗ E_{0}) → Ω^{1}(S^{3}, W_{0}^{−} ⊗ E_{0}) are defined by the one-parameter family of g_{r}-compatible Clifford maps γ_{r} : T^{∗}S^{3} → End(W_{0}^{+}), the one-parameter family of U(2) connections on W_{0}^{+} × S^{3}, the one-parameter family of determinant connections on det E_{0} × S^{3}, and the product SO(3) connection on 𝔰𝔲(E_{0}) × S^{3}. Then, by [57, Proposition II.5.3] we have

⟨Dψ, ψ′⟩ = ⟨ψ, Dψ′⟩ + div_{g_{r}} ξ,

where ξ(r) is the one-parameter family of vector fields on S^{3} defined by α(ξ) = −⟨ψ, γ(α)ψ′⟩, for all α ∈ Ω^{1}(S^{3}). Hence, the divergence theorem for the metric g_{r} implies that D is self-adjoint with respect the the family of L^{2} inner products on Ω^{1}(S^{3}, W_{0}^{+} ⊗ E_{0}) defined by g_{r}:

∫_{S^{3}}⟨Dψ, ψ′⟩ d vol_{r} = ∫_{S^{3}}⟨ψ, Dψ′⟩ d vol_{r} .

The Agmon-Nirenberg theorem is not immediately applicable to the ordinary differential equation (5.16) since the differential operator P_{r} := − ∗_{g_{r}} d ⊕ D is only self-adjoint on the Hilbert space

ℌ_{r} := L^{2}(S^{3}, Λ^{1} ⊗ 𝔰𝔲(E_{0})) ⊕ L^{2}(S^{3}, W_{0}^{+} ⊗ E_{0}),

with L^{2} inner products (· , ·)_{r} defined by the family of metrics g_{r} on S^{3}, rather than a fixed inner product.

If d vol_{r} is the volume form on S^{3} defined by the metric g_{r}, then we may write

d vol_{r} = h_{r}^{2}d vol, r ∈ (0, ϱ),

for some positive function h_{r} on S^{3}, where d vol is the volume form on S^{3} defined by the standard metric. However,

Q_{r} := h_{r}P_{r}h_{r}^{−1}, r ∈ (0, ϱ), (5.17)

is a differentiable path of self-adjoint, first-order, elliptic differential operators on the fixed, real Hilbert space underlying

ℌ := L^{2}(S^{3}, Λ^{1} ⊗ 𝔰𝔲(E_{0})) ⊕ L^{2}(S^{3}, W_{0}^{+} ⊗ E_{0}),

with L^{2} inner product (· , ·) defined by the standard metric on S^{3}. Indeed, if we define a Hilbert-space isomorphism ℌ_{r} → ℌ by (b, ψ) ↦ (β, φ) := h_{r}(b, ψ), then

∫_{S^{3}}⟨Q_{r}(β, φ), (β′, φ′)⟩ d vol = ∫_{S^{3}}⟨h_{r}P_{r}h_{r}^{−1}h_{r}(b, ψ), h_{r}(b′, ψ′)⟩ d vol
= ∫_{S^{3}}⟨P_{r}(b, ψ), (b′, ψ′)⟩ d vol_{r}
= ∫_{S^{3}}⟨(b, ψ), P_{r}(b′, ψ′)⟩ d vol_{r}
= ∫_{S^{3}}⟨h_{r}(b, ψ), h_{r}P_{r}h_{r}^{−1}h_{r}(b′, ψ′)⟩ d vol,

and therefore,

∫_{S^{3}}⟨Q_{r}(β, φ), (β′, φ′)⟩ d vol = ∫_{S^{3}}⟨(β, φ), Q_{r}(β′, φ′)⟩ d vol, r ∈ (0, ϱ).

The operators Q_{r} have dense domain L_{1}^{2}. Since (b, ψ) = h_{r}^{−1}(β, φ), we see that

\frac{d(b, ψ)}{dr} = −h_{r}^{−2} \frac{dh_{r}}{dr} (β, φ) + h_{r}^{−1} \frac{d(β, φ)}{dr},

and so, subsituting into (5.16) gives

\begin{matrix}\frac{d(β, φ)}{dr} = Q_{r}(β, φ) + h_{r}Z_{r}h_{r}^{−1}(β, φ) + h_{r}^{−1} \frac{dh_{r}}{dr} (β, φ), \\ r ∈ (0, ϱ), \end{matrix} (5.18)

with (β, φ) a path in ℌ.

5.3.5. Verification of the Agmon-Nirenberg conditions. We can now conclude the proof of our unique continuation result for reducible PU(2) monopoles:

Completion of proof of Proposition 5.19. The estimates (5.10) and (5.12) are, in principle, straightforward to check since we only need them for r varying in the compact interval [\frac{1}{2}r_{0}, r_{1}]; the second condition, (5.12), requires a little more explanation since we need an estimate for dQ/dr. We first check condition (5.10). Comparing (5.15), (5.16), and (5.18), we find that

∥\frac{d(β, φ)}{dr} − Q_{r}(β, φ)∥ ≤ c_{1}∥(β, φ)∥, \frac{1}{2}r_{0} ≤ r ≤ \frac{1}{2}ϱ,

for some positive constant c_{1} = c_{1}(g, r_{0}, A, Φ), and so the ordinary differential equation (5.18) obeys the estimate (5.10) on the interval [\frac{1}{2}r_{0}, r_{1}]. To check condition (5.12), observe that the definition of Q_{r} in (5.16) and (5.17) yields the pointwise bound

|\frac{dQ_{r}}{dr} (β, φ)| ≤ C (|(∇β, ∇φ)| + |(β, φ)|) ,

where \frac{1}{2}r_{0} ≤ r ≤ \frac{1}{2}ϱ, for some positive constant C = C(g, τ, A_{W} , r_{0}), and ∇ denotes covariant derivatives on Λ^{1} ⊗ 𝔰𝔲(E_{0}) and W_{0}^{+} ⊗ E_{0} × S^{3} which are independent of r. Thus, using the standard elliptic estimate for Q_{r} we obtain the L^{2} bound

∥\frac{dQ_{r}}{dr} (β, φ)∥ ≤ C (∥Q_{r}(β, φ)∥ + ∥(β, φ)∥) ,

and so, for \frac{1}{2}r_{0} ≤ r ≤ \frac{1}{2}ϱ,

|((β, φ), \frac{dQ_{r}}{dr} (β, φ))| ≤ C (∥Q_{r}(β, φ)∥(β, φ)∥ + ∥(β, φ)∥^{2}) .

Therefore, (5.12) is obeyed with c_{2} = c_{3} = C on the interval [\frac{1}{2}r_{0}, r_{1}]. By Theorem 5.25 and Remark 5.26, the solution (β(r), φ(r)) vanishes for r ∈ (0, r_{1}) since it vanishes for r ∈ (0, r_{0}). Hence, (a(r), ϕ(r)) = 0 for r ∈ (0, r_{1}) and û(A, Φ) = (A, Φ) on the ball B(x_{0}, r_{1}). This completes the proof of Proposition 5.19. q.e.d.

Appendix A. Holonomy perturbations and regularity

When defining our holonomy perturbations in §2.5.2 we deferred a detailed discussion of several important regularity issues which arise in their construction. The first concerns the regularity of sections of E and 𝔰𝔲(E) which are constructed by parallel transport via Sobolev connections and is described in §A.1. The second concerns the regularity of the G_{E} equivariant maps 𝔥_{j,l,α} : A_{E}(X) → L_{k+1}^{2}(X, 𝔰𝔲(E)) and we show in §A.2 that these maps are C^{∞}. (All of the G_{E} equivariant maps discussed here are ^{∘}G_{E} equivariant since S_{Z}^{1} acts trivially on connections in A_{E}; if the connection on det E is not fixed, then all of the G_{E} equivariant maps are Aut E equivariant.)

The definition of the maps 𝔪_{j,l,α} also makes use of the existence of a locally finite, C^{∞} ‘positive partition’ on a paracompact manifold modelled on a separable Hilbert space, in the sense of Proposition A.12. The differentials of the cutoff functions in these partitions need not be bounded a priori for the usual reason that in an infinite-dimensional Hilbert manifold, closed and bounded sets are not compact. Nonetheless, as we shall see in §A.3, it is possible to modify the standard proof [55] of the existence of a C^{∞} partition of unity on a paracompact C^{∞} manifold to produce C^{∞} cutoff functions all of whose differentials are bounded. The sums defining the perturbations τ⃗ · 𝔪⃗ and ϑ⃗ · 𝔪⃗ of §2.5.2 are finite when restricted to any of the open neighborhoods (π ∘ r_{Y_{j}})^{−1}(U_{j,α}) ⊂ A_{E}^{∗}(X), where r_{Y_{j}} : A_{E}(X) → A_{E}(Y_{j}) is the restriction map and π : A_{E}(X) → B_{E}(Y_{j}) is the canonical projection. However, the number of terms in these sums may be infinite in the neighborhood of a reducible connection: it is for this reason that they are required to converge with respect to a suitable choice of weights, as described in §2.5.2. We specify the choice of weights in §A.4 and explain why the G_{E} equivariant maps τ⃗ · 𝔪⃗ and ϑ⃗ · 𝔪⃗ are C^{∞} on A_{E}(X).

A.1. Parallel transport for Sobolev connections. The fact that parallel transport is well-defined for L_{2}^{2} connections has been pointed out in [61]. However, to construct L_{k+1}^{2} sections using parallel transport via L_{k}^{2} connections with k ≥ 2, we need to regularize the sections, taking care to do this gauge equivariantly. (Given an L_{k}^{2} connection A, it does not follow that local sections constructed by Aparallel transport are in L_{k+1}^{2}.) If we ignored the issue of equivariance, then it would suffice to use the standard smoothing kernel described in [33, §7.2 & §7.3], yielding a C^{∞} section irrespective of whether the connection is L_{k}^{2} or C^{∞}. Instead, we use the kernel

K_{t}(A)(x, y) ∈ Hom(𝔰𝔲(E)|_{y}, 𝔰𝔲(E)|_{x}),

t ∈ (0, ∞), of the heat operator

K_{t}(A) = exp(−td_{A}^{∗}d_{A}) : L^{2}(X, 𝔰𝔲(E)) → L_{k+1}^{2}(X, 𝔰𝔲(E)),

defined by the C^{∞} metric g on T X and L_{k}^{2} connection A on E [34, §1.6]; related constructions are described in [84, p. 339], [85, p. 177]. The key well-known properties [12], [34, §1.6] of the heat kernel which we need are summarized below:

Lemma A.1. Continue the above notation and suppose that ζ ∈ L^{2}(X, 𝔰𝔲(E)).

(1) As t → 0, the heat operator exp(−td_{A}^{∗}d_{A}) converges to the L^{2} orthogonal projection onto Ker(d_{A}^{∗}d_{A})^{⊥}.

(2) If A on 𝔰𝔲(E) is irreducible, then K_{t}(A)(x, ·) converges to the Dirac delta distribution δ(x, ·).

(3) If A is L_{k}^{2}, then K_{t}(A)ζ is L_{k+1}^{2}; if A is C^{∞}, then K_{t}(A)ζ is C^{∞}.

(4) If ζ ∈ C^{0}(Ũ, 𝔰𝔲(E)), for an open set Ũ ⊂ X, and A is irreducible on 𝔰𝔲(E), then K_{t}(A)ζ → ζ in C^{0}(U, 𝔰𝔲(E)) as t → 0, for any open subset U ⋐ Ũ.

Lemma A.1 continues to hold for unitary connections A on Hermitian bundles E over compact C^{∞} manifolds with boundary \overline{Y} = Y ∪∂Y; we use the Neumann boundary conditions of [20, p. 192], [87, Proposition 2.1] to obtain an L^{2} self-adjoint Laplacian d_{A}^{∗}d_{A}. Our main application will be to sections defined by parallel translation:

Lemma A.2. Let k ≥ 2 be an integer and let A be an L_{k}^{2} unitary connection on a Hermitian two-plane bundle E over X. Let x_{0} be a point in X, let ζ_{0} ∈ 𝔰𝔲(E)|_{x_{0}}, and let B(x_{0}, r) be a geodesic ball with center x_{0} and radius r > 0.

(1) Parallel transport with respect to A is well-defined along C^{∞} paths in X.

(2) If ζ is the section of 𝔰𝔲(E)|_{B(x_{0},r)} obtained by A-parallel transport of ζ_{0} along radial geodesics originating at x_{0}, and A is C^{∞}, then ζ is C^{∞} on B(x_{0}, r). If A is L_{k}^{2}, then ζ is C^{0} on B(x_{0}, r) and its mollification K_{t}(A|_{B(x_{0},r)})ζ is L_{k+1}^{2} on B(x_{0}, r), for any t > 0.

Proof. Let γ : [0, 1] → X be a C^{∞} path such that γ(0) = x_{0} and let U ⊂ X be an open neighborhood of γ([0, 1]). We may suppose without loss of generality that A ∈ L_{k}^{2}(U, Λ^{1} ⊗ 𝔲(2)) is a connection matrix and consider the parallel transport of η_{0} ∈ ℂ^{2}. Note that we have a continuous Sobolev embedding L_{2}^{2}(U) → L^{2}([0, 1]) [2, Theorem V.4]. We wish to solve the ordinary differential equation

\frac{dη}{dt} + aη = 0, t ∈ [0, 1], (A.1)

where a = γ^{∗}A ∈ L^{2}([0, 1], 𝔲(2)), with initial condition η(0) = η_{0}, for a solution η ∈ (L_{1}^{2} ∩ C^{0})([0, 1], ℂ^{2}).

If A is C^{∞} then there is a unique solution η ∈ C^{∞}([0, 1], ℂ^{2}) to (A.1) (see, for example, [36]) and it obeys the inequality [36, Lemma IV.4.1]

(A.2) |η(t)| ≤ |η_{0}| exp (∫_{0}^{t} |a(s)| ds)
≤ |η_{0}| exp (∥a∥_{L^{1}([0,1])}) ≤ |η_{0}| exp(c∥A∥_{L_{2}^{2}(U)}),

for t ∈ [0, 1]. If A is only L_{2}^{2}, we may choose any sequence {A_{α}} of C^{∞} connections which converge to A in L_{2}^{2} and let {η_{α}} be the corresponding sequence of C^{∞} solutions to (A.1). From (A.2) it follows that {η_{α}} is C^{0}-Cauchy, and then (A.1) implies that {η_{α}} is L_{1}^{2}-Cauchy with limit η ∈ (L_{1}^{2} ∩ C^{0})([0, 1], ℂ^{2}) solving (A.1). This proves Assertion (1).

From the proof of Assertion (1) we see that ζ ∈ C^{0}(U, 𝔰𝔲(2)). If the connection A is C^{∞} on B(x_{0}, r), then by differentiating (A.1) with respect to local coordinates (regarded as parameters [36, Chapter 5]) on B(x_{0}, r) and solving the resulting first order ordinary differential equations for the derivatives, it follows that the section ζ is C^{∞} on B(x_{0}, r). The rest of Assertion (2) is an easy consequence of the properties of the smoothing kernel K_{t}(A)(x, y) given in Lemma A.1. q.e.d.

A.2. Regularity of holonomy maps. We consider the regularity of the holonomy maps from A ∈ A_{E} to h_{γ,x_{0}}(A) ∈ 𝔲(E)|_{x_{0}}, to 𝔥̂_{γ}(A) ∈ C^{0}(B(x_{0}, r_{0}), 𝔰𝔲(E)), and to 𝔥_{γ}(A) ∈ L_{k+1}^{2}(B(x_{0}, r_{0}), 𝔰𝔲(E)). See also [8, §3.3], [29, Lemma 1b.1], [86, §8a], and [90, §5A] for related calculations.

Let γ : [0, 1] → X denote a parametrization for an oriented path γ ⊂ X with γ(0) = x_{0}. The general Sobolev embedding [2, Theorem V.4] implies that we have a continuous restriction map L_{k}^{2}(X, Λ^{1} ⊗𝔰𝔲(E)) → L_{k−2}^{2}(γ, Λ^{1} ⊗ 𝔰𝔲(E)), for k ≥ 2. We fix an isomorphism E|_{γ} ≃ γ × ℂ^{2} of U(2) bundles and denote the connection matrix corresponding to a unitary connection A on E again by A ∈ L_{k−2}^{2}(γ, Λ^{1} ⊗ 𝔲(2)). Let A + sa, s ∈ ℝ, be a one-parameter family of nearby connections on E, with connection matrices A(t) + sa(t) ∈ L_{k−2}^{2}(γ, Λ^{1} ⊗ 𝔲(2)). Let ξ_{0} ∈ ℂ^{2} correspond to a point in the fiber E|_{x_{0}} and, with respect to the connection A + sa, let ξ(t; s) be the parallel transport of ξ_{0} along γ(t), so ξ(t; s) solves

(\frac{d}{dt} + A(t) + sa(t)) ξ(t; s) = 0, ξ(0; s) = ξ_{0}. (A.3)

Thus, if P_{γ}(t; A) ∈ Isom(E|_{γ(0)}, E_{γ(t)}) ≃ U(2) denotes parallel transport along γ from γ(0) to γ(t) with respect to the connection A, we have

ξ(t; s) = P_{γ}(t; A + sa)ξ_{0}, t ∈ [0, 1].

Therefore, our task is to compute

(DP_{γ}(t; ·))_{A}(a)ξ_{0} = \frac{d}{ds}P_{γ}(t; A + sa)ξ_{0}|_{s=0} = \frac{dξ}{ds}(t; s)|_{s=0} , t ∈ [0, 1].

Differentiating (A.3) with respect to s, we see that dξ(t; s)/ds solves

(\frac{d}{dt} + A(t) + sa(t)) \frac{dξ}{ds} = −a(t)ξ(t; s), \frac{dξ}{ds}(0; s) = 0. (A.4)

Let Y (t; s) be the fundamental matrix solution for the linear differential operator on the left-hand side. Then the solution dξ(t; s)/ds can be written as [36, Corollary IV.2.1]

\frac{dξ}{ds}(t; s) = −Y (t; s) ∫_{0}^{t} Y^{−1}(τ; s)a(τ)ξ(τ; s) dτ.

Since ξ(t; s) = Y (t; s)ξ_{0} = P_{γ}(t; A + sa)ξ_{0} for any ξ_{0} ∈ E|_{γ(0)}, setting s = 0 above gives

(DP_{γ}(t; ·))_{A}(a) = −P_{γ}(t; A) ∫_{0}^{t} P_{γ}^{−1}(τ; A)a(τ)P_{γ}(τ; A) dτ, (A.5) and so, by the Sobolev embedding theorem, a derivative bound

|(DP_{γ}(t; ·))_{A}(a)| ≤ ∥a∥_{L^{1}(γ)} ≤ c∥a∥_{L_{2,A}^{2}(X)}, t ∈ [0, 1]. (A.6)

The estimate (A.6) implies that the right-hand side of (A.5) is welldefined whenever k ≥ 2. In particular, when t = 1 we have P_{γ}(1; A) = h_{γ,x_{0}}(A), and so (A.5) gives

(Dh_{γ,x_{0}})_{A}(a) = −h_{γ,x_{0}}(A) ∫_{γ} P_{γ}^{−1}(A)aP_{γ}(A). (A.7)

Thus, we have a well-defined differential

(Dh_{γ,x_{0}})_{A} : L_{k}^{2}(X, Λ^{1} ⊗ 𝔲(E)) → T_{h_{γ,x_{0}}(A)} U(E)|_{x_{0}} ≃ 𝔲(E)|_{x_{0}}

of the holonomy map h_{γ,x_{0}} : A_{E}(X) → U(E)|_{x_{0}} at the point A ∈ A_{E}(X).

A similar argument shows that all higher derivatives exist by repeatedly applying the derivative formulas (A.5) and (A.7) (see, for example, [86, §8a]) and so we may conclude:

Lemma A.3. For k ≥ 2, the holonomy map

h_{γ,x_{0}} : A_{E}(X) → U(E)|_{x_{0}}

is C^{∞}.

In the same vein, the holonomy map h_{γ,x_{0}} : A_{E}(X) → SO(𝔰𝔲(E))|_{x_{0}} is C^{∞} and we have:

Lemma A.4. For k ≥ 2, the following holonomy maps are C^{∞}:

A_{E}(X) ∋ A ↦ 𝔥̂_{γ}(A) ∈ C^{0}(B(x_{0}, r_{0}), 𝔰𝔲(E)),
A_{E}(X) ∋ A ↦ 𝔥_{γ}(A) ∈ L_{k+1}^{2}(B(x_{0}, r_{0}), 𝔰𝔲(E)).

Proof. The fact that 𝔥̂_{γ} is a C^{∞} map follows from the proof of Lemma A.3. The map 𝔥_{γ} is defined by 𝔥_{γ}(A) = K_{t}(A)𝔥̂_{γ}(A) and so the fact that the map

A_{E}(X) ∋ A ↦ K_{t}(A) ∈ Hom(L^{2}(B(x_{0}, r_{0}), 𝔰𝔲(E)),
L_{k+1}^{2}(B(x_{0}, r_{0}), 𝔰𝔲(E)))

is C^{∞} yields the second conclusion. q.e.d.

A.3. Positive partitions and cutoff functions with bounded differentials on Hilbert manifolds. We modify Lang’s proof of the existence of a C^{∞} partition of unity on a paracompact C^{∞} manifold modelled on a separable Hilbert space [55, Corollary II.3.8] to produce a C^{∞} positive partition whose cutoff functions have bounded differentials of all orders; see Proposition A.12. Recall:

Proposition A.5 [55, Corollary II.3.8]. Let X be a paracompact C^{∞} manifold modelled on a separable Hilbert space ℌ. Then X admits locally finite, C^{∞} partitions of unity: For any open cover of X there is a countable, locally finite open subcover {U_{α}}_{α=1}^{∞} and a family of C^{∞} functions ψ_{α} : X → [0, 1] such that

• supp ψ_{α} ⊂ U_{α},

• ∑_{α=1}^{∞} ψ_{α}(x) = 1 for all x ∈ X.

We shall need the following generalizations to a Banach space setting of the analogous familiar facts from analysis on finite-dimensional spaces.

Lemma A.6 [1, Proposition 1.3.10]. Let M be a topological space and let (N, d) be a complete metric space. Then the set C(M, N) of all bounded continuous maps is a complete metric space with respect to the metric D(f, g) := sup{d(f(x), g(x)) : x ∈ M}.

Let E, F be Banach spaces and let U ⊂ E be an open subset. By analogy with the usual definitions in finite dimensions, we let C^{s}(U, F) be the set of C^{s} maps f : U → F with norm

∥f∥_{C^{s}(U)} := max_{0≤p≤s} sup_{x∈U} ∥(D^{p}f)_{x}∥ < ∞,

where ∥(D^{p}f)_{x}∥ is the norm of (D^{p}f)_{x} ∈ Hom(⊗^{p}E, F). Lemma A.6 implies that C^{0}(U, F) is a Banach space. In general we have:

Lemma A.7. For any integer s ≥ 0, the set C^{s}(U, F) is a Banach space.

Proof. For the case s = 1 this follows from Lemma A.6 and [37, Theorem 1.1.5]. The general case [1, pp. 113–114] is proved in the same way using Taylor’s Theorem. q.e.d.

By analogy with the finite-dimensional case we set C^{∞}(U, F) = ∩_{s≥0}C^{s}(U, F).

The C^{∞} cutoff functions produced by Proposition A.5 may not necessarily have bounded differentials on Hilbert manifolds, as their supports are not compact, so we now describe a modified procedure which does produce bounded cutoff functions.

Lemma A.8 [55, Lemma II.3.5]. Let M be a metric space and let {B_{α}}_{α=1}^{∞} be a covering of a subset W ⊂ M by open balls. Then there exists a locally finite open covering {V_{α}}_{α=1}^{∞} of W such that V_{α} ⊂ B_{α} for all α and

V_{α} = B_{α} ∩^{c}B̄(x_{α,1}, r_{α,1}) ∩ ⋯ ∩^{c}B̄(x_{α,α−1}, r_{α,α−1}),

where ^{c}B̄ = M − B̄.

Lemma A.9. Let B_{0}, B_{1} …, B_{m} be open balls in a Hilbert space ℌ and let V be a scalloped open subset:

V = B_{0} ∩^{c}B̄_{1} ∩ ⋯ ∩^{c}B̄_{m}.

Then there exists a C^{∞} function ω : ℌ → [0, 1] such that ω(x) > 0 if x ∈ V and ω(x) = 0 otherwise, while the differentials D^{p}ω are bounded for all p ∈ ℕ, with bounds depending only on p, r_{0}, …, r_{m}.

Proof. Let φ : ℝ → [0, 1] be a C^{∞} function such that φ(t) = 1 for t ≤ 1, 0 < φ(t) < 1 for 1 < t < 2, and φ(t) = 0 for t ≥ 2. Suppose B_{α} = B(x_{α}, r_{α}) for α = 0, …, m. Set

φ_{α}(x) := φ(r_{α}^{−2}∥x − x_{α}∥^{2}), x ∈ ℌ, α = 1, …, m,

so 0 ≤ φ_{α} < 1 on^{c}B̄_{α} and φ_{α} = 1 on B̄_{α}. Set φ_{0}(x) = φ(2r_{0}^{−2}∥x−x_{0}∥^{2}), so 0 < φ_{0} ≤ 1 on B_{0} and φ_{0} = 0 on ^{c}B_{0}. Then

ω := φ_{0} ∏_{α=1}^{m}(1 − φ_{α})

is positive on V and zero on ^{c}V = ℌ − V, while

∥D^{p}ω∥ ≤ C,

for some positive constant C = C(φ, p, r_{0}, …, r_{m}) and all p ∈ ℕ. q.e.d.

Lemma A.10. Let A_{1}, A_{2} be non-empty, closed disjoint subsets of a separable Hilbert space ℌ. Then there exists a C^{∞} function χ : ℌ → [0, 1] such that χ(x) = 0 if x ∈ A_{1} and χ(x) > 0 if x ∈ A_{2}, with bounded differentials of all orders.

Proof. By Lindelöf’s Theorem there is a countable collection of balls {B_{α}}_{α=1}^{∞} covering A_{2} such that B_{α} ⊂ ^{c}A_{1} = ℌ−A_{1}. Let W = ∪_{α}B_{α} and find a locally finite refinement {V_{α}} of scalloped open subsets V_{α} ⊂ B_{α}, using Lemma A.8, which covers W. Using Lemma A.9 we find a C^{∞} function ω_{α} : ℌ → ℝ such that ω_{α} is positive on V_{α}, zero on ^{c}V_{α}, and has bounded differentials of all orders. The sum

χ := ∑_{α=1}^{∞} \frac{2^{−α}ω_{α}}{1 + ∥ω_{α}∥_{C^{α}(ℌ)}}

is finite on any neighborhood V_{α}. The function χ : ℌ → ℝ is positive on ∪_{α}V_{α} = W ⊃ A_{2} and zero on A_{1}. Moreover, χ has bounded differentials of all orders, as desired. q.e.d.

Remark A.11. It is at this point that the usual proof of existence of a partition of unity encounters difficulties if we wish to ensure that the cutoff function χ has bounded differentials of all orders. In the proof of Theorem II.3.7 of [55], Lang first constructs a function ω obeying the conclusions of Lemma A.10 and then a C^{∞} function σ : ℌ → ℝ such that σ > 0 on the closed set ^{c}U disjoint from A_{2}, where U is the open set where ω > 0, and σ = 0 on A_{2}. He then defines χ = ω/(ω + σ), so χ = 1 on A_{2}. In finite dimensions, the compactness of closed bounded sets ensures that χ will have bounded differentials of all orders if A_{2} is bounded, but this obviously fails in infinite dimensions.

Proposition A.12. Let X be a paracompact C^{∞} manifold modelled on a separable Hilbert space ℌ. Then X admits a locally finite, C^{∞} positive partition: For any open cover of X there is a countable, locally finite open subcover by parametrizations {U_{α}, π_{α}}_{α=1}^{∞}, where

π_{α} : ℌ ⊃ U_{α} ≃ U_{α} ⊂ X ,

and a family of C^{∞} functions χ_{α} : X → [0, 1] such that

• supp χ_{α} ⊂ U_{α},

• ∑_{α=1}^{∞} χ_{α}(x) > 0 for all x ∈ X,

• χ_{α} ∘ π_{α} : ℌ → [0, 1] has bounded differentials of all orders.

Proof. Let {B_{x}} be an open covering of X by balls and, using paracompactness and Lemma A.8, let {U_{α}} be a countable, locally finite refinement such that each open subset U_{α} is contained in some B_{x(α)}, where π_{α}^{−1} : X ⊃ B_{x(α)} → ℌ is a coordinate chart and π_{α}^{−1}(U_{α}) = U_{α}. We now find open refinements {V_{α}} and then {W_{α}} such that

\overline{W}_{α} ⊂ V_{α} ⊂ \overline{V}_{α} ⊂ U_{α},

the bar denoting closure in X. For each α, Lemma A.10 and the identification π_{α} : ℌ ⊃ U_{α} ≃ U_{α} ⊂ X provide a C^{∞} cutoff function χ_{α} : X → ℝ such that χ_{α} > 0 on \overline{W}_{α} and χ_{α} = 0 on X − V_{α}. The C^{∞} map χ_{α} ∘ π_{α} : U_{α} → [0, 1] extends by zero to a C^{∞} map χ_{α} ∘ π_{α} : ℌ → [0, 1] having bounded differentials of all orders. Hence, {U_{α}, χ_{α}} is the desired positive partition. q.e.d.

Let \overline{Y} = Y ∪ ∂Y ⊂ X be a smooth submanifold-with-boundary. For any open subset U ⊂ A_{E}(Y) and C^{s} function f : U → ℂ, we define

∥f∥_{C^{s}(U)} := sup_{A∈U} sup_{\begin{matrix}1≤i≤s \\ ∥a_{i}∥_{L{2 \atop k,A}(Y )}≤1 \end{matrix}} |(D^{s}f)_{A}(a_{1}, …, a_{s})|. (A.8)

Let {B([A_{α}], r_{α})} be a countable covering of B_{E}^{∗}(Y) by L_{k}^{2} balls with subordinate locally finite subcover {U_{α}} and C^{∞} positive partition {χ_{α}}, with supp χ_{α} ⊂ U_{α}. We may suppose that B([A_{α}], r_{α}) = π(B(A_{α}, r_{α})), where B(A_{α}, r_{α}) ⊂ A_{E}^{∗}(Y) is an open L_{k}^{2} ball in the Coulomb slice A_{α} + Ker d_{A_{α}}^{∗} with center A_{α} and radius r_{α}, and π : A_{E}^{∗}(Y) → B_{E}^{∗}(Y) is the canonical projection. Let U_{α} = π^{−1}(U_{α}) ∩ B(A_{α}, r_{α}). Then U_{α} ⊂ A_{α} + Ker d_{A_{α}}^{∗} and

supp χ_{α} ∘ π ⊂ π^{−1}(U_{α}) = G_{E}(Y) · U_{α} ≃ G_{E}(Y) × U_{α}.

Now χ_{α}∘π(u(A)) = χ_{α}∘π(A) for all u ∈ G_{E}(Y), and by the construction of Proposition A.12 the C^{∞} map

χ_{α} ∘ π : A_{α} + Ker d_{A_{α}}^{∗} ⊂ L_{k+1}^{2}(Y, 𝔰𝔲(E)) → [0, 1] (A.9)

has bounded differentials of all orders with respect to the fixed L_{k+1,A_{α}}^{2} norm on L_{k+1}^{2}(Y, 𝔰𝔲(E)), that is, it has bounded differentials of all orders in the sense of Proposition A.12. Now for any A ∈ U_{α}, the L_{k+1,A_{α}}^{2} and L_{k+1,A}^{2} norms on L_{k+1}^{2}(Y, 𝔰𝔲(E)) compare uniformly with constants depending only on r_{α} ≥ ∥A − A_{α}∥_{L_{k;Aα}^{2}} (since k ≥ 2), and so the maps (A.9) have bounded differentials of all orders in the sense of equation (A.8). Moreover, the same holds for the maps

χ_{α} ∘ π : A_{E}^{∗}(Y) → [0, 1].

Finally, we have a restriction map r_{Y} : A_{E}^{∗}(X) → A_{E}^{∗}(Y) given by A ↦ A|_{Y} and the composition

χ_{α} ∘ π ∘ r_{Y} : A_{E}^{∗}(X) → [0, 1]

again has bounded differentials of all orders in the sense of equation (A.8).

A.4. Uniform convergence of holonomy perturbations on neighborhoods of reducibles. In §2.5.2 we constrained the sequences perturbations τ⃗ and ϑ⃗ to vary in certain weighted ℓ_{δ}^{1} spaces. We now show that a sequence of positive weights δ ∈ ℓ^{∞}(ℝ^{+}) may be chosen in such a way that the sums τ⃗ · 𝔪⃗ and ϑ⃗ · 𝔪⃗ and all their differentials converge uniformly on A_{E}(X).

For any open subset U ⊂ A_{E}(X) and C^{s} map

𝔣 : U → L_{k+1}^{2}(X, 𝔰𝔲(E)),

we define

∥𝔣∥_{C^{s}(U)} := sup_{A∈U} sup_{\begin{matrix}1≤i≤s \\ ∥a_{i}∥_{L{2 \atop k,A}(X)}≤1 \end{matrix}} ∥(D^{s}𝔣)_{A}(a_{1}, …, a_{s})∥_{L_{k+1,A}^{2}(X)}. (A.10)

We then have:

Proposition A.13. Continue the notation of §2.5.2 and let k ≥ 2 be an integer. Then there exists a sequence δ = (δ_{α})_{α=1}^{∞} ∈ ℓ^{∞}((0, 1]) of positive weights such that the G_{E} equivariant maps

τ⃗ · 𝔪⃗ : A_{E}(X) → L_{k+1}^{2}(X, 𝔤𝔩(Λ^{+}) ⊗_{ℝ} 𝔰𝔬(𝔰𝔲(E))),
ϑ⃗ · 𝔪⃗ : A_{E}(X) → L_{k+1}^{2}(X, Hom(W^{+}, W^{−}) ⊗_{ℂ} 𝔰𝔩(E)),

are C^{∞}, with uniformly bounded differentials of all orders in the sense of (A.10). In particular, they satisfy the following C^{0} estimates for r ≥ k + 1,

sup_{A∈A_{E}} ∥ϑ⃗ · 𝔪⃗(A)∥_{L_{k+1,A}^{2}(X)} ≤ C∥ϑ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))},
sup_{A∈A_{E}} ∥τ⃗ · 𝔪⃗(A)∥_{L_{k+1,A}^{2}(X)} ≤ C∥τ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))},

for some positive constant C = C(g, k), and more generally they satisfy the C^{s} estimates of (A.11) and (A.12), for every integer s ≥ 0.

Remark A.14. As should be clear from their definition, the maps τ⃗ · 𝔪⃗ and ϑ⃗ · 𝔪⃗ are not analytic, although this will cause no difficulty in practice.

Proof. It suffices to consider ϑ⃗ · 𝔪⃗, since the argument for τ⃗ · 𝔪⃗ is obviously identical. We first observe that the sum ϑ⃗ · 𝔪⃗(A) is finite for each connection A ∈ A_{E}(X), and so defines an element of L_{k+1}^{2}(X, Hom(W^{+}, W^{−}) ⊗ 𝔤𝔩(E)), while it is identically zero if A is reducible on X. However, on any open neighborhood of a reducible connection A ∈ A_{E}(X), the number of terms in the sum ϑ⃗ · 𝔪⃗ may be infinite, and so our task is to choose a sequence of weights δ such that this sum and all its differentials converge uniformly on A_{E}(X). The sum ϑ⃗·𝔪⃗ will then define a C^{∞} map. We begin with a couple of preparatory lemmas.

Lemma A.15. Let \overline{Y} = Y ∪ ∂Y ⊂ X be a smooth submanifold with boundary, let B(A_{0}, r_{0}) ⊂ A_{E}^{∗}(Y) be an open L_{k}^{2} ball with k ≥ 2, and let η ∈ L^{2}(Y, 𝔰𝔲(E)). Then the maps

B(A_{0}, r_{0}) ∋ A ↦ 𝔣(A) := e^{−td_{A}^{∗}d_{A}}η ∈ L_{k+1}^{2}(Y, 𝔰𝔲(E))

have bounded differentials of all orders, in the sense of (A.10), with constants c = c(k, r_{0}, s, t):

∥𝔣∥_{C^{s}(B(A_{0},r_{0}))} ≤ c∥η∥_{L^{2}(Y)}.

Proof. The Sobolev multiplication theorems and the fact that k ≥ 2 imply that the norms

∥ζ∥_{L_{k+1,A}^{2}(Y)} and ∥ζ∥′_{L_{k+1,A}^{2}(Y)} := ∥(1 + Δ_{A})^{(k+1)/2}ζ∥_{L^{2}(Y)}

on L_{k+1}^{2}(Y, 𝔰𝔲(E)) are equivalent for A ∈ B(A_{0}, r_{0}) ⊂ A_{E}^{∗}(Y), with constants depending at most on k, r_{0}.

Since Δ_{A} = d_{A}^{∗}d_{A}, its derivative with respect to A in the direction δA = a is given by δΔ_{A} = [a, ·]^{∗}d_{A} + d_{A}^{∗}[a, ·]; we use the abbreviation δΔ_{A} = (DΔ_{(·)})_{A}(a). The connection A_{0} is irreducible by hypothesis and so for a small enough open L_{k}^{2} ball B(A_{0}, r_{0}), there is a positive constant λ_{0} > 0 such that λ[A] ≥ λ_{0} > 0 for all A ∈ B(A_{0}, r_{0}), where λ[A] is the least eigenvalue of Δ_{A}. Thus,

Spec(Δ_{A}) ⊂ [λ_{0}, ∞), A ∈ B(A_{0}, r_{0}),

and, for any holomorphic function f on an open neighborhood Ω ⊂ ℂ with Spec(Δ_{A}) ⊂ Ω and Γ any contour that surrounds Spec(Δ_{A}) in Ω, we have [76]

f(Δ_{A}) = \frac{1}{2πi} ∮_{Γ} f(λ)(λ − Δ_{A})^{−1} dλ,
(Df)_{Δ_{A}}(δΔ_{A}) = \frac{1}{2πi} ∮_{Γ} f(λ)(λ − Δ_{A})^{−1}(δΔ_{A})(λ − Δ_{A})^{−1} dλ,

and similarly for all higher-order derivatives. Note that

(1 + Δ_{A})^{(k+1)/2}f(Δ_{A})

= \frac{1}{2πi} ∮_{Γ}(1 + λ)^{(k+1)/2}f(λ)(λ − Δ_{A})^{−1} dλ,

(1 + Δ_{A})^{(k+1)/2}(Df)_{Δ_{A}}(δΔ_{A})

= \frac{1}{2πi} ∮_{Γ}(1 + λ)^{(k+1)/2}f(λ)(λ − Δ_{A})^{−1}(δΔ_{A})(λ − Δ_{A})^{−1} dλ.

We can fix Ω and Γ ⊂ Ω such that dist(Γ, Spec(Δ_{A})) ≥ d_{0} > 0, for some positive constant d_{0} and all A ∈ B(A_{0}, r_{0}) (see [34, §1.6]). Now choose f(z) = e^{−tz} and f_{k+1}(z) = (1 + z)^{(k+1)/2}f(z), and estimate as in [34, p. 53]:

∥e^{−tΔ_{A}}η∥_{L_{k+1,A}^{2}(Y)} ≤ c∥(1 + Δ_{A})^{(k+1)/2}e^{−tΔ_{A}}η∥_{L^{2}(Y)}

≤ c ∮_{Γ} |(1 + λ)^{(k+1)/2}e^{−tλ} dλ| sup_{λ∈Γ} ∥(λ − Δ_{A})^{−1}η∥_{L^{2}(Y)}

≤ c∥η∥_{L^{2}(Y)},

which gives the desired C^{0} bound

sup_{A∈B(A_{0},r_{0})} ∥e^{−tΔ_{A}}η∥_{L_{k+1,A}^{2}(Y)} ≤ c∥η∥_{L^{2}(Y)},

where c = c(d_{0}, k, r_{0}, t).

To obtain the C^{1} bound, observe that

∥(Df)_{Δ_{A}}(δΔ_{A})η∥_{L_{k+1,A}^{2}(Y)}

≤ c∥(1 + Δ_{A})^{(k+1)/2}(Df)_{Δ_{A}}(δΔ_{A})η∥_{L^{2}(Y)}

≤ c ∮_{Γ} |(1 + λ)^{(k+1)/2}f(λ) dλ| sup_{λ∈Γ} ∥(λ − Δ_{A})^{−1}(δΔ_{A})(λ − Δ_{A})^{−1}η∥_{L^{2}(Y)}

≤ sup_{λ∈Γ} c∥(λ − Δ_{A})^{−1}(δΔ_{A})(λ − Δ_{A})^{−1}η∥_{L^{2}(Y)}.

Our expression for δΔ_{A} gives

∥(λ − Δ_{A})^{−1}(δΔ_{A})(λ − Δ_{A})^{−1}η∥_{L^{2}(Y)}

≤ ∥(λ − Δ_{A})^{−1}([a, ·]^{∗}d_{A} + d_{A}^{∗}[a, ·])(λ − Δ_{A})^{−1}η∥_{L^{2}(Y)}

≤ c (∥[a, ·]^{∗}d_{A}(λ − Δ_{A})^{−1}η∥_{L^{2}(Y)} + ∥d_{A}^{∗}[a, ·](λ − Δ_{A})^{−1}η∥_{L^{2}(Y)})

≤ c∥a∥_{L_{2;A}^{2}(Y)}∥(λ − Δ_{A})^{−1}η∥_{L_{2;A}^{2}(Y)}

≤ c∥a∥_{L_{k,A}^{2}(Y)}∥η∥_{L^{2}(Y)},

where c = c(r_{0}, d_{0}). Combining these estimates yields

∥δ(e^{−tΔ_{A}})η∥_{L_{k+1,A}^{2}(Y)} ≤ c∥a∥_{L_{k,A}^{2}(Y)}∥η∥_{L^{2}(Y)},

and so we have the desired C^{1} bound

sup_{A∈B(A_{0},r_{0})} sup_{∥a∥_{L{2 \atop k,A}(X)}≤1} ∥(De^{−tΔ_{(·)}})_{A}(a)η∥_{L_{k+1,A}^{2}(Y)} ≤ c∥η∥_{L^{2}(Y)},

for some c = c(d_{0}, k, r_{0}, t). The analysis can be repeated, essentially unchanged, for all higher differentials and is left to the reader. q.e.d.

Lemma A.16. The G_{E} equivariant holonomy maps

𝔪_{j,l,α} : A_{E}(X) → L_{k+1}^{2}(X, 𝔰𝔲(E))

of (2.20) are C^{∞} with bounded differentials of all orders in the sense of (A.10).

Proof. Recall from (2.17) and (2.20) that

𝔪_{j,l,α}(A) = β_{j}[A]χ_{j,α}[A|_{B(x_{j},2R_{0})}]φ_{j}𝔥_{γ_{j,l,α}}(A),
φ_{j}𝔥_{γ_{j,l,α}}(A) = φ_{j}K_{t}(A|_{B(x_{0},2R_{0})})𝔥̂_{γ_{j,l,α}}(A).

The cutoff functions χ_{j,α} ∘ π ∘ r_{B(x_{j},2R_{0})} : A_{E}(X) → [0, 1] have bounded differentials of all orders, in the sense of (A.8), by the remarks following Proposition A.12; moreover, they are supported in A_{E}^{∗}(X). The functions β_{j} : A_{E}(X) → [0, 1] have bounded differentials of all orders, as is clear from their definition in (2.18). Finally, from the proofs of Lemmas A.3 and A.4, together with Lemma A.15, one can see that the maps

φ_{j}K_{t}(·|_{B(x_{0},2R_{0})})𝔥̂_{γ_{j,l,α}} : A_{E}^{∗}(X) → L_{k+1}^{2}(X, 𝔰𝔲(E))

have bounded differentials of all orders in the sense of (A.10) on open subsets G_{E} · B(A_{0}, r_{0}) ⊂ A_{E}^{∗}(X), where B(A_{0}, r_{0}) is an open L_{k}^{2} ball in K_{A_{0}} ⊂ A_{E}^{∗}(X). q.e.d.

Given Lemma A.16, we have

M_{α,s} := max_{j,l} ∥𝔪_{j,α,l}∥_{C^{s}(A_{E})} < ∞,

where we recall that 1 ≤ l ≤ 3 and 1 ≤ j ≤ N_{b}. Now choose a sequence of positive weights δ = (δ_{α})_{α=1}^{∞} ∈ ℓ^{∞}(ℝ^{+}) by setting

δ_{α} := (1 + max_{0≤s≤α} M_{α,s})^{−1} , α ∈ ℕ,

and suppose ϑ⃗ ∈ ℓ_{δ}^{1}(𝔸, C^{r}(X)). Then

∥ϑ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))} = ∑_{j,l,α} δ_{α}^{−1}∥ϑ_{j,l,α}∥_{C^{r}(X)} < ∞,

and therefore,

∥ϑ⃗ · 𝔪⃗∥_{C^{s}(A_{E})} ≤ ∑_{j,l,α} ∥ϑ_{j,l,α}𝔪_{j,l,α}∥_{C^{s}(A_{E})}

≤ c ∑_{j,l,α} ∥ϑ_{j,l,α}∥_{L_{k+1}^{2}(X)}∥𝔪_{j,l,α}∥_{C^{s}(A_{E})} by (A.10)

≤ c ∑_{j,l,α} ∥ϑ_{j,l,α}∥_{C^{r}(X)}M_{α,s},

since r ≥ k +1. Here, c = c(g, k) is a universal positive constant coming from the continuous Sobolev multiplication L_{k+1}^{2} × L_{k+1,A}^{2} → L_{k+1,A}^{2}. Hence, using the facts that M_{α,s} ≤ 1 + max_{0≤t≤α} M_{α,t} = δ_{α}^{−1} for α ≥ s and 1 ≤ δ_{α}^{−1} for 1 ≤ α ≤ s − 1 , we get

∥ϑ⃗ · 𝔪⃗∥_{C^{s}(A_{E})} ≤ c ∑_{j,l} (∑_{α=1}^{s−1} M_{α,s}∥ϑ_{j,l,α}∥_{C^{r}(X)} + ∑_{α≥s} δ_{α}^{−1}∥ϑ_{j,l,α}∥_{C^{r}(X)})

≤ c (1 + max_{1≤α≤s−1} M_{α,s}) ∑_{j,l} (∑_{α=1}^{s−1} ∥ϑ_{j,l,α}∥_{C^{r}(X)}

(A.11)  + ∑_{α≥s} δ_{α}^{−1}∥ϑ_{j,l,α}∥_{C^{r}(X)})

≤ c (1 + max_{1≤α≤s−1} M_{α,s}) ∑_{j,l,α} δ_{α}^{−1}∥ϑ_{j,l,α}∥_{C^{r}(X)}

= C∥ϑ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))} < ∞,

where C = C(g, s, k) is defined by the last equality above. In particular, we see that ϑ⃗ · 𝔪⃗ is a C^{s} map on A_{E}(X), for every integer s ≥ 0. The same argument gives

∥τ⃗ · 𝔪⃗∥_{C^{s}(A_{E})} ≤ C∥τ⃗∥_{ℓ_{δ}^{1}(C^{r}(X))} < ∞, (A.12)

and τ⃗ · 𝔪⃗ is a C^{s} map on A_{E}(X), for every integer s ≥ 0. q.e.d.

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