Geometry of the Moduli Space of Self-Dual Connections over the Four-Sphere

Paul Matthew Niall Feehan

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

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Paul Matthew Niall Feehan All Rights Reserved

ABSTRACT

Geometry of the Moduli Space of Self-Dual Connections over the Four-Sphere

Paul Matthew Niall Feehan

A Riemannian metric on a compact four-manifold induces a natural L^{2} metric on the corresponding moduli space of (anti-) self-dual connections on a principal Gbundle P. When the bundle structure group G is SU(2) and −c_{2}(P) = k, Groisser, Parker and others have found explicit formulas for the components of the L^{2} metric on the moduli space M_{k} when k = 1 and the four-manifold is the sphere 𝕊^{4} or the complex projective space ℂℙ^{2}. The moduli space M_{1}(𝕊^{4}) is diffeomorphic to the open five-ball, while M_{1}(ℂℙ^{2}) is diffeomorphic to the open cone over ℂℙ^{2}: these moduli spaces have finite volume and diameter with respect to the L^{2} metric.

Donaldson, Groisser, and Parker have conjectured that the moduli space M_{k} has finite volume and diameter with respect to the L^{2} metric for any integer k. We consider the case where the four-manifold is the sphere 𝕊^{4} with its standard round metric, the group G is SU(2), and k = 2. We obtain estimates for the components of the L^{2} metric on the moduli space M_{2}(𝕊^{4}) of self-dual SU(2)-connections over the four-sphere, a non-compact 13-dimensional manifold which is homotopic to the Grassman manifold of real 2-planes in ℝ^{5}. As an application, we show that the space M_{2}(𝕊^{4}) has finite volume and diameter with respect to the L^{2} metric.

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . iii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter I. Moduli Space of Self-Dual Connections

§1.1. Preliminaries on Gauge Theory . . . . . . . . . . . . . . . . 10 §1.2. Moduli Space of Self-Dual Connections . . . . . . . . . . . . . 15 §1.3. L^{2} Metric on Moduli Space . . . . . . . . . . . . . . . . . . 25 §1.4. Families of Connections . . . . . . . . . . . . . . . . . . . . 27 §1.5. Infinitesimal Deformations and Horizontal Projections . . . . . . 33 §1.6. Classifying Maps and Canonical Connections . . . . . . . . . . 37

Chapter II. Stable Vector Bundles of Rank Two

§2.1. Atiyah-Ward Correspondence . . . . . . . . . . . . . . . . . 41 §2.2. Stable Vector Bundles on Complex Projective Space . . . . . . . 46 §2.3. Real Structures and Jumping Lines . . . . . . . . . . . . . . . 50 §2.4. Moduli Space of Rank Two Stable Bundles . . . . . . . . . . . 52 §2.5. Moduli Space of Rank Two Instanton Bundles . . . . . . . . . . 56 §2.6. Real Structures on Complex Moduli Spaces . . . . . . . . . . . 58

Chapter III. Atiyah-Drinfeld-Hitchin-Manin Construction

§3.1. ADHM Construction of Instanton Bundles . . . . . . . . . . . 65 §3.2. Global Sections of Twisted Instanton Bundles . . . . . . . . . . 72 §3.3. Conics in Complex Projective Space . . . . . . . . . . . . . . 77 §3.4. Two Constructions of Instanton Bundles . . . . . . . . . . . . 79

Chapter IV. Parametrization of Self-Dual Connections

§4.1. Parametrization of Connection One-forms . . . . . . . . . . . . 82 §4.2. Parametrization of Connection One-forms with k = 2 . . . . . . . 89 Chapter V. Asymptotic Behaviour of the L^{2} Metric

§5.1. Local Coordinate Patches on Moduli Space . . . . . . . . . . . 99 §5.2. Local Coordinate Patches on the Four-Sphere . . . . . . . . . 101 §5.3. Tangent Vectors to Moduli Space . . . . . . . . . . . . . . 103 §5.4. Estimates of Tangent Vector Norms. I . . . . . . . . . . . . 109 §5.5. Estimates of Tangent Vector Norms. II . . . . . . . . . . . . 120 §5.6. Diameter and Volume of the Moduli Space . . . . . . . . . . 131

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

ACKNOWLEDGEMENTS

It is with the greatest pleasure that I thank my advisor, Duong Phong, for his encouragement and help throughout my years at Columbia. His energy and his enthusiasm for Mathematics and Physics have been a great source of inspiration and I offer him my warmest thanks for his generosity, his many helpful suggestions and ideas, and his mathematical insight and guidance.

I would like to express my great appreciation to the Mathematics Department at Columbia University for providing a very friendly and active research environment, and for financial support. I am especially grateful to Robert Friedman, John Morgan, Ngaiming Mok, Deane Yang, and Velayudhan Nair for their helpful and generous suggestions. I would like to take this opportunity to thank them, and Henry Pinkham and Joan Birman, for their inspiring lectures. I would also like to express my sincere appreciation and thanks to Hervé Jacquet, Patrick Gallagher, and Masatake Kuranishi for their kindness, assistance, and interest.

I am very grateful to Peter Woit, Roberto Silvotti, Huai-Dong Cao, Oisin McGuinness, Johan Tysk, Siye Wu, Mike Heumos, and Xiaochun Rong for their friendship and conversations about Mathematics and Physics. I would especially like to thank Oisin for both his friendship and his generous computer help.

I would particularly like to thank Adebisi Agboola for many entertaining discussions, mathematical and otherwise, and for his steadfast friendship over the years. I am deeply grateful for the shared mathematical ideas, friendship, and support of all my graduate student colleagues: I would especially like to thank Richard Wentworth, Tom Leness, Hong-Jie Yang, Tom Graham, David Gomprecht, Haru Pinson, John Smyrnakis, Lisa Carbone, Liz Finkelstein, Arthur Robb, Yuhan Lim, Ted Stanford, George Pappas and Effie Kalfagianni, Keith Pardue, Cormac Herley, Wing-Keung To, I-Hsun Tsai, Sai-Kee Yeung, Wei-Ping Li, and many others to whom I hasten to offer my apologies for any inadvertent omissions!

I would very much like to express my thanks and appreciation to Francine Brown, for her thoughtful assistance and kindness, and to the Mathematics Department Staff. I am expecially grateful to Dolores Cea and Mary Young for their warm and patient assistance, and to Fred Johnson.

I am most grateful the Mathematics Library staff for their help with library matters. I would particularly like to thank the Research Librarians Mei-Ling Lo and Mary Kay for their cheerful and efficient assistance. I would also like to thank Beatrice Terrien-Somerville, Roger Bagnall, Russell Berg, and Ward Dennis of the Graduate School of Arts and Sciences for their help during my years at Columbia. I would like to thank Roger Bagnall for a rewarding opportunity to serve as the Mathematics representative on the Graduate Student Advisory Council.

I would also like to express my thanks and appreciation to Robert Penner, Paul Yang, Russell Johnson, Francis Bonahon, Dennis Estes, Gary Rosen, and the Mathematics Department at the University of Southern California, for their encouragement and assistance. I gratefully acknowledge the financial support provided by the University of Southern California prior to my arrival at Columbia. I would particularly like to thank Robert Penner, Russell Johnson, and Paul Yang for their patient and friendly encouragement during the earlier years of my mathematical training.

Finally, I can only begin to acknowledge the wonderful support and trust of my Mother and late Father and the other members of my family over many years. I am deeply indebted to them for their encouragement and constant faith.

To My Mother and the Memory of My Father v

INTRODUCTION

A Riemannian metric on a compact four-manifold M induces a natural L^{2} metric on the corresponding moduli space M_{k}^{∗} of (anti-) self-dual connections (or kinstantons) on a principal SU(n)-bundle with second Chern number −k [G-P1]. This metric has been studied by Groisser and Parker and others, and explicit formulas have been found for the metric components when k is 1, the four-manifold is the sphere 𝕊^{4} or the complex projective space ℂℙ^{2}, and the bundle structure group is SU(2) [G], [G-P1, 2], [Hab], [D-M-M], [I]. The space M_{1}^{∗}(𝕊^{4}, SU(2)) is diffeomorphic to the open five-ball [A-H-S], [Har], and M_{1}^{∗}(ℂℙ^{2}, SU(2)) is diffeomorphic to the open cone over ℂℙ^{2} [Bu], [D2], [G]. With respect to the L^{2} metric, these moduli spaces have finite volume and diameter. These finiteness results were extended by Groisser and Parker to the case where M is any compact, oriented, simply-connected, Riemannian four-manifold with positive definite intersection form [G-P3]. In contrast, relatively little is known about either the geometry or topology of the moduli spaces M_{k}^{∗} when |k| > 1.

Donaldson and Groisser-Parker have conjectured that the moduli space of multi-instantons M_{k}^{∗} has finite volume and diameter with respect to the L^{2} metric for any integer k [D4, 5], [G], [G-P2]. One of the motivations for studying the asymptotic behaviour of the L^{2} metric arises in physics [G-P3], [O]. Path integrals over the infinite-dimensional space of connections modulo gauge transformations arising in quantum Yang-Mills theory are thought to be well-approximated by integrals over the finite-dimensional moduli spaces with respect to an appropriate quantum measure. The standard definitions of this measure involve the L^{2} metric [G-P3]. Another motivation comes from the definition of Donaldson invariants arising in the differential topology of four-manifolds [D6], [D-K], [F-M]. For even-dimensional moduli spaces, the Donaldson polynomial invariants may be expressed — at least formally — as integrals of certain differential forms over the moduli space [D-K], [D5], [Wi]. Donaldson has then posed the problem of showing that these integrals converge and that their values coincide with the polynomial invariants as defined in [D6].

We have obtained estimates for the components of the L^{2} metric on the moduli space M_{2}^{∗}(𝕊^{4}, SU(2)) of self-dual SU(2)-connections over the four-sphere. This moduli space is a non-compact 13-dimensional manifold homotopy equivalent to the Grassman manifold G_{2}(ℝ^{5}) [Au-Do], [Har2], [Hat] [S-T]. With regard to its geometry, we have the

Theorem A. The moduli space M_{2}^{∗}(𝕊^{4}, SU(2)) of self-dual SU(2)-connections over the four-sphere 𝕊^{4} with its standard round metric g_{0} has finite volume and diameter with respect to the naturally induced L^{2} metric.

The result also holds for any metric g on 𝕊^{4} which is globally conformally equivalent to the standard round metric. We require the metric on 𝕊^{4} to be conformally equivalent to the standard round metric g_{0} in order to apply the ADHM construction, and we need compactness of the base four-manifold M in order to achieve finiteness of volume and diameter of the moduli space M_{2}^{∗}(𝕊^{4}, SU(2)).

The author is currently investigating the problems of (1) extending these finiteness results to more general four-manifolds and arbitrary values of k, and (2) examining whether the Donaldson invariants may be represented by integrals of differential forms over moduli space.

In order to describe our results in more detail, we recall the definition of the L^{2} metric on Yang-Mills moduli spaces [G-P1]. We let (M, g) be a compact, oriented, Riemannian four-manifold and let P be a principal bundle over M with structure group G. The space of smooth connections A on the G-bundle P → M is an infinite-dimensional affine space whose tangent space may be identified with the Hilbert space completion of Ω^{1}(M, ad P), with respect to a suitable Sobolev L_{s}^{2} norm. Here, Ω^{p}(M, ad P) is the bundle of smooth p-forms on M with values in the associated vector bundle ad P = P ×_{ad} 𝔤, where 𝔤 is the Lie algebra of G. We assume that G is a compact, semi-simple Lie group, and that the vector bundles Ω^{p}(M, ad P) have fibre metrics ⟨ , ⟩ induced by the Riemannian metric g on M and a constant negative multiple of the Cartan-Killing form on G. The L^{2} inner product of such forms defines an inner product on A which is invariant under the action of the group of gauge transformations G — the group of automorphisms covering the identity of M, and so induces a metric on the infinite-dimensional space B^{∗} = A^{∗}/G, of irreducible connections modulo gauge transformations. For G = SU(2), the space B^{∗} is a smooth Hilbert manifold.

The tangent space T_{[ω]}B^{∗} is identified with the horizontal subspace Ker d_{ω}^{∗} of T_{ω}A^{∗}. Let α, β be tangent vectors in T_{ω}A^{∗} and define

g(α, β) = ∫_{M}⟨h_{ω}α, h_{ω}β⟩\sqrt{g} dx, h_{ω} = I − d_{ω}G_{ω}^{0}d_{ω}^{∗},

where d_{ω} is the exterior covariant derivative associated with the connection ω, h_{ω} is the L^{2} orthogonal projection onto the horizontal subspace Ker d_{ω}^{∗}, and G_{ω}^{0} is the Green’s operator corresponding to the Laplacian Δ_{ω}^{0} = d_{ω}^{∗}d_{ω} on Ω^{0}(M, ad P). One obtains a well-defined, Riemannian metric g on the quotient space B^{∗} = A^{∗}/G [G-P1], [I].

For convenience we assume G = SU(2) and let P be a principal G-bundle with topology fixed by choosing the second Chern number c_{2}(P) = −k. A connection ω on P whose curvature 2-form F_{ω} satisfies ∗F_{ω} = F_{ω} or ∗F_{ω} = −F_{ω}, is called self-dual or anti-self-dual, respectively, with respect to the metric g; the self-dual connections have k > 0 and the anti-self-dual connections have k < 0. Here, ∗ denotes the Hodge star-operator. The self-dual connections on P are solutions of the Yang-Mills equations, d_{ω}^{∗}F_{ω} = 0. Under a variety of different combinations of conditions on (M, g), group G, and topology of P — see for example [A-H-S], [T1], [T2], [F-U], [D-K] — the moduli space M_{k}^{∗} of gauge-equivalence classes of self-dual connections is non-empty and is a smooth, finite-dimensional, noncompact submanifold of B^{∗}. The tangent space T_{[ω]}M_{k}^{∗} = Ker d_{ω}^{−}/Im d_{ω} = H_{ω}^{1} may be identified with the harmonic space H_{ω}^{1} = {α ∈ Ω^{1}(M, ad P) : d_{ω}^{∗}α = 0 and d_{ω}^{−}α = 0}, where d_{ω}^{−}α = p_{−}d_{ω}α, with p_{−} denoting the projection onto antiself-dual 2-forms. Exactly analogous statements hold for the moduli space of antiself-dual connections. Applying an orientation-reversing map to M identifies the moduli space of self-dual connections over M with the moduli space of anti-self-dual connections over \overline{M}, where the manifold \overline{M} denotes M with the opposite orientation. In either case, one obtains the L^{2} metric g on M_{k}^{∗} by restriction [G-P1], [I].

One of the first difficulties one encounters when attempting to evaluate components of g is that of finding a suitable basis for the harmonic subspace H_{w}^{1} corresponding to an appropriate choice of local coordinates on the moduli space. When G is SU(2), the space M_{1}^{∗}(𝕊^{4}, SU(2)) has a global coordinate system given essentially by the centre and scale of the self-dual connection and the required horizontal projections may be computed explicitly [D-M-M], [G-P1], [Hab]. However, the space M_{2}^{∗}(𝕊^{4}, SU(2)) has a more complicated topology and a good system of local coordinates is not so readily apparent. Moreover, it no longer seems possible to perform the horizontal projections explicitly and hence obtain a basis for H_{w}^{1}. Also, as |k| increases, the topology of the moduli spaces becomes more difficult to identify [Bo-Ma], [Hur].

The moduli spaces M_{k}^{∗} have topological compactifications \overline{M_{k}^{∗}} as subsets of the space of ideal connections, which may be viewed as connections with curvature densities possibly having δ-measure concentrations at up to k points of M [D-K, p. 157]. The ideal boundary connections in ∂M_{k}^{∗} are then self-dual connections ω with (normalized) curvature density of mass k − l, where 1 ≤ l ≤ k, together with ‘curvature densities’ given by δ-measures of mass k_{i} at m points of M, where 1 ≤ m ≤ l, so that ∑_{i=0}^{m} k_{i} = l and the curvature density of the ideal boundary connection has total mass k. Neighbourhoods of the boundary corresponding to l distinct points with δ-measures of mass 1 may be described explicitly with the aid of Taubes’ gluing construction [T1, 2], [D3]. Using gluing maps onto neighbourhoods of the l distinct centres, one attaches concentrated self-dual connections with k_{i} = 1 over 𝕊^{4} to a backround self-dual connection on a bundle P′ → M with −c_{2}(P′) = k − l, producing a family of almost self-dual connections on a bundle P with −c_{2}(P) = k. One then perturbs this family slightly to produce a family of self-dual connections on P → M parametrizing an open neighbourhood of the ideal boundary. A difficulty now arises when one wishes to explicitly parametrize neighbourhoods of the moduli space boundary corresponding to configurations of points in M and associated δ-measures with at least one of the masses greater than 1. One must now attach concentrated self-dual connections over 𝕊^{4} with k_{i} > 1. Thus, a better understanding of the ends of the moduli space M_{k}^{∗}(𝕊^{4}) for k > 1 would aid our understanding of the ends of M_{k}^{∗} of more general four-manifolds — for example, compact, oriented, simply-connected, Riemannian four-manifolds with positive definite intersection form.

To determine whether the Riemannian manifold (M_{k}^{∗}, g) has finite diameter and volume, we examine the asymptotic behaviour of the L^{2} metric g as one approaches the boundary of moduli space. When k is 2, M is 𝕊^{4}, and G is SU(2), our approach to this problem makes use of an explicit parametrization of the moduli space.

When M is 𝕊^{4} and G is one of the classical Lie groups O(n), SU(n), or Sp(n), then the moduli space of self-dual connections may be parametrized, at least in principle, using the Atiyah-Drinfeld-Hitchin-Manin correspondence [A-D-H-M]. This correspondence gives a diffeomorphism from the space of ADHM matrices, consisting of isomorphism classes of solutions to the non-linear ADHM matrix equations, and isomorphism classes of solutions to the non-linear self-duality equations. When G is SU(2) and −c_{2}(P) = k, the moduli space is an (8k − 3)-dimensional manifold [A-H-S]. As is well-known, the construction of the connection 1-forms provided by the ADHM map becomes very unwieldy when |k| is greater than 1, and it has proved difficult to study the moduli space for arbitrary values of k by appealing directly to the ADHM correspondence [A].

We employ the parametrization, due to Hartshorne [Har] and Jackiw-Nohl-Rebbi [J-N-R], of the moduli space M_{2}^{∗}(𝕊^{4}, SU(2)) in terms of three distinct points P_{i} in 𝕊^{4} and three positive weights λ_{i} (up to a common rescaling). We let T̃_{2} denote the space of unordered pairs (P_{i}, λ_{i}), with P_{i} ∈ 𝕊^{4} distinct and (λ_{0}, λ_{1}, λ_{2}) ∼ ν(λ_{0}, λ_{1}, λ_{2}) for ν > 0. The space T̃_{2} is a 14-dimensional non-compact manifold, while M_{2}^{∗}(𝕊^{4}, SU(2)) has dimension 13. There is an induced action of the group of gauge transformations on T̃_{2}: this action has been interpreted as a motion of the points P_{i} around the circle in 𝕊^{4} determined by those points [Har2, 3], [J-N-R]. This gauge equivalence is most easily understood in the context of algebraic geometry. The space M_{2}^{∗}(𝕊^{4}, SU(2)) is connected, with fundamental group ℤ_{2} [Au-Do], [Har2], [Hur].

The parametrization of Hartshorne and Jackiw-Nohl-Rebbi corresponds to a particularly convenient choice of ADHM matrices. Our principal SU(2)-bundles P with −c_{2}(P) = 2 are the pull-backs of the quaternionic Hopf bundle 𝕊^{11} → ℍℙ^{2}, via suitable classifying maps provided by the ADHM construction from 𝕊^{4} = ℍℙ^{1} into the quaternionic projective space ℍℙ^{2}. For the moduli space M_{2}^{∗}(𝕊^{4}, SU(2)), we obtain the following

Theorem B. Let M_{2}^{∗}(𝕊^{4}, SU(2)) denote the moduli space of self-dual connections on a principal SU(2)-bundle P over the sphere 𝕊^{4}, where −c_{2}(P) = 2 and 𝕊^{4} has its standard round metric g_{0}. Let these connections be parametrized by the space T̃_{2} of unordered pairs (P_{i}, λ_{i}), where λ_{0}, λ_{1}, λ_{2} are positive weight parameters satisfying the scaling condition λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2} = 1, and P_{0}, P_{1}, P_{2} are distinct points in 𝕊^{4}. Let c_{i}^{μ} denote the standard inhomogeneous coordinates of the point P_{i} ∈ 𝕊^{4} = ℍℙ^{1}, for i = 0, 1, 2, μ = 0, …, 3, so that c_{i} = a_{i} if P_{i} = [a_{i}, 1], lying in the southern hemisphere, or c_{i} = b_{i} if P_{i} = [1, b_{i}], lying in the northern hemisphere. With respect to this choice of parameters, we have the following estimates for the corresponding components of the L^{2} metric g:

g_{λ_{i}λ_{i}} = g (\frac{∂ω}{∂λ_{i}} , \frac{∂ω}{∂λ_{i}} ) ≤ C \{1 + log (\frac{1}{λ_{i}} ) + ∑_{j≠i} log (1 + \frac{1}{|c_{i} − c_{j}|})\} ,

g_{c_{i}^{μ}c_{i}^{μ}} = g (\frac{∂ω}{∂c_{i}^{μ}} , \frac{∂ω}{∂c_{i}^{μ}} ) ≤ C \{1 + log (\frac{1}{λ_{i}} ) + ∑_{j≠i} log (1 + \frac{1}{|c_{i} − c_{j}|})\} ,

where C is a universal constant independent of moduli parameters, and ω denotes the family of self-dual connections parametrized by T̃_{2}.

Again, the estimates continue to hold for any metric g on 𝕊^{4} which is globally conformally equivalent to the standard round metric, the constant C now depending on g: it is just the compactness of (𝕊^{4}, g) which is required when deriving the estimates. We need a metric g on 𝕊^{4} in the conformal class [g_{0}] in order to apply the ADHM construction. The tangent space T_{[ω]}M_{2}^{∗}(𝕊^{4}, SU(2)) is spanned by the vectors ∂ω/∂λ_{i}, ∂ω/∂c_{i}^{μ}, for i = 0, 1, 2 and μ = 0, …, 3. Non-diagonal components of g may be estimated via the Schwarz inequality. By seeking upper bounds on the metric components, we avoid the difficulties associated with explicitly computing the horizontal projections appearing in the definition of the metric components — apparently a difficult problem when |k| is greater than 1.

We now provide an outline of the remaining chapters. In Chapter I we establish our notation and conventions, and we review some of the basic concepts of gauge theory. We describe the moduli space of self-dual connections and define the induced L^{2} metric. The ADHM construction produces a family of self-dual connections on a family of principal G-bundles over 𝕊^{4}, parametrized by the space of ADHM matrices. So we review the concept of a family of connections and discuss some related issues concerning infinitesimal deformations and horizontal projections. The ADHM construction defines certain classifying maps from 𝕊^{4} to a classifying space BG, and so we obtain self-dual connections on G-bundles P → 𝕊^{4} by pulling back a canonical connection on a universal G-bundle EG → BG.

In Chapter II we review the correspondence between the moduli space of selfdual SU(2)-connections over 𝕊^{4} and the moduli space of instanton bundles over ℂℙ^{3}, which are stable, holomorphic, rank 2 vector bundles over ℂℙ^{3} satisfying certain technical conditions. This allows us to relate the picture of M_{2}^{∗}(𝕊^{4}, SU(2)) in terms of distinct points P_{i} in 𝕊^{4} and positive weights λ_{i} due to Jackiw, Nohl, and Rebbi, with that of Hartshorne, where the instanton bundles E correspond to curves Y in ℂℙ^{3}. The curve Y associated to a bundle E arises as the zero-set (s)_{0} of a global section s of the twisted bundle E(1) = E ⊗ O_{ℂℙ^{3}}(1).

In Chapter III, we describe the ADHM construction of self-dual SU(2)- connections over 𝕊^{4}. As an application, we explicitly construct the sections s ∈ H^{0}(E(1)) arising in Hartshorne’s study of M_{2}^{∗}(𝕊^{4}, SU(2)).

In Chapters IV and V, we come to our main goal, which is to examine the asymptotic behaviour of the L^{2} metric g on M_{2}^{∗}(𝕊^{4}, SU(2)). In Chapter IV, we describe in detail the parametrization of the self-dual SU(2)-connection 1-forms for k = 2. In Chapter V, we give the local 1-forms representing the tangent vectors to the moduli space M_{2}^{∗}(𝕊^{4}, SU(2)). We avoid the difficulty of computing horizontal projections and obtaining harmonic representatives of the tangent space by instead deriving estimates for the L^{2} norms of the tangent vectors and hence the corresponding components of the L^{2} metric. With these estimates at hand, we are then able to show that the space (M_{2}^{∗}(𝕊^{4}, SU(2)), g) has finite volume and diameter.

CHAPTER I MODULI SPACE OF SELF-DUAL CONNECTIONS

We review the standard description of the moduli space of self-dual connections on a principal G-bundle π : P → M, where G denotes a compact, semi-simple Lie group with Lie algebra 𝔤, and (M, g) is a compact, connected, oriented, Riemannian, smooth four-manifold without boundary. We define the L^{2} metric on the moduli space and discuss related issues of deformation theory and problems arising in the computation of the L^{2} metric components. General references for this chapter are [A-B], [A-H-S], [D-K], [F-U], [G-P1, 2], [I], and [L].

§1.1. Preliminaries on Gauge Theory

We recall some aspects of gauge theory which will later prove useful and establish our notation. General references for this section are [Fi], [F-U], [Hu], [K-N], [Mo], [M-V], and [St].

Definition 1.1.1. We define the following infinite-dimensional topological groups:

(i) Aut(P) = {f ∈ Diff(P) : f(pa) = f(p)a for all p ∈ P, a ∈ G, and π ∘ f = π}; (ii) C_{Ad}^{∞}(P, G) = {φ ∈ C^{∞}(P, G) : φ(pa) = a^{−1}φ(p)a, for all p ∈ P and a ∈ G},

where C^{∞}(P, G) denotes the space of smooth maps from P to G; (iii) G(P) = Γ(Ad P), where Ad P denotes the bundle of groups P ×_{Ad} G and

Γ(Ad P) denotes the space of smooth sections of Ad P.

As topological groups, Aut(P), C_{Ad}^{∞}(P, G), and G(P) are naturally isomorphic and comprise alternative descriptions of the group of gauge transformations, which we will usually denote simply by G(P) — unless we wish to emphasize a particular description [Fi], [Mo]. If f ∈ Aut(P), then the corresponding element φ ∈ C_{Ad}^{∞}(P, G) is defined by letting φ(p) be the unique element of G such that f(p) = pφ(p), for each p ∈ P. Conversely, a map φ ∈ C_{Ad}^{∞}(P, G) determines an element f ∈ Aut(P) by setting f(p) = pφ(p).

To describe a gauge transformation in terms of a section in G(P), let {O_{α}} be an open cover of M and let σ_{α} : O_{α} → P be a system of local sections, with corresponding transition functions T_{βα} : O_{α} ∩ O_{β} → G defined by

σ_{β} = σ_{α}T_{αβ} on O_{α} ∩ O_{β},

and satisfying the cocycle condition

T_{αβ}T_{βγ}T_{γα} = 1 on O_{α} ∩ O_{β} ∩ O_{γ}.

The corresponding local trivialisations τ_{α} : P |_{O_{α}} → O_{α} × G, p ↦ (π(p), φ_{α}(p)), may be defined by setting τ_{α}(σ_{α}(x)a) = (x, a), for (x, a) ∈ O_{α} × G. As usual, φ_{α}(p)φ_{β}(p)^{−1} = T_{αβ}(x), for any p ∈ π^{−1}(x), and τ_{α} ∘ τ_{β}^{−1}(x, a) = (x, T_{αβ}(x)a).

If f ∈ Aut(P), we obtain induced maps g_{α} : O_{α} → G defined by f(σ_{α}(x)) =

σ_{α}(x)g_{α}(x). Hence,

g_{β} = T_{αβ}^{−1}g_{α}T_{αβ} = Ad(T_{αβ}^{−1})g_{α} on O_{α} ∩ O_{β},

and so we may glue the g_{α} together along overlaps to give a global section g ∈ Γ(Ad P). If φ ∈ C_{Ad}^{∞}(P, G), then the section g ∈ Γ(Ad P) may be defined by setting g_{α} = φ ∘ σ_{α}. Conversely, a section g ∈ Γ(Ad P) determines elements f ∈ Aut(P) and φ ∈ C_{Ad}^{∞}(P, G).

More generally, let E → M, E′ → M be fibre bundles with base M, fibre F, and structure group G. Let T_{αβ}, T′_{αβ} be the transition functions of the bundles E, E′ with respect to the open cover {O_{α}} of M. Then, there exists a fibre bundle isomorphism f : E → E′ if and only if there exist maps g_{α} : O_{α} → G such that

T′_{αβ} = g_{α}^{−1}T_{αβ}g_{β} on O_{α} ∩ O_{β},

where f(σ_{α}) = σ′_{α}g_{α} on O_{α} [St, p. 12], [Hu, p. 61].

Let ϱ : G → GL(V) be a representation defining a left action of G on a vector space V and let E = E(ϱ) be the associated vector bundle P ×_{ϱ} V. When V = 𝔤 and ϱ = Ad , then E(ϱ) is denoted ad P. Where convenient, we let ad E ≃ ad P denote the subbundle of End(E) whose sections have values ψ(x) ∈ 𝔤 ⊂ End(E_{x}) for all x ∈ M.

Let C_{ϱ}^{∞}(P, V) denote the space of smooth maps Φ : P → V such that Φ(pa) = ϱ(a^{−1})p for all a ∈ G and p ∈ P. There is a bijection C_{ϱ}^{∞}(P, V) → Γ(E) defined by Φ → ϕ, with ϕ(x) = [p, Φ(p)] for x ∈ M and any p ∈ π^{−1}(x), where [p, ξ] = {(pa, ϱ(a^{−1})ξ) : a ∈ G} [Hu, p. 46]. A section ϕ ∈ Γ(E) may then be represented locally by maps ϕ_{α} : O_{α} → V, with ϕ_{α} = Φ ∘ σ_{α}, so that

ϕ_{β} = ϱ(T_{αβ}^{−1})ϕ_{α} on O_{α} ∩ O_{β}.

Hence, E has corresponding transition functions ϱ(T_{βα}). The action of Aut (P) on P induces an action on the associated vector bundle E: if g ∈ G(P) is represented locally by σ_{α} ↦ σ_{α}g_{α}, then ϕ_{α} ↦ ϱ(g_{α}^{−1})ϕ_{α} on O_{α}.

Definition 1.1.2. We define the following bundle-valued forms:

(i) Ω^{q}(P, 𝔤) = Γ(∧^{q}(T^{∗}P) ⊗ 𝔤); (ii) Ω^{q}(M, E) = Γ(∧^{q}(T^{∗}M) ⊗ E); (iii) Ω^{q}(M, ad P) = Γ(∧^{q}(T^{∗}M) ⊗ ad P).

The action of Aut (P) on P induces a right action Ω^{q}(P, 𝔤)×Aut (P) → Ω^{q}(P, 𝔤) by pull-back, (ω, f) ↦ f^{∗}ω, where we will often denote f^{∗}ω by ω^{f} or f(ω) for convenience. Moreover, there is an induced right action of Aut (P) on the bundlevalued q-forms Ω^{q}(M, E) and Ω^{q}(M, ad P): the action of G(P) on Ω^{q}(M, E) may be represented locally by ω_{α} ↦ ϱ(g_{α}^{−1})ω_{α}, where ω_{α} ∈ Ω^{q}(O_{α}, V) [Mo], [K-N1, p. 75]. Where convenient, we denote the corresponding global form by ω^{g} or g(ω).

Let A(P) ⊂ Ω^{1}(P, 𝔤) denote the affine space of connection 1-forms on P and let ω_{α} = σ_{α}^{∗}ω ∈ Ω^{1}(O_{α}, 𝔤) be the corresponding local connection 1-forms [K-N1, p. 64-66]. Then

ω_{β} = Ad(T_{αβ}^{−1})ω_{α} + T_{αβ}^{−1}dT_{αβ} on O_{α} ∩ O_{β},

where T_{αβ}^{−1}dT_{αβ} = T_{αβ}^{∗}θ, if θ denotes the Maurer-Cartan form of G. If we fix any connection ω_{0} ∈ Ω^{q}(P, 𝔤) and let ω ∈ Ω^{q}(P, 𝔤), then the difference w − ω_{0} defines an element δ of Ω^{q}(M, ad P) by setting δ_{α} = σ_{α}^{∗}(ω − ω_{0}) ∈ Ω^{q}(O_{α}, 𝔤) [Mo].

The group Aut (P) induces a right action on A(P) × Aut (P) → A(P) by pull-back, (ω, f) ↦ f^{∗}ω [Mo]. Denoting f^{∗}ω by ω′, the corresponding action of φ ∈ C_{Ad}^{∞}(P, G) on A(P) is represented by

ω′ = Ad (φ^{−1})ω + φ^{∗}θ ∈ Ω^{1}(P, 𝔤).

Denoting ω′_{α} = σ_{α}^{∗}ω′, the corresponding action of g ∈ G(P) is represented locally by

ω′_{α} = Ad(g_{α}^{−1})ω_{α} + g_{α}^{−1}dg_{α} ∈ Ω^{1}(O_{α}, 𝔤),

where g_{α}^{−1}dg_{α} = g_{α}^{∗}θ [Fi, p. 239].

Next we consider the tangent space T_{ω_{0}}A(P) to A(P) at a connection ω_{0}. If ω(t) is a smooth curve in A(P) with ω(0) = ω_{0}, then ω̇(0) ∈ T_{ω_{0}}A(P). If ω_{α}(t) = σ_{α}^{∗}ω(t) are the local connection 1-forms, then

ω̇_{α}(0) = \frac{d}{dt}σ_{α}^{∗}ω(t)|_{t=0} = σ_{α}^{∗}ω̇(0) ∈ Ω^{q}(O_{α}, 𝔤).

Then ω̇_{β}(0) = Ad (T_{αβ}^{−1})ω̇_{α}(0), and so ω̇(0) defines an element of Ω^{q}(M, ad P). This gives the standard identification of T_{ω_{0}}A(P) with Ω^{q}(M, ad P).

Let ω ∈ A(P) and let ∇^{ω} : Ω^{0}(M, E) → Ω^{1}(M, E) be the covariant derivative on an associated vector bundle E(ϱ) [Mo], [K-N1]. The exterior covariant derivative d_{ω} : Ω^{q}(M, E) → Ω^{q+1}(M, E) is defined by requiring that

(i) d_{ω} is ℝ-linear; (ii) d_{ω}(ϕ ⊗ ψ) = ∇^{ω}ϕ ∧ ψ + ϕ ⊗ dψ for all ϕ ∈ Γ(E) and ψ ∈ Ω^{q}(M). Consequently, d_{ω}(ϕ ∧ ψ) = d_{ω}ϕ ∧ ψ + (−1)^{q}ϕ ∧ dψ for all ϕ ∈ Ω^{q}(M, E) and ψ ∈ Ω^{p}(M). The formal adjoint, δ_{ω} : Ω^{q}(M, E) → Ω^{q−1}(M, E), of d_{ω} with respect to the metric g on M is defined by δ_{ω} = d_{ω}^{∗} = − ∗ d_{ω}∗. The connection ∇^{ω} on E naturally induces a connection on End(E) by setting [K-N1, p. 124]:

(∇^{ω}ψ)(ϕ) = ∇^{ω}(ψ(ϕ)) − ψ(∇^{ω}ϕ), for all ψ ∈ Γ(End(E)), ϕ ∈ Γ(E).

The curvature F = F_{ω} ∈ Ω^{2}(M, End(E)) may be defined invariantly by d_{ω} ∘ d_{ω}ϕ = F_{ω}ϕ for all ϕ ∈ Γ(E) and defined locally by

F_{α} = dω_{α} + ω_{α} ∧ ω_{α} ∈ Ω^{2}(O_{α}, 𝔤),

with F_{β} = Ad (T_{βα})F_{α} on O_{α} ∩O_{β}, with values in 𝔤 ⊂ End(V), so that {F_{α}} defines an element of Ω^{2}(M, End(E)), [G-H], [K-N1]. More explicitly, a calculation shows [F-U, p. 31]:

\begin{aligned}F_{α,ij} & = \frac{∂ω_{α,j}}{∂x_{i}} − \frac{∂ω_{α,i}}{∂x_{j}} + [ω_{α,i}, ω_{α,j}], where \\ F_{α} & = \frac{1}{2} ∑_{i,j} F_{α,ij}dx_{i} ∧ dx_{j} = ∑_{i<j} F_{α,ij}dx_{i} ∧ dx_{j}. \end{aligned}

If G = SU(n), ϱ is the fundamental representation on ℂ^{n}, and E = P ×_{ϱ} ℂ^{n}, then the instanton number of P may be computed by

k = −c_{2}(E)[Mo] = −\frac{1}{8π^{2}} ∫_{M} tr (F_{ω} ∧ F_{ω}).

Moreover, c_{1}(E)[M] = 0, and SU(2) bundles on any closed, oriented, four-manifold M are classified topologically by c_{2}(E) ∈ H^{4}(M, ℤ) ≃ ℤ [F-U, p. 179].

§1.2. Moduli Space of Self-dual connections

We review the description of the moduli space of self-dual connections. General references for this section are [K-D], [F-U], [M].

We fix the topology of a principal G-bundle P → M and denote A(P) and G(P) by A and G, respectively. The metric on Ω^{q}(M, ad P) is induced by the Riemannian metric g of M and a constant negative multiple of the Cartan-Killing form K on G: if X, Y ∈ 𝔰𝔲(n), we have K(X, Y) = tr (XY) for the fundamental representation of 𝔰𝔲(n). We recall that ω is a Yang-Mills connection if it satisfies the conformally invariant Yang-Mills equations: d_{ω}F_{ω} = 0 and d_{ω}^{∗}F_{ω} = 0. A connection ω is self-dual (SD) (respectively, anti-self-dual (ASD)), if it satisfies the conformally invariant equation ∗F_{ω} = F_{ω} (respectively, ∗F_{ω} = −F_{ω}), and consequently is a solution of the Yang-Mills equations [F-U, p. 35].

A G-connection ω is reducible if its holonomy group Φ_{ω} is a proper subgroup of G, and is irreducible otherwise [D-K, p. 131]. For any connection ω, if Γ_{ω} = {g ∈ G : g(ω) = ω} denotes the isotropy group of ω, then Γ_{ω} is isomorphic to {g ∈ G : ghg^{−1} = h for all h ∈ Φ_{ω}}, the centralizer of Φ_{ω} in G, where we view Γ_{ω} and Φ_{ω} as subgroups of Aut (E_{x}) for some fixed x ∈ M [D-K, p. 131]. Let Z = Z(G) denote the centre of the gauge group G and let Z = Z(G) denote the centre of G. We recall that Z = Γ(P ×_{Ad} Z) ≃ Z. For example, if G = SU(2), we have Z = {±1} and so Z ≃ {±1}.

When G = SU(2), ϱ is the fundamental representation of SU(2) on ℂ^{2}, and E = P ×_{ϱ} ℂ^{2}, the following are equivalent [F-U, p. 47]:

(i) The connection ω is reducible; (ii) The connection d_{ω} and bundle E split, so that d_{ω} = d_{1} ⊕ d_{2} on E = L_{1} ⊕ L_{2}; (iii) Γ_{ω}/{±1} ≃ U(1); (iv) Ker {d_{ω} : Ω^{0}(E) → Ω^{1}(E)} ≠ (0).

We note that if H^{2}(M, ℤ) = (0), (for example, if M = 𝕊^{4}), then there are no reducible connections [F-U, p. 33]. The affine space of irreducible connections on P is denoted by A^{∗}. Fix a Sobolev index s ≥ 2 and define the following Hilbert manifolds [F-U, p. 46]:

(i) Let A_{s} = ω_{0} + L_{s}^{2}Ω^{1}(M, ad P), the affine space A_{s} of Sobolev L_{s}^{2}-connections, where ω_{0} ∈ A is a fixed basepoint connection;

(ii) Let ϱ : G → Aut (V) be a faithful representation, so that G ⊂ Γ(P ×_{ϱ} Aut (V)) and G_{s+1} ⊂ L_{s+1}^{2}(P ×_{ϱ} Aut (V)). Then G_{s+1} is an infinite-dimensional Lie group with Lie algebra L_{s+1}^{2}Ω^{0}(M, ad E);

(iii) Let G̃ ⊂ Γ(P ×_{Ad} Aut (𝔤)) and G̃_{s+1} ⊂ L_{s+1}^{2}(P ×_{Ad} Aut (𝔤)). Then G̃_{s+1} = G_{s+1}/Z and G̃_{s+1} is an infinite-dimensional Lie group with Lie algebra L_{s+1}^{2}Ω^{0}(M, ad P). We denote the space of all irreducible Sobolev connections by A_{s}^{∗}. For s ≥ 2,

there is a smooth action A_{s}^{∗} ×G_{s+1} → G_{s+1}. Let B_{s}^{∗} denote the orbit space A_{s}^{∗}/G̃_{s+1} endowed with the quotient topology. The topology of the moduli space of self-dual connections M^{∗} ⊂ B_{s}^{∗} is independent of s ≥ 2, so we will henceforth omit Sobolev subscripts and denote A_{s}^{∗}, B_{s}^{∗}, G̃_{s+1}, and G_{s+1}, by A^{∗}, B^{∗}, G̃, and G respectively.

We recall the construction of local coordinate maps for the principal G̃-bundle π : A^{∗} → B^{∗}, ω ↦ [ω] [F-U, p. 48]. If ω is irreducible, then its holonomy group is Φ_{ω} = G, and Γ_{ω} = Z. Then G/Z = G̃ acts freely on A^{∗} and B^{∗} = A^{∗}/G̃. Fix a connection ω ∈ A^{∗} and let G · ω denote the orbit of the gauge group through ω. We have smooth maps:

0 ⟶ G ⟶^{·ω} G · ω ⟶^{⊂} A^{∗}

Computing differentials gives:

0 ⟶ T_{1}G ⟶^{−d_{ω}} T_{ω}(G · ω) ⟶^{⊂} T_{ω}A^{∗}

where T_{1}G = Ω^{0}(M, ad P) is the Lie algebra of G and T_{ω}A^{∗} = Ω^{1}(M, ad P). In particular,

T_{ω} = (G · ω) = Im {d_{ω} : Ω^{0}(M, ad P) → Ω^{1}(M, ad P)},

which we denote by V_{ω} = Im d_{ω}, the vertical subspace through ω. Then Ker {d_{ω}^{∗} : Ω^{1}(M, ad P) → Ω^{0}(M, ad P)} is the L^{2}-orthogonal complement of Im d_{ω}, which we denote by Ker d_{ω}^{∗} = H_{ω}, the horizontal subspace through ω. This gives the L^{2}-orthogonal decomposition

T_{ω}A^{∗} = H_{ω} ⊕ V_{ω}

for each ω ∈ A^{∗}. The corresponding horizontal and vertical projection operators on T_{ω}A^{∗} are:

h_{ω} = 1 − d_{ω}G_{ω}^{0}d_{ω}^{∗}  and  v_{ω} = d_{ω}G_{ω}^{0}d_{ω}^{∗},

where G_{ω}^{0} is the Green’s operator for the Laplacian Δ_{ω}^{0} = d_{ω}^{∗}d_{ω} on Ω^{0}(M, ad P). There is an open neighbourhood O_{ω} of ω in A^{∗}, and a diffeomorphism

Φ : A^{∗} ⊃ O_{ω} ⟶ H_{ω} × G, β ⟼ (φ(β), g_{β})

onto an open neighbourhood of (0, 1) ∈ H_{ω} × G. The corresponding open neighbourhood S_{ω} of 0 ∈ H_{ω} is a slice for the action of G on A^{∗} [F-U, p. 49]. The map g : O_{ω} → G, β → g_{β} is obtained by solving

d_{ω}^{∗}(g_{β}^{∗}(β) − α) = 0,

so that g_{β}^{∗}(β) ∈ ω + H_{ω} for ω ∈ O_{ω}; the submersion φ : O_{ω} → S_{ω} is then given by β ↦ g_{β}^{∗}(β) − ω. In particular, O_{ω} may be chosen to be G-invariant, and (Φ, g) is G-equivariant, in the sense that

(φ(f^{∗}β), g_{f^{∗}β}) = (φ(β), f^{−1} ∘ g_{β}) for f ∈ G.

Hence, we get an induced diffeomorphism

φ : B^{∗} ⊃ O_{ω}/G̃ ⟶ S_{ω} ⊂ H_{ω},

with inverse π : S_{ω} → O_{ω}/G̃, the restriction of the natural projection π : A^{∗} → B^{∗}. In particular, B^{∗} is a smooth Hilbert manifold, π : A^{∗} → B^{∗} is a principal G̃-bundle, and the maps φ : π(O_{ω}) → S_{ω} provide a system of local coordinate charts on B^{∗} [F-U, p. 50], [L, p. 33]. Computing differentials at [ω] ∈ B^{∗} gives an isomorphism φ_{∗} : T_{[ω]}B^{∗} → H_{ω}, and this provides the standard identification of the tangent space T_{[ω]}B^{∗} with H_{ω}.

We consider the tangent space to B^{∗} in more detail and compute the differential φ_{∗}_{[ω]} : T_{[ω]}B^{∗} → H_{ω}. First, we examine the action of the gauge group on paths of connections and their tangent vectors. A calculation yields [I, p. 16]:

Lemma 1.2.1. Let ω(t) be a smooth path of connections in A^{∗} through ω(0) = ω, with ω̇(0) ∈ T_{ω}A^{∗}. If g(t) is a smooth path of gauge transformations in G through g(0) = g, we obtain a new path ω^{g}(t) in A^{∗} through ω^{g}(0) = ω^{g}. Then

ω̇^{g}(0) = Ad (g^{−1})(ω̇(0)) + d_{ω^{g}} (g^{−1}ġ(0)) ∈ Ω^{1}(M, ad P).

Proof. Calculation. □

Lemma 1.2.2. Let g ∈ G, ψ ∈ Ω^{q}(M, ad P) and ω ∈ A^{∗}. Then

(i) g(d_{ω}ψ) = d_{g(ω)}g(ψ); (ii) g(d_{ω}^{∗}ψ) = d_{g(ω)}^{∗}g(ψ); (iii) g(Δ_{ω}ψ) = Δ_{g(ω)}g(ψ).

Proof. Calculation. □

(We recall that G acts on A^{∗} by ω ↦ g(ω) = ω^{g} = Ad (g^{−1})ω + g^{−1}dg, and on Ω^{q}(M, ad P), by ψ ↦ g(ψ) = Ad (g^{−1})ψ.)

Lemma 1.2.3. Let g ∈ G, α ∈ Ω^{1}(M, ad P) and ω ∈ A^{∗}. Then

h_{g(ω)}g(α) = g(h_{ω}α).

Proof. Calculation. □

Let ω(t) be a path in A^{∗} through ω(0) = ω, with ω̇(0) ∈ T_{ω}A^{∗}. Let g(t) be a path in G through g(0) = 1, with ġ(0) = ϕ, for some ∈ Ω^{0}(M, ad P). Then ω^{g}(t) is a path in A^{∗} through ω^{g}(0) = ω, with tangent vector at ω given by

ω̇^{g}(0) = ω̇(0) + d_{ω}ϕ ∈ T_{ω}A^{∗}.

Choosing the path g(t) such that d_{ω}^{∗}(ω̇^{g}(t) − ω) = 0, ensures ω̇^{g}(t) − ω ∈ H_{ω} (for small t), so φ(ω(t)) is a path in the horizontal slice S_{ω}. Moreover,

φ_{∗}(ω̇(0)) = ω̇^{g}(0) = ω̇(0) + d_{ω}ϕ ∈ H_{ω}.

Enforcing the horizontality condition, d_{ω}^{∗}(ω̇^{g}(0)) = 0, gives ϕ = −G_{ω}^{0}d_{ω}^{∗}β̇(0). Hence,

\begin{aligned}ω̇^{g}(0) & = ω̇(0) − d_{ω}G_{ω}^{0}d_{ω}^{∗}ω̇(0) \\ & = h_{ω}ω̇(0) ∈ H_{ω}, \end{aligned}

and so ω̇^{g}(0) is the horizontal projection of ω̇(0).

Lemma 1.2.4. The differential of φ : O_{ω} → S_{ω} at ω ∈ A^{∗} is given by the horizontal projection operator

φ_{∗}_{ω} : T_{ω}A^{∗} ⟶ H_{ω}, α ⟼ h_{ω}α.

Proof. This follows from the above calculations, since φ_{∗}_{ω}(α) = h_{ω}α where ω(t) is a path of conections with ω(0) = ω, α = ω̇(0) and g(t) is a path of gauge transformations chosen as above so that g(0) = 1, ω̇^{g}(0) = h_{ω}α. □

Next, we observe that the paths ω^{g}(t) through ω^{g}(0) = ω^{g}, corresponding to different choices of g(t), g(0) = g, all project to the same path [ω^{g}(t)] through [ω] in B^{∗}, for each g ∈ G. We have exact sequences,

0 ⟶ T_{1}G ⟶^{−d_{ω^{g}}} T_{ω^{g}}A^{∗} ⟶^{π_{∗}} T_{[ω]}B^{∗} ⟶ 0,

so that the tangent space to B^{∗} at [ω] is given by

T_{[ω]}B^{∗} = \frac{Ω^{1}(M, ad P)}{Im \{d_{ω} : Ω^{0}(M, ad P) → Ω^{1}(M, ad P)\}}
≃ Ker {d_{ω}^{∗} : Ω^{1}(M, ad P) → Ω^{0}(M, ad P)} = H_{ω}.

We have canonical isomorphisms T_{ω}A^{∗}/Im d_{ω} = T_{ω^{g}}A^{∗}/Im d_{ω^{g}} and H_{ω} ≃ H_{ω^{g}} for any g ∈ G. The spaces T_{[ω]}B^{∗} = T_{ω}A^{∗}/Im d_{ω} and H_{ω} are identifed via the induced isomorphism:

φ_{∗} : T_{[ω]}B^{∗} ⟶ H_{ω}, [α] ⟼ h_{ω}α,

where [α] ∈ T_{[ω]}B^{∗} = T_{ω}A^{∗}/Im d_{ω}.

Finally, we define the self-dual moduli space M by

M = {[ω] ∈ B : F_{w}^{−} ≡ 0}.

We recall the Proposition 1.2.5. [D-K, p. 138] If ω is a self-dual G-connection over M, then a neighbourhood of [ω] in the moduli space M is modelled on a quotient f^{−1}(0)/Γ_{ω} where

f : Ker d_{ω}^{∗} ⟶ Cokerd_{ω}^{−}

is a Γ_{ω}-equivariant map and

Ker d_{ω}^{∗} = Ker {d_{ω}^{∗} : Ω^{1}(M, ad P) → Ω^{0}(M, ad P)},
Cokerd_{ω}^{−} = \frac{Ω_{−}^{2}(M, ad P)}{Im \{d_{ω}^{−} : Ω^{1}(M, ad P) → Ω_{−}^{2}(M, ad P)\}}.

Here, d_{ω}^{−} = p_{−}d_{ω} and p_{−} denotes the projection Ω^{2}(M, ad P) → Ω_{−}^{2}(M, ad P) onto anti-self-dual 2-forms, with p_{−} = \frac{1}{2}(1 − ∗) in terms of the Hodge star operator.

Indeed, f is induced by the map ψ(α) = F_{ω+α}^{−} = d_{ω}^{−}α + (α ∧ α)^{−} for α in an open ball around 0 ∈ Ker d_{ω}^{∗} [D-K, p. 134]. Hence, a neighbourhood of [ω] in M has a local model:

{α ∈ Ω^{1}(M, ad P) : d_{ω}^{∗}α = 0 and d_{ω}^{−}α + (α ∧ α)^{−} = 0}/Γ_{ω}

The first equation d_{ω}^{∗}α = 0 defines the construction of a local slice through the G-orbit; the second equation is the self-duality equation F_{ω+α}^{−} = 0. The self-duality condition ensures that d_{ω}^{−} ∘ d_{ω} = 0 and so one obtains an elliptic deformation complex:

0 ⟶ Ω^{0}(M, ad P) ⟶^{d_{ω}} Ω^{1}(M, ad P) ⟶^{d_{ω}^{−}} Ω_{−}^{2}(M, ad P) ⟶ 0

with associated cohomology groups H_{ω}^{0}, H_{ω}^{1}, H_{ω}^{2}, where

H_{ω}^{1} = \frac{Ker \{d_{ω}^{−} : Ω^{1}(M, ad P) → Ω_{−}^{2}(M, ad P)\}}{Im \{d_{ω} : Ω^{0}(M, ad P) → Ω^{1}(M, ad P)\}},
H_{ω}^{2} = \frac{Ω_{−}^{2}(M, ad P)}{Im \{d_{ω}^{−} : Ω^{1}(M, ad P) → Ω_{−}^{2}(M, ad P)\}},

while H_{ω}^{0} is the Lie algebra of Γ_{ω}. The negative Euler characteristic

s = − dim H_{ω}^{0} + dim H_{ω}^{1} − dim H_{ω}^{2}

gives the virtual dimension of the moduli space. By the Hodge theory for this complex we have natural isomorphisms:

H_{ω}^{1} ≃ H_{ω}^{1} = {α ∈ Ω^{1}(M, ad P) : d_{ω}^{−}α = 0 and d_{ω}^{∗}α = 0},
H_{ω}^{2} ≃ H_{ω}^{2} = {α ∈ Ω_{−}^{2}(M, ad P) : d_{ω}^{−}α = 0}.

We recall that an irreducible self-dual connection is regular if H_{ω}^{2} = 0 and the moduli space M^{∗} is regular if all its points are regular points [D-K, p. 146]. The moduli space of regular, irreducible connections M^{∗}′ is a smooth manifold of dimension s. For G = SU(2), the space M^{∗} coincides with M^{∗}′ for generic metrics g on M, and so is a smooth manifold [F-U, p. 61], [D-K, p. 149]. Alternatively, under a certain curvature condition on (M, g), we have H_{ω}^{2} = 0 and so the moduli space M^{∗} will be a smooth manifold [A-H-S]. This curvature condition is satisfied by 𝕊^{4} with its standard round metric.

Indeed, a computation shows that the tangent space at any regular point [ω] ∈ M^{∗} is given by [G-P1, p. 670]:

T_{[ω]}M^{∗} = H_{ω}^{1} ≃ H_{ω}^{1}.

The description of T_{[ω]}M^{∗} as the cohomology group H_{ω}^{1} will be more useful for our purposes, due to the difficulty in finding a suitable basis for the harmonic space H_{ω}^{1}.

§1.3. L^{2} Metric on Moduli Space

We describe the construction of the natural Riemannian metric on the quotient space B^{∗} and the moduli space M^{∗} ↪ B^{∗} of self-dual connections. General references for this section are [G-P1, 2], [I].

The affine space A^{∗} has a (weak) Riemannian metric via the identification T_{ω}A^{∗} = Ω^{1}(M, ad P) and the L^{2}-inner product on Ω^{1}(M, ad P):

(α_{1}, α_{2}) = ∫_{M}⟨α_{1}, α_{2}⟩\sqrt{g} dx for α_{1}, α_{2} ∈ Ω^{1}(M, ad P),

where ⟨ , ⟩ denotes the fibre metric on T^{∗}M ⊗ ad P. The gauge group G acts isometrically on A^{∗}, preserving the L^{2}-orthogonal splitting of each tangent space T_{ω}A^{∗}. The orbit space B^{∗} inherits a (weak) Riemannian metric by requiring that the natural projection π : A^{∗} → B^{∗} be a Riemannian submersion [C-E, p. 65]. Hence, the differential π_{∗}_{ω} : H_{ω} → T_{[ω]}B^{∗} is required to be an isometry, while π_{∗}_{ω} is zero on V_{ω}. The moduli space M^{∗} is a smoothly embedded submanifold of B^{∗}, and so there is an induced smooth Riemannian metric on M^{∗} [G-P1, p. 671].

In order to obtain a more explicit expression for the metric on B^{∗}, choose a representative ω ∈ [ω] and observe that, by definition, the Hilbert space isomorphism

φ_{∗}_{[ω]} = (π_{∗}_{ω})^{−1} : T_{[ω]}B^{∗} ⟶ H_{ω} ⊂ Ω^{1}(M, ad P)

is an isometry. Hence, T_{[ω]}B^{∗} acquires an inner product by pulling back the natural L^{2} inner product on Ω^{1}(M, ad P). Then, the L^{2} metric on B^{∗} is given explicitly by:

g_{[ω]}([α_{1}], [α_{2}]) = φ_{[ω]}^{∗}([α_{1}], [α_{2}])
= (φ_{∗}_{[ω]}[α_{1}], φ_{∗}_{[ω]}[α_{2}])
= (h_{ω}α_{1}, h_{ω}α_{2}).

Observe that for a given [ω] ∈ B^{∗}, we may choose different coordinate charts φ : B^{∗} ⊃ π(S_{ω^{g}}) → S_{ω^{g}}, where S_{ω^{g}} are the horizontal slices through ω^{g} ∈ A^{∗} for each g ∈ G, and the inner product apparently depends on this choice. However, we have the Lemma 1.3.1. [I, p. 16] The above expression for the L^{2} inner product on T_{[ω]}B^{∗} is independent of the choice of slice neighbourhood, and so we obtain a well-defined (weak) Riemannian metric on B^{∗}.

Proof. Let S_{ω}, S_{ω^{g}} be two slice neighbourhoods with associated local coordinate charts

\begin{matrix}φ : B^{∗} ⊃ π(S_{ω}) → S_{ω} ⊂ H_{ω}, \\ φ′ : B^{∗} ⊃ π(S_{ω^{g}}) → S_{ω^{g}} ⊂ H_{ω^{g}}. \end{matrix}

Choose tangent vectors [α_{i}] ∈ T_{[ω]}B^{∗}, with α_{i} ∈ T_{ω}A^{∗}. Let ω_{i}(t) be paths of connections through ω_{i}(0) = ω, with tangent vectors ω̇_{i}(0) = α_{i}, for i = 1, 2, and let g(t) be any path of gauge transformations through g(0) = g. Then ω_{i}^{g}(t) = g(t)^{∗}ω_{i}(t) are paths through ω_{i}^{g}(0) = ω^{g}, and the tangent vectors at ω^{g} are given by

ω̇_{i}^{g}(0) = Ad (g^{−1})ω̇_{i}(0) + d_{ω^{g}} (g^{−1}ġ(0)) for i = 1, 2.

Computing differentials, we get

φ_{∗}_{[ω]}[α_{i}] = h_{ω}ω̇_{i}(0),
φ′_{∗}_{[ω]}[α_{i}] = h_{ω^{g}}ω̇_{i}^{g}(0)
= Ad (g^{−1})h_{ω}ω̇_{i}(0) for i = 1, 2.

Now computing inner products, we obtain

(φ′)_{[ω]}^{∗}([α_{1}], [α_{2}]) = (φ′_{∗}_{[ω]}[α_{1}], φ′_{∗}_{[ω]}[α_{2}])
= (Ad (g^{−1})h_{ω}α_{1}, Ad (g^{−1})h_{ω}α_{2})
= (h_{ω}α_{1}, h_{ω}α_{2}),
= φ_{[ω]}^{∗}([α_{1}], [α_{2}]),

making use of the Ad -invariance of the L^{2} inner product on Ω^{q}(M, ad P). Thus, g_{[ω]}([α_{1}], [α_{2}]) is independent of the choice of slice neighbourhood. □

Hence, the L^{2} metric g as described above is well-defined. We restrict g to M^{∗} ↪ B^{∗} and this defines a smooth Riemannian metric on the moduli space [G-P1, p. 671].

§1.4. Families of Connections

In our description of the moduli space of self-dual connections on a principal Gbundle P, we assumed that the bundle P was fixed. However, in our discussion of the ADHM construction, we will be required to consider the more general situation — familiar from the Kodaira-Spencer deformation theory [Ko] — of a family of connections defined on a family of principal G-bundles. We review this idea and describe the implications for the definition of the L^{2} metric g on M^{∗}. General references for this section are [D-K], [Ko], [M], [A-J].

Definition 1.4.1. A family of connections in a principal G-bundle P → M parametrized by a topological space T is a G-bundle

\underline{P} ⟶ M × T

with the property that each slice P_{t} = \underline{P} |_{M×\{t\}} is isomorphic to P, together with a connection ω_{t} in P_{t} for each t, forming a family of connections \underline{ω} = \{ω_{t} : t ∈ T \} [D-K, p. 173 & p. 237], [Ko, p. 124 & p. 324]. We will generally assume that T is a smooth manifold.

By analogy with [Ko, p. 61 & p. 192], we say that two families (\underline{P} , \underline{ω}, T) and (\underline{P}′, \underline{ω}′, T) are equivalent if there exists a G-bundle map Φ so that the following diagram commutes,

\underline{P}  ⟶^{Φ}  \underline{P}′
↓  ↓
M × T ⟶ M × T

where ω_{t} = Φ_{t}^{∗}ω′_{t} for all t ∈ T, and Φ_{t} = Φ|_{P ×{t}}.

Remark 1.4.2. It is not necessarily true that \underline{P} ≃ P × T.

It is useful to consider framed families of connections. We recall the definition of a framed connection [D-K, p. 173]:

Definition 1.4.3. Let x_{0} be a basepoint in M. A framed connection in a G-bundle P → M is a pair (ω, φ), where ω is a connection and φ is an isomorphism of Gspaces, φ : G → P_{x_{0}}.

The gauge group G acts naturally on framed connections, and we denote the quotient by

B̃ = (ω × Hom_{G}(G, P_{x_{0}})) /G.

Alternatively, if we fix a framing ϕ and define the group of based gauge transformations,

G_{0} = {g ∈ G : g(x_{0}) = 1},

then the isotropy groups Γ_{0,ω} = {g ∈ G_{0} : g(ω) = ω} are trivial for all ω ∈ A, G_{0} acts freely on A, and B̃ = A/G_{0}. There is a natural map B̃ → B which forgets the framing. Alternatively, B is the quotient by the residual of the gauge group:

B = A/G ≃ (A/G_{0}) / (G/G_{0}) = B̃/ (G/G_{0}) , where G/G_{0} ≃ Aut (P_{x_{0}}) ≃ G.

Restricting to irreducible connections, we recall that G̃ = G/Z acts freely on A^{∗}. We have

B^{∗} = A^{∗}/G̃ ≃ B̃/ (G̃/G_{0}) , where G̃/G_{0} ≃ G/Z.

Then B̃^{∗} → B^{∗} is a principal G^{ad} -bundle, where G^{ad} = G/Z. (We note that G^{ad} is isomorphic to the image of G under the adjoint representation Ad : G → Aut (𝔤).) For example, if G = SU(2), then Z = {±1}, G^{ad} = SO(3), and B̃^{∗} → B^{∗} is a principal SO(3)-bundle.

We have a framed family of connections if there is an isomorphism:

\underline{φ} : \underline{P} |_{\{x_{0}\}×T} ⟶ T × G.

Then for each t, the pair (ω_{t}, φ_{t}) is a framed connection. We recall the construction of the universal family of framed connections parametrized by B̃ [D-K, p. 175]. Let π_{1} : M × A → M be the projection onto the first factor and let π_{1}^{∗}P → M × A be the pull-back bundle, so π_{1}^{∗}P = P × A. The G bundle P × A → M × A carries a tautological family of connections \underline{ω}, in which the connection on P_{ω} = (P × A)|_{M×{A}} is π_{1}^{∗}ω. The group G_{0} acts freely on M × A as well as on P × A, by (p, ω) ↦ (pg, g(ω)) for g ∈ G_{0}, and so there is a quotient bundle

\begin{aligned}& ℙ̃ ⟶ M × B̃, \\ where & ℙ̃ = (P × A)/G_{0} = P ×_{G_{0}} A. \end{aligned}

The family of connections \underline{ω} and the framing φ are preserved by G_{0}, so ℙ̃ carries an inherited family of connections (𝔸̃, \underline{φ}). This is the universal framed family in P → M parametrized by B̃.

If (\underline{ω}, \underline{φ}) is a framed family of connections parametrized by a space T and carried by a bundle \underline{P} → M × T, there is an associated map

f : T ⟶ B̃, t ⟼ [ω_{t}, φ_{t}].

Conversely, given a map f : T → B̃, there is a corresponding pull-back family of connections carried by (1_{M} × f)^{∗}ℙ̃:

(1_{M} × f)^{∗}ℙ̃ ⟶  ℙ̃
↓  ↓
M × T  ⟶^{1_{M} ×f} M × B̃

These two constructions are inverses of one another:

Lemma 1.4.4. [D-K, p. 175] The maps f : T → B̃ are in one-to-one correspondence with framed families of connections on M parametrized by T, and this correspondence is obtained by pulling back from the universal framed family, (𝔸̃, ℙ̃, \underline{φ}).

Lemma 1.4.5. [D-K, p. 175] The homotopy classes of maps [T, B̃] parametrize isomorphism classes of pairs (\underline{P} , \underline{φ}), where

(i) \underline{P} → M × T is a G-bundle with P_{t} ≃ P for all t; (ii) \underline{φ} : \underline{P} |_{\{x_{0}\}×T} → G × T is a trivialization.

Restricting to irreducible connections on P → M, we let P × A^{∗} → M × A^{∗} denote the pull-back G-bundle, carrying the tautological family of connections. Now taking quotients by G, we obtain a bundle

ℙ^{ad} ⟶ M × B^{∗},
where ℙ^{ad} = (P × A^{∗})/G.

Then ℙ^{ad} → M × B^{∗} is a principal G^{ad} -bundle. For example, if G = SU(2) = Spin(3), then ℙ^{ad} → M × B^{∗} is an SO(3)-bundle carrying a family of connections (without framing) for the SO(3)-bundle P/{±1} → M parametrized by B^{∗}. This lifts to an SU(2)-family if and only if the second Stiefel-Whitney class w_{2}(ℙ^{ad}) in H^{2}(M × B^{∗}, ℤ_{2}) is zero.

Example 1.4.6. Suppose \underline{ω} is a family of G-connections in a G bundle P → M carried by a G-bundle \underline{P} → M × T, where T is a contractible space. Then the identity map 1_{T} : T → T is homotopic to a constant map t_{0} : T → {t_{0}} ⊂ T, and we obtain the pull-back bundle (1_{M} × t_{0})^{∗}\underline{P} = P × T, where P = P_{t_{0}}:

P × T ⟶  \underline{P}
↓  ↓
M × T ⟶^{1_{M} ×t_{0}} M × T

Pulling back by the identity map 1_{M} ×1_{T} gives (1_{M} ×1_{T})^{∗}\underline{P} = \underline{P}, and since 1_{M} ×1_{T} is homotopic to 1_{M} × t_{0}, then \underline{P} ≃ P × T as G bundles [St, p. 49].

A more direct argument that \underline{P} ≃ P × T when T is contractible, may be given by analogy with [Ko, p. 66 & p. 327]. The essential point is that when T is contractible, we may assume that connections ω_{t} are connections on a fixed bundle P → M. Indeed, if we denote the above bundle isomorphism by Φ : P ×T → \underline{P}, then we have isomorphisms Φ(·, t) : P → P_{t}, and we may replace the family {ω_{t} : t ∈ T } by the equivalent family {Φ(·, t)^{∗}ω_{t} : t ∈ T } on the fixed bundle P.

Remark 1.4.7. If T ↪ B^{∗} is contractible and M is spin, then ℙ^{ad} |_{M×T} lifts to an SU(2)-bundle ℙ|_{M×T}, since w_{2}(P)[M] will be zero in H^{2}(M, ℤ_{2}).

Example 1.4.8. Suppose T is the moduli space M_{k}^{∗}(𝕊^{4}, SU(2)), or the framed moduli space M̃_{k}^{∗}(𝕊^{4}, SU(2)), of self-dual connections on an SU(2)-bundle P → 𝕊^{4} with topology fixed by −c_{2}(P)[M] = k. When k = 1, we have M_{1}^{∗}(𝕊^{4}, SU(2)) ≃ 𝔹^{5}, and so T is contractible, with \underline{P} ≃ P × T. The family of connections parametrized by T are connections on a fixed SU(2)-bundle defined by the quaternionic Hopf fibration 1 → Sp(1) → 𝕊^{7} → ℍℙ^{1} → 1. However, for k > 1, we have the fundamental groups [Hur]:

π_{1} (M̃_{k}^{∗}(𝕊^{4}, SU(2))) = ℤ_{2} for all k;
π_{1} (M_{k}^{∗}(𝕊^{4}, SU(2))) = \{ \begin{matrix}0 & if k odd; \\ ℤ_{2} & if k even. \end{matrix}

Hence, at least for k even, we should not expect \underline{P} to be isomorphic to P × T. For k = 2, we have that π_{1} (M_{2}^{∗}(𝕊^{4}, SU(2))) = ℤ_{2} and moreover, that this moduli space is homotopy equivalent to the Grassmann manifold G_{2}(ℝ^{5}) [S-T, p. 342], [Au]. Hence, we should not expect that the family of connections parametrized by T = M_{2}^{∗}(𝕊^{4}, SU(2)) are carried by a bundle \underline{P} = P × T.

§1.5. Infinitesimal Deformations and Horizontal Projections

We now consider infinitesimal deformations of a family of irreducible connections ω(t) = ω_{t} on P (t) = P_{t} → M, where t varies in a parameter space T. The topology of a principal G-bundle on M, for simple, simply-connected Lie groups G, is fixed by specifying −c_{2}(P)[M] = k, so the G-bundles P (t) are all isomorphic to some fixed G-bundle P [T1, p. 168].

For any two isomorphisms f_{t} : P → P_{t}, h_{t} : P → P_{t}, we have h_{t}^{−1} ∘ f_{t} ∈ G(P), and consequently, the quotient spaces B^{∗}(P), B^{∗}(P_{t}) may be canonically identified [T2, p. 528], [T3, p. 344]. Suppose I ↪ T is a small interval around 0 ∈ ℝ. We may choose a smooth family of isomorphisms f(t) = f_{t} : P → P (t), t ∈ I, where P (0) = P, f(0) = 1_{P}, the identity automorphism of P, and pull back the connection ω(t) on P (t) → M to give a connection ω^{f}(t) = f(t)^{∗}ω(t) on the fixed G-bundle P → M. Then ω^{f}(t) is a family of connections in the fixed bundle P → M parametrized by the interval I, with ω^{f}(0) = ω = ω(0). We may then, as usual, compute the derivative ω̇^{f}(0) ∈ T_{ω}A^{∗}(P) = Ω^{1}(M, ad P).

Lemma 1.5.1. h_{ω}ω̇^{f}(0) is independent of the path of isomorphisms f(t), and so ω̇^{f}(0) defines an element in T_{[ω]}B^{∗}(P) ≃ H_{ω} independent of f.

Proof. If k(t) is another choice, then ω^{f}(t) is gauge equivalent to ω^{k}(t), using the gauge transformation k_{t}^{−1} ∘ f_{t} ∈ G(P), and so ω̇^{f}(0), ω̇^{k}(0) define the same element in the tangent space T_{[ω]}B^{∗}(P) ≃ H_{ω}. Hence, h_{ω}ω̇^{f}(0) is independent of f. □

We consider some of the consequences for local calculations. The isomorphism f(t) : P (0) → P (t) is represented locally by f_{t}(σ_{α}(0)) = σ_{α}(t)g_{α}(t), where we choose {O_{α}} to be a fixed open cover of M, the corresponding local sections are σ_{α}(t) ∈ Γ(O_{α}, P_{t}), and the mapping transformations g_{α}(t) : O_{α} → G relate the transition functions of P and P_{t} by:

T_{αβ}(t) = g_{α}(t)^{−1}T_{αβ}(0)g_{β}(t) on O_{α} ∩ O_{β}.

A calculation shows that

ω_{α}^{f}(t) = Ad (g_{α}(t)^{−1})ω_{α}(t) + g_{α}(t)^{−1}dg_{α}(t) ∈ Ω^{1}(O_{α}, 𝔤),

relating the local connection 1-forms of ω^{f}(t) ∈ A^{∗}(P) and ω(t) ∈ A^{∗}(P_{t}). Computing derivatives, we get:

ω̇_{α}^{f}(0) = Ad (g_{α}(0)^{−1}) ω̇_{α}(0) + d_{ω^{f} (0)} (g_{α}(0)^{−1}ġ_{α}(0)) ∈ Ω^{1}(O_{α}, 𝔤).

Since f(0) = 1_{P}, we have g_{α}(0) = 1, and so

ω̇_{α}^{f}(0) = ω̇_{α}(0) + d_{ω} (ġ_{α}(0)) ∈ Ω^{1}(O_{α}, 𝔤).

We cannot immediately take horizontal projections, since ω̇_{α}(0), ġ_{α}(0) do not transform as global sections of Ω^{1}(M, ad P) and Ω^{0}(M, ad P), respectively. Unlike the cases considered earlier — where f ∈ Aut (P) was a gauge transformation — we now have f_{t} ∈ Iso(P, P_{t}), and consequently the local maps

ġ_{α}(0) : O_{α} → 𝔤

do not necessarily transform correctly to give a global section ϕ in Ω^{0}(M, ad P). Indeed, using g_{β}(t) = T_{αβ}(0)^{−1}g_{α}(t)T_{αβ}(t), we may compute ġ_{β}(0):

ġ_{β}(0) = Ad (T_{αβ}(0)^{−1})ġ_{α}(0) + T_{αβ}(0)^{−1}Ṫ_{αβ}(0).

A similar calculation shows that the local 1-forms ω̇_{α}(0), do not necessarily transform correctly to give a global section of Ω^{1}(M, ad P).

In order to take care of this difficulty, we consider the calculation of horizontal projections in more detail. Suppose we have a local map ψ_{α} ∈ Ω^{q}(O′_{α}, 𝔤), where O′_{α} is a coordinate patch in M and P |O′_{α} ≃ O′_{α} × G. If O_{α} ⋐ O′_{α}, we obtain a smooth map by restricting ψ_{α} to the closed set \overline{O}_{α}. Lifting this restricted map to an element in Ω^{q}(\overline{O}_{α}, ad P) via the trivialization, the local section extends to a global section, denoted ψ^{ext,α} in Ω^{q}(M, ad P) [K-N1, p. 58], [St, p. 55].

Lemma 1.5.2. Let {O_{α}} be a refinement of the cover {O′_{α}} of M, with O_{α} ⋐ O′_{α} and P locally trivial over O′_{α}. For each patch O′_{α}, let ġ^{ext,α}(0) be a global extension of ġ_{α}(0) on \overline{O}_{α} to a section in Ω^{0}(M, ad P). Define corresponding global sections ω̇^{ext,α}(0) by setting

ω̇^{ext,α}(0) = ω̇^{f}(0) − d_{ω}ġ^{ext,α}(0) ∈ Ω^{1}(M, ad P).

Then:

(i) Locally, ω̇_{α}^{ext,α}(0) = ω̇_{α}(0) on O_{α}; (ii) Globally,

h_{ω}ω̇^{f}(0) = h_{ω}ω̇^{ext,α}(0) for all α.

Proof. By definition, ġ_{α}^{ext,α}(0) = ġ_{α}(0) on O_{α}, and so

\begin{aligned}ω̇_{α}^{ext,α}(0) & = ω̇_{α}^{f}(0) − d_{ω}ġ_{α}(0) \\ & = ω̇_{α}^{f}(0) on O_{α}, \end{aligned}

since ω̇^{α}(0) = ω̇_{α}^{f}(0) − ġ^{α}(0) on O_{α} and this gives (i). Moreover,

ω̇^{f}(0) = ω̇^{ext,α}(0) + d_{ω}ġ^{ext,α}(0) ∈ Ω^{1}(M, ad P),

and taking horizontal projections gives (ii). □ We obtain the following useful consequence:

Lemma 1.5.3. Let the coverings {O_{α}} and {O′_{α}} be as in the previous lemma. Then, we have the estimate:

∥h_{ω}ω̇^{f}(0)∥_{L^{2}(Ω^{1}(O_{α},ad P))} ≤ ∥ω̇(0)∥_{L^{2}(Ω^{1}(O_{α},ad P))}.

Proof. Observe that h_{ω}ω̇^{f}(0) = h_{ω}ω̇^{ext,α}(0) by the previous lemma, and so

∥h_{ω}ω̇^{f}(0)∥_{L^{2}(Ω^{1}(O_{α},ad P))} = ∥h_{ω}ω̇^{ext,α}(0)∥_{L^{2}(Ω^{1}(O_{α},ad P))}
≤ ∥ω̇^{ext,α}(0)∥_{L^{2}(Ω^{1}(O_{α},ad P))}
= ∥ω̇_{α}(0)∥_{L^{2}(Ω^{1}(O_{α},𝔤))},

and this gives the desired inequality. □

Remark 1.5.4. The essential point is that we can estimate h_{ω}ω̇^{f}(0) without any explicit knowledge of the 1-parameter family of isomorphisms f(t).

§1.6. Classifying Maps and Canonical Connections

We review some examples of classifying spaces for G bundles and canonical connections which we will later use in our application of the ADHM construction. General references for this section are [F-U], [Hu], [M-S], and [K-N].

If G is a Lie group, we let BG denote the corresponding classifying space and let EG denote the total space of the universal G-bundle 1 → G → EG → BG → 1. For G = SU(2) = Sp(1), we have BSU(2) = ℍℙ^{∞} and ESU(2) = 𝕊^{∞}, and if f : M → BSU(2) is a map, then we obtain a principal SU(2)-bundle on M by pull-back:

P ⟶ 𝕊^{∞}
↓  ↓
M ⟶^{f} ℍℙ^{∞}

In particular, all G-bundles P → M arise this way and two classifying maps are homotopic if and only if they induce isomorphic bundles [F-U], [Hu].

We review the construction of the canonical SU(2)-connection γ_{0} on the SU(2) bundle

𝕊^{4k+3} ⟶ ℍℙ^{k},

where 𝕊^{4k+3} = {q ∈ ℍ^{k+1} : |q| = 1} [K-N2, p. 6]. We consider ℍ^{k+1} as a right vector space over ℍ and define the right quaternionic projective space ℍℙ^{k} as {(q_{0}, …, q_{k}) ∈ ℍ^{k+1} \ {0} : (q_{0}, …, q_{k}) ∼ (q_{0}q, …, q_{k}q), q ∈ ℍ^{∗}}, where ℍ^{∗} = ℍ\{0} and k ≥ 1. The symplectic group Sp(k) may be defined by {Q ∈ GL(k, ℍ) : (Qp)^{†}(Qq) = p^{†}q\}, where p^{†}q = ∑_{i=0}^{k} \overline{p}_{i}q_{i} denotes the standard symplectic scalar product on ℍ^{k}, for any k ≥ 1. Then, ℍℙ^{k} and 𝕊^{4k+3} may be viewed as homogeneous spaces:

ℍℙ^{k} = Sp(k + 1)/Sp(k) × Sp(1),
𝕊^{4k+3} = Sp(k + 1)/Sp(k).

We have the following principal bundles over ℍℙ^{k}:

(i) An Sp(k) × Sp(1)-bundle Sp(k + 1) → ℍℙ^{k}, denoted by Q; (ii) An Sp(1)-bundle 𝕊^{4k+3} → ℍℙ^{k}, denoted by Q_{0}.

The bundle 𝕊^{4k+3} → ℍℙ^{k} is the quaternionic Hopf fibration, with Sp(1) acting on 𝕊^{4k+3} by right scalar multiplication.

Let Θ be the Maurer-Cartan form of Sp(k + 1) with values in 𝔰𝔭(k + 1) [K-N1, p. 41]. Let γ be the 𝔰𝔭(k)+𝔰𝔭(1) component of Θ with respect to the decomposition

𝔰𝔭(k + 1) = 𝔰𝔭(k) + 𝔰𝔭(1) + 𝔤(k, 1),

where 𝔤(k, 1) is the orthogonal complement of 𝔰𝔭(k)+𝔰𝔭(1) in 𝔰𝔭(k+1) with respect to the Cartan-Killing form of Sp(k+1). Then, γ defines an Sp(k)+Sp(1) connection on Sp(k + 1) → ℍℙ^{k} by [K-N1, p. 103], and γ is the canonical connection for this bundle.

Let π_{0} : Sp(k + 1) → Sp(k + 1)/Sp(k) = 𝕊^{4k+3} denote the natural projection. Then π_{0} defines a bundle map Q → Q_{0}, and by [K-N1, p. 79], there is a unique connection γ_{0} on Q_{0} → ℍℙ^{k} such that γ = (π_{0})^{∗}γ_{0}, and γ_{0} is the canonical Sp(1)- connection for this bundle. With respect to the coordinates q_{0}, …, q_{k} on ℍ^{k+1}, we have

γ_{0} = ∑_{i=0}^{k} \overline{q}_{i}dq_{i} = q^{†}dq ∈ Ω^{1}(𝕊^{4k+3}, 𝔰𝔭(1)),

where q^{†} denotes the quaternionic conjugate transpose of q. If E_{0} → ℍℙ^{k} is the universal quaternionic line bundle, then Q_{0} is the associated principal Sp(1)-bundle, and so E_{0} inherits a covariant derivative ∇^{γ_{0}}. This covariant derivative may be obtained by orthogonal projection of the standard flat connection on the trivial bundle ℍℙ^{k} × ℍ^{k+1} → ℍℙ^{k}.

Finally, if f : M → ℍℙ^{k} is a classifying map inducing an Sp(1)-bundle P → M by pull-back,

P ⟶^{f′} 𝕊^{4k+3}
↓  ↓
M ⟶^{f} ℍℙ^{k}

then P inherits a connection ω = (f′)^{∗}γ_{0}, by pulling back the canonical Sp(1)-connection on 𝕊^{4k+3} → ℍℙ^{k}. The ADHM method constructs the classifying maps f. Note that the total space P is

P = {([x, y], p) ∈ ℍP^{1} × 𝕊^{4k+3} : f(x, y) = π(p)},

and if π_{2} : ℍP^{1} × 𝕊^{4k+3} → 𝕊^{4k+3} denotes the projection onto the second factor, then the induced map f′ on total spaces is just f′ = π_{2}|_{P} : P → 𝕊^{4k+3}. Then

ω = (f′)^{∗}γ_{0} = (f′)^{†}df′ ∈ Ω^{1}(P, 𝔤)

is the corresponding SU(2)-connection 1-form on P.

CHAPTER II STABLE VECTOR BUNDLES OF RANK TWO

It will be useful — especially in view of our later discussion of the moduli space M_{2} of self-dual SU(2)-connections with k = 2 over 𝕊^{4} — to consider the correspondence between anti-self-dual (ASD) connections and holomorphic vector bundles. This correspondence was employed by Hartshorne to discuss certain families of ASD SU(2)-connections discovered by Jackiw, Nohl, and Rebbi using techniques of algebraic geometry [Har2], [J-N-R]. General references for this chapter are [A], [A-H-S], [A-W], [Har2, 3], [O-S-S], [Wa-We].

§2.1. Atiyah-Ward Correspondence

The Atiyah-Ward correspondence which gives a bijection between the moduli space of ASD SU(2)-connections with second Chern number k over 𝕊^{4} (endowed with its standard round metric g_{0}) and the moduli space of instanton bundles over ℂℙ^{3}, which are rank 2 holomorphic vector bundles over ℂℙ^{3} satisfying certain technical conditions. We summarise the main features of this correspondence. General references for this section are [A], [A-H-S], [A-W], [Har2], [Wa-We].

Let ℍ denote the space of quaternions, with basis 1, i, j, k over ℝ, and let ℍℙ^{1} = 𝕊^{4} denote the (left) quaternionic projective space. As left complex vector spaces, we identify ℂ^{2} with ℍ = ℂ ⊕ ℂj by (z_{0}, z_{1}) ↦ z_{0} + z_{1}j. More generally, we have the following isomorphisms of left complex vector spaces, ℂ^{2n} → ℍ^{n} = ℂ^{n} ⊕ ℂ^{n}j, given by

(z_{0}, z_{1}, …, z_{2n−2}, z_{2n−1}) ⟼ (z_{0} + z_{1}j, …, z_{2n−2} + z_{2n−1}j).

In particular, we have a fibre bundle,

π : ℂℙ^{3} ⟶ ℍℙ^{1}, [z_{0}, z_{1}, z_{2}, z_{3}] ⟼ [z_{0} + z_{1}j, z_{2} + z_{3}j]

with fibre ℂℙ^{1}.

Definition 2.1.1. Let X be a complex algebraic variety. A real structure on X is a conjugate-linear map σ_{X} : X → X with σ_{X}^{2} = 1_{X}. The fixed points (if any) of σ_{X} are the real points of X with respect to the real structure σ_{X}. If there is no ambiguity, we often denote σ_{X} simply by σ.

Left multiplication by j on ℍ^{2} induces a conjugate-linear map,

σ : ℂ^{4} ⟶ ℂ^{4}, (z_{0}, z_{1}, z_{2}, z_{3}) ⟼ (−\overline{z}_{1}, \overline{z}_{0}, −\overline{z}_{3}, \overline{z}_{2}),

and this induces a map σ : ℂℙ^{3} → ℂℙ^{3}. Note that σ^{2} = −1_{ℂ^{4}} on ℂ^{4}, and so ℂ^{4} and ℍ^{2} are identified as quaternionic vector spaces. We have σ^{2} = 1_{ℂℙ^{3}} on ℂℙ^{3} and hence σ defines a real structure on ℂℙ^{3}, preserving the fibration π : ℂℙ^{3} → ℍℙ^{1}. The map σ : ℂℙ^{3} → ℂℙ^{3} has no fixed points, but the fixed lines of σ are precisely the fibres π^{−1}(x) ≃ ℂℙ^{1}, x ∈ ℍℙ^{1}. These are the real lines of ℂℙ^{3} with respect to the real structure σ.

Definition 2.1.2. Let X be a complex algebraic variety with a real structure σ_{X} and let E be a vector bundle over X. An anti-linear map σ_{E} : E → E covering σ_{X} : X → X, or equivalently a bundle ismorphism σ_{E} : E → σ^{∗}\overline{E}, is a real structure if σ_{E}^{2} = 1 and a symplectic structure if σ_{E}^{2} = −1.

Two real or symplectic structures σ_{E} and σ′_{E} are equivalent if σ′_{E} = λσ_{E}, for some λ ∈ ℂ, |λ| = 1. If there is no ambiguity, we often denote σ_{E} simply by σ. We recall the fundamental

Theorem 2.1.3. (Atiyah-Ward) Let E → ℍℙ^{1} denote a smooth, complex, rank 2 vector bundle with Hermitian metric h, and let g_{0} denote the standard metric on ℍℙ^{1}. There is a natural bijection between

(i) Isomorphism classes of g_{0}-ASD SU(2)-connections on E → ℍℙ^{1}, with c_{2}(E) = k > 0; and

(ii) Isomorphism classes of rank 2 holomorphic vector bundles E → ℂℙ^{3}, with c_{2}(E) = k > 0, such that:

(a) E has a holomorphic bundle isomorphism b : E → E^{∗} covering the identity 1_{ℂℙ^{3}} and defining a symplectic form b( , ) on E;

(b) E has a conjugate-linear isomorphism σ_{E} : E → E, covering σ : ℂℙ^{3} → ℂℙ^{3}, such that σ_{E}^{2} = −1_{E} and σ_{E} is compatible with the symplectic form, in the sense that b(σξ, ση) = \overline{b(ξ, η)};

(c) E is holomorphically trivial when restricted to real lines L ⊂ ℂℙ^{3} and the Hermitian form h on O(E|L), defined by h(ξ, η) = b(ξ, ση) for ξ ∈ E_{z}, η ∈ E_{σz}, is positive definite.

Proof. See [A-W, p. 119], [A, p. 49], [A-H-S, p. 441], or [Wa-We, p. 390]. □

The rank 2 holomorphic vector bundles E on ℂℙ^{3} arising from ASD SU(2)- connections over 𝕊^{4} as above are known as instanton bundles. We describe briefly how one passes from ASD SU(2)-connections over 𝕊^{4} to the associated instanton bundle E on ℂℙ^{3} via the Atiyah-Ward correspondence. If ω is an ASD SU(2)- connection on E, let E = π^{∗}E with pull-back connection w̃ = π^{∗}w. Let Ω^{p}(ℂℙ^{3}, E) be the space of smooth p-forms on ℂℙ^{3} with values in E and let Ω^{(p,q)}(E) be the space of smooth (p, q)-forms on ℂℙ^{3} with values in E. Then we have the associated covariant derivative

dω̃ : Ω^{0}(ℂℙ^{3}, E) ⟶ Ω^{1}(ℂℙ^{3}, E),

and using the splitting

Ω^{1}(ℂℙ^{3}, E) = Ω^{(1,0)}(ℂℙ^{3}, E) ⊕ Ω^{(0,1)}(ℂℙ^{3}, E),

we let dω̃ = ∂ω̃ + \overline{∂}ω̃. Then we have

\overline{∂}ω̃ : Ω^{0}(ℂℙ^{3}, E) ⟶ Ω^{(0,1)}(ℂℙ^{3}, E),

and the connection ω is ASD if and only if the curvature Fω̃ of the pullback connection is a (1,1)-form, so that Fω̃ ∈ Ω^{(1,1)}(ℂℙ^{3}, End(E)) and hence, \overline{∂}ω̃ defines a holomorphic structure on E [A, p. 48], [A-H-S, p. 441].

The following slight reformulation of the Atiyah-Ward correspondence shows that the identfication of the moduli space of ASD SU(2)-connections over 𝕊^{4} may be viewed as a problem in algebraic geometry:

Theorem 2.1.4. (Atiyah-Ward) Let E → ℍℙ^{1} denote a smooth, complex, rank 2, Hermitian vector bundle with Hermitian metric h, and let g_{0} denote the standard metric on 𝕊^{4}. There is a natural bijection between

(i) Isomorphism classes of g_{0}-ASD SU(2)-connections on E → ℍℙ^{1} with c_{2}(E) = k > 0; and

(ii) Isomorphism classes of holomorphic rank 2 vector bundles E → ℂℙ^{3} together with a map σ_{E} : E → E (up to multiplication by λ ∈ ℂ^{∗}, |λ| = 1), such that:

(a) c_{1}(E) = 0, c_{2}(E) = k > 0;

(b) E is stable;

(c) For each x ∈ ℍℙ^{1}, E|π^{−1}(x) is holomorphically trivial — so E has no real jumping lines;

(d) The map σ_{E} : E → E is conjugate-linear isomorphism, covering σ : ℂℙ^{3} → ℂℙ^{3}, such that σ_{E}^{2} = −1, and so E has a symplectic structure.

Proof. [Har2, p. 3], [A, p. 51], [A-W, p. 119]. □

Definition 2.1.5. We have the following moduli spaces:

(i) The moduli space N (0, k) of stable holomorphic rank 2 vector bundles over ℂℙ^{3} with c_{1}(E) = 0 and c_{2}(E) = k;

(ii) The moduli space N_{𝕀}(0, k) of instanton bundles over ℂℙ^{3} satisfying the condi-

tions of the Atiyah-Ward correspondence; (iii) The moduli space M_{k} of g_{0}-ASD SU(2)-connections on E → 𝕊^{4} with c_{2}(E) =

k.

The problem of identifying the moduli space N_{𝕀}(0, k) of instanton bundles may then be solved in two steps. One first identifies the moduli space N (0, k) of holomorphic rank 2 stable vector bundles on ℂℙ^{3} with c_{1}(E) = 0 and c_{2}(E) = k. One then identifies the points in N (0, k) corresponding to bundles with symplectic structure and having no real jumping lines — this gives N_{𝕀}(0, k) ⊂ N (0, k).

§2.2. Stable Vector Bundles on Complex Projective Space

We describe Hartshorne’s approach to the problem of identifying the moduli space N (0, k) of holomorphic, stable, rank 2 vector bundles over ℂℙ^{3} with c_{1} = 0 and c_{2} = k, employing the correspondence between rank 2 bundles over ℂℙ^{3} and certain algebraic curves in ℂℙ^{3} [O-S-S, p. 90], [Har2, 3].

Definition 2.2.1. A holomorphic, rank 2 vector bundle E → ℂℙ^{n} is stable (respectively, semistable) if for every line bundle L ⊂ E, we have c_{1}(L) < c_{1}(E)/2 (respectively, ≤).

Let O_{ℂℙ^{3}}(1) denote the hyperplane bundle on ℂℙ^{3} and let E(m) denote the twisted bundle E ⊗ O_{ℂℙ^{3}}(1)^{⊗m}. Then a bundle E is stable if and only if E(m) is stable, for any m ∈ ℤ [Har3, p. 241]. Moreover, if c_{1}(E) = 0 or −1, then E is stable if and only if H^{0}(ℂℙ^{3}, E) = 0 [Har3, p. 241].

By a theorem of Serre, H^{0}(ℂℙ^{3}, E(m)) ≠ 0 for m sufficiently large, and so one may choose a non-zero global section s ∈ E(m) [G-H, p. 700], [O-S-S, p. 9]. Let Y = (s)_{0} ⊂ ℂℙ^{3} be the zero-set of that section: for sufficiently general s, Y will be an algebraic curve in ℂℙ^{3} [Har3, p. 234]. If c_{1}(E) = 0 and c_{2}(E) = k, then [Har2, p. 4]:

degree(Y) = k + m^{2} and genus(Y) = max{(k + m^{2})(m − 2) + 1, 0}.

(See also [Har3, p. 236] and [O-S-S, pp. 90-110].) The canonical bundle K_{Y} on Y is isomorphic to O_{Y} (2m − 4) and is the restriction of a line bundle on ℂℙ^{3} [Har2, p. 4]. The precise correspondence between rank 2 bundles over ℂℙ^{3} and curves Y in ℂℙ^{3} is given by the following

Theorem 2.2.2. [Har2, p. 4], [Har3, p. 232] A curve Y in ℂℙ^{3} is the scheme of zeros (s)_{0} of a section s of a holomorphic rank 2 vector bundle E → ℂℙ^{3} if and only if Y is a locally complete intersection and K_{Y} is isomorphic to the restriction to Y of some line bundle on ℂℙ^{3}. More precisely, for any fixed line bundle L → ℂℙ^{3} there is a bijection between (i) and (ii):

(i) The set of triples ⟨E, s, φ⟩ modulo the equivalence relation ∼, where (a) E is a rank 2 vector bundle on ℂℙ^{3};

(b) s ∈ H^{0}(ℂℙ^{3}, E) is a global section whose scheme of zeros (s)_{0} has codimension 2;

(c) φ : det E → L is an isomorphism; and ⟨E, s, φ⟩ ∼ ⟨E′, s′, φ′⟩ if there is an isomorphism ψ : E → E′ and λ ∈ ℂ^{∗} such that

s′ = λψ(s)  and  φ′ = λ^{2}φ ∘ (det ψ)^{−1},

where det E = ∧^{2}E and det ψ = ∧^{2}ψ. (ii) The set of pairs ⟨Y, ξ⟩, where

(a) Y is a locally complete intersection curve in ℂℙ^{3}; and

(b) ξ : L ⊗ K_{ℂℙ^{3}} ⊗ O_{Y} → K_{Y} is an isomorphism.

Remark 2.2.3. By a curve, we mean a 1-dimensional closed subscheme of ℂℙ^{3} which may be reducible, disconnected, and may have nilpotent elements.

In particular, one has a criterion for distinct sections of a bundle to have to the same scheme of zeros:

Proposition 2.2.4. [Har3, p. 234] Let E be a rank 2 bundle on ℂℙ^{3} and assume that for every nonzero s ∈ H^{0}(ℂℙ^{3}, E), the scheme of zeros (s)_{0} has codimension 2. (This will be the case if H^{0}(ℂℙ^{3}, E(−1)) = 0.) Then two non-zero sections s, s′ have the same scheme of zeros if and only if s′ = λs for some λ ∈ ℂ^{∗}.

We recall the criterion for the scheme of zeros of a section to be non-singular:

Proposition 2.2.5. [Har3, p. 234] Let E be a rank 2 bundle on ℂℙ^{3}. If E is generated by global sections, then for all sufficiently general s ∈ H^{0}(ℂℙ^{3}, E), the scheme of zeros (s)_{0} will be non-singular.

By a theorem of Maruyama, the set of stable, holomorphic, rank 2 vector bundles over ℂℙ^{3} with given Chern classes c_{1} and c_{2} has a coarse moduli scheme N (c_{1}, c_{2}) which is separated and of finite type [Har3, p. 245], [Mu-Fo].

Proposition 2.2.6. [Har3, p. 245] Let E be a stable bundle over a non-singular projective variety X. Then H^{1}(X, End E) is naturally isomorphic to the Zariski tangent space of the moduli scheme N at the point corresponding to E. If H^{2}(X, End E) = 0, then N is nonsingular at that point and its dimension is equal to dim H^{1}(X, End E).

Proposition 2.2.7. [Har3, p. 245] Let E be a stable bundle over ℂℙ^{3}, with c_{1}E) = 0 and c_{2}(E) = k. Then

dim H^{1}(X, End E) − dim H^{2}(X, End E) = 8k − 3.

If c_{1}(E) = 0, then det E ≃ O_{ℂℙ^{3}} and det E(1) ≃ O_{ℂℙ^{3}}(2). One obtains a bijection between

(i) The set of pairs ⟨E, s⟩, where E is a rank 2 vector bundle on ℂℙ^{3} with c_{1}(E) = 0, and s ∈ H^{0}(ℂℙ^{3}, E(m)) is a non-zero section for some m; and

(ii) The set of curves Y in ℂℙ^{3}, together with a given isomorphism K_{Y} ≃ O_{Y} (2m− 4). Furthermore, the bundle is stable if and only if the curve Y is not contained in any surface of degree ≤ m [Har2, p. 4], Har3, p. 241]. The curve Y obtained in this correspondence depends on the section s ∈

H^{0}(ℂℙ^{3}, E(m)), as well as the bundle E. To obtain the moduli space for the bundles E, one needs to eliminate the ambiguity introduced by s.

We next describe Hartshorne’s example of the family of holomorphic rank 2 bundles over ℂℙ^{3}. When these are endowed with a symplectic structure, this example corresponds to the Jackiw-Nohl-Rebbi family of ASD SU(2)-connections over 𝕊^{4}.

Example 2.2.8. [Har2, p. 5], [Har3, p. 242 & p. 247] Let Y = Y_{0} ∪ … ∪ Y_{k} be a disjoint union of k + 1 lines Y_{i} ≃ ℂℙ^{1} in ℂℙ^{3}, with k ≥ 1. For ℂℙ^{n}, the canonical bundle K_{ℂℙ^{n}} = O_{ℂℙ^{n}}(−n − 1) and so the canonical bundle of a line is K_{ℂℙ^{1}} = O_{ℂℙ^{1}}(−2) [G-H, p. 146]. Thus, K_{Y} ≃ O_{Y} (−2), and this isomorphism is determined by k + 1 non-zero complex numbers ζ_{0}, …, ζ_{k} ∈ ℂ^{∗}. Taking m = 1, we have H^{0}(ℂℙ^{3}, E(1)) ≠ 0 [Har3, p. 263] and we obtain a bundle E for each choice of Y and each choice of isomorphism K_{Y} ≃ O(−2). Since degree(Y) = k + 1, we have c_{1}(E) = 0 and c_{2}(E) = k. For k ≥ 2, Y is not contained in a plane, therefore E(1) and hence E will be stable [Har3, p. 241]. By [Har3, p. 246], H^{2}(ℂℙ^{3}, End E) = 0 and the moduli space N (0, k) is non-singular of dimension 8k − 3.

One can then compute the dimension of this family of bundles: the Grassman variety G(1, 3) of lines ℂℙ^{1} in ℂℙ^{3} has dimension dim_{ℂ} G(1, 3) = 4: hence the choice of Y ⊂ ℂℙ^{3} requires 4(k+1) complex parameters. The choice of isomorphism K_{Y} ≃ O_{Y} (−2) depends on the k + 1 complex parameters ζ_{0}, …, ζ_{k}. Hence, the pair ⟨E, s⟩ depends on 5k + 5 parameters. One then needs to subtract dim_{ℂ} H^{0}(ℂℙ^{3}, E(1)), the number of parameters used in the choice of s ∈ H^{0}(ℂℙ^{3}, E(1)):

dim_{ℂ} H^{0}(ℂℙ^{3}, E(1)) = \{ \begin{matrix}5 & if k = 1; \\ 2 & if k = 2; \\ 1 & if k ≥ 3. \end{matrix}

One then finds that the bundles constructed by this method form an irreducible algebraic family parametrized by a non-singular variety T of complex dimension

dim_{ℂ} T = \{ \begin{matrix}5 & if k = 1; \\ 13 & if k = 2; \\ 5k + 4 & if k ≥ 3. \end{matrix}

Further details may be found in [Har3] — in particular, see [Har3, p. 238].

§2.3. Real Structures and Jumping Lines

The moduli space of instanton bundles N_{𝕀}(0, k) is obtained as an open subset of the set of real points of the moduli space N (0, k) of holomorphic rank 2 stable bundles over ℂℙ^{3} with c_{1} = 0 and c_{2} = k. In order to make this more precise, we first need to specify the required real structures and recall the concept of a jumping line [Har2], [O-S-S].

Proposition 2.3.1. [Har2, p. 9] If n is even, then ℂℙ^{n} has a unique real structure, the standard one given by complex conjugation of the coordinates. If n is odd, then ℂℙ^{n} has two possible real structures: the standard one, and another, with no real points. In the latter case, O_{ℂℙ^{3}}(1) has a symplectic structure and one can choose homogeneous coordinates z_{0}, z_{1}, …, z_{n} such that σ is given by σ(z_{0}, z_{1}, …, z_{n−1}, z_{n}) = (−\overline{z}_{1}, \overline{z}_{0}, …, −\overline{z}_{n}, \overline{z}_{n−1}).

For example, the non-standard real structure σ on ℂℙ^{3} canonically induces a map σ_{m} : O_{ℂℙ^{3}}(m) → σ^{∗}\overline{O_{ℂℙ^{3}}(m)}, with σ_{m}^{2} = (−1)^{m}, so that O_{ℂℙ^{3}}(m) has a real structure for m even and a symplectic structure for m odd. Also, if a bundle E → ℂℙ^{3} has a real structure, then E ⊗ O_{ℂℙ^{3}}(1) has a symplectic structure and vice versa [Har2, p. 9], [A-W], [S-T, p. 341].

We consider ℂℙ^{3} with its (non-standard) real structure σ. Via the Plücker embedding, G(1, 3) may be viewed as a quadric hypersurface in ℂℙ^{5}. The real structure σ on ℂℙ^{3} induces a real structure on G(1, 3) and the (standard) real structure on the ℂℙ^{5} in which it is embedded. In particular, the set of real points of G(1, 3) is ℍℙ^{1}, corresponding precisely with the real lines of ℂℙ^{3}.

For any holomorphic rank 2 vector bundle E → ℂℙ^{3}, consider its restriction E|L to a line L ⊂ ℂℙ^{3}: then E|L ≃ O_{L}(a) ⊕ O_{L}(b) for some a, b ∈ ℤ, and if c_{1}(E) = 0, then a + b = 0. Furthermore, if E is stable, then a = b = 0 for lines L corresponding to an open dense subset of the Grassman variety G(1, 3) and E|L for such an L will be holomorphically trivial. A line for which a, b ≠ 0 is called a jumping line of E. The set of jumping lines of a given bundle E corresponds to a divisor Z ⊂ G(1, 3) of degree c_{2}(E) [Har2, p. 10].

The problem of identifying the moduli space N_{𝕀}(0, k) of instanton bundles over ℂℙ^{3} may then be solved in the following steps [Har2, p. 10]:

(i) Identify the moduli space N (0, k) of stable, rank 2 vector bundles on ℂℙ^{3} with c_{1}(E) = 0 and c_{2}(E) = 2;

(ii) Identify the real structure on N (0, k) induced by σ, and find its real points N_{ℝ}(0, k); (iii) Among the real points N_{ℝ}(0, k), identify the points N_{ℝ}^{+}(0, k) corresponding

to bundles E with a real structure, and the points N_{ℝ}^{−}(0, k) corresponding to bundles with a symplectic structure.

(iv) Among the points N_{ℝ}^{−}(0, k), identify those corresponding to bundles whose divisor Z ⊂ G(1, 3) of jumping lines has no real points. This will be the moduli space N_{𝕀}(0, k) of instanton bundles and will be an open subset of N_{ℝ}(0, k). In the following sections, we will review Hartshorne’s construction of the moduli

space N_{𝕀}(0, 2).

§2.4. Moduli Space of Rank Two Stable Bundles

We review Hartshorne’s construction of the moduli space N (0, 2) of stable bundles on ℂℙ^{3} with c_{1} = 0 and c_{2} = 2 [Har3], [S-T].

If c_{1}(E) = 0 and c_{2}(E) = 2, then H^{0}(ℂℙ^{3}, E(1)) ≠ 0, while H^{0}(ℂℙ^{3}, E) = 0 since E is stable [Har3, p. 263]. Let 0 ≠ s ∈ H^{0}(ℂℙ^{3}, E(1)) and let Y = (s)_{0} denote the zero set of s. Then Y will be a curve of degree 3 such that K_{Y} ≃ O_{Y} (−2). One has the

Proposition 2.4.1. [Har3, p. 267] Let Y be a curve of degree 3 in ℂℙ^{3} such that K_{Y} ≃ O_{Y} (−2). Then Y consists of either 3 skew lines, a single line plus a double line, or a single line of multiplicity 3.

Lemma 2.4.2. [Har3, p. 268] If Y denotes any curve of degree 3 in ℂℙ^{3} and K_{Y} ≃ O_{Y} (−2), then Y is contained in a unique nonsingular quadric surface Q ⊂ ℂℙ^{3}.

Recall that any non-singular quadric surface Q ⊂ ℂℙ^{3} is isomorphic to ℂℙ^{1} × ℂℙ^{1} [G-H, p. 478]. The curve Y is a divisor of type (3,0) on Q: the divisor class group of Q is ℤ ⊕ ℤ, generated by a line in each of the two rulings and the type refers to the class in ℤ ⊕ ℤ [Har3, p. 268].

Lemma 2.4.3. [Har3, p. 268] Let E be a holomorphic rank 2 vector bundle on ℂℙ^{3} with c_{1}(E) = 0 and c_{2}(E) = 2, and let s ∈ H^{0}(ℂℙ^{3}, E(1)) be a section with zero set (s)_{0} = Y. Then the nonsingular quadric surface Q containing Y depends only on E, and not on the choice of s. There is a linear map

H^{0}(ℂℙ^{3}, E(1)) ⟶ H^{0}(O_{Q}(3, 0)), s ⟼ s|Q,

so that as s varies in H^{0}(ℂℙ^{3}, E(1)), Y cuts out a linear system on Q of type (3,0) and dimension 1.

Lemma 2.4.4. [Har3, p. 269] With E and Q as in the previous lemmas, the linear system of curves Y on Q obtained by varying s ∈ H^{0}(ℂℙ^{3}, E(1)) is a linear system without basepoints.

One then has the important Corollary 2.4.5. [Har3, p. 270] For c_{1}(E) = 0 and c_{2}(E) = 2, there exists a section s ∈ H^{0}(ℂℙ^{3}, E(1)) whose zero set (s)_{0} is three skew lines.

Remark 2.4.6. Thus, the rank 2 vector bundle construction described earlier gives all stable bundles with c_{1}(E) = 0 and c_{2}(E) = 2.

The linear system of curves Y of type (3,0) on Q ≃ ℂℙ^{1} × ℂℙ^{1} is induced by a linear system g_{3}^{1}, without basepoints, of degree 3 and dimension 1 on one of the factors ℂℙ^{1}. The general member of the g_{3}^{1} will consist of three distinct points and the corresponding curve Y on Q will be three skew lines [Har3, p. 270].

In summary, a stable bundle E over ℂℙ^{3} with c_{1}(E) = 0 and c_{2}(E) = 2 corresponds to a curve Y of degree 3 with K_{Y} ≃ O_{Y} (−2), where Y consists of 3 skew lines in ℂℙ^{3}. A set of 3 skew lines in ℂℙ^{3} determines a unique nonsingular quadric surface Q in ℂℙ^{3}, where the quadric may be constructed as the union of all other lines which meet each of the given lines [G-H, p. 478]. The points of the ℂℙ^{1} factors correspond to the lines on the quadric surface Q. As the section s ∈ H^{0}(ℂℙ^{3}, E(1)) varies, the curves Y move in a linear system on the same quadric Q. Each divisor Y consists of three lines in one of the two families of lines on Q. This selects one of the two factors ℂℙ^{1} of Q and the linear system of curves Y then corresponds to a linear system g_{3}^{1} on this ℂℙ^{1}, without basepoints, of degree 3 and dimension 1. Collecting all these observations gives the following

Theorem 2.4.7. [Har2, p. 13], [Har3, p. 270] A stable holomorphic bundle E on ℂℙ^{3} with c_{1}(E) = 0, c_{2}(E) = 2 determines

(i) A nonsingular quadric surface Q ⊂ ℂℙ^{3}; (ii) A choice of one of the two factors in the isomorphism Q ≃ ℂℙ^{1} × ℂℙ^{1}; (iii) A linear system g_{3}^{1} of degree 3 and dimension 1 on the selected ℂℙ^{1}, without basepoints. Conversely, any such data arise from a unique such bundle E.

In particular, the moduli space N (0, 2) has a description as a fibre space [Har2, p. 13], [Har3, p. 271], [S-T, p. 336]. The quadric surfaces in ℂℙ^{3} are parametrized by ℂℙ^{9}: we let Δ ⊂ ℂℙ^{9} be the subset corresponding to the singular or degenerate quadric surfaces. Then N (0, 2) is fibred over ℂℙ^{9} − Δ:

\begin{matrix}N (0, 2) \\ ↓ & U⊔U \\ ℂℙ^{9} − Δ \end{matrix}

The fibre is the disjoint union of two copies of the variety U wich parametrizes the set of possible g_{3}^{1} without basepoints on ℂℙ^{1}. A g_{3}^{1} is determined by a 2-dimensional subspace of the 4-dimensional vector space H^{0}(ℂℙ^{1}, O_{ℂℙ^{1}}(3)). Thus, the set of all possible g_{3}^{1} is parametrized by G(1, 3) and those without basepoints form an open subset U.

Corollary 2.4.8. [Har3, p. 271] The moduli space N (0, 2) is an irreducible nonsingular complex variety of dimension 13.

A stable, holomorphic, rank 2, vector bundle E on ℂℙ^{3} with c_{1}(E) = 0 and c_{2}(E) = 2, determines a unique point in N (0, 2). The effect of varying s ∈ ℙ(H^{0}(ℂℙ^{3}, E(1))) may be described as follows. Recall that E determines a quadric Q in ℂℙ^{3} and a choice of factor ℂℙ^{1} of Q. The points of ℂℙ^{1} correspond to lines in Q, which in turn correspond to points of the Grassman variety G(1, 3) of lines in ℂℙ^{3}. As p ∈ ℂℙ^{1} varies, its image in G(1, 3) ⊂ ℂℙ^{5} describes a conic γ in G(1, 3). The g_{3}^{1} on Q induces a g_{3}^{1} on γ. So as s varies in ℙ(H^{0}(ℂℙ^{3}, E(1))), the corresponding divisor D_{s} = Y_{1} + Y_{2} + Y_{3} varies in the g_{3}^{1} on the conic γ ⊂ G(1, 3) [Har2, p. 14], [Har3, p. 276].

§2.5. Moduli Space of Rank Two Instanton Bundles

We review Hartshorne’s description of the open set N_{𝕀}(0, 2) ⊂ N (0, 2) parametrizing isomorphism classes of instanton bundles [Har2], [S-T].

Considering the above fibration N (0, 2) → ℂℙ^{9} − Δ, the induced real structure on ℂℙ^{9} is the standard one and its real points are ℝℙ^{9}. The real points of Δ are given by Δ_{ℝ} which is isomorphic to the quotient space ℂℙ^{3}/σ, a compact real 6-manifold. Next one considers the fibre U ⊔ U. A point in the base ℝℙ^{9} − Δ_{ℝ} corresponds to a nonsingular quadric Q with a real structure σ. It will be convenient to label the ℂℙ^{1} factors in the quadric ℂℙ^{1} × ℂℙ^{1} as ℂℙ_{α}^{1} × ℂℙ_{β}^{1}. Then, Q ≃ ℂℙ_{α}^{1} × ℂℙ_{β}^{1} and σ leaves each factor fixed so that each ℂℙ^{1} has a real structure with one factor — say ℂℙ_{α}^{1} — having the standard real structure and the other — say ℂℙ_{β}^{1} — having the non-standard structure. The G(1, 3) of which U is an open subset has the standard structure in one case (as desired) and the non-standard one in the other case. Bundles E with a symplectic structure (as desired) correspond to the choice of factor ℂℙ_{α}^{1} with the standard real structure; those E with a real structure correspond to the other factor, ℂℙ_{β}^{1}. The required U in the fibre U ⊔ U is then the one which is an open subset of G(1, 3) with its standard real structure.

A triple ⟨E, s, φ⟩ consisting of a rank 2 bundle E with c_{1}(E) = 0, c_{2}(E) = 0, a suitably chosen section s ∈ H^{0}(ℂℙ^{3}, E(1)), and an isomorphism φ : det E(1) ≃ O_{ℂℙ^{3}}(2), corresponds to three skew lines Y_{i} ⊂ ℂℙ^{3} and three complex weights ζ_{i} ∈ ℂ^{∗}. If E is endowed with a real structure, then the lines Y_{i} are real lines (corresponding to points in 𝕊^{4}) and the weights ζ_{i} are real. Finally, the requirement that the divisor Z ⊂ G(1, 3) of jumping lines has no real points is equivalent to the requirement that the real weights ζ_{0}, ζ_{1}, ζ_{2} are positive [Har2, p. 13-15], [Wa-We, p. 412]. The moduli space N_{𝕀}(0, 2) of instanton bundles is the total space of a fibre bundle over ℝℙ^{9} − Δ_{ℝ}, with fibre an open connected real 4-manifold [Har2, p. 13], [S-T, p. 340].

Theorem 2.5.1. [Har2, p. 12] The moduli space N_{𝕀}(0, 2) of instanton bundles is a connected, but not simply connected, real 13-dimensional manifold. Every instanton bundle E → ℂℙ^{3} may obtained by the construction of Jackiw-Nohl-Rebbi corresponding to 3 points P_{0}, P_{1}, P_{2} in 𝕊^{4} and 3 positive numbers ζ_{0}, ζ_{1}, ζ_{2}.

Remark 2.5.2. The analogous theorem holds for the moduli space N_{𝕀}(0, 1) ≃ ℝ^{5}, the instanton bundles then corresponding to 2 points P_{0}, P_{1} in 𝕊^{4} and 2 positive numbers ζ_{0}, ζ_{1} > 0.

As observed by Hartshorne, it is difficult to use this method to explicitly construct either the holomorphic bundles E → ℂℙ^{3} or the corresponding complex bundles E → 𝕊^{4} and their associated ASD SU(2)-connections [Har2, p. 4]. The construction of the corresponding ASD SU(2)-connections over 𝕊^{4} is outlined in [A-W, p. 122] and described in detail in [Wa-We, p. 398 & 412]. We will instead use the ADHM method to construct bundles E → 𝕊^{4} and all ASD SU(2)-connections with c_{2}(E) = 2. We describe this relationship between the two methods in the next chapter.

§2.6. Real Structures on Complex Moduli Spaces

We discuss the parametrization of N (0, 2) in terms of skew lines Y_{i} in ℂℙ^{3} and complex weights ζ_{i}, and the corresponding parametrization of N_{𝕀}(0, 2) in terms of real skew lines Y_{i} in ℂℙ^{3} and positive weights ζ_{i}; we provide proofs for some assertions whose proofs were omitted earlier. General references for this section are [A-W], [Bu], [Har2, 3], [Wa-We].

The moduli space N (0, k) parametrizes the set of isomorphism classes [E] of rank 2, holomorphic, stable vector bundles E → ℂℙ^{3}, with c_{1}(E) = 0 and c_{2}(E) = k > 0. Recall [A-W, p. 120] that the (non-standard) real structure σ on ℂℙ^{3} induces a real structure σ_{N} on the moduli space N (0, k):

σ_{N} : N (0, k) ⟶ N (0, k), [E] ⟼ [σ^{∗}\overline{E}].

Lemma 2.6.1. Let N_{ℝ}(0, k) be the real points of N (0, k) with respect to the real structure σ_{N}. Then N_{ℝ}(0, k) = {[E] : [E] ∈ N (0, k) and σ_{E} is a real or symplectic structure on E, with σ_{E} ∼ σ′_{E} if and only if λ ∈ ℂ, |λ| = 1}.

Proof. Suppose [E] ∈ N (0, k) and that σ_{E} is a real or symplectic structure. Then σ_{E} : E → σ^{∗}\overline{E} is a bundle isomorphism. Hence

σ_{N} [E] = [σ^{∗}\overline{E}] = [E],

and so [E] ∈ N_{ℝ}(0, k).

Conversely, suppose [E] ∈ N_{ℝ}(0, k). Then

σ_{N} [E] = [σ^{∗}\overline{E}] = [E],

and so there exists a bundle isomorphism σ_{E} : E → σ^{∗}\overline{E}. Hence, we obtain a bundle automorphism σ_{E}^{2} : E → E. The holomorphic vector bundle E → ℂℙ^{3} is stable and therefore simple [O-S-S, p. 172]. Since E is simple, we have Aut (E) = ℂ^{∗} [O-S-S, p. 74]. Hence, σ_{E}^{2} = μ · 1_{E} for some μ ∈ ℂ^{∗}. We claim that μ ∈ ℝ^{∗}. Suppose U ⊂ ℂℙ^{3} is open, E|U holomorphically trivial, and {z, σz} ⊂ U for some z. (This will certainly be true for instanton bundles E, since we assume that E|L is trivial for all real lines L ⊂ ℂℙ^{3}.) Let {ϕ_{1}, ϕ_{2}} ∈ O(E)(U) be a local frame. Then,

\begin{aligned}σ_{E} : E|_{U} ⟶ E|_{σ(U)}, & ϕ_{i}(z) ⟼ ∑_{j=1}^{2} λ_{ji}(σz)ϕ_{j}(σz), \\ & ϕ_{i}(σz) ⟼ ∑_{j=1}^{2} λ_{ji}(z)ϕ_{j}(z), \end{aligned}

and consequently, we have

\begin{aligned}σ_{E}^{2} : E ⟶ E, & ϕ_{i}(z) ⟼ ∑_{j,k=1}^{2} \overline{λ}_{ji}(σz)λ_{kj}(z)ϕ_{k}(z), \\ & ϕ_{i}(σz) ⟼ ∑_{j,k=1}^{2} \overline{λ}_{ji}(z)λ_{kj}(σz)ϕ_{k}(σz). \end{aligned}

But σ_{E}^{2} = μ · 1_{E} and so

\begin{aligned}μ & = ∑_{j=1}^{2} \overline{λ}_{j1}(σz)λ_{1j}(z) = ∑_{j=1}^{2} \overline{λ}_{j1}(z)λ_{1j}(σz) \\ & = ∑_{j=1}^{2} \overline{λ}_{j2}(σz)λ_{2j}(z) = ∑_{j=1}^{2} \overline{λ}_{j2}(z)λ_{2j}(σz) \end{aligned}

Then,

4μ = ∑_{j,k=1}^{2} \overline{λ}_{ji}(σz)λ_{kj}(z) + ∑_{j,k=1}^{2} \overline{λ}_{ji}(z)λ_{kj}(σz),

and so μ ∈ ℝ^{∗}. By replacing σ_{E} with |μ|^{−1/2}σ_{E}, we may assume σ_{E}^{2} = ±1_{E}, corresponding to a real or a symplectic structure, respectively. □

Remark 2.6.2. (i) The condition σ_{E}^{2} = 1 or σ_{E}^{2} = −1 is constant on connected components of N_{ℝ}(0, k). Indeed, N_{ℝ}(0, k) = N_{ℝ}^{+}(0, k) ⊔ N_{ℝ}^{−}(0, k), where N_{ℝ}^{−}(0, k) represent the required bundles with a symplectic structure and N_{ℝ}^{+}(0, k) represent bundles with a real structure.

(ii) The condition that E|L be trivial for any real line L ⊂ ℂℙ^{3} is an open condition. Hence, the moduli space of instanton bundles N_{𝕀}(0, k) is an open subset of N_{ℝ}^{−}(0, k).

Lemma 2.6.3. Let Y ⊂ ℂℙ^{3} consist of k + 1 skew lines Y_{0}, …, Y_{k} and let K_{Y} denote the canonical bundle of Y. Then K_{Y} ≃ O_{Y} (−2).

Proof. From [Har1, p. 182] we have

K_{Y} ≃ K_{ℂℙ^{3}} ⊗ det N_{Y/ℂℙ^{3}},

where K_{ℂℙ^{3}} ≃ O_{ℂℙ^{3}}(−4), N_{Y/ℂℙ^{3}} = (I_{Y} /I_{Y}^{2} )^{∗} is the normal bundle of Y ⊂ ℂℙ^{3}, and I_{Y} is the ideal sheaf of Y ⊂ ℂℙ^{3}. By assumption, Y = ⋃_{i=0}^{k} Y_{i}, with each line Y_{i} ≃ ℂℙ^{1} an intersection of hyperplanes Y_{i} = H_{i} ∩ Ĥ_{i}, where H_{i} = {z ∈ ℂℙ^{3} : f_{i}(z) = 0}, Ĥ_{i} = {z ∈ ℂℙ^{3} : h_{i}(z) = 0}, for i = 0, …, k. Let {U_{0}, …, U_{k}} be an open cover of ℂℙ^{3} with Y_{i} ⊂ U_{i} and E|U_{i} holomorphically trivial. Then

I_{Y} |U_{i} = (f_{i}, h_{i})O_{ℂℙ^{3}}|U_{i},

and consequently, as O_{Y} = O_{ℂℙ^{3}}/I_{Y},

(I_{Y} /I_{Y}^{2} )|U_{i} = (f_{i}O_{Y} ⊕ h_{i}O_{Y})|U_{i}.

Taking determinants, we have

det (I_{Y} /I_{Y}^{2} )|U_{i} = (f_{i} ∧ h_{i})O_{Y} |U_{i}.

An ideal sheaf I that locally has a single generator is locally free of rank one. If D is the corresponding divisor in ℂℙ^{3}, then D = supp (O_{ℂℙ^{3}}/I) and is an analytic subvariety of ℂℙ^{3}. Then denoting I by I_{D} and O_{ℂℙ^{3}}/I by O_{D}, we have

\begin{aligned}I_{D} & = O_{ℂℙ^{3}}(−D), \\ I_{D}/I_{D}^{2} & = I_{D} ⊗ O_{D}, \end{aligned}

where O_{ℂℙ^{3}}(−D) = O_{ℂℙ^{3}}([−D]) and [−D] denotes the line bundle on ℂℙ^{3} corresponding to the divisor −D. If U ⊂ ℂℙ^{3} is an open subset and D ∩ U = {z ∈ ℂℙ^{3} : f(z) = 0}, then I_{D}|U = fO_{ℂℙ^{3}}|U [G-H, p. 138 & p. 698], [O-S-S, p. 4].

Returning to the proof of the lemma, we have

\begin{aligned}f_{i}O_{ℂℙ^{3}}|U_{i} & = O_{ℂℙ^{3}}(−H_{i})|U_{i} \\ h_{i}O_{ℂℙ^{3}}|U_{i} & = O_{ℂℙ^{3}}(−Ĥ_{i})|U_{i}, \end{aligned}

and so

\begin{aligned}f_{i}O_{Y} |U_{i} & = O_{Y} (−H_{i})|U_{i}, \\ h_{i}O_{Y} |U_{i} & = O_{Y} (−Ĥ_{i})|U_{i}. \end{aligned}

Hence,

I_{Y} /I_{Y}^{2} |U_{i} = O_{Y} (−H_{i}) ⊕ O_{Y} (−Ĥ_{i})|U_{i},

and taking determinants, we obtain

\begin{aligned}det (I_{Y} /I_{Y}^{2} )|U_{i} & ≃ O_{Y} (−H_{i}) ⊗ O_{Y} (−Ĥ_{i})|U_{i} \\ & ≃ O_{Y} (−2)|U_{i}, for i = 0, …, k. \end{aligned}

Therefore, det (I_{Y} /I_{Y}^{2} ) ≃ O_{Y} (−2), and

det N_{Y/ℂℙ^{3}} = det (I_{Y} /I_{Y}^{2} )^{∗} = O_{Y} (2).

The canonical line bundle K_{Y} is then given by

K_{Y} ≃ K_{ℂℙ^{3}} ⊗ O_{Y} ⊗ O_{ℂℙ^{3}}(2)
≃ O_{ℂℙ^{3}}(−4) ⊗ O_{Y} ⊗ O_{ℂℙ^{3}}(2)
≃ O_{Y} (−2),

as required. □

Lemma 2.6.4. Let E → ℂℙ^{3} be a, holomorphic, rank 2 vector bundle with c_{1}(E) = 0 and c_{2}(E) = k. Then c_{1}(E(1)) = 2 and c_{2}(E(1)) = k + 1.

Proof. Immediate from the general formulas for c(∧^{p}E) and c(E ⊗ L), where L is a line bundle over ℂℙ^{3} [O-S-S, p. 16]. □

Lemma 2.6.5. [Har3, p. 232], [O-S-S, p. 93] Let the triple ⟨E(1), s, φ⟩ be as in the statement of Theorem 1.1. [Har3, p. 232], with c_{1}(E) = 0, c_{2}(E) = k, line bundle L = O_{ℂℙ^{3}}(2), and fixed isomorphism φ : det E(1) ≃ L. Then the triple ⟨E(1), s, φ⟩ canonically determines an isomorphism ξ : O_{Y} (−2) ≃ K_{Y} and corresponding element ζ ∈ H^{0}(Y, O_{Y}).

Proof. Recall from Theorem 1.1. [Har3, p. 232] that the triple ⟨E(1), s, φ⟩ canonically determines an isomorphism ξ : L ⊗ K_{ℂℙ^{3}} ⊗ O_{Y} ≃ K_{Y}. Since L = O_{ℂℙ^{3}}(2) and K_{ℂℙ^{3}} = O_{ℂℙ^{3}}(−4), we then have ξ : O_{Y} (−2) ≃ K_{Y}. From [Har3, p. 232], [O-S-S, p. 90], we have a locally free resolution of I_{Y} given by the Koszul complex for s:

0 ⟶ det E^{∗} ⟶ E^{∗} ⟶^{s} I_{Y} ⟶ 0

The bundle map φ induces an isomorphism det E^{∗} → L^{∗}, and so we have an exact sequence

0 ⟶ L^{∗} ⟶ E^{∗} ⟶^{s} I_{Y} ⟶ 0

Since L^{∗} = O_{ℂℙ^{3}}(−2), this global extension of E^{∗} determines an element of Ext^{1}(ℂℙ^{3}; I_{Y} , O_{ℂℙ^{3}}(−2)) [G-H, p. 725]. Proceeding as in [O-S-S, p. 97] or [Har3, p. 233], we obtain a canonical isomorphism

Ext^{1}(ℂℙ^{3}; I_{Y} , O_{ℂℙ^{3}}(−2)) ≃ H^{0}(Y, O_{Y}),

as required. □

Lemma 2.6.6. Let E → ℂℙ^{3} be a stable, holomorphic, rank 2 vector bundle with c_{1}(E) = 0 and c_{2}(E) = k. Let E be endowed with a real or symplectic structure σ_{E} and fix an isomorphism det E(1) ≃ O_{ℂℙ^{3}}(2). Suppose s ∈ H^{0}(ℂℙ^{3}, E(1)) and that ⟨E, s⟩ corresponds to k + 1 skew lines Y = Y_{0} ∪ ⋯ ∪ Y_{k} ⊂ ℂℙ^{3} and a choice of isomorphism ξ : O_{Y} (−2) ≃ K_{Y} specifed by ζ_{0}, …, ζ_{k} ∈ ℂ^{∗}. Then:

(i) The lines Y_{i} are real lines; (ii) The weights ζ_{i} are real.

Proof. (i) Let s ∈ H^{0}(ℂℙ^{3}, E(1)) with s^{−1}(0) = Y. We have an isomorphism σ_{E} : E → σ^{∗}\overline{E}, and so the pair ⟨σ^{∗}\overline{E}, σ^{∗}\overline{s}⟩ determines Y also. Hence,

Y = (\overline{s} ∘ σ)^{−1}(0) = (s ∘ σ)^{−1}(0) = σ^{−1}(Y)

and so σ(Y) = Y. Thus Y is preserved by σ and so consists of k + 1 real lines. This proves (i).

(ii) Recall that the real structure σ on ℂℙ^{3} induces a real or symplectic structure on any object that is functorially associated with ℂℙ^{3} [Har2, p. 8]. For example, a real structure is induced on N (0, k) by requiring that bundles E and σ^{∗}\overline{E} be isomorphic, the isomorphism being given by σ_{E} : E → σ^{∗}\overline{E}. Hence, the bundles E and σ^{∗}\overline{E} determine the same isomorphism ξ : O_{Y} (−2) ≃ K_{Y}, these isomorphisms being parametrized by H^{0}(Y, O_{Y}) = ℂ^{k+1}. The space H^{0}(Y, O_{Y}) inherits a real structure, so that H^{0}(Y, O_{Y}) = ℝ^{k+1} and hence the weights ζ_{i} are real. □

Lemma 2.6.7. With the hypotheses of the previous lemma, assume further that E is an instanton bundle with c_{2}(E) = 2. Then the weights ζ_{i} are positive.

Proof. See [Har2, p. 12], [Wa-We, p. 413]. □

CHAPTER III ATIYAH-DRINFELD-HITCHIN-MANIN CONSTRUCTION

We outline the ADHM monad construction of rank 2 instanton bundles over the complex projective space ℂℙ^{3} and the corresponding complex vector bundles over the four-sphere 𝕊^{4} with anti-self-dual SU(2)-connections. We then have two methods of constructing these holomorphic rank 2 vector bundles over ℂℙ^{3}, namely the method of curves and the ADHM monad construction, and so we describe the explicit correspondence between these two methods. General references for this chapter are [A], [A-D-H-M], [B-H], [Bo-Ma], [D-K], [D-M,1-4], [G-H], [O-S-S], [Ra2], [Sa].

§3.1. ADHM Construction of Instanton Bundles

We recall the construction of instanton bundles E → ℂℙ^{3} corresponding to ASD SU(2)-connections on a smooth, complex, Hermitian, rank 2 bundle E → 𝕊^{4} with topology fixed by c_{2}(E) = k.

Definition 3.1.1. Let X be a compact, complex manifold. A monad over X is a complex

0 ⟶ A ⟶^{α} B ⟶^{β} C ⟶ 0

of holomorphic vector bundles over X such that βα = 0. The holomorphic vector bundle E = Ker β/Im α over X is called the cohomology of the monad.

Lemma 3.1.2. [O-S-S, p. 240] If E → X is the cohomology of a monad as above, then the rank rk E and total Chern class c(E) are given by

rk E = rk B − rk A − rk C,
c(E) = c(B)c(A)^{−1}c(C)^{−1}.

We next describe Horrocks’ monad construction of rank 2 instanton bundles on ℂℙ^{3} [A], [Wa-We]. The monad construction may be described in terms of the following data [A, p. 59], [Wa-We, p. 415]:

Data 3.1.3. The linear algebra data for the monad construction of instanton bundles E → ℂℙ^{3} corresponding to ASD SU(2)-connections on a smooth complex rank 2 bundle E → 𝕊^{4} with c_{2}(E) = k is given by the following:

(i) A map σ : ℂ^{4} → ℂ^{4} defined by σ(z_{0}, z_{1}, z_{2}, z_{3}) = (−z̄_{1}, z̄_{0}, −z̄_{3}, z̄_{2}). (ii) A complex vector space W, with dim_{ℂ} W = k and a conjugate-linear map

σ_{W} : W → W, σ_{W}^{2} = 1. (When there is no ambiguity, σ_{W} is denoted by σ.) (iii) A complex vector space V, with dim_{ℂ} V = 2k + 2, with a symplectic form b

and conjugate-linear map σ_{V} : V → V, so that σ_{V}^{2} = −1, and satisfying:

(a) the form b is compatible with σ_{V}, in the sense that b(σu, σv) = \overline{b(u, v)};

(b) the induced Hermitian form h(u, v) = b(u, σv) is required to be positive definite. (When there is no ambiguity, σ_{V} is denoted by σ.)

(iv) A linear map A(z) : W → V, depending linearly on z = (z_{0}, z_{1}, z_{2}, z_{3}), so A(z) = ∑_{α=0}^{3} A_{α}z_{α}, where A_{α} : W → V are constant linear maps satisfying:

(a) (Rank or non-degeneracy condition.) For all z ≠ 0, dim_{ℂ} Im A(z) = k;

(b) (Isotropy condition.) For all z ≠ 0, Im A(z) is an isotropic subspace of V, so that Im A(z) ⊂ (Im A(z))^{∘};

(c) (Compatibility with σ.) For all z ∈ ℂ^{4}, w ∈ W, then σA(z)w = A(σz)σw. (If U is any subspace of V, then U^{∘} = {v ∈ V : b(u, v) = 0 for all u ∈ U}

denotes the polar subspace corresponding to U.)

The symplectic form b : V ⊗ V → ℂ induces an isomorphism b : V → V^{∗} given by v ↦ b(v) = b(·, v). Since A^{∗}(z) : V^{∗} → W^{∗}, we obtain a map A^{∗}(z)b : V → W^{∗} defined by

(A^{∗}(z)b(v)) (w) = b(A(z)w, v) for v ∈ V, w ∈ W.

We then have the corresponding monad:

0 ⟶ W (−1) ⟶^{A} \underline{V} ⟶^{A^{∗}b} W^{∗}(1) ⟶ 0

where \underline{V} denotes the trivial bundle V × ℂℙ^{3} → ℂℙ^{3}. We use the symplectic form b to define a conjugate-linear isomorphism b : V → V^{∗}, v ↦ b(·, v). The bundle E → ℂℙ^{3} is then defined as Ker (A^{∗}b)/Im A, with fibres

E_{z} = Ker (A^{∗}(z)b)/Im A(z) = (Im A(z))^{∘}/Im A(z) for z ∈ ℂℙ^{3}.

If we let O_{ℂℙ^{3}}(−1) denote the tautological line bundle on ℂℙ^{3} and recall that V ≃ ℂ^{2k+2}, W ≃ ℂ^{k}, then we see that the above monad is equivalent to

0 ⟶ O_{ℂℙ^{3}}^{k}(−1) ⟶^{A} O_{ℂℙ^{3}}^{2k+2} ⟶^{A^{∗}b} ⊗ O_{ℂℙ^{3}}^{k}(1) ⟶ 0

One can then verify that the bundle E constructed from this data is indeed an instanton bundle [A], [Wa-We].

To explicitly construct E, one chooses bases on ℂ^{4}, W, and V, and one defines maps σ, σ_{W}, σ_{V}, and b as follows:

(i) The map σ : ℂ^{4} → ℂ^{4} corresponds to left multiplication j : ℍ^{2} → ℍ^{2}. (ii) Let w_{1}, …, w_{k} be a real basis for W, so that the map σ_{W} : W → W is complex

conjugation after identifying W with ℂ^{k} and recalling that σw_{m} = w_{m}, for m = 1, …, k. Then the vector space W with dim_{ℂ} W = k, may be viewed as the complexification of the real vector space W_{ℝ} left fixed by σ, so that W = ℂ ⊗ W_{ℝ}.

(iii) Let {v_{0}, …, v_{k}, σv_{0}, …, σv_{k}} be an orthogonal basis for V with respect to the Hermitian inner product. Then the complex vector space V with dim_{ℂ} V = 2k + 2 may be viewed as a left quaternion vector space with dim_{ℍ} V = k + 1, with left multiplication by j on ℍ^{k+1} corresponding to σ_{V} : V → V. The symplectic form on V is now represented by a (2k + 2) × (2k + 2) complex matrix,

J = ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) ,

where I is the k × k identity matrix. Then if u, v are row vectors in ℂ^{2k+2}, we have b(u, v) = uJv^{t}. The Hermitian form h(u, v) = uv^{†}, the standard positive definite Hermitian scalar product on ℂ^{2k+2}. The quaternionic basis {v_{0}+σv_{0}, …, v_{k} +σv_{k}} is orthogonal with respect to the standard quaternionic scalar product q(·, ·) on ℍ^{k+1} given by q(ξ, η) = ξη^{†}. Note that q(·, ·) = h(·, ·) + b(·, ·)j on ℍ^{k+1} = ℂ^{k+1} ⊕ ℂ^{k+1}j.

(iv) After identifying V with ℂ^{2k+2}, the map σ_{V} : V → V is given by σv = Jv̄. In matrix form, the isotropy condition is equivalent to

A(z)^{t}JA(z) = 0 for all z ∈ ℂ^{4}.

The matrix representation of A(z) = ∑_{α=0}^{3} A_{α}z_{α} is then defined by

A(z)w_{m} = ∑_{α=0}^{3} ∑_{n=0}^{k} z_{α}A_{mn}^{α}v_{n} + z_{α}A_{m,k+1+n}^{α}σv_{n} for m = 1, …, k.

Then A_{α} may be written as a block matrix A_{α} = (A′_{α}, A′′_{α}) for α = 0, …, 3, where A′_{α}, A′′_{α} are complex k ×(k +1) matrices. The reality condition σA(z)w = A(σz)σw now becomes

\begin{aligned}A′_{1} & = −\overline{A′′_{0}}, & A′_{3} & = −\overline{A′′_{2}}, \\ A′′_{1} & = \overline{A′_{0}}, & A′′_{3} & = \overline{A′_{2}}. \end{aligned}

Alternatively, we may write the reality condition as J\overline{A(z)} = A(σz), and in terms of the matrices A_{0}, …, A_{3}, this becomes

\begin{aligned}J\overline{A_{0}} & = A_{1}, & J\overline{A_{2}} = A_{3}, \\ J\overline{A_{1}} & = −A_{0}, & J\overline{A_{3}} = −A_{2}, \end{aligned}

and decomposing the matrices A_{α} into blocks (A′_{α}, A′′_{α}), we obtain the previous matrix conditions. Letting C = A′_{0} + A′′_{0}j, D = A′_{2} + A′′_{2}j, we observe that

A(z) = z_{0}A_{0} + z_{1}A_{1} + z_{2}A_{2} + z_{3}A_{3} = xC + yD = A(x, y),

where x = z_{0} + z_{1}j, y = z_{2} + z_{3}j. This gives the convenient k × (k + 1) quaternion matrix representation of A.

The quaternion formulation may be seen a little more directly as follows. We have ℂ^{4} ⊗_{ℂ} W ≃ ℍ^{2} ⊗_{ℝ} W_{ℝ}, and the induced map σ on ℂ^{4} ⊗_{ℂ} W corresponds to left multiplication by j on the left quaternion vector space ℍ^{2} ⊗_{ℝ} W_{ℝ}. The complex linear map A : ℂ^{4} ⊗_{ℂ} W → V may now be viewed as a map

A : ℍ^{2} ⊗_{ℝ} W_{ℝ} ⟶ V,

and compatibility of A(z) with σ is equivalent to requiring that A be quaternion linear. If C = (C_{1}, …, C_{k})^{t} and D = (D_{1}, …, D_{k})^{t}, where C_{i} and D_{i} denote the rows of C and D respectively, then

\begin{aligned}C_{i} & = A((1, 0) ⊗ w_{i}), \\ D_{i} & = A((0, 1) ⊗ w_{i}) for i = 1, …, k. \end{aligned}

The non-degeneracy or rank condition on A is equivalent to requiring that the quaternion matrix A(x, y) have rank k for all (x, y) ≠ 0. The isotropy condition is equivalent to requiring that A(x, y)A(x, y)^{†} be real for all (x, y), where A^{†} denotes the quaternionic-conjugate matrix transpose of A. Let ℍℙ^{1} = 𝕊^{4} denote the left quaternionic projective space and recall that Sp(1) = SU(2). The Sp(1)-bundle E → ℍℙ^{1} is obtained by setting E_{(x,y)} = (Im A(x, y))^{⊥} ⊂ ℍ^{k+1}, for [x, y] ∈ ℍℙ^{1}, and the ASD Sp(1)-connection is obtained by orthogonal projection from \underline{ℍ}^{k+1} to E, where \underline{ℍ}^{k+1} denotes the trivial bundle ℍℙ^{1} × ℍ^{k+1} → ℍℙ^{1}. where ℍℙ^{1} = 𝕊^{4} denotes the left quaternionic projective space. We now have a quaternionic monad over ℍℙ^{1} given by

0 ⟶ E ⟶ \underline{ℍ}^{k+1} ⟶ kL ⟶ 0

where L → ℍℙ^{1} denotes the tautological quaternionic line bundle and kL = L ⊕ ⋯⊕L. The k×(k+1) quaternion matrix A is now required to satisfy the conditions:

(i) A(x, y) has quaternion rank k for all (x, y) ∈ ℍ^{2} \ (0, 0); (ii) A(x, y)A(x, y)^{†} is real for all (x, y) ∈ ℍ^{2}.

In terms of moduli spaces, the main result is the following:

Theorem 3.1.4. [B-H, p. 19], [A] There is a bijection between

(i) Isomorphism classes of instanton bundles E → ℂℙ^{3}; and (ii) Isomorphism classes of holomorphic bundles E → ℂℙ^{3} which arise from the Horrocks construction via a linear map A(z) : W → V.

Remark 3.1.5. Two bundles E ∼ (A, W, V), E′ ∼ (A′, W′, V′) arising from the Horrocks construction are isomorphic if and only if there are complex vector space isomorphisms W → W′, V → V′ preserving structures and taking A to A′. Hence, A and A′ give isomorphic bundles E and E′ if and only if

A′(z) = QA(z)R, for Q ∈ Sp(k + 1), R ∈ GL(k, ℝ).

Here, Sp(k + 1) denotes the subgroup of GL(2k + 2, ℂ) preserving the symplectic form b and real form σ_{V}, and GL(k, ℝ) denotes the subgroup of GL(k, ℂ) preserving the real form σ_{W} [A, p. 62], [A-D-H-M, p. 186], [Wa-We, p. 417], [B-H, p. 19], [D-M4, p. 847].

In particular, if two triples (A, W, V), (A′, W′, V′) are isomorphic as above, we then obtain isomorphisms between the complex vector bundles E, E′ over 𝕊^{4} and their associated principal bundles P, P′, sending the connection ω to ω′. So if f : P → P′ is the induced isomorphism of SU(2) bundles over 𝕊^{4}, then ω = f^{∗}w′.

Let T_{k} denote the space of quaternionic k ×(k +1) ADHM matrices C, D, such that A(x, y) has rank k and A(x, y)A(x, y)^{†} is real for all (x, y) ∈ ℍ^{2} \(0, 0), modulo the action of Sp(k + 1) × GL(k, ℝ). Then dim_{ℝ} T_{k} = 8k − 3 [A, p. 26], coinciding with dim_{ℝ} M_{k}, where M_{k} is the moduli space of ASD SU(2)-connections over 𝕊^{4}. In particular, the ADHM construction gives a diffeomorphism T_{k} → M_{k}.

§3.2. Global Sections of Twisted Instanton Bundles

In the next chapter, we construct a bundle E → 𝕊^{4} with −c_{2}(E) = k using the ADHM monad construction with a choice of ADHM matrices corresponding to k + 1 distinct points P_{i} in 𝕊^{4} and k + 1 positive weights λ_{i}. When E has an ASD SU(2)-connection, this construction produces a stable, holomorphic, rank 2 vector bundle E over ℂℙ^{3} with c_{1}(E) = 0 and c_{2}(E) = k. We exhibit a section s ∈ H^{0}(ℂℙ^{3}, E(1)) with zero set (s)_{0} given precisely by the real lines Y_{i} = π^{−1}(P_{i}). Fixing a choice of isomorphism φ : det E(1) ≃ O_{ℂℙ^{3}}(2), we see that a choice of complex weights or ‘residues’ ζ_{0}, …, ζ_{k} corresponds to a choice of isomorphism K_{Y} ≃ O_{Y} (−2). If the bundle E has a real or symplectic structure, then the ζ_{i} are required to be real. Finally, the requirement that E has no real jumping lines is equivalent to the condition ζ_{i} > 0 [Wa-We, p. 413]. The bundle E constructed explicitly from ADHM data corresponding to a choice of points P_{i} in 𝕊^{4} and weights λ_{i} > 0 gives ζ_{i} = λ_{i}^{2}. This shows the relationship between the ADHM monad construction of instanton bundles over ℂℙ^{3} and Hartshorne’s construction of the same bundles using the method of curves [A], [Har2, 3], [Wa-We]. Moreover, we obtain an alternative verification that our ad hoc choice of ADHM matrices gives all ASD SU(2)-connections with k = 2 on 𝕊^{4}.

For simplicity, we assume k = 2. Let A(z) = ∑_{α=0}^{3} z_{α}A_{α} = xC + yD, where the A_{α} are 2 × 6 complex matrices, and C,D are 2 × 3 quaternionic matrices. Let \underline{ℍℙ}^{1} denote the right projective space. For points \underline{P}_{i} with homogeneous coordinates [a_{i}, b_{i}] in \underline{ℍℙ}^{1}, for i = 0, 1, 2, the corresponding matrices C, D may be chosen to be

C = ( \begin{matrix}b_{0} & −b_{1} & 0 \\ b_{0} & 0 & −b_{2} \end{matrix} )  and  D = ( \begin{matrix}−a_{0} & a_{1} & 0 \\ −a_{0} & 0 & a_{2} \end{matrix} ) ,

and so

A(x, y) = ( \begin{matrix}xb_{0} − ya_{0} & −xb_{1} + ya_{1} & 0 \\ xb_{0} − ya_{0} & 0 & −xb_{2} + ya_{2} \end{matrix} ) .

Remark 3.2.1. If [x, y] denote homogeneous coordinates for the left projective space ℍℙ^{1} and [a, b] denote homogeneous coordinates for the right projective space \underline{ℍℙ}^{1}, then xb − ya = 0 if and only if [x, y] = [b^{−1}, a^{−1}] = P ∈ ℍℙ^{1}. We have a map, denoted \underline{ℍℙ}^{1} → ℍℙ^{1}, \underline{P} → P, given by [a, 1] ↦ [a, 1], [1, b] ↦ [1, b], so that on overlapping patches [a, b] ↦ [b^{−1}, a^{−1}].

Lemma 3.2.2. Suppose [a, b] ∈ \underline{ℍℙ}^{1} (right projective space). Then xb − ya = 0 or [x, y] = [b^{−1}, a^{−1}], [x, y] ∈ ℍℙ^{1} (left projective space) if and only if z ∈ H ∩ Ĥ, where H = {z ∈ ℂℙ^{3} : f(z) = 0}, Ĥ = {z ∈ ℂℙ^{3} : h(z) = 0},

\begin{aligned}f(z) & = z_{0}b′ − z_{1}\overline{b}′′ − z_{2}a′ + z_{3}\overline{a}′′, \\ h(z) & = z_{0}b′′ + z_{1}\overline{b}′ − z_{2}a′′ − z_{3}\overline{a}′, \end{aligned}

z = [z_{0}, z_{1}, z_{2}, z_{3}], with x = z_{0} + z_{1}j, y = z_{2} + z_{3}j, a = a′ + a′′j, and b = b′ + b′′j under the standard identification ℍ = ℂ ⊕ ℂj.

Proof. If π : ℂℙ^{3} → ℍℙ^{1} is the standard projection, then [x, y] = π[z_{0}, z_{1}, z_{2}, z_{3}], where x = z_{0} + z_{1}j, y = z_{2} + z_{3}j, and we observe that

xb − ya = (z_{0}, z_{1}) ( \begin{matrix}b′ & b′′ \\ −\overline{b}′′ & \overline{b}′ \end{matrix} ) − (z_{2}, z_{3}) ( \begin{matrix}a′ & a′′ \\ −\overline{a}′′ & \overline{a}′ \end{matrix} ) ,

under the identification ℍ = ℂ ⊕ ℂj. □

Let a_{i} = a′_{i} + a′′_{i}j and b_{i} = b′_{i} + b′′_{i}j, for i = 0, 1, 2. Then the relationship between the quaternionic matrices C, D and the complex matrices A′_{α}, A′′_{α} is given by

C = A′_{0} + A′′_{0}j, D = A′_{2} + A′′_{2}j,

so that the corresponding matrices A_{α} are given by

\begin{aligned}A_{0} & = (A′_{0}, A′′_{0}),  A_{1} = (−\overline{A′′_{0}}, \overline{A′_{0}}), \\ A_{2} & = (A′_{2}, A′′_{2}),  A_{3} = (−\overline{A′′_{2}}, \overline{A′_{2}}). \end{aligned}

Then, the matrix A(z) has the following form:

A(z) = ( \begin{matrix}f_{0}(z) & f_{1}(z) & 0 & h_{0}(z) & h_{1}(z) & 0 \\ f_{0}(z) & 0 & f_{2}(z) & h_{0}(z) & 0 & h_{2}(z) \end{matrix} ) .

Recall that E = Ker (A^{t}J)/Im A, where

0 ⟶ O_{ℂℙ^{3}}^{2}(−1) ⟶^{A} O_{ℂℙ^{3}}^{6} ⟶^{A^{t}J} O_{ℂℙ^{3}}^{2}(1) ⟶ 0

and so the fibres are given by E_{z} = Ker (A^{t}(z)J)/Im A(z), for z ∈ ℂℙ^{3}. Let {e_{0}, e_{1}, e_{2}, ê_{0}, ê_{1}, ê_{2}} be the standard ordered basis of ℂ^{6}, so that e_{0} = (1, 0, 0, 0, 0, 0), etc. We let H_{i}, Ĥ_{i} ⊂ ℂℙ^{3} be the associated hyperplanes

H_{i} = {z ∈ ℂℙ^{3} : f_{i}(z) = 0} and Ĥ_{i} = {z ∈ ℂℙ^{3} : h_{i}(z) = 0},

and so the lines Y_{i} are the complete intersections Y_{i} = H_{i} ∩ Ĥ_{i} for i = 0, 1, 2. We define sections s_{i} ∈ H^{0}(ℂℙ^{3}, O_{ℂℙ^{3}}^{6}(1)) by setting

s_{i}(z) = f_{i}(z)e_{i} + h_{i}(z)ê_{i} for z = [z_{0}, z_{1}, z_{2}, z_{3}] ∈ ℂℙ^{3}.

Then, s_{0} is a section of O_{ℂℙ^{3}}^{6}(1) with zero set Y_{0} = H_{0} ∩ Ĥ_{0}. Moreover,

A(z)Js_{0}^{t}(z) = A(z) ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) s_{0}^{t}(z) = 0 z ∈ ℂℙ^{3},

so that s_{0}(z) ∈ Ker A^{t}(z)J, and hence s_{0} defines a section in H^{0}(ℂℙ^{3}, E(1)). Then, s_{0}(z) = 0 ∈ E_{z} if and only if z ∈ Y_{0} or s_{0}(z) ∈ Im A(z), where Im A(z) denotes the span of the rows of the matrix A(z) in ℂ^{6}. If s_{0}(z) = 0 ∈ E_{z} and z ≠ Y_{0}, then f_{0}(z) ≠ 0 or h_{0}(z) ≠ 0, and it follows that either z ∈ Y_{1} = H_{1} ∩ Ĥ_{1}, or z ∈ Y_{2} = H_{2} ∩ Ĥ_{2}. We observe that the sections s_{i} ∈ H^{0}(ℂℙ^{3}, O_{ℂℙ^{3}}^{6}(1)) all project to the same section s ∈ H^{0}(ℂℙ^{3}, E(1)). We have the Lemma 3.2.3. Let P_{0}, P_{1}, P_{2} be distinct points in ℍℙ^{1} (left projective space), with homogeneous coordinates given by \underline{P}_{i} = [a_{i}, b_{i}] in \underline{ℍℙ}^{1} (right projective space). Let s ∈ H^{0}(ℂℙ^{3}, E(1)) be the corresponding section constructed as above. Then (s)_{0} = Y, where Y consists of three skew lines Y_{0}, Y_{1}, Y_{2} in ℂℙ^{3} given by Y_{i} = π^{−1}(P_{i}).

Fix an isomorphism det E ≃ O_{ℂℙ^{3}}, and hence an isomorphism φ : det E(1) → O_{ℂℙ^{3}}(2). Note that e_{i}, ê_{i} ∈ ℂ^{6} project to elements in E_{z} when z ∈ Y_{i} and f_{i}(z) = h_{i}(z) = 0. Moreover, e_{i} ∧ ê_{i} = e_{i}Jê_{i}^{t} = 1. For z in a neighbourhood of Y_{i}, choose ε_{i}(z), ε̂_{i}(z) vanishing along Y_{i}, so that θ_{i}(z) = e_{i} + ε_{i}(z), θ̂_{i}(z) = ε̂_{i} + ε̂_{i}(z) give local frames for E. Then

\begin{aligned}s_{i}(z) & = f_{i}(z)e_{i} + h_{i}(z)ê_{i} = f̃_{i}(z)θ_{i}(z) + h̃_{i}(z)θ̂_{i}(z), \\ ∧^{2}s_{i}(z) & = f̃_{i}(z)h̃_{i}(z)θ_{i}(z) ∧ θ̂_{i}(z), \end{aligned}

and the ∧^{2}s_{i} are sections of det E = O_{ℂℙ^{3}}(2). Fixing a non-zero section θ ∈ H^{0}(ℂℙ^{3}, O_{ℂℙ^{3}}(2)), we have

∧^{2}s_{i} = ζ_{i}^{−1}θ ∈ H^{0}(ℂℙ^{3}, O_{ℂℙ^{3}}^{∗}) = ℂ^{∗}.

Indeed, if (∧^{2}s_{i}) = θ and f_{i}, h_{i} are replaced by λ_{i}^{−1}f_{i}, λ_{i}^{−1}h_{i}, where λ_{i} ∈ ℂ^{∗}, we have ζ_{i} = λ_{i}^{2} for i = 0, 1, 2. Since E^{∗} is an instanton bundle, the constants ζ_{i} are required to be positive, and hence the λ_{i} are real.

The analogous relationship between the monad construction of rank 2 holomorphic bundles over ℂℙ^{2} and the construction of bundles corresponding to configurations of points in ℂℙ^{2} and complex residues at those points is described in [D-K, p. 397].

§3.3. Conics in Complex Projective Space

Recall that the curve Y = Y_{0} ∪ Y_{1} ∪ Y_{2} ⊂ ℂℙ^{3} arises as the zero set of a section s ∈ H^{0}(ℂℙ^{3}, E(1)). As s varies in ℙ(H^{0}(ℂℙ^{3}, E(1))) ≃ ℂℙ^{1}, then the corresponding points Y_{0}, Y_{1}, Y_{2} move in a conic γ ⊂ G(1, 3), inducing a linear system g_{3}^{1} without basepoints on ℂℙ_{α}^{1}. General references for this section are [Har, 1-3], [G-H], [J-N-R], [N-T, 1-3], [S-T].

In order to see the consequences for the space N_{𝕀}(0, 2), it is useful to follow the approach of Singhof and Trautman in constructing this moduli space as the total space of a certain fibre bundle [S-T, p. 336-342]. Fix a complex vector space U ≃ ℂ^{2} and let V = U × U, with ℂℙ^{1} = ℙ(U) and ℂℙ^{3} = ℙ(V). Let Q ⊂ ℂℙ^{3} be a non-singular quadric surface, so that Q ≃ ℂℙ^{1} × ℂℙ^{1}. For convenience, we use the labelling ℂℙ_{α}^{1} × ℂℙ_{β}^{1}. Let g ∈ GL(V) and let σ denote the Segré embedding

σ : ℂℙ^{1} × ℂℙ^{1} ↪ ℂℙ^{3}, ([s_{0}, s_{1}], [t_{0}, t_{1}]) ⟼ [s_{0}t_{0}, s_{0}t_{1}, s_{1}t_{0}, s_{1}t_{1}].

Let g̃ ∈ PGL(V) be the induced automorphism of ℂℙ^{3} and let

φ_{g} = g̃ ∘ σ : ℂℙ^{1} × ℂℙ^{1} ↪ ℂℙ^{3}

be the corresponding embedding. Let Q_{g} = Im φ_{g} ≃ ℂℙ^{1} × ℂℙ^{1}, with Q_{g} ⊂ ℂℙ^{3}, and note that every non-singular quadric arises via this construction [S-T], [G-H].

For convenience, let G = GL(V) and let H ⊂ G be the subgroup H = GL(U)× GL(U). On U we have the real forms c, j, where c^{2} = 1_{U} (corresponding to complex conjugation, the standard real structure) and j^{2} = −1_{U} (the non-standard real or symplectic structure). Let U_{ℝ} denote the real vector space fixed by c, so that U ≃ U_{ℝ} ⊗_{ℝ} ℂ. These structures are inherited by ℂℙ^{1}, so ℙ(U_{ℝ}) = ℝℙ^{1} denotes the real points of ℂℙ^{1} with respect to c. Moreover, U becomes a 1-dimensional left ℍ-vector space by setting j · u = j(u), and V becomes a 2-dimensional left ℍ-vector space using J = c ⊕ j. Let G_{J} = Aut_{ℍ}(V) ⊂ GL(V) be the subgroup of complex automorphisms φ such that φ(Jv) = Jφ(v), and hence G_{J} = GL(2, ℍ). Then for any g ∈ G_{J}, the corresponding quadric Q_{g} is invariant under J. Let H_{J} = H ∩ G_{J}, and note that

H_{J} = GL_{ℝ}(U_{ℝ}) × GL_{ℍ}(U) ≃ GL(2, ℝ) × GL(1, ℍ).

As usual, π : ℂℙ^{3} → ℍℙ^{1} denotes the standard projection, taking ℂ-lines in V to the corresponding ℍ-lines in V. Then the J-invariant lines in ℂℙ^{3} are precisely the fibres of π.

Fix a point t ∈ ℂℙ_{α}^{1}. Then {t}×ℂℙ_{α}^{1} ⊂ ℂℙ_{α}^{1} ×ℂℙ_{β}^{1}, and so we have an induced map

{t} × ℂℙ_{α}^{1} ⟶^{φ_{g}} ℂℙ^{3} ⟶^{π} 𝕊^{4}, ⟨t, u⟩ ⟼ πφ_{g}(t, u),

for any u ∈ ℂℙ_{β}^{1}. We recall from our discussion of Hartshorne’s description of the space N_{𝕀}(0, 2), that the factor ℂℙ_{α}^{1} had its standard real structure: the corresponding real points in ℂℙ_{α}^{1} are given by 𝕊^{1} ≃ ℝℙ^{1} ⊂ ℂℙ_{α}^{1}. If t ∈ ℝℙ^{1} ⊂ ℂℙ_{α}^{1}, then φ_{g} maps {t} × ℂℙ_{β}^{1} to a real line in ℂℙ^{3}, and the projection π maps this real line to a point P ∈ 𝕊^{4}. If we choose any fixed u ∈ ℂℙ_{β}^{1}, we then have an embedding (independent of the choice of u)

ℝℙ^{1} ↪ 𝕊^{4}, t ⟼ πφ_{g}(t, u)

of 𝕊^{1} ≃ ℝℙ^{1} into a circle in 𝕊^{4} [S-T, p. 342]. So as t varies in 𝕊^{1} ≃ ℝℙ^{1} ⊂ ℂℙ_{α}^{1}, the corresponding point P_{t} = πφ_{g}(t, u) moves in a circle in 𝕊^{4}.

If t varies in ℂℙ^{1}, then the corresponding point Y_{t} = πφ_{g}(t, u) in G(1, 3) (or line in ℂℙ^{3}) moves in the conic γ in G(1, 3) which we described earlier. Recall that the curve Y = Y_{0} ∪ Y_{1} ∪ Y_{2} ⊂ ℂℙ^{3} arises as the zero set of a section s ∈ H^{0}(ℂℙ^{3}, E(1)). As s varies in ℙ(H^{0}(ℂℙ^{3}, E(1))) ≃ ℂℙ^{1}, then the corresponding points Y_{0}, Y_{1}, Y_{2} move in the conic γ ⊂ G(1, 3). Consequently, if the lines Y_{i} ⊂ ℂℙ^{3} are real, then the corresponding points P_{0}, P_{1}, P_{2} — where P_{i} = π(Y_{i}) ∈ 𝕊^{4} — move in a circle in 𝕊^{4} as s varies in ℙ(H^{0}(ℂℙ^{3}, E(1)))_{ℝ} ≃ ℝℙ^{1}. Of course, the rank 2 holomorphic bundle E is fixed and so the pair ⟨E, s⟩ determines the same equivalence class of ASD SU(2)-connections while s varies as above. This phenomenon was described in [J-N-R] using different techniques.

§3.4. Two Constructions of Instanton Bundles

We have described how one may use the ADHM method to construct complex, rank 2 bundles with ASD SU(2)-connection over 𝕊^{4}, and then stable, holomorphic, rank 2 bundles over ℂℙ^{3} with c_{1} = 0 and c_{2} = k, corresponding to configurations of points P_{i} in 𝕊^{4} and positive weights λ_{i}. In this section, we provide an outline of how one may construct the ASD SU(2) connections, if we are given an instanton bundle over ℂℙ^{3} corresponding to a curve Y in ℂℙ^{3} consisting of k + 1 real skew lines. General references for this section are [A-W], [D-K], [Wa-We].

If E → ℂℙ^{3} is a rank 2 instanton bundle corresponding to a curve Y consisting of k + 1 skew lines Y_{i} in ℂℙ^{3}, then E(1) has a section s vanishing only along the curve Y = Y_{0} ∪ … ∪ Y_{k} ⊂ ℂℙ^{3}. Hence, s is nowhere vanishing along the complement Y′ = ℂℙ^{3} − Y, and so generates a trivial line bundle on Y′. Noting that I_{Y} |Y′ = O_{ℂℙ^{3}}|Y′, the section s induces an exact sequence [O-S-S, p. 93], [G-H, p. 726]:

0 ⟶ O_{ℂℙ^{3}}|Y′ ⟶^{s} E(1)|Y′ ⟶ O_{ℂℙ^{3}}(2)|Y′ ⟶ 0

and so E(1)|Y′ is an extension of L_{2} = O_{ℂℙ^{3}}(2)|Y′ by L_{1} = O_{ℂℙ^{3}}|Y′. Such extensions are classified by

Ext^{1}(Y′; L_{2}, L_{1}) ≃ H^{1}(Hom(L_{2}, L_{1}))
≃ H^{1}(Y′, L_{1} ⊗ L_{2}^{−1})
≃ H^{1}(Y′, O_{ℂℙ^{3}}(−2)),

as discussed in [Wa-We, p. 412], [D-K, p. 388], [G-H, p. 725]. Thus, such an extension naturally corresponds to a cocycle Γ ∈ H^{1}(Y′, O_{ℂℙ^{3}}(−2)), and this cocycle may be readily computed from the transition functions of E(1)|Y′ — see, for example, [A-W, p. 122], [Wa-We, p. 399], [D-K, p. 389], [Gu]. Indeed, the 2 × 2 transition matrices may be put in a standard upper triangular form and the cocycle Γ may then be identified with the upper-right entry. If

Γ = ∑_{i=0}^{k} ζ_{i}Γ_{i}, where Γ_{i} ∈ H^{1}(ℂℙ^{3} − Y_{i}, O_{ℂℙ^{3}}(−2)),

we note that Γ determines a singular potential ϕ [Wa-We, p. 388-409]:

ϕ(x) = ∑_{i=0}^{k} \frac{ζ_{i}}{|x − a_{i}|^{2}} x ∈ ℍ.

Here, a_{i} = π(Y_{i}) ∈ ℍ ∪ {∞}, and the reality condition on E is equivalent to the condition that the weights ζ_{i} are real and that the lines Y_{i} are real lines — corresponding to points a_{i} ∈ ℍ ∪ {∞}. The requirement that E has no real jumping lines is equivalent to the condition ϕ(x) > 0 for x ∈ ℍ ∪ {∞}, and so ζ_{i} > 0 [Wa-We, p. 412]. Finally, the potential ϕ determines a singular, local, anti-self-dual, SU(2)-connection 1-form:

ω(x) = ∑_{i=0}^{k} \frac{ζ_{i}}{|x − a_{i}|^{4}ρ}Im \{(x − a_{i})d\overline{x}\} , where ρ(x) = ∑_{i=0}^{k} \frac{ζ_{i}}{|x − a_{i}|^{2}} ,

[Wa-We, p. 388-409]. This is the local connection 1-form described in [J-N-R], corresponding to a choice of positive weights λ_{i}, with ζ_{i} = λ_{i}^{2} above, and points P_{i} ∈ 𝕊^{4}, with P_{i} = [a_{i}, 1] above.

Conversely, given distinct points P_{i} ∈ 𝕊^{4} and positive weights λ_{i}, with ζ_{i} = λ_{i}^{2}, we may construct an instanton bundle E via an appropriate choice of ADHM matrices. By explicitly constructing the section s ∈ H^{0}(ℂℙ^{3}, E(1)) and employing the correspondence between rank 2 holomorphic bundles over ℂℙ^{3} and curves in ℂℙ^{3} as discussed in [Har2, 3], [O-S-S], and [Wa-We], we recover the original data consisting of distinct points P_{i} and positive weights λ_{i}. This gives the correspondence between the two methods of constructing instanton bundles, and moreover, demonstrates that our ad hoc choice of ADHM matrices does indeed yield all anti-self-dual SU(2)-connections over 𝕊^{4} with k = 2.

CHAPTER IV PARAMETRIZATION OF SELF-DUAL CONNECTIONS

We first describe the ADHM construction of all self-dual connection 1-forms on an SU(2)-bundle P → 𝕊^{4} with −c_{2}(P)[𝕊^{4}] = k > 0. We then discuss the parametrization of all self-dual connection 1-forms with k = 2. In the following, the moduli space M_{k}^{∗}(𝕊^{4}, SU(2)) of self-dual SU(2)-connections over 𝕊^{4} simply by M_{k}. General references for chapter are [A], [C-G-F-T], [C-W-S], [D1], [D-K], [Wa-We].

§4.1. Parametrization of Connection One-forms

We review the ADHM construction of the classifying maps f : 𝕊^{4} → ℍℙ^{k} which provide all self-dual SU(2)-connections on P → 𝕊^{4} by pulling back the canonical SU(2)-connection on 𝕊^{4k+3} → ℍℙ^{k}. General references for section are [A], [Bo-Ma], [R2], [Sa].

We consider self-dual rather than anti-self-dual connections and so all quaternionic vector spaces are now assumed to be right vector spaces, and quaternioniclinear maps act by left matrix multiplication. We now let ℍℙ^{n} denote the right quaternionic projective space:

ℍℙ^{n} = {(q_{0}, …, q_{n}) ∈ ℍ^{n+1} \ {0} : (q_{0}, …, q_{n}) ∼ (q_{0}a, …, q_{n}a), a ∈ ℍ^{∗}}.

As before, G_{m}(ℍ^{n}) denotes the Grassmann manifold of m-dimensional quaternionic subspaces of ℍ^{n}. The symplectic group Sp(n) is defined by:

Sp(n) = {Q ∈ GL(n, ℍ) : (Qp)^{†}(Qq) = p^{†}q},

where p^{†}q = ∑_{i=1}^{n} \overline{p}_{i}q_{i} denotes the symplectic scalar product on ℍ^{n}.

Let A(x, y) = Cx + Dy, where C and D are (k + 1) × k quaternionic ADHM matrices chosen so that A(x, y) satisfies the following conditions:

(i) Rank_{ℍ} A(x, y) = k, for all (x, y) ∈ ℍ^{2} \ (0, 0); (ii) A^{†}(x, y)A(x, y) is real for all (x, y) ∈ ℍ^{2}.

Hence, we obtain a map A : ℍℙ^{1} → G_{k}(ℍ^{k+1}), and A(x, y) now acts on ℍ^{k+1} by left matrix multiplication. Define u : ℍℙ^{1} → ℍℙ^{k} by requiring that:

\begin{aligned}u(x, y)^{†}A(x, y) & = 0, \\ u^{†}(x, y)u(x, y) & = 1, \end{aligned}

for all (x, y) ∈ ℍ^{2} \ (0, 0). The map u is a quaternionic rational map, in the sense that it may appear to have singularities at a finite number of points of ℍℙ^{1}. However, these are always removable — we will see this explicitly for the special cases considered in the next section — and so u may be assumed to be smooth [A, p. 26]. Let E = u^{∗}E′ → ℍℙ^{1} denote the pull-back of the universal quaternionic line bundle E′ → ℍℙ^{k}, and let P = u^{∗}𝕊^{4k+3} → ℍℙ^{1} denote the pull-back of the universal principal Sp(1)-bundle 𝕊^{4k+3} → ℍℙ^{k}.

We first compute the transition functions of the bundle π : 𝕊^{4k+3} → ℍℙ^{k}. Let {O′_{i}} denote the standard open cover of ℍℙ^{k}, so that O′_{i} = {[q_{0}, …, q_{k}] : q_{i} ≠ 0}, i = 0, …, k, where q = (q_{0}, …, q_{k}) are coordinates on ℍ^{k+1}. Define a system of local sections σ′_{i} : O′_{i} → 𝕊^{4k+3}, by setting

σ′_{i}(q) = \frac{q}{|q|} \frac{\overline{q}_{i}}{|q_{i}|} on O′_{i}.

The corresponding transition functions T′_{ij} : O′_{i}∩O′_{j} → Sp(1) are defined by σ′_{j}(q) = σ′_{i}(q)T′_{ij}(q), so that

T′_{ij}(q) = \frac{q_{i}}{|q_{i}|} \frac{\overline{q}_{j}}{|q_{j}|} on O′_{i} ∩ O′_{j}.

Alternatively, we may define local trivializations τ_{i} : π^{−1}(O′_{i}) → O′_{i} × Sp(1), with τ_{i}(p) = (π(p), φ_{i}(p)), where π(p) = [p] and

φ : π^{−1}(O′_{i}) → Sp(1), (p_{0}, …, p_{k}) ⟼ \frac{p_{i}}{|p_{i}|}.

The transition functions are then given by T′_{ij}(q) = φ_{i}(q)φ_{j}(q)^{−1}, just as before.

Let {O_{i}} be the induced open cover of ℍℙ^{1} obtained by setting O_{i} = u^{−1}(O′_{i}) for i = 0, …, k, and let σ_{i} = u^{∗}σ′_{i} : O_{i} → P be the corresponding local sections:

σ_{i} = \frac{u}{|u|} \frac{\overline{u}_{i}}{|u_{i}|} on O_{i}.

The induced transition functions T_{ij} = u^{∗}T′_{ij} : O_{i} ∩ O_{j} → Sp(1) of P are:

T_{ij} = \frac{u_{i}}{|u_{i}|} \frac{\overline{u}_{j}}{|u_{j}|} on O_{i} ∩ O_{j}.

Recall that the total space P is

P = {([x, y], p) ∈ ℍℙ^{1} × 𝕊^{4k+3} : u(x, y) = π(p)},

and if π_{2} : ℍℙ^{1} × 𝕊^{4k+3} → 𝕊^{4k+3} denotes the projection onto the second factor, then the induced map u′ on total spaces is given by u′ = π_{2}|_{P} : P → 𝕊^{4k+3}.

If γ ∈ Ω^{1}(𝕊^{4k+3}, 𝔤) denotes the canonical Sp(1)-connection of 𝕊^{4k+3} → ℍℙ^{k}, let ω denote the pull-back connection on P → 𝕊^{4}. Then:

\begin{aligned}γ(q) & = q^{†}dq, q ∈ ℍ^{k+1}, \\ ω & = (u′)^{∗}γ ∈ Ω^{1}(P, 𝔤), \\ ω_{i} & = σ_{i}^{∗}ω = σ_{i}^{†}dσ_{i} ∈ Ω^{1}(O_{i}, 𝔤). \end{aligned}

We have the corresponding local curvature 2-forms:

\begin{aligned}F_{i} & = dω_{i} + ω_{i} ∧ ω_{i} \\ & = dσ_{i}^{†} ∧ dσ_{i} + σ_{i}^{†}dσ_{i} ∧ σ_{i}^{†}dσ_{i} ∈ Ω^{2}(O_{i}, 𝔤). \end{aligned}

Remark 4.1.1. We recall that ADHM matrices A and A′ give isomorphic Sp(1)- bundles P, P′ with connections ω, ω′ if and only if A′ = QAR, where Q ∈ Sp(k+1), R ∈ GL(k, ℝ) [A, p. 62].

Remark 4.1.2. The conformal transformations of 𝕊^{4}, Conf_{0}(𝕊^{4}) = SO(5, 1) ≃ SL(2, ℍ)/{±1}, lift to isomorphisms of the total space P. If the conformal map φ acts on ℍℙ^{1} by φ : [x, y] ↦ [ax + by, cx + dy], then φ^{∗}A(x, y) = (Ca + Dc)x + (Cb + Dd)y, so that C ↦ Ca + Dc and D ↦ Cb + Dd. Hence, given any fixed self-dual connection with ADHM matrices C, D, we may generate a family of self-dual connections parametrized by SO(5, 1). Indeed, the moduli space M_{1} is diffeomorphic to SO(5, 1)/SO(5) and may be generated by pulling back the canonical connection via conformal maps [A-H-S], [D-M-M], [Hab], [Har2], [G-P1], [F-U].

Coordinate patches for the sphere ℍℙ^{1} (or ℍ ∪ {∞}) are given by O_{s} = {[x_{0}, x_{1}] : x_{1} ≠ 0} covering the south pole [0, 1] (or 0), and O_{n} = {[x_{0}, x_{1}] : x ≠ 0} covering the north pole [1, 0] (or ∞), with the standard local coordinate maps:

\begin{aligned}ψ_{s} & : O_{s} → ℍ,  [x_{0}, x_{1}] ↦ x = x_{0}x_{1}^{−1}, \\ ψ_{n} & : O_{n} → ℍ,  [x_{0}, x_{1}] ↦ y = x_{1}x_{0}^{−1}. \end{aligned}

We let O_{si} = O_{s} ∩ O_{i}, O_{ni} = O_{n} ∩ O_{i} denote the refined covering of ℍℙ^{1} with corresponding coordinate maps ψ_{si} = ψ_{s}|O_{si}, ψ_{ni} = ψ_{n}|O_{ni}, respectively.

Example 4.1.3. We describe the more explicit standard form of the ADHM matrix equations, and corresponding local connection 1-forms [A], [C-W-S], [C-G-F-T], [Sa]. By making use of gauge equivalence, the matrices C and D can be chosen to have the following form:

C = ( \begin{matrix}0 \\ −I \end{matrix} ) ,  and  D = ( \begin{matrix}λ \\ B \end{matrix} ) ,

where B is a k × k quaternionic matrix and λ is a k × 1 quaternionic vector. Then in local coordinates, the map A : O_{s} → G_{k}(ℍ^{k+1}) becomes

A(x) = ( \begin{matrix}λ \\ B − xI \end{matrix} ) , x ∈ ℍ.

In terms of B and λ, the conditions on A are now:

(i) B is symmetric and B^{†}B + λ^{†}λ is real;

(ii) Rank_{ℍ} ( \begin{matrix}λ \\ B − x \end{matrix} ) = k, for all x ∈ ℍ.

We choose the section σ_{s} : O_{s} → P to be

σ_{s}(x) = \frac{1}{\sqrt{ρ(x)}} ( \begin{matrix}−1 \\ U(x) \end{matrix} ) ,

where ρ_{s}(x) = 1 + |U(x)|^{2}, and setting U(x) = {λ(B − xI)^{−1}}^{†}. Hence,

ω_{s}(x) = σ_{s}^{†}(x)dσ_{s}(x) = \frac{Im \{U^{†}(x)dU(x)\}}{1 + |U(x)|^{2}} ∈ Ω^{1}(O_{s}, 𝔤),

where σ_{s}, and hence ω_{s}, may have singularities. In particular, every self-dual connection ω on a principal SU(2)-bundle P → 𝕊^{4} with −c_{2}(P)[𝕊^{4}] = k arises from the parameters λ, B satisfying (i) and (ii), the local connection 1-form ω_{s} being given by the above formula. The connections defined by (λ, B) and (λ′, B′) are gauge equivalent if and only if λ′ = aλT, B′ = T^{−1}BT with a ∈ Sp(1) and T ∈ O(k, ℝ) [A, p. 26].

Example 4.1.5. If k = 1, the corresponding local connection 1-form is

ω_{s} = \frac{λ^{2}}{|x − b|^{2}(λ^{2} + |x − b|^{2})}Im \{(\overline{x − b})dx\} ,

where λ > 0 and b ∈ ℍ, so that ω_{s} is specified by 5 real parameters. Setting λ = 1 and b = 0, we obtain the canonical Sp(1)-connection on the universal quaternionic line bundle L → ℍℙ^{1}, with local connection and curvature forms:

ω_{s} = \frac{1}{|x|^{2}(1 + |x|^{2})}Im \{\overline{x}dx\}  and  ω_{n} = \frac{1}{(1 + |y|^{2})}Im \{d\overline{y}\} ,
F_{s} = \frac{dx ∧ d\overline{x}}{(1 + |x|^{2})}  and  F_{n} = \frac{dy ∧ d\overline{y}}{(1 + |y|^{2})}.

The Chern-Weil formula gives −c_{2}(L)[𝕊^{4}] = 1. By reversing the orientation on ℍℙ^{1}, we obtain a quaternionic line bundle L → ℍℙ^{1} with −c_{2}(L)[𝕊^{4}] = −1.

Remark 4.1.6. For the case k = 2, the pair (λ, B) provides 20 real parameters, but the corresponding formulas for the local connection 1-forms are much less convenient for computational purposes.

Example 4.1.7. Let b_{1}, …, b_{k} be distinct centres in ℍ ⊂ 𝕊^{4} and let λ_{1}, …, λ_{k} in (0, ∞) be positive scales. By choosing B = diag (b_{1}, …, b_{k}) and λ = (λ_{1}, …, λ_{k}), with λ_{i} ∈ (0, ∞), we obtain the relatively simple t’Hooft multi-instanton solution [A], [C-W-S], [J-N-R]:

ω_{s} = ∑_{i=1}^{k} \frac{λ_{i}^{2}}{|x − b_{i}|^{4}ρ}Im \{(\overline{x − b}_{i})dx\} , where ρ(x) = 1 + ∑_{i=1}^{k} \frac{λ_{i}^{2}}{|x − b_{i}|^{2}} .

Unfortunately, this only gives a 5k-parameter family of self-dual SU(2)-connections and hence does not give a parametrization of M_{k} for k > 1 since the moduli space M_{k} has dimension 8k − 3 [A-H-S].

Example 4.1.8. The set of t’Hooft multi-instantons described above is not closed under the action of the conformal group, and we may generate new self-dual connections by applying conformal transformations to 𝕊^{4} = ℍ∪{∞}. A (5k+4)-parameter family of self-dual connections on 𝕊^{4} for any k ≥ 1 was discovered in this way by Jackiw, Nohl and Rebbi [J-N-R]. Let {a_{0}, …, a_{k}} be distinct points in ℍ ⊂ 𝕊^{4}, and let {λ_{0}, …, λ_{k}} be positive weights in (0, ∞). The corresponding Jackiw-Nohl-Rebbi (JNR) local connection 1-form is given by:

ω_{s} = ∑_{i=0}^{k} \frac{λ_{i}^{2}}{|x − a_{i}|^{4}ρ}Im \{(\overline{x − a}_{i})dx\} , where ρ(x) = ∑_{i=0}^{k} \frac{λ_{i}^{2}}{|x − a_{i}|^{2}} .

The expression for ω_{s} is clearly homogeneous in the λ_{i}’s and so one parameter is removable by a common rescaling or by setting one of the λ_{i}’s equal to 1. For k ≥ 3, one obtains a (5k + 4)-dimensional family of self-dual connections with −c_{2}(P)[𝕊^{4}] = k, so these solutions do not give all self-dual connections when k ≥ 3 [J-N-R]. Hartshorne has proven that the JNR family of solutions give all self-dual SU(2)-connections when k = 1 or 2 [Har2].

The space M_{1} is 5-dimensional, and so four of the JNR parameters must be removable by gauge equivalence. The space M_{2} is a 13-dimensional manifold [A-H-S], and so one of the JNR parameters must be removable by gauge equivalence. The residual gauge symmetry has been interpreted in [J-N-R] as a motion of the points a_{0}, a_{1}, a_{2} around the circle in ℍ ∪ {∞} determined by those points [J-N-R]. In the next section, we will discuss the case k = 2 in more detail.

§4.2. Parametrization of Connection One-forms with k = 2

We recall how the JNR family of self-dual SU(2)-connections may be obtained from the ADHM construction by making a suitable choice of matrices. We will then be able to obtain the non-singular local connection 1-forms using the ADHM construction as a guide. For k = −c_{2} = 2, the JNR family gives all self-dual SU(2)-connections over 𝕊^{4}. General references for this section are [A-W], [C-G-F-T], [J-N-R], [O], [Sa], [Sc], [Wa-We].

We want to choose ADHM matrices corresponding to the JNR family of selfdual connections on an SU(2)-bundle P → 𝕊^{4} with −c_{2}(P) = k. This family is parametrized by the following data: k + 1 distinct points {P_{0}, …, P_{k}} in 𝕊^{4} and k + 1 positive weights {λ_{0}, …, λ_{k}} in (0, ∞).

Let \underline{ℍℙ}^{1} = \{(x, y) ∈ ℍ^{2} \ (0, 0) : (x, y) ∼ (ax, ay), a ∈ ℍ^{∗}\} denote the left projective space, while ℍℙ^{1} denotes the right projective space. (We have a map \underline{ℍℙ}^{1} → ℍℙ^{1}, [a, b] → [b^{−1}, a^{−1}]. If we replace ℍℙ^{1} by ℂℙ^{1}, then this map reduces to the identity map on ℂℙ^{1}.) The ADHM matrices C and D which give rise to the JNR SU(2)-connections are easily identifed [C-G-F-T, p. 41], [O, p. 413]. We choose C and D to be of the following form:

C = ( \begin{matrix}b_{0} & b_{0} & ⋯ & b_{0} \\ −b_{1} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & ⋯ & −b_{k} \end{matrix} ) , D = ( \begin{matrix}−a_{0} & −a_{0} & ⋯ & −a_{0} \\ a_{1} & 0 & ⋯ & 0 \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & ⋯ & a_{k} \end{matrix} ) ,

where the (a_{i}, b_{i}) are distinct points in ℍ^{2}\(0, 0). Hence, we have the corresponding matrix A(x, y) = Cx + Dy:

A(x, y) = ( \begin{matrix}b_{0}x − a_{0}y & b_{0}x − a_{0}y & ⋯ & b_{0}x − a_{0}y \\ −b_{1}x + a_{1}y & 0 & ⋯ & 0 \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & ⋯ & −b_{k}x + a_{k}y \end{matrix} ) .

Then, the corresponding points in 𝕊^{4} are given by \underline{P}_{i} = [a_{i}, b_{i}] ∈ \underline{ℍℙ}^{1}, and the weights are given by λ_{i} = (|a_{i}|^{2} + |b_{i}|^{2})^{−1/2}. We see that A^{†}(x, y)A(x, y) is real and Rank_{ℍ}A(x, y) = k for all (x, y) ∈ ℍ^{2} \ (0, 0), provided the points P_{i} are all distinct.

It is more convenient to define a matrix A(x, y) corresponding to parameters κ_{i} and (α_{i}, β_{i}), i = 0, …, k, as follows:

A(x, y) = ( \begin{matrix}κ_{0}^{−1}(β_{0}x − α_{0}y) & κ_{0}^{−1}(β_{0}x − α_{0}y) & ⋯ & κ_{0}^{−1}(β_{0}x − α_{0}y) \\ −κ_{1}(β_{1}x − α_{1}y) & 0 & ⋯ & 0 \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & ⋯ & −κ_{k}^{−1}(β_{k}x − α_{k}y) \end{matrix} ) .

Then, if we are given \underline{P}_{i} = [a_{i}, b_{i}] ∈ \underline{ℍℙ}^{1} and λ_{i} ∈ (0, ∞), we may choose:

\begin{aligned}κ_{i} & = λ_{i}, \\ (α_{i}, β_{i}) & = \frac{(a_{i}, b_{i})}{\sqrt{|a_{i}|^{2} + |b_{i}|^{2}}} ∈ 𝕊^{7}, \end{aligned}

The matrix A(x, y) will now be determined up to left multiplication by an element of Sp(k + 1): if (a_{i}, b_{i}) ↦ (q_{i}a_{i}, q_{i}b_{i}), with q_{i} ∈ ℍ^{∗}, then A ↦ QA, where Q = diag (q_{0}/|q_{0}|, …, q_{k}/|q_{k}|) ∈ Sp(k + 1). Hence, the data λ_{0}, …, λ_{k} ∈ (0, ∞) and P_{0}, …, P_{k} ∈ 𝕊_{4} determines A up to isomorphism via the above assignment.

Example 4.2.1. We recover the JNR local connection 1-form given previously, by choosing (α_{i}, β_{i}) = (1, a_{i}) and κ_{i} = λ_{i}, for i = 0, …, k, where the a_{i} are distinct points in ℍ and the λ_{i} are positive weights.

Remark 4.2.2. We recall that if φ in SL(2, ℍ)/{±1} is a conformal transformation of ℍℙ^{1} represented by φ : [x, y] → [ax + by, cx + dy], then φ^{∗}A(x, y) = C′x + D′y, where C′ = Ca + Dc, D′ = Cb + Dd. In particular, we see that (a_{i}, b_{i}) ↦ (a′_{i}, b′_{i}) = (a_{i}c − b_{i}a, −a_{i}d + b_{i}b), and so the set of matrices for the JNR solutions is closed under the induced action of SL(2, ℍ).

The classifying map u : ℍℙ^{1} → ℍℙ^{k} is defined by the following conditions:

\begin{aligned}u(x, y)^{†}A(x, y) & = 0, \\ u(x, y)^{†}u(x, y) & = 1, for all (x, y) ∈ ℍ^{2} \ (0, 0). \end{aligned}

Hence,

u(x, y) = \frac{1}{\sqrt{ρ(x, y)}} ( κ_{0}(\overline{β_{0}x − α_{0}y})^{−1} ; κ_{1}(\overline{β_{1}x − α_{1}y})^{−1} ; ⋮ ; κ_{k}(\overline{β_{k}x − α_{k}y})^{−1} ) , ρ(x, y) = ∑_{m=0}^{k} \frac{κ_{m}^{2}}{|β_{m}x − α_{m}y|^{2}} .

Denoting u = [u_{0}, …, u_{k}] ∈ ℍℙ^{k}, we have

\begin{aligned}u_{i}(x, y) & = \frac{κ_{i}(β_{i}x − α_{i}y)}{\sqrt{ρ(x, y)}|β_{i}x − α_{i}y|^{2}} \\ & = \frac{κ_{i}(β_{i}x − α_{i}y)}{|β_{i}x − α_{i}y|} (κ_{i}^{2} + ∑_{m≠i} κ_{m}^{2} \frac{|β_{i}x − α_{i}y|^{2}}{|β_{m}x − α_{m}y|^{2}} )^{−1/2} , \end{aligned}

for i = 0, …, k. Clearly, u has singularities at the points {P_{0}, …, P_{k}}, but we can now see that these are removable [G-H, p. 490]. Define an open cover of ℍℙ^{1} by choosing O_{i} = ℍℙ^{1} \ {P_{0}, …, P̂_{i}, …, P_{k}} for i = 0, …, k, where the caret denotes an omission. Then, each patch O_{i} covers the point P_{i} and each u_{i} has singularities at P_{j} for j = 0, …, k.

We extend u on the patches O_{i}, for i = 0, …, k, by defining: ũ|O_{i} = u\overline{g}_{i}, where g_{i} is defined by

g_{i} : π^{−1}(O_{i} \ \{P_{i}\}) ⟶ Sp(1), (x, y) ⟼ \frac{u_{i}(x, y)}{|u_{i}(x, y)|} = \frac{β_{i}x − α_{i}y}{|β_{i}x − α_{i}y|},

and π : ℍ^{2} \ (0, 0) → ℍℙ^{1} is the natural projection. Then, the extension ũ is welldefined on overlapping patches, since [ũ] = [u\overline{g}_{i}] = [u\overline{g}_{j}] on O_{i} ∩ O_{j}. Hence, we obtain a smooth extension ũ : ℍℙ^{1} → ℍℙ^{k}. If there is no ambiguity, we will often denote the extension ũ simply by u.

Let O′_{i} = {[q_{0}, …, q_{k}] ∈ ℍℙ^{k} : q_{i} ≠ 0}, for i = 0, …, k, denote the collection of standard coordinate patches for ℍℙ^{k}. More explicitly, we have for j ≠ i:

ũ_{j}|O_{i} = \frac{κ_{j}(β_{j}x − α_{j}y)(\overline{β_{i}x − α_{i}y})}{|β_{j}x − α_{j}y|^{2}} (κ_{i}^{2} + ∑_{m≠i} κ_{m}^{2} \frac{|β_{i}x − α_{i}y|^{2}}{|β_{m}x − α_{m}y|^{2}} )^{−1/2} ,

which is smooth at P_{i} — and so smooth on O_{i} — with singularities at the points P_{j}, for j ≠ i. Next, we have:

\begin{aligned}ũ_{i}|O_{i} & = κ_{i} (κ_{i}^{2} + ∑_{m≠i} κ_{m}^{2} \frac{|β_{i}x − α_{i}y|^{2}}{|β_{m}x − α_{m}y|^{2}} )^{−1/2} and, \\ ũ_{i}|O_{j} & = \frac{κ_{i}(β_{i}x − α_{i}y)(\overline{β_{j}x − α_{j}y})}{|β_{i}x − α_{i}y|^{2}} (κ_{j}^{2} + ∑_{m≠j} κ_{m}^{2} \frac{|β_{j}x − α_{j}y|^{2}}{|β_{m}x − α_{m}y|^{2}} )^{−1/2} , \end{aligned}

for j ≠ i. Then ũ_{i} is smooth at P_{i} and zero at all points P_{m}, for m ≠ i. Hence, ũ^{−1}(O′_{i}) = {[x, y] ∈ ℍℙ^{1} : u_{i}(x, y) ≠ 0} = O_{i}, where we recall that O_{i} ⊂ ℍℙ^{1} was defined by O_{i} = ℍℙ^{1} \ {P_{0}, …, P̂_{i}, …, P_{k}}.

Next, we give the transition functions, T_{ij} = u^{∗}T′_{ij}, and local sections, σ_{i} = u^{∗}σ′_{i}, of the pullback bundle P = u^{∗}𝕊^{4k+3}. We have

\begin{aligned}T_{ij}(x, y) & = \frac{u_{i}(x, y)}{|u_{i}(x, y)|} \frac{\overline{u}_{j}(x, y)}{|u_{j}(x, y)|} \\ & = \frac{(β_{i} − α_{i}y)}{|β_{i} − α_{i}y|} \frac{(\overline{β_{j} − α_{j}y})}{|β_{j} − α_{j}y|} on O_{i} ∩ O_{j}. \end{aligned}

The local sections σ_{i} : O_{i} → P are

σ_{i}(x, y) = u(x, y)\overline{g}_{i}(x, y) = u(x, y)\frac{\overline{u}_{i}(x, y)}{|u_{i}(x, y)|} on O_{i},

and σ_{j} = σ_{i}T_{ij} on overlaps. The local connection 1-forms are then given by

ω_{i} = σ_{i}^{∗}ω = σ_{i}^{†}dσ_{i} ∈ Ω^{1}(O_{i}, 𝔤).

We again note that the effect of replacing (α_{i}, β_{i}) by (q_{i}α_{i}, q_{i}β_{i}), for i = 0, …, k, where q_{i} are constant elements of Sp(1), is simply to replace the transition functions T_{ij} by q_{i}T_{ij}q_{j}^{−1} and this corresponds to an Sp(1)-bundle isomorphism. Hence, the the bundle P and connection ω corresponding to the points (α_{i}, β_{i}) in 𝕊^{7}, are equivalent to the bundle with connection corresponding to the points (q_{i}α_{i}, q_{i}β_{i}).

With respect to the cover {O_{si}, O_{ni}} of ℍℙ^{1}, we have 2k + 2 smooth local connection 1-forms:

\begin{aligned}ω_{si} & = σ_{si}^{∗}ω = σ_{si}^{†}dσ_{si} ∈ Ω^{1}(O_{si}, 𝔤), \\ ω_{ni} & = σ_{ni}^{∗}ω = σ_{ni}^{†}dσ_{ni} ∈ Ω^{1}(O_{ni}, 𝔤), \end{aligned}

where σ_{si} = σ_{i}|O_{si} and σ_{ni} = σ_{i}|O_{ni}.

Lemma 4.2.3. The local 1-form ω_{si} is given by:

\begin{matrix}ω_{si}(x) = \frac{ρ_{i}}{ρ|β_{i}x − α_{i}|^{2}} Im \{(β_{i}x − α_{i})\overline{β_{i}dx}\} \\ + ∑_{{j=0 \atop j≠i}}^{k} \frac{κ_{j}^{2}}{ρ|β_{j}x − α_{j}|^{4}|β_{i}x − α_{i}|^{2}} (β_{i}x − α_{i})Im \{(\overline{β_{j}x − α_{j}})β_{j}dx\} (\overline{β_{i}x − α_{i}}), \end{matrix}

where

ρ_{i}(x) = ∑_{{j=0 \atop j≠i}}^{k} \frac{κ_{j}^{2}}{|β_{j}x − α_{j}|^{2}} ,

and ρ(x) = ρ(x, 1). The local 1-form ω_{ni} is given by:

\begin{matrix}ω_{ni}(y) = \frac{−ρ_{i}}{ρ|β_{i} − α_{i}y|^{2}} Im \{(β_{i} − α_{i}y)\overline{α_{i}dy}\} \\ + ∑_{{j=0 \atop j≠i}}^{k} \frac{κ_{j}^{2}}{ρ|β_{j} − α_{j}y|^{4}|β_{i} − α_{i}y|^{2}} (β_{i} − α_{i}y)Im \{(\overline{β_{j} − α_{j}y})α_{j}dy\} (\overline{β_{i} − α_{i}y}), \end{matrix}

where

ρ_{i}(y) = ∑_{{j=0 \atop j≠i}}^{k} \frac{κ_{j}^{2}}{|β_{j} − α_{j}y|^{2}} ,

and ρ(y) = ρ(1, y).

Proof. Calculation. □

Let O′ = ℍℙ^{1} \ {P_{0}, …, P_{k}} and let O′_{s} = O_{s} ∩ O′, O′_{n} = O_{n} ∩ O′, and define local sections σ_{s} : O′_{s} → P, σ_{n} : O′_{n} → P by:

\begin{aligned}σ_{s}(x, y) & = u(x, y)\frac{\overline{x}}{|x|}  on O′_{s}, \\ σ_{n}(x, y) & = u(x, y)\frac{\overline{y}}{|y|}  on O′_{n}. \end{aligned}

Let ω_{s} = σ_{s}^{∗}ω ∈ Ω^{1}(O′_{s}, 𝔤) and ω_{n} = σ_{n}^{∗}ω ∈ Ω^{1}(O′_{n}, 𝔤) denote the corresponding local connection 1-forms.

Lemma 4.2.4. The local connection 1-forms ω_{s}, ω_{n} are given by:

ω_{s} = ∑_{i=0}^{k} \frac{κ_{i}^{2}}{ρ|β_{i}x − α_{i}|^{4}} Im \{(\overline{β_{i}x − α_{i}})β_{i}dx\} ,

where ρ(x) = ρ(x, 1) ∑_{i=0}^{k} \frac{κ_{i}^{2}}{|β_{i}x − α_{i}|^{2}} ,

ω_{n} = ∑_{i=0}^{k} \frac{−κ_{i}^{2}}{ρ|β_{i} − α_{i}y|^{4}} Im \{(\overline{β_{i} − α_{i}y})α_{i}dy\} ,

where ρ(y) = ρ(1, y) ∑_{i=0}^{k} \frac{κ_{i}^{2}}{|β_{i} − α_{i}y|^{2}} .

Proof. Calculation. □

The connection 1-forms ω_{s}, ω_{n} have simple real poles at the points {P_{0}, …, P_{k}}. In order to estimate the lengths of tangent vectors in T M_{2}, we need smooth local connection 1-forms corresponding to a cover of 𝕊^{4}.

Lemma 4.2.5. The local curvature 2-form F_{s} = dw_{s} + ω_{s} ∧ ω_{s} ∈ Ω^{1}(O′_{s}, 𝔤) corresponding to the parameters κ_{i}, (α_{i}, 1) is given by:

\begin{aligned}F_{s} = & ∑_{i≠j} \frac{κ_{i}^{2}κ_{j}^{2}}{ρ^{2}|x − α_{i}|^{6}|x − α_{j}|^{2}} (\overline{x − α_{i}})dx ∧ d\overline{x}(x − α_{i}) \\ & + ∑_{i≠j} \frac{κ_{i}^{2}κ_{j}^{2}}{ρ^{2}|x − α_{i}|^{4}|x − α_{j}|^{4}} (\overline{x − α_{i}})dx ∧ d\overline{x}(x − α_{j}), \end{aligned}

and similarly for F_{n} ∈ Ω^{2}(O′_{n}, 𝔤).

Remark 4.2.6. There is no direct relationship between the k + 1 weights λ_{i} and k + 1 points P_{i} arising in the JNR family of self-dual connections and the scales and centres of curvature-density concentration arising in the Donaldson-Uhlenbeck description of neighbourhoods of the boundary of moduli space [D-K, p. 156] Definition 4.2.7. We let T̃_{k} denote the space of unordered pairs (λ_{i}, P_{i}), i = 0, …, k, where λ_{i} ∈ (0, ∞), P_{i} ∈ 𝕊^{4}, and (λ_{0}, …, λ_{k}) ∼ ν(λ_{0}, …, λ_{k}), for ν > 0.

The spaces T̃_{k} parametrize the family of JNR connections. We recall the

Theorem 4.2.8. (Hartshorne) The spaces T̃_{k} parametrize the moduli spaces M_{k} of self-dual SU(2)-connections on 𝕊^{4} for k = 1, 2.

Proof. [Har2, 3], [A-W], [Wa-We]. □

Remark 4.2.9. In particular, when k = 2, the induced ADHM map from T̃_{2} to M_{2} is surjective.

If we normalize the weights λ_{i} so that λ_{0}^{2} +λ_{1}^{2} +λ_{2}^{2} = 1 and λ_{i} ∈ (0, 1), then one approaches the boundary of T̃_{2} by letting P_{i} → P_{j} or λ_{i} → 0. For comparison, we recall the description of the Uhlenbeck-Donaldson topological compactification of M_{k} [D-K, p. 156]. An ideal self-dual connection on an SU(2)-bundle P → 𝕊^{4} with −c_{2}(P) = k is a pair ([ω], x_{1}, …, x_{l}), where [ω] is a point in M_{k−l} and (x_{1}, …, x_{l}) is an unordered l-tuple of points of 𝕊^{4}. The curvature density of ([ω], x_{1}, …, x_{l}) is the measure

|F_{ω}|^{2} + 8π^{2} ∑_{i=1}^{l} δ_{x_{i}}.

By defining weak convergence of gauge equivalence classes [ω_{α}] to an ideal selfdual connection ([ω], x_{1}, …, x_{l}) in terms of convergence of the curvature densities as measures, one may topologize the set of ideal self-dual connections IM_{k}. The moduli space M_{k} is embedded as an open subset of of IM_{k} and its closure \overline{M}_{k} is then compact. The space IM_{k} has the following stratification:

IM_{k} = M_{k} ∪ M_{k−1} × 𝕊^{4} ∪ M_{k−2} × s^{2}(𝕊^{4}) ∪ … ∪ M_{1} × s^{k−1}(𝕊^{4}) ∪ M_{0} × s^{k}(𝕊^{4}),

where s^{i}(𝕊^{4}), and M_{0} consists of a single point corresponding to the product connection on the trivial bundle 𝕊^{4} × Sp(1) → 𝕊^{4}.

CHAPTER V ASYMPTOTIC BEHAVIOUR OF THE L^{2} METRIC

Using the parametrization of M_{2} by the parameter space T̃_{2} due to Hartshorne and Jackiw, Nohl, and Rebbi, we compute the corresponding tangent vectors ∂ω/∂t_{α}. We then derive estimates for the components of the L^{2} metric g, the principal question being their asymptotic behaviour as one approaches the boundary of moduli space. With these estimates at hand, it is then easy to see that the space M_{2} has finite volume with respect to its L^{2} metric g, and in particular, that the distance to the boundary is finite.

To obtain the required estimates, we first need to choose appropriate local coordinate maps, for the base manifold 𝕊^{4}, and for the parameter space T̃_{2}.

§5.1. Local Coordinate Patches on Moduli Space

The parameter space T̃_{k} is the space of unordered pairs (λ_{0}, P_{0}), …, (λ_{k}, P_{k}), with P_{i} ∈ 𝕊^{4} distinct points and weights λ_{i} ∈ (0, ∞), modulo rescaling. The parameter space T_{k} denotes the space of ADHM matrices modulo Sp(k+1)×GL(k, ℝ) and the ADHM map provides a diffeomorphism T_{k} → M_{k}. For k = 2, we have submersions T̃_{2} → T_{2} and T̃_{2} → M_{2}.

Given points \underline{P}_{i} with homogeneous coordinates [a_{i}, b_{i}] ∈ \underline{ℍℙ}^{1}, we select corresponding representatives (α_{i}, β_{i}) in 𝕊^{7}. Cover \underline{ℍℙ}^{1} with the overlapping hemispheres

\begin{aligned}\underline{O}_{s}(R_{s}) & = \{[a, b] ∈ \underline{ℍℙ}^{1} : b ≠ 0, |b^{−1}a| < R_{s}\}, \\ \underline{O}_{n}(R_{n}) & = \{[a, b] ∈ \underline{ℍℙ}^{1} : a ≠ 0, |a^{−1}b| < R_{n}\}, \end{aligned}

where 1 < R_{s}, R_{n} ≤ ∞ and \underline{O}_{s}(∞), \underline{O}_{n}(∞) are simply denoted by \underline{O}_{s}, \underline{O}_{n}. Usually we take R_{s} = R_{n} = 2. The equator of \underline{ℍℙ}^{1} is the sphere 𝕊^{3} given by \{[a, b] ∈ \underline{ℍℙ}^{1} : a ≠ 0, |a^{−1}b| = 1} and similarly for ℍℙ^{1}. We have the standard local coordinate maps

\begin{matrix}\underline{O}_{s} ∋ [a, b] ⟼ b^{−1}a ∈ 𝔹(0, R_{s}), \\ \underline{O}_{n} ∋ [a, b] ⟼ a^{−1}b ∈ 𝔹(0, R_{n}), \end{matrix}

where 𝔹(c, R) denotes the Euclidean ball {x ∈ ℍ : |x − c| < R}. We define sections of the Sp(1)-bundle 𝕊^{7} → \underline{ℍℙ}^{1} by

\begin{matrix}\underline{O}_{s} ∋ [a, b] ⟼ \frac{\overline{b}}{|b|} \frac{(a, b)}{\sqrt{|a|^{2} + |b|^{2}}} ∈ 𝕊^{7}|\underline{O}_{s}, \\ \underline{O}_{n} ∋ [a, b] ⟼ \frac{\overline{a}}{|a|} \frac{(a, b)}{\sqrt{|a|^{2} + |b|^{2}}} ∈ 𝕊^{7}|\underline{O}_{n}. \end{matrix}

Combining these coordinate maps and local sections, we have maps

\begin{matrix}𝔹(0, R_{s}) ∋ a ⟼ \frac{(a, 1)}{\sqrt{1 + |a|^{2}}} = (α, β) ∈ 𝕊^{7}|\underline{O}_{s}, \\ 𝔹(0, R_{n}) ∋ b ⟼ \frac{(1, b)}{\sqrt{1 + |b|^{2}}} = (α, β) ∈ 𝕊^{7}|\underline{O}_{n}. \end{matrix}

We let {c_{i}^{μ} : i = 0, 1, 2, μ = 0, …, 3} denote local coordinates for the points P_{i} in 𝕊^{4}, the corresponding tangent vectors in T M_{2} being denoted by ∂ω/∂c_{i}^{μ}, where c_{i}^{μ} = a_{i}^{μ} on the southern hemisphere and c_{i}^{μ} = b_{i}^{μ} on the northern hemisphere.

The positive weights λ_{0}, λ_{1}, λ_{2} lie in the positive cone in ℝℙ^{2} and so we may assume for convenience that (λ_{0}, λ_{1}, λ_{2}) lies in the unit sphere 𝕊^{2}. We then obtain local coordinates on the space T̃_{2}; the map T̃_{2} → T_{2} has already been described, and the ADHM map provides a diffeomorphism T_{2} → M_{2}.

§5.2. Local Coordinate Patches on the Four-Sphere

We consider the definition of the L^{2} inner product on Ω^{1}(𝕊^{4}, ad P) over an appropriate system of coordinate patches. We have assumed that the Riemannian metric g of the four-sphere 𝕊^{4} is globally conformally equivalent to the standard round metric g_{0}. Let d_{g}( , ) denote the distance function on the four-sphere corresponding to the metric g, and let d( , ) be the distance function for the round metric g_{0}. By compactness, we may fix a constant K_{g} ≥ 1 such that

K_{g}^{−1}d(Q, Q′) ≤ d_{g}(Q, Q′) ≤ K_{g}d(Q, Q′) for all Q, Q′ ∈ 𝕊^{4}.

Our metric component estimates are most conveniently expressed in terms of the weight parameters λ_{i} and the distances between distinct points P_{i}, P_{j}. Since the distance functions d_{g}, d are equivalent and our estimate calculations are now entirely local, we may for convenience assume at the outset that the four-sphere has its standard round metric g_{0} with radius 1.

The k + 1 distinct points P_{i} may be covered by disjoint small balls

B(P_{i}, ε_{i}) = \{Q ∈ ℍℙ^{1} : d(Q, P_{i}) < ε_{i}\}, where ε_{i} = \frac{1}{2} min_{j≠i}\{1, d(P_{i}, P_{j})\}.

The coordinate maps ψ_{s}, ψ_{n} pull back the sphere metric g to define a metric on the Euclidean balls 𝔹(0, R_{s}), 𝔹(0, R_{n}) in ℍ which is equivalent to the standard flat metric. Hence, if z = ψ(Q), with ψ = ψ_{s} or ψ_{n}, then the distance function d satisfies

K^{−1}|z − z′| ≤ d(Q, Q′) ≤ K|z − z′|,

where K is a constant.

Let {O_{si}, O_{ni}} be the open cover of ℍℙ^{1} described earlier, with O_{si} = O_{s} ∩ O_{i}, O_{ni} = O_{n} ∩ O_{i}, and corresponding local coordinate maps ψ_{si} = ψ_{s}|O_{si}, ψ_{ni} = ψ_{n}|O_{ni}. The patches O_{s}, O_{n} are obtained from the four-sphere by deleting the north and south poles respectively, and the patches O_{i} are obtained by deleting all points P_{j} with j ≠ i. We choose a partition of unity {χ_{i}} on ℍℙ^{1} subordinate to the cover {O_{i}} so that

(i) supp χ_{i} ⊂ B(P_{i}, K^{−2}ε_{i}) and χ_{i} ≡ 1 on \overline{B}(P_{i}, \frac{1}{2}K^{−2}ε_{i}) for i = 0, …, k; (ii) supp χ_{k+1} ⊂ ℍℙ^{1} \ ⋃_{i=0}^{k} B(P_{i}, \frac{1}{2}K^{−2}ε_{i}) and χ_{k+1} ≡ 1 on

ℍℙ^{1} \ ⋃_{i=0}^{k} B(P_{i}, K^{−2}ε_{i}); (iii) ∑_{i=0}^{k+1} χ_{i} = 1.

Suppose the points Q, P_{i} lie in the southern hemisphere O_{s} and ψ_{s}(Q) = x, ψ_{s}(P_{i}) = a_{i}. If Q ∈ supp χ_{i}, then x ∈ 𝔹(a_{i}, δ_{i}), where δ_{i} = \frac{1}{2} min_{j≠i}\{1, |a_{i}−a_{j}|\} and i = 0, …, k. Similarly, if Q ∈ supp χ′, then |x − a_{j}| ≥ \frac{1}{2}K^{−4}δ_{j} for all j = 0, …, k. The analogous statements hold if we assume that the points Q, P_{i} lie in the northern hemisphere O_{n}. Finally, we let χ_{s}, χ_{n} be a partition of unity subordinate to the cover {O_{s}, O_{n}}, so that supp χ_{s} ⊂ O_{s}, supp χ_{n} ⊂ O_{n}, and χ_{s} + χ_{n} = 1. We then set χ_{si} = χ_{s}χ_{i}, χ_{ni} = χ_{n}χ_{i}, to give the required partition of unity for ℍℙ^{1}.

Given tangent vectors [α_{1}], [α_{2}] ∈ T_{[ω]}M_{k}, their inner product with respect to the Riemannian metric g on M_{k} is computed by

g_{[ω]}([α_{1}], [α_{2}])

= ∫_{𝕊^{4}}⟨h_{ω}α_{1}, h_{ω}α_{2}⟩\sqrt{g}dx

= ∑_{i=0}^{k+1} ∫_{O_{si}} χ_{si}⟨h_{ω}α_{1}, h_{ω}α_{2}⟩\sqrt{g}dx + ∑_{i=0}^{k+1} ∫_{O_{ni}} χ_{ni}⟨h_{ω}α_{1}, h_{ω}α_{2}⟩\sqrt{g}dx.

The tangent space T_{[ω]}M_{2} is spanned by the representatives ∂ω/∂λ_{i} and ∂ω/∂c_{j}^{μ} for i, j = 0, 1, 2, and μ = 0, …, 3. These representatives are not necessarily horizontal and it appears to be difficult to explicitly compute the horizontal projections when k is greater than 1. Nonetheless, we can still obtain upper bounds for the metric components, and that will be our strategy in the sequel.

§5.3. Tangent Vectors to Moduli Space

We obtain formulas for the local 1-forms in Ω^{1}(𝕊^{4}, ad P) representing tangent vectors to the moduli space M_{2}. In the following lemmas, we record our formulas for the local 1-forms representing the tangent vectors corresponding to the parameters κ_{i} > 0 and (α_{i}, β_{i}) ∈ ℍ^{2} \ (0, 0).

Lemma 5.3.1. The 1-form ∂ω_{s}/∂κ_{i} is smooth on ℍ:

\frac{∂ω_{s}}{∂κ_{i}} = \frac{2κ_{i}ρ_{i}}{ρ^{2}|β_{i}x − α_{i}|^{4}} Im \{(\overline{β_{i}x − α_{i}})β_{i}dx\}

− ∑_{{j=0 \atop j≠i}}^{k} \frac{2κ_{i}κ_{j}^{2}}{ρ^{2}|β_{i}x − α_{i}|^{2}|β_{j}x − α_{j}|^{4}} Im \{(\overline{β_{j}x − α_{j}})β_{j}dx\} ,

with ρ_{i}(x) = ∑_{{j=0 \atop j≠i}}^{k} \frac{κ_{j}^{2}}{|β_{j}x − α_{j}|^{2}} and ρ = ρ(x, 1).

Proof. Calculation. □ Lemma 5.3.2. The 1-form ∂ω_{n}/∂κ_{i} is smooth on ℍ:

\frac{∂ω_{n}}{∂κ_{i}} = −\frac{2κ_{i}ρ_{i}}{ρ^{2}|β_{i} − α_{i}y|^{4}} Im \{(\overline{β_{i} − α_{i}y})α_{i}dy\}

+ ∑_{{j=0 \atop j≠i}}^{k} \frac{2κ_{i}κ_{j}^{2}}{ρ^{2}|β_{i} − α_{i}y|^{2}|β_{j} − α_{j}y|^{4}} Im \{(\overline{β_{j} − α_{j}y})α_{j}dy\} ,

with ρ_{i}(y) = ∑_{{j=0 \atop j≠i}}^{k} \frac{κ_{j}^{2}}{|β_{j} − α_{j}y|^{2}} and ρ = ρ(1, y).

Proof. Calculation. □ Lemma 5.3.3. The 1-forms ∂ω_{s}/∂α_{i}^{μ} and ∂ω_{n}/∂α_{i}^{μ} are smooth on ℍ \ {β_{i}^{−1}α_{i}}, with real poles at x = β_{i}^{−1}α_{i}:

(i)  \frac{∂ω_{s}}{∂α_{i}^{μ}} = \frac{4κ_{i}^{2}(β_{i}x − α_{i})^{μ}}{ρ|β_{i}x − α_{i}|^{6}} Im \{(\overline{β_{i}x − α_{i}})β_{i}dx\}

− \frac{κ_{i}^{2}}{ρ|β_{i}x − α_{i}|^{4}} Im \{\overline{e}_{μ}β_{i}dx\}

− ∑_{j=0}^{k} \frac{2κ_{i}^{2}κ_{j}^{2}(β_{i}x − α_{i})^{μ}}{ρ^{2}|β_{i}x − α_{i}|^{4}|β_{j}x − α_{j}|^{4}} Im \{(\overline{β_{j}x − α_{j}})β_{j}dx\} .

(ii)  \frac{∂ω_{s}}{∂β_{i}^{μ}} = −\frac{4κ_{i}^{2}Re\{(\overline{β_{i}x − α_{i}})e_{μ}x\}}{ρ|β_{i}x − α_{i}|^{6}}Im \{(\overline{β_{i}x − α_{i}})β_{i}dx\}

+ \frac{κ_{i}^{2}}{ρ|β_{i}x − α_{i}|^{4}} Im \{(2β_{i}^{μ}\overline{x} − \overline{α}_{i}e_{μ})dx\}

+ ∑_{j=0}^{k} \frac{2κ_{i}^{2}κ_{j}^{2}Re\{(\overline{β_{i}x − α_{i}})e_{μ}x\}}{ρ^{2}|β_{i}x − α_{i}|^{4}|β_{j}x − α_{j}|^{4}}Im \{(\overline{β_{j}x − α_{j}})β_{j}dx\} .

Proof. Calculation. □ Lemma 5.3.4. The 1-forms ∂ω_{n}/∂α_{i}^{μ} and ∂ω_{n}/∂α_{i}^{μ} are smooth on ℍ \ {β_{i}^{−1}α_{i}}, with real poles at y = α_{i}^{−1}β_{i}:

(i)  \frac{∂ω_{n}}{∂α_{i}^{μ}} = −\frac{4κ_{i}^{2}Re\{(\overline{β_{i} − α_{i}y})e_{μ}y\}}{ρ|β_{i} − α_{i}y|^{6}}Im \{(\overline{β_{i} − α_{i}y})α_{i}dy\}

− \frac{κ_{i}^{2}}{ρ|β_{i} − α_{i}y|^{4}} Im \{(\overline{β}_{i}e_{μ} − 2α_{i}^{μ}\overline{y})dy\}

+ ∑_{j=0}^{k} \frac{2κ_{i}^{2}κ_{j}^{2}Re\{(\overline{β_{i} − α_{i}y})e_{μ}y\}}{ρ^{2}|β_{i} − α_{i}y|^{4}|β_{j} − α_{j}y|^{4}}Im \{(\overline{β_{j} − α_{j}y})β_{j}dy\} .

(ii)  \frac{∂ω_{n}}{∂β_{i}^{μ}} = \frac{4κ_{i}^{2}(β_{i} − α_{i}y)^{μ}}{ρ|β_{i} − α_{i}y|^{6}} Im \{(\overline{β_{i} − α_{i}y})α_{i}dy\}

− \frac{κ_{i}^{2}}{ρ|β_{i} − α_{i}y|^{4}} Im \{\overline{e}_{μ}α_{i}dy\}

− ∑_{j=0}^{k} \frac{2κ_{i}^{2}κ_{j}^{2}(β_{i} − α_{i}y)^{μ}}{ρ^{2}|β_{i} − α_{i}y|^{4}|β_{j} − α_{j}y|^{4}} Im \{(\overline{β_{j} − α_{j}y})α_{j}dy\} .

Proof. Calculation. □ Lemma 5.3.5. The 1-form ∂ω_{si}/∂α_{i}^{μ} has real poles at points x = β_{j}^{−1}α_{j}, j ≠ i, but is smooth elsewhere:

\frac{∂ω_{si}}{∂α_{i}^{μ}} = \frac{2ρ_{i}^{2}(β_{i}x − α_{i})^{μ}}{ρ^{2}|x − α_{i}|^{4}} Im \{(β_{i}x − α_{i})\overline{β_{i}dx}\} − \frac{ρ_{i}}{ρ|β_{i}x − α_{i}|^{2}} Im \{e_{μ}\overline{β_{i}dx}\}

+ ∑_{{j=0 \atop j≠i}}^{k}\{\frac{2κ_{j}^{2}ρ_{i}(β_{i}x − α_{i})^{μ}}{ρ^{2}|β_{i}x − α_{i}|^{4}|β_{j}x − α_{j}|^{4}} (β_{i}x − α_{i})Im \{(\overline{β_{j}x − α_{j}})β_{j}dx\} (\overline{β_{i}x − α_{i}})

− \frac{κ_{j}^{2}}{ρ|β_{i}x − α_{i}|^{2}|β_{j}x − α_{j}|^{4}} e_{μ}Im \{(\overline{β_{j}x − α_{j}})β_{j}dx\} (\overline{β_{i}x − α_{i}})

− \frac{κ_{j}^{2}}{ρ|β_{i}x − α_{i}|^{2}|β_{j}x − α_{j}|^{4}} (β_{i}x − α_{i})Im \{(\overline{β_{j}x − α_{j}})β_{j}dx\} \overline{e}_{μ}\}.

Proof. Calculation. □ Lemma 5.3.6. The 1-form ∂ω_{si}/∂β_{i}^{μ} has real poles at points x = β_{j}^{−1}α_{j}, j ≠ i, but is smooth elsewhere:

\frac{∂ω_{si}}{∂β_{i}^{μ}} = −\frac{2ρ_{i}^{2}Re\{(\overline{β_{i}x − α_{i}})e_{μ}x\}}{ρ^{2}|β_{i}x − α_{i}|^{4}}Im \{(β_{i}x − α_{i})\overline{β_{i}dx}\}

+ \frac{ρ_{i}}{ρ|β_{i}x − α_{i}|^{2}} Im \{e_{μ}x\overline{β_{i}dx} + (β_{i}x − α_{i})\overline{e_{μ}dx}\}

+ ∑_{{j=0 \atop j≠i}}^{k}\{−\frac{2κ_{j}^{2}ρ_{i}Re\{(β_{i}x − α_{i})e_{μ}x\}}{ρ^{2}|β_{i}x − α_{i}|^{4}|β_{j}x − α_{j}|^{4}}(β_{i}x − α_{i})Im \{(\overline{β_{j}x − α_{j}})β_{j}dx\} (\overline{β_{i}x − α_{i}})

+ \frac{κ_{j}^{2}}{ρ|β_{i}x − α_{i}|^{2}|β_{i}x − α_{j}|^{4}} (e_{μ}x)Im \{(\overline{β_{j}x − α_{j}})β_{j}dx\} (\overline{β_{i}x − α_{i}})

+ \frac{κ_{j}^{2}}{ρ|β_{i}x − α_{i}|^{2}|β_{j}x − α_{j}|^{4}} (β_{i}x − α_{i})Im \{(\overline{β_{j}x − α_{j}})β_{j}dx\} \overline{e_{μ}x}\}.

Proof. Calculation. □

Lemma 5.3.7. The 1-form ∂ω_{ni}/∂α_{i}^{μ} has real poles at points y = α_{j}^{−1}β_{j}, j ≠ i, but is smooth elsewhere:

\frac{∂ω_{ni}}{∂α_{i}^{μ}} = −\frac{2ρ_{i}^{2}Re\{(\overline{β_{i} − α_{i}y})e_{μ}y\}}{ρ^{2}|β_{i}x − α_{i}|^{4}}Im \{(\overline{β_{i} − α_{i}y})α_{i}dy\}

− \frac{ρ_{i}}{ρ|β_{i} − α_{i}y|^{2}} Im \{(β_{i} − α_{i}y)\overline{e_{μ}dy} − e_{μ}y\overline{α_{i}dy}\}

+ ∑_{{j=0 \atop j≠i}}^{k}\{−\frac{2κ_{j}^{2}ρ_{i}Re\{(\overline{β_{i} − α_{i}y})e_{μ}y\}}{ρ^{2}|β_{i} − α_{i}y|^{4}|β_{j} − α_{j}y|^{4}}(β_{i} − α_{i}y)Im \{(\overline{β_{j} − α_{j}y})α_{j}dy\} (\overline{β_{i} − α_{i}y})

+ \frac{κ_{j}^{2}}{ρ|β_{i} − α_{i}y|^{2}|β_{i} − α_{j}y|^{4}} (e_{μ}y)Im \{(\overline{β_{j} − α_{j}y})α_{j}dy\} (\overline{β_{i} − α_{i}y})

+ \frac{κ_{j}^{2}}{ρ|β_{i} − α_{i}y|^{2}|β_{j} − α_{j}y|^{4}} (β_{i} − α_{i}y)Im \{(\overline{β_{j} − α_{j}y})α_{j}dy\} \overline{e_{μ}y}\}.

Proof. Calculation. □

Lemma 5.3.8. The 1-form ∂ω_{ni}/∂β_{i}^{μ} has real poles at points y = α_{j}^{−1}β_{j}, j ≠ i, but is smooth elsewhere:

\frac{∂ω_{ni}}{∂β_{i}^{μ}} = \frac{2ρ_{i}^{2}(β_{i} − α_{i}y)^{μ}}{ρ^{2}|β_{i} − α_{i}y|^{4}} Im \{(β_{i} − α_{i}y)\overline{α_{i}dy}\} − \frac{ρ_{i}}{ρ|β_{i} − α_{i}y|^{2}} Im \{e_{μ}\overline{α_{i}dy}\}

+ ∑_{{j=0 \atop j≠i}}^{k}\{\frac{2κ_{j}^{2}ρ_{i}(β_{i} − α_{i}y)^{μ}}{ρ^{2}|β_{i} − α_{i}y|^{4}|β_{j} − α_{j}y|^{4}} (β_{i} − α_{i}y)Im \{(\overline{β_{j} − α_{j}y})α_{j}dy\} (\overline{β_{i} − α_{i}y})

− \frac{κ_{j}^{2}}{ρ|β_{i} − α_{i}y|^{2}|β_{i} − α_{j}y|^{4}} e_{μ}Im \{(\overline{β_{j} − α_{j}y})α_{j}dy\} (\overline{β_{i} − α_{i}y})

− \frac{κ_{j}^{2}}{ρ|β_{i} − α_{i}y|^{2}|β_{j} − α_{j}y|^{4}} (β_{i} − α_{i}y)Im \{(\overline{β_{j} − α_{j}y})α_{j}dy\} \overline{e}_{μ}\}.

Proof. Calculation. □

Recall that the points \underline{P}_{i} ∈ \underline{ℍℙ}^{1} may be assigned representatives (α_{i}, β_{i}) in 𝕊^{7} ⊂ ℍ^{2}. If c_{i} = a_{i} or b_{i} in ℍ denote the inhomogeneous local coordinates for the point P_{i}, then

\begin{matrix}a_{i} & ⟼ (α_{i}, β_{i}) = \frac{(a_{i}, 1)}{\sqrt{|a_{i}|^{2} + 1}}  in 𝕊^{7}|O_{s}, \\ b_{i} & ⟼ (α_{i}, β_{i}) = \frac{(1, b_{i})}{\sqrt{1 + |b_{i}|^{2}}}  in 𝕊^{7}|O_{n}. \end{matrix}

Then, derivatives of the connection 1-forms with respect to the coordinates c_{i}^{μ} = a_{i}^{μ} are given by:

\frac{∂ω}{∂a_{i}^{μ}} = ∑_{ν=0}^{3} \frac{∂ω}{∂α_{i}^{ν}} \frac{∂α_{i}^{ν}}{∂a_{i}^{μ}} + \frac{∂ω}{∂β_{i}^{ν}} \frac{∂β_{i}^{ν}}{∂a_{i}^{μ}} .

Note that with respect to our choice of coordinate patches, for (α_{i}, β_{i}) ∈ O_{s} the factors

\begin{aligned}\frac{∂α_{i}^{ν}}{∂a_{i}^{μ}} & = \frac{a_{i}^{ν}a_{i}^{μ}}{(|a_{i}|^{2} + 1)^{3/2}} + \frac{δ_{μν}}{(|a_{i}|^{2} + 1)^{1/2}} , \\ \frac{∂β_{i}^{ν}}{∂a_{i}^{μ}} & = \frac{a_{i}^{μ}δ_{0μ}}{(|a_{i}|^{2} + 1)^{3/2}} , \end{aligned}

are bounded, and similarly for derivatives of α_{i}^{ν}, β_{i}^{ν} with respect to c_{i}^{μ} = b_{i}^{μ}, where (α_{i}, β_{i}) ∈ O_{n}.

In order to estimate the L^{2} norms of the tangent vectors in T M_{2}, we need to examine the behaviour of the corresponding 1-forms as the moduli parameters approach the boundary of T̃_{2}. It is useful to consider two distinct cases: Case 1: Three points approaching each other. The points P_{0}, P_{1}, P_{2} all lie in either the southern extended hemisphere O_{s}(2) or the northern extended hemisphere O_{n}(2). If the points lie in O_{s}(2), then our definition of the partition of unity on ℍℙ^{1} is completed by choosing:

supp χ_{s} ⊂ O_{s}(8),
supp χ_{n} ⊂ O_{n}(3/8).

In the equivalent configuration, where the three points points lie in O_{n}(2), we simply interchange the support radii. Case 2: Two points approaching each other. The points P_{i}, P_{j} lie below the equator of ℍℙ^{1}, and one point P_{m} lies in O_{n}(2) \ O_{s}(2). We choose:

supp χ_{s} ⊂ O_{s}(7/4),
supp χ_{n} ⊂ O_{n}(4/5),

and for the equivalent configuration, where one point lies in O_{s}(2) \ O_{n}(2) and two points lie above the equator, the support radii are interchanged.

These choices for the partition of unity χ_{s}, χ_{n} allow us to obtain estimates for the norms of the tangent vectors for all possible configurations of points P_{0}, P_{1}, P_{2}.

§5.4. Estimates of Tangent Vector Norms. I

We derive estimates for the L^{2} norms of the tangent vectors ∂ω/∂λ_{i} in terms of the weight parameters λ_{i} and coordinates of the points P_{i}.

For notational convenience, we assume i = 0. We observe that it is enough to consider Case 1 — where three points are approaching each other — since the estimates obtained in Case 2 will certainly be sharper that those obtained in Case 1. We further assume, again for notational convenience, that the three points P_{i} all lie in the southern hemisphere O_{s}(2) ⊂ ℍℙ^{1}. Hence, the points \underline{P}_{i} have coordinates [a_{i}, 1], with |a_{i}| < 2 for all i. Moreover, if x = x_{0}x_{1}^{−1} is a local coordinate on the patch O_{s} = {[x_{0}, x_{1}] : |x_{0}x_{1}^{−1}| < 8} and y = x_{1}x_{0}^{−1} is a local coordinate on the patch O_{n} = {[x_{0}, x_{1}] : |x_{1}x_{0}^{−1}| < 3/8} in ℍℙ^{1}, then |x| < 8 and |y| < 3/8.

The connection 1-forms are expressed in terms of parameters κ_{i} and (α_{i}, β_{i}). For weights λ_{i} and points \underline{P}_{i} = [a_{i}, 1], we would usually choose

\begin{aligned}κ_{i} & = λ_{i}, \\ (α_{i}, β_{i}) & = \frac{(a_{i}, 1)}{\sqrt{1 + |a_{i}|^{2}}} for i = 0, 1, 2. \end{aligned}

However, it is more convenient to instead express the 1-forms in terms of the following choices for κ_{i} and (α_{i}, β_{i}):

\begin{aligned}κ_{i} & = λ_{i}\sqrt{1 + |a_{i}|^{2}} \\ (α_{i}, β_{i}) & = (a_{i}, 1) for i = 0, 1, 2. \end{aligned}

Both choices of parameters give the same ADHM matrices and so the same family of self-dual connections. We recall that the weights λ_{i} are always normalized so that λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2} = 1. We shall occasionally use η_{ij} = |a_{i} − a_{j}| to denote the Euclidean distance beteen the coordinates a_{i} and a_{j}. We use C to denote a generic positive constant, which is frequently updated, but always independent of moduli parameters.

We record the formulas for the required local 1-forms when k = 2 in the following lemmas:

Lemma 5.4.1. The local tangent vector 1-form ∂ω_{s}/∂λ_{0} on O_{s} ⊂ ℍℙ^{1} is given by:

\frac{∂ω_{s}}{∂λ_{0}} = \frac{∂ω_{s}}{∂κ_{0}} \frac{∂κ_{0}}{∂λ_{0}} = \sqrt{1 + |a_{0}|^{2}} \frac{∂ω_{s}}{∂κ_{0}} ,
\frac{∂ω_{s}}{∂κ_{0}} = \frac{2κ_{0}ρ_{0}}{ρ^{2}|x − a_{0}|^{4}} Im \{(\overline{x − a_{0}})dx\} − \frac{2κ_{0}κ_{1}^{2}}{ρ^{2}|x − a_{0}|^{2}|x − a_{1}|^{4}} Im \{(\overline{x − a_{1}})dx\} ,

− \frac{2κ_{0}κ_{2}^{2}}{ρ^{2}|x − a_{0}|^{2}|x − a_{2}|^{4}} Im \{(\overline{x − a_{2}})dx\} ,

where

\begin{aligned}ρ(x) & = \frac{κ_{0}^{2}}{|x − a_{0}|^{2}} + \frac{κ_{1}^{2}}{|x − a_{1}|^{2}} + \frac{κ_{2}^{2}}{|x − a_{2}|^{2}} , \\ ρ_{0}(x) & = \frac{κ_{1}^{2}}{|x − a_{1}|^{2}} + \frac{κ_{2}^{2}}{|x − a_{2}|^{2}} , x ∈ ℍ ≃ O_{s}. \end{aligned}

Proof. Apply the previous formulas with k = 2, weights κ_{i} = λ_{i}\sqrt{1 + |a_{i}|^{2}}, and (α_{i}, β_{i}) = (1, a_{i}) for i = 0, 1, 2. □

Lemma 5.4.2. The local tangent vector 1-form ∂ω_{n}/∂λ_{0} on O_{n} ⊂ ℍℙ^{1} is given by:

\frac{∂ω_{n}}{∂λ_{0}} = \frac{∂ω_{n}}{∂κ_{0}} \frac{∂κ_{0}}{∂λ_{0}} = \sqrt{1 + |a_{0}|^{2}} \frac{∂ω_{n}}{∂κ_{0}} ,
\frac{∂ω_{n}}{∂κ_{0}} = −\frac{2κ_{0}ρ_{0}}{ρ^{2}|1 − a_{0}y|^{4}} Im \{(\overline{1 − a_{0}y})a_{0}dy\}

+ \frac{2κ_{0}κ_{1}^{2}}{ρ^{2}|1 − a_{0}y|^{2}|1 − a_{1}y|^{4}} Im \{(\overline{1 − a_{1}})a_{1}dy\} ,

+ \frac{2κ_{0}κ_{2}^{2}}{ρ^{2}|1 − a_{0}y|^{2}|1 − a_{2}y|^{4}} Im \{(\overline{1 − a_{2}y})a_{2}dy\} ,

where

\begin{aligned}ρ(y) & = \frac{κ_{0}^{2}}{|1 − a_{0}y|^{2}} + \frac{κ_{1}^{2}}{|1 − a_{1}y|^{2}} + \frac{κ_{2}^{2}}{|1 − a_{2}y|^{2}} \\ ρ_{0}(y) & = \frac{κ_{1}^{2}}{|1 − a_{1}y|^{2}} + \frac{κ_{2}^{2}}{|1 − a_{2}y|^{2}} , y ∈ ℍ ≃ O_{n}. \end{aligned}

Proof. Again, apply the previous formulas with k = 2, weights κ_{i} = λ_{i}\sqrt{1 + |a_{i}|^{2}}, and (α_{i}, β_{i}) = (1, a_{i}) for i = 0, 1, 2. □

The corresponding norms are:

∥\frac{∂ω}{∂λ_{0}} ∥^{2} = ∥\frac{∂ω_{s}}{∂λ_{0}} \sqrt{χ_{s}}∥^{2} + ∥\frac{∂ω_{n}}{∂λ_{0}} \sqrt{χ_{n}}∥^{2} ,

where

\begin{aligned}∥\frac{∂ω_{s}}{∂λ_{0}} \sqrt{χ_{s}}∥^{2} & = ∫_{O_{s}} χ_{s} |\frac{∂ω_{s}}{∂λ_{0}} |^{2} \sqrt{g}dx, \\ ∥\frac{∂ω_{n}}{∂λ_{0}} \sqrt{χ_{n}}∥^{2} & = ∫_{O_{n}} χ_{n} |\frac{∂ω_{n}}{∂λ_{0}} |^{2} \sqrt{g}dy, \end{aligned}

and | · | denotes the fibre metric on Ω^{1}(ℍℙ^{1}, ad P), and g is the metric on ℍℙ^{1}.

For convenience, we record some elementary integration formulas and inequalities that we will require for our estimate calculations.

Lemma 5.4.3. Let a, b, c, m, n, R, t, δ be real numbers, with 0 < t < 1, 0 < δ ≤ 1, and R ≥ 1.

∫_{m}^{n} \frac{r}{a + r^{2}b} dr = \frac{1}{2b} log (\frac{a + n^{2}b}{a + m^{2}b}); (i)

∫_{m}^{n} \frac{r^{3}}{a + r^{2}b} dr = \frac{1}{2b^{2}} \{(n^{2} − m^{2})b − a log (\frac{a + n^{2}b}{a + m^{2}b})\}; (ii)

(iii) \frac{1}{1 − t} log (\frac{t + R(1 − t)}{t}) ≤ 2 + 2 log (\frac{R}{t}) , 0 < t < 1;

(iv) \frac{1}{1 − t} log (\frac{t + R(1 − t)}{t + δ(1 − t)}) ≤ 2R + 2 log (\frac{R}{δ}) , 0 < t < 1;

(v) ∫ \frac{1}{ar^{2} + br + c} dr = \frac{2}{\sqrt{4ac − b^{2}}} arctan (\frac{2ar + b}{\sqrt{4ac − b^{2}}} ) if b^{2} − 4ac < 0;

∫ \frac{r}{ar^{2} + br + c} dr = \frac{1}{2a} log (ar^{2} + br + c) − \frac{b}{2a} ∫ \frac{1}{ar^{2} + br + c}. (vi)

Proof. Calculation. □

Lemma 5.4.4. Let a, b ∈ ℍ, R > 0, and let |a − b| denote the Euclidean distance between a and b. Then:

∫_{|x|<R} \frac{1}{|x − a|^{2}|x − b|^{2}} dx ≤ C \{1 + log (1 + \frac{R}{|a − b|})\} ,

where dx denotes the standard Euclidean measure on ℍ = ℝ^{4}, and C is a constant independent of a, b, and R.

Proof. The Euclidean ball 𝔹(0, R) is clearly contained in the union of the following two annuli centred at a and b:

𝔹(0, R) ⊂ {x : η/2 ≤ |x − a| < R′} ⋃{x : η/2 ≤ |x − b| < R′},

where R′ = R + |a| + |b| and η = |a − b|. Then, our integal may be estimated by:

\begin{matrix}∫_{|x|<R} \frac{1}{|x − a|^{2}|x − b|^{2}} dx \\ ≤ ∫_{{|x−a|<R′ \atop |x−b|≥η/2}} \frac{1}{|x − a|^{2}|x − b|^{2}} dx + ∫_{{|x−b|<R′ \atop |x−a|≥η/2}} \frac{1}{|x − a|^{2}|x − b|^{2}} dx. \end{matrix}

We choose two spherical polar coordinate systems, one centred at x = a, with r = |x − a|, and the other centred at x = b, with s = |x − b|. Then dx = r^{3}drdθ or dx = s^{3}dsdθ, where dθ denotes the standard Riemannian measure on the 3-sphere 𝕊^{3} of unit radius. In the first integal, we have both |x−b| ≥ η/2 and |x−b| ≥ |r−η|, so that:

|x − b|^{2} ≥ (η^{2}/4 + (r − η)^{2}) /2.

Similarly, in the second integral we have |x − a|^{2} ≥ (η^{2}/4 + (s − η)^{2}) /2. Using these inequalities we obtain:

\begin{aligned}∫_{|x|<R} \frac{1}{|x − a|^{2}|x − b|^{2}} dx & ≤ 2 ∫_{0}^{R′} \frac{r}{η^{2}/4 + (r − η)^{2}} dr + 2 ∫_{0}^{R′} \frac{s}{η^{2}/4 + (s − η)^{2}} ds \\ & = 4 ∫_{0}^{R′} \frac{r}{5η^{2}/4 − 2ηr + r^{2}} dr. \end{aligned}

Integrating, we get:

∫_{|x|<R} \frac{1}{|x − a|^{2}|x − b|^{2}} dx ≤ 2 log (1 + \frac{4R^{′2}}{5η^{2}} − \frac{8R′}{5η} ) + 8 arctan (\frac{2R′ − 2η}{η} )
− 8 arctan(−2)

≤ 8π + 4 log (1 + \frac{R}{η}) ,

and this gives the required estimate. □

Lemma 5.4.5. The norms of the three terms in ∂ω_{s}/∂λ_{0} have the following upper bounds:

(i)  ∥\frac{∂ω_{s}^{(1)}}{∂λ_{0}} \sqrt{χ_{s}}∥^{2} ≤ C \{1 + log (\frac{1}{λ_{0}} )\} ,

(ii)  ∥\frac{∂ω_{s}^{(2)}}{∂λ_{0}} \sqrt{χ_{s}}∥^{2} ≤ C \{1 + log (1 + \frac{1}{|a_{0} − a_{1}|})\} ,

(iii)  ∥\frac{∂ω_{s}^{(3)}}{∂λ_{0}} \sqrt{χ_{s}}∥^{2} ≤ C \{1 + log (1 + \frac{1}{|a_{0} − a_{2}|})\} .

Consequently,

∥\frac{∂ω_{s}}{∂λ_{0}} \sqrt{χ_{s}}∥^{2} ≤ C \{1 + log (\frac{1}{λ_{0}} ) + log (1 + \frac{1}{|a_{0} − a_{1}|}) + log (1 + \frac{1}{|a_{0} − a_{2}|})\} .

Proof. We consider each of the three norms in turn, noting that integration over supp χ_{s} ⊂ O_{s} in ℍℙ^{1} corresponds to integration over the Eulidean ball {x ∈ ℍ : |x| < 8} ⊂ ℍ. For the first term in ∂ω_{s}/∂λ_{0}, we have

∥\frac{∂ω_{s}^{(1)}}{∂λ_{0}} \sqrt{χ_{s}}∥^{2} ≤ C ∫_{|x|<8} \frac{κ_{0}^{2}ρ_{0}^{2}}{ρ^{4}|x − a_{0}|^{6}} dx.

Now using the inequalities ρ_{0}(x) ≤ ρ(x) and

\frac{κ_{0}^{2}}{|x − a_{0}|^{2}} ≤ ρ(x),

we see that

∫_{|x|<8} \frac{κ_{0}^{2}ρ_{0}^{2}}{ρ^{4}|x − a_{0}|^{6}} dx ≤ C ∫_{|x|<8} \frac{1}{ρ|x − a_{0}|^{4}} dx.

We choose spherical polar coordinates on ℍ = ℝ^{4} with r = |x − a_{0}|, so that dx = r^{3}drdθ. Since |x| < 8, |a_{i}| < 2, we have |x − a_{i}| < 10, for i = 0, 1, 2, and so we get the following lower bounds for ρ(x):

\begin{aligned}ρ(x) & = \frac{κ_{0}^{2}}{|x − a_{0}|^{2}} + \frac{κ_{1}^{2}}{|x − a_{1}|^{2}} + \frac{κ_{2}^{2}}{|x − a_{2}|^{2}} , \\ & ≥ C (\frac{κ_{0}^{2}}{|x − a_{0}|^{2}} + κ_{1}^{2} + κ_{2}^{2}) . \end{aligned}

Then, as κ_{i} = λ_{i}\sqrt{1 + |a_{i}|^{2}} ≥ λ_{i} for all i, we have

ρ(x) ≥ C (\frac{λ_{0}^{2}}{r^{2}} + λ_{1}^{2} + λ_{2}^{2}) .

Returning to our integral estimate, the ball {x : |x| < 8} is contained in {x : |x − a_{0}| < 10}, and so

∫_{|x|<8} \frac{1}{ρ|x − a_{0}|^{4}} dx ≤ ∫_{|x−a_{0}|<10} \frac{1}{ρ|x − a_{0}|^{4}} dx

≤ C ∫_{0}^{10} \frac{r^{3}}{(r^{−2}λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2})r^{4}} dr

= C ∫_{0}^{10} \frac{r}{λ_{0}^{2} + r^{2}(λ_{1}^{2} + λ_{2}^{2})} dr

= \frac{C}{λ_{1}^{2} + λ_{2}^{2}} log (\frac{λ_{0}^{2} + 100(λ_{1}^{2} + λ_{2}^{2})}{λ_{0}^{2}}) .

Since λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2} = 1, we may set t = λ_{0}^{2}, 1 − t = λ_{1}^{2} + λ_{2}^{2}, with 0 < t < 1, and applying one of our log inequalities, we obtain

\frac{1}{λ_{1}^{2} + λ_{2}^{2}} log (\frac{λ_{0}^{2} + 100(λ_{1}^{2} + λ_{2}^{2})}{λ_{0}^{2}}) ≤ C \{1 + log (\frac{1}{λ_{0}} )\} ,

and combining the above inequalities gives estimate (i).

Considering the second term, we have

\begin{aligned}∥\frac{∂ω_{s}^{(2)}}{∂λ_{0}} \sqrt{χ_{s}}∥^{2} & ≤ C ∫_{|x|<8} \frac{κ_{0}^{2}κ_{1}^{4}}{ρ^{4}|x − a_{0}|^{4}|x − a_{1}|^{6}} dx \\ & ≤ C ∫_{|x−a_{0}|<8} \frac{1}{ρ|x − a_{0}|^{2}|x − a_{1}|^{2}} dx \end{aligned}

where we use the inequalities

\frac{κ_{i}^{2}}{|x − a_{i}|^{2}} ≤ ρ(x) for i = 0, 1.

As before, κ_{i} ≥ λ_{i} and |x − a_{i}| < 10 for i = 0, 1, 2. Then, we have

ρ(x) ≥ C (κ_{0}^{2} + κ_{1}^{2} + κ_{2}^{2})
≥ C (λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2})
= C,

since λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2} = 1, and so ρ(x) is bounded below by some positive constant. Hence, we see that

\begin{aligned}∫_{|x−a_{0}|<8} \frac{1}{ρ|x − a_{0}|^{2}|x − a_{1}|^{2}} dx & ≤ C ∫_{|x−a_{0}|<10} \frac{1}{|x − a_{0}|^{2}|x − a_{1}|^{2}} dx \\ & ≤ C \{1 + log (1 + \frac{1}{|a_{0} − a_{1}|})\} , \end{aligned}

by a previous integration lemma. This gives estimate (ii).

Considering the third term, we have:

∥\frac{∂ω_{s}^{(3)}}{∂λ_{0}} \sqrt{χ_{s}}∥^{2} ≤ C ∫_{|x|<8} \frac{κ_{0}^{2}κ_{2}^{4}}{ρ^{4}|x − a_{0}|^{4}|x − a_{2}|^{6}} dx,

and so estimate (iii) follows by symmetry. □

Lemma 5.4.6. The norms of the three terms in ∂ω_{n}/∂λ_{0} are bounded by a constant C, independent of moduli parameters:

∥\frac{∂ω_{n}^{(j)}}{∂λ_{0}} \sqrt{χ_{n}}∥^{2} ≤ C, for j = 1, 2, 3.

Consequently,

∥\frac{∂ω_{n}}{∂λ_{0}} \sqrt{χ_{n}}∥^{2} ≤ C.

Proof. We note that integration over supp χ_{n} ⊂ O_{n} in ℍℙ^{1} corresponds to integration over the Eulidean ball {y ∈ ℍ : |y| < 3/8} ⊂ ℍ. Then, considering the first term in ∂ω_{n}/∂λ_{0}, we have

∥\frac{∂ω_{n}^{(1)}}{∂λ_{0}} \sqrt{χ_{n}}∥^{2} ≤ C ∫_{|y|<3/8} \frac{κ_{0}^{2}ρ_{0}^{2}|a_{0}|^{2}}{ρ^{4}|1 − a_{0}y|^{6}} dy.

Now |a_{0}| < 2, ρ_{0}(y) ≤ ρ(y), and

\frac{κ_{0}^{2}}{|1 − a_{0}y|^{2}} ≤ ρ(y).

Hence, we see

∫_{|y|<3/8} \frac{κ_{0}^{2}ρ_{0}^{2}|a_{0}|^{2}}{ρ^{4}|1 − a_{0}y|^{6}} dy ≤ C ∫_{|y|<3/8} \frac{1}{ρ|1 − a_{0}y|^{4}} dy.

Next, we observe that

\begin{aligned}|1 − a_{i}y| & ≥ 1 − |a_{i}y| > 1/4, \\ |1 − a_{i}y| & ≤ 1 + |a_{i}y| < 7/4 for i = 0, 1, 2, \end{aligned}

since |a_{i}| < 2 and |y| < 3/8. So |1 − a_{0}y|^{4} is bounded below by a positive constant, and moreover

ρ(y) = \frac{κ_{0}^{2}}{|1 − a_{0}y|^{2}} + \frac{κ_{1}^{2}}{|1 − a_{1}y|^{2}} + \frac{κ_{2}^{2}}{|1 − a_{2}y|^{2}}
≥ C (κ_{0}^{2} + κ_{1}^{2} + κ_{2}^{2})
≥ C (λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2})
= C,

as κ_{i} ≥ λ_{i}, for i = 0, 1, 2, and λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2} = 1. We now obtain

\begin{aligned}∫_{|y|<3/8} \frac{1}{ρ|1 − a_{0}y|^{4}} dy & ≤ C ∫_{|y|<3/8} dy \\ & = C, \end{aligned}

and combining these inequalities gives the required estimate.

Considering the second term, we see

∥\frac{∂ω_{n}^{(2)}}{∂λ_{0}} \sqrt{χ_{n}}∥^{2} ≤ C ∫_{|y|<3/8} \frac{κ_{0}^{2}κ_{1}^{4}|a_{1}|^{2}}{ρ^{4}|1 − a_{0}y|^{4}|1 − a_{1}y|^{6}} dy

≤ C ∫_{|y|<3/8} \frac{1}{ρ|1 − a_{0}y|^{2}|1 − a_{1}y|^{2}} dy

≤ C ∫_{|y|<3/8} dy

= C,

by similar arguments. The estimate for the third term is obtained by symmetry. □

Proposition 5.4.7. For i = 0, 1, 2 we have the following upper bounds:

∥\frac{∂ω}{∂λ_{i}} ∥^{2} ≤ C \{1 + log (\frac{1}{λ_{i}} ) + ∑_{{j=0 \atop j≠i}}^{2} log (1 + \frac{1}{|a_{i} − a_{j}|})\} .

Proof. One makes the obvious changes in the estimates obtained when i = 0 to get estimates for i = 1 and 2. We then combine these inequalities to give the required upper bound. □

§5.5. Estimates of Tangent Vector Norms. II

In this section we estimate the norms of the tangent vectors ∂ω/∂c_{i}^{μ}. For notational convenience we assume that i = 0. As in the previous section, it suffices to consider Case 1, and we may again assume, for notational convenience, that the three points P_{i} lie in the southern hemisphere O_{s} ⊂ ℍℙ^{1}. Hence, the points \underline{P}_{i} have coordinates [a_{i}, 1], with |a_{i}| < 2 for i = 0, 1, 2.

We just need to estimate the L^{2} norms of the local 1-forms

\frac{∂ω_{si}}{∂a_{i}^{μ}} , \frac{∂ω_{s}}{∂a_{i}^{μ}} , , \frac{∂ω_{n}}{∂a_{i}^{μ}} ,

on the coordinate patches O_{si}, O_{s} \ \{\underline{P}_{i}\}, and O_{n}, since these patches obviously cover ℍℙ^{1}. Consequently, we have

\begin{aligned}∥\frac{∂ω}{∂a_{i}^{μ}} ∥^{2} & = ∫_{O_{s}} |\frac{∂ω_{si}}{∂a_{i}^{μ}} |^{2} χ_{s}χ_{i}\sqrt{g}dx + ∫_{O_{s}} |\frac{∂ω_{s}}{∂a_{i}^{μ}} |^{2} χ_{s}(1 − χ_{i})\sqrt{g}dx \\ & + ∫_{O_{n}} |\frac{∂ω_{n}}{∂a_{i}^{μ}} |^{2} χ_{n}\sqrt{g}dy. \end{aligned}

As in the previous section, the connection 1-forms are parametrized by κ_{i} and (α_{i}, β_{i}), where we choose

\begin{aligned}κ_{i} & = λ_{i}\sqrt{1 + |a_{i}|^{2}}, \\ (α_{i}, β_{i}) & = (a_{i}, 1), \end{aligned}

and λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2} = 1. We recall that with this choice of parametrization, we have:

\frac{∂ω}{∂a_{i}^{μ}} = \frac{∂ω}{∂κ_{i}} \frac{∂κ_{i}}{∂a_{i}^{μ}} + \frac{∂ω}{∂α_{i}^{μ}} |_{α_{i}=a_{i},β_{i}=1}.

We obtained the estimates for the L^{2} norms of ∂ω/∂κ_{i} in the previous section, and so it suffices to consider the L^{2} norms of ∂ω/∂α_{i}^{μ}. We record our formulas for the local 1-forms representing the required tangent vectors in the following series of lemmas.

Lemma 5.5.1. The local tangent vector 1-form ∂ω_{s0}/∂α_{0}^{μ} on O_{s0} is given by

\frac{∂ω_{s0}}{∂α_{0}^{μ}} |_{α_{i}=a_{i},β_{i}=1} = \frac{2ρ_{0}^{2}(x − a_{0})^{μ}}{ρ^{2}|x − a_{0}|^{4}} Im \{(x − a_{0})d\overline{x}\} − \frac{ρ_{0}}{ρ|x − a_{0}|^{2}} Im \{e_{μ}d\overline{x}\}

+ \frac{2κ_{1}^{2}ρ_{0}(x − a_{0})^{μ}}{ρ^{2}|x − a_{0}|^{4}|x − a_{1}|^{4}} (x − a_{0})Im \{(\overline{x − a_{1}})dx\} (\overline{x − a_{0}})

− \frac{κ_{1}^{2}}{ρ|x − a_{0}|^{2}|x − a_{1}|^{4}} e_{μ}Im \{(\overline{x − a_{1}})dx\} (\overline{x − a_{0}})

− \frac{κ_{1}^{2}}{ρ|x − a_{0}|^{2}|x − a_{1}|^{4}} (x − a_{0})Im \{(\overline{x − a_{1}})dx\} \overline{e}_{μ}

+ \frac{2κ_{2}^{2}ρ_{0}(x − a_{0})^{μ}}{ρ^{2}|x − a_{0}|^{4}|x − a_{2}|^{4}} (x − a_{0})Im \{(\overline{x − a_{2}})dx\} (\overline{x − a_{0}})

− \frac{κ_{2}^{2}}{ρ|x − a_{0}|^{2}|x − a_{2}|^{4}} e_{μ}Im \{(\overline{x − a_{2}})dx\} (\overline{x − a_{0}})

− \frac{κ_{2}^{2}}{ρ|x − a_{0}|^{2}|x − a_{2}|^{4}} (x − a_{0})Im \{(\overline{x − a_{2}})dx\} \overline{e}_{μ}.

Proof. Apply the previous formulas with k = 2, weights κ_{i} = λ_{i}\sqrt{1 + |a_{i}|^{2}}, and (α_{i}, β_{i}) = (1, a_{i}) for i = 0, 1, 2. □

Lemma 5.5.2. The local tangent vector 1-form ∂ω_{s}/∂α_{0}^{μ} on O_{s} \ {P_{0}} is given by

\frac{∂ω_{s}}{∂α_{0}^{μ}} |_{α_{i}=a_{i},β_{i}=1} = \frac{4κ_{0}^{2}(x − a_{0})^{μ}}{ρ|x − a_{0}|^{6}} Im \{(\overline{x − a_{0}})dx\}

− \frac{κ_{0}^{2}}{ρ|x − a_{0}|^{4}} Im \{\overline{e}_{μ}dx\}

− \frac{2κ_{0}^{4}(x − a_{0})^{μ}}{ρ^{2}|x − a_{0}|^{8}} Im \{(\overline{x − a_{0}})dx\}

− \frac{2κ_{0}^{2}κ_{1}^{2}(x − a_{0})^{μ}}{ρ^{2}|x − a_{0}|^{4}|x − a_{1}|^{4}} Im \{(\overline{x − a_{1}})dx\}

− \frac{2κ_{0}^{2}κ_{2}^{2}(x − a_{0})^{μ}}{ρ^{2}|x − a_{0}|^{4}|x − a_{2}|^{4}} Im \{(\overline{x − a_{2}})dx\} .

Proof. Again, apply the previous formulas with k = 2, weights κ_{i} = λ_{i}\sqrt{1 + |a_{i}|^{2}}, and (α_{i}, β_{i}) = (1, a_{i}) for i = 0, 1, 2. □

Lemma 5.5.3. The local tangent vector 1-form ∂ω_{n}/∂α_{0}^{μ} on O_{n} is given by

\frac{∂ω_{n}}{∂α_{0}^{μ}} |_{α_{i}=a_{i},β_{i}=1} = −\frac{4κ_{0}^{2}Re\{(\overline{1 − a_{0}y})e_{μ}y\}}{ρ|1 − a_{0}y|^{6}}Im \{(\overline{1 − a_{0}y})dy\}

− \frac{κ_{0}^{2}}{ρ|1 − a_{0}y|^{4}} Im \{(\overline{e}_{μ} − 2a_{0}^{μ}\overline{y})dy\}

+ \frac{2κ_{0}^{4}Re\{(\overline{1 − a_{0}y})e_{μ}y\}}{ρ^{2}|1 − a_{0}y|^{8}}Im \{(\overline{1 − a_{0}y})dy\}

+ \frac{2κ_{0}^{2}κ_{1}^{2}Re\{(\overline{1 − a_{0}y})e_{μ}y\}}{ρ^{2}|1 − a_{0}y|^{4}|1 − a_{1}y|^{4}}Im \{(\overline{1 − a_{1}y})dy\}

+ \frac{2κ_{0}^{2}κ_{2}^{2}Re\{(\overline{1 − a_{0}y})e_{μ}y\}}{ρ^{2}|1 − a_{0}y|^{4}|1 − a_{2}y|^{4}}Im \{(\overline{1 − a_{2}y})dy\} .

Proof. Apply the previous formulas with k = 2, weights κ_{i} = λ_{i}\sqrt{1 + |a_{i}|^{2}}, and (α_{i}, β_{i}) = (1, a_{i}), for i = 0, 1, 2. □

Lemma 5.5.4. The norms of the eight terms in ∂ω_{s0}/∂α_{0}^{μ} have the following upper bounds:

(i)  ∫_{O_{s}} |\frac{∂ω_{s0}^{(j)}}{∂α_{0}^{μ}} |^{2} χ_{s}χ_{0}\sqrt{g}dx ≤ C \{1 + log (\frac{1}{λ_{0}} )\} , for j = 1, 2.

(ii)  ∫_{O_{s}} |\frac{∂ω_{s0}^{(j)}}{∂α_{0}^{μ}} |^{2} χ_{s}χ_{0}\sqrt{g}dx ≤ C, for j = 3, …, 8.

Consequently,

∫_{O_{s}} |\frac{∂ω_{s0}}{∂α_{0}^{μ}} |^{2} χ_{s}χ_{0}\sqrt{g}dx ≤ C \{1 + log (\frac{1}{λ_{0}} )\} .

Proof. Since we are integrating over the support of χ_{s}χ_{0} in ℍℙ^{1}, we can assume |x − a_{0}| < δ_{0} and |x| < 8, for x ∈ ℍ. Commencing with the first term in ∂ω_{s0}/∂α_{0}^{μ}, we have

∫_{O_{s}} |\frac{∂ω_{s0}^{(1)}}{∂α_{0}^{μ}} |^{2} χ_{s}χ_{0}\sqrt{g}dx ≤ C ∫_{|x−a_{0}|<δ_{0}} \frac{ρ_{0}^{4}}{ρ^{4}|x − a_{0}|^{4}} dx

≤ C ∫_{|x−a_{0}|<δ_{0}} \frac{ρ_{0}}{ρ|x − a_{0}|^{4}} dx

= C ∫_{|x−a_{0}|<δ_{0}} \frac{ρ_{0}}{(λ_{0}^{2} + |x − a_{0}|^{2}ρ_{0})|x − a_{0}|^{2}} dx.

Since δ_{0} = \frac{1}{2} min\{1, η_{01}, η_{02}\}, we have |x − a_{j}| ≥ η_{0j}/2 for j ≠ 0 and |x − a_{0}| < δ_{0}, where η_{0j} = |a_{0} − a_{j}|. Then

ρ_{0}(x) = \frac{λ_{1}^{2}}{|x − a_{1}|^{2}} + \frac{λ_{2}^{2}}{|x − a_{2}|^{2}} ≤ 4 (\frac{λ_{1}^{2}}{η_{01}^{2}} + \frac{λ_{2}^{2}}{η_{02}^{2}} ) ,

and so

\frac{ρ_{0}}{λ_{0}^{2} + |x − a_{0}|^{2}ρ_{0}} ≤ \frac{4(λ_{1}^{2}/η_{01}^{2} + λ_{2}^{2}/η_{02}^{2})}{λ_{0}^{2} + 4|x − a_{0}|^{2}(λ_{1}^{2}/η_{01}^{2} + λ_{2}^{2}/η_{02}^{2})}.

We employ spherical polar coordinates on ℍ, so that dx = r^{3}drdθ and r = |x − a_{0}|. Hence,

\begin{aligned}∫_{O_{s}} |\frac{∂ω_{s0}^{(1)}}{∂α_{0}^{μ}} |^{2} χ_{s}χ_{0}\sqrt{g}dx & ≤ C ∫_{0}^{δ_{0}} \frac{r(λ_{1}^{2}/η_{01}^{2} + λ_{2}^{2}/η_{02}^{2})}{λ_{0}^{2} + r^{2}(λ_{1}^{2}/η_{01}^{2} + λ_{2}^{2}/η_{02}^{2})} dr \\ & = C log (\frac{λ_{0}^{2} + δ_{0}^{2}(λ_{1}^{2}/η_{01}^{2} + λ_{2}^{2}/η_{02}^{2})}{λ_{0}^{2}}) . \end{aligned}

Since δ_{0} ≤ η_{01}/2, η_{02}/2, we have

\begin{aligned}∫_{O_{s}} |\frac{∂ω_{s0}^{(1)}}{∂α_{0}^{μ}} |^{2} χ_{s}χ_{0}\sqrt{g}dx & ≤ C log (\frac{λ_{0}^{2} + (λ_{1}^{2} + λ_{2}^{2})/4}{λ_{0}^{2}}) \\ & = C log (\frac{1}{λ_{0}} ) , \end{aligned}

since λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2} = 1. For the second term, we have

\begin{aligned}∫_{O_{s}} |\frac{∂ω_{s0}^{(2)}}{∂α_{0}^{μ}} |^{2} χ_{s}χ_{0}\sqrt{g}dx & ≤ C ∫_{|x−a_{0}|<δ_{0}} \frac{ρ_{0}^{2}}{ρ^{2}|x − a_{0}|^{4}} dx \\ & ≤ C ∫_{|x−a_{0}|<δ_{0}} \frac{ρ_{0}}{ρ|x − a_{0}|^{4}} dx, \end{aligned}

with the same bound as the first term, and this establishes (i). Examining the third term in ∂ω_{s0}/∂α_{0}^{μ}, we see

\begin{aligned}∫_{O_{s}} |\frac{∂ω_{s0}^{(3)}}{∂α_{0}^{μ}} |^{2} χ_{s}χ_{0}\sqrt{g}dx & ≤ C ∫_{|x−a_{0}|<δ_{0}} \frac{κ_{1}^{4}ρ_{0}^{2}}{ρ^{4}|x − a_{0}|^{2}|x − a_{1}|^{6}} dx \\ & ≤ C ∫_{|x−a_{0}|<δ_{0}} \frac{1}{|x − a_{0}|^{2}|x − a_{1}|^{2}} dx. \end{aligned}

Since |x − a_{1}| ≥ η_{01}/2 and δ_{0} ≤ η_{01}/2, we obtain

\begin{aligned}∫_{O_{s}} |\frac{∂ω_{s0}^{(3)}}{∂α_{0}^{μ}} |^{2} χ_{s}χ_{0}\sqrt{g}dx & ≤ C ∫_{0}^{δ_{0}} \frac{r}{η_{01}^{2}} dr \\ & ≤ C. \end{aligned}

For the fourth term, we have

\begin{aligned}∫_{O_{s}} |\frac{∂ω_{s0}^{(4)}}{∂α_{0}^{μ}} |^{2} χ_{s}χ_{0}\sqrt{g}dx & ≤ C ∫_{|x−a_{0}|<δ_{0}} \frac{κ_{1}^{4}}{ρ^{2}|x − a_{0}|^{2}|x − a_{1}|^{6}} dx \\ & ≤ C ∫_{|x−a_{0}|<δ_{0}} \frac{1}{|x − a_{0}|^{2}|x − a_{1}|^{2}} dx, \end{aligned}

which is bounded above by the same constant C, and similarly for term five. The upper bounds for terms six, seven, and eight follow immediately by symmetry. □

Lemma 5.5.5. The norms of the five terms in ∂ω_{s}/∂α_{0}^{μ} have the following upper bound:

(i) ∫_{O_{s}} |\frac{∂ω_{s}^{(j)}}{∂α_{0}^{μ}} |^{2} χ_{s}(1 − χ_{0})\sqrt{g}dx ≤ C \{1 + log (1 + \frac{1}{|a_{0} − a_{1}|} + \frac{1}{|a_{0} − a_{2}|})\};
for j = 1, 2, 3.
(ii) ∫_{O_{s}} |\frac{∂ω_{s}^{(4)}}{∂α_{0}^{μ}} |^{2} χ_{s}(1 − χ_{0})\sqrt{g}dx ≤ C \{1 + log (1 + \frac{1}{|a_{0} − a_{1}|})\};

(iii) ∫_{O_{s}} |\frac{∂ω_{s}^{(5)}}{∂α_{0}^{μ}} |^{2} χ_{s}(1 − χ_{0})\sqrt{g}dx ≤ C \{1 + log (1 + \frac{1}{|a_{0} − a_{2}|})\} .

Consequently,

∫_{O_{s}} |\frac{∂ω_{s}}{∂α_{0}^{μ}} |^{2} χ_{s}(1 − χ_{0})\sqrt{g}dx ≤ \{1 + log (1 + \frac{1}{|a_{0} − a_{1}|}) + log (1 + \frac{1}{|a_{0} − a_{2}|})\} .

Proof. We note that |x| < 8 and |x − a_{0}| ≥ \frac{1}{2}K^{−4}δ_{0} since we are only integrating over the support of χ_{s}(1 − χ_{0}). Commencing with the first term in ∂ω_{s}/∂a_{0}^{μ}, and employing spherical polar coordinates on ℍ with r = |x−a_{0}|, dx = r^{3}drdθ, we have

∫_{O_{s}} |\frac{∂ω_{s}^{(1)}}{∂α_{0}^{μ}} |^{2} χ_{s}(1 − χ_{0})\sqrt{g}dx ≤ C ∫_{{|x|<8 \atop |x−a_{0}|≥ \frac{1}{2} K^{−4}δ_{0}}} \frac{κ_{0}^{4}}{ρ^{2}|x − a_{0}|^{8}} dx

≤ C ∫_{\frac{1}{2} K^{−4}δ_{0}≤|x−a_{0}|<10} \frac{1}{|x − a_{0}|^{4}} dx

≤ C ∫_{\frac{1}{2} K^{−4}δ_{0}}^{10} \frac{1}{r} dr

≤ C \{1 + log (\frac{1}{δ_{0}} )\} ,

and recalling that δ_{0} = \frac{1}{2} min\{1, η_{01}, η_{02}\}, we obtain the upper bound in (i). Considering the second term, we have

∫_{O_{s}} |\frac{∂ω_{s}^{(2)}}{∂α_{0}^{μ}} |^{2} χ_{s}(1 − χ_{0})\sqrt{g}dx ≤ C ∫_{{|x|<8 \atop |x−a_{0}|≥ \frac{1}{2} K^{−4}δ_{0}}} \frac{κ_{0}^{4}}{ρ^{2}|x − a_{0}|^{8}} dx,

leading to the same upper bound as obtained for the first term. For the third term, we have

\begin{aligned}∫_{O_{s}} |\frac{∂ω_{s}^{(3)}}{∂α_{0}^{μ}} |^{2} χ_{s}(1 − χ_{0})\sqrt{g}dx & ≤ C ∫_{{|x|<8 \atop |x−a_{0}|≥ \frac{1}{2} K^{−4}δ_{0}}} \frac{κ_{0}^{8}}{ρ^{4}|x − a_{0}|^{12}} dx \\ & ≤ C ∫_{\frac{1}{2} K^{−4}δ_{0}≤|x−a_{0}|<10} \frac{1}{|x − a_{0}|^{4}} dx, \end{aligned}

which again leads to the same upper bound. For the fourth term, we have

∫_{O_{s}} |\frac{∂ω_{s}^{(4)}}{∂α_{0}^{μ}} |^{2} χ_{s}(1 − χ_{0})\sqrt{g}dx ≤ C ∫_{{|x|<8 \atop |x−a_{0}|≥ \frac{1}{2} K^{−4}δ_{0}}} \frac{κ_{0}^{4}κ_{1}^{4}}{ρ^{4}|x − a_{0}|^{6}|x − a_{1}|^{6}} dx

≤ C ∫_{|x|<8} \frac{1}{|x − a_{0}|^{2}|x − a_{1}|^{2}} dx

≤ C \{1 + log (\frac{1}{|a_{0} − a_{1}|})\} ,

applying a previous integration lemma to obtain the last inequality, and this gives estimate (ii). Estimate (iii) follows by symmetry. □

Lemma 5.5.6. The norms of the five terms in ∂ω_{n}/∂α_{0}^{μ} are bounded by a constant:

∫_{O_{n}} |\frac{∂ω_{n}^{(j)}}{∂α_{0}^{μ}} |^{2} χ_{n}\sqrt{g}dx ≤ C for j = 1, …, 5.

Consequently,

∫_{O_{n}} |\frac{∂ω_{n}}{∂α_{0}^{μ}} |^{2} χ_{n}\sqrt{g}dy ≤ C.

Proof. Since we are integrating over the support of χ_{n}, we have |y| < 3/8, and as |a_{i}| < 2, then |1 − a_{i}y| > 1/4 for i = 0, 1, 2. Considering the first term in ∂ω_{n}/∂α_{0}^{μ}, we have

∫_{O_{n}} |\frac{∂ω_{n}^{(1)}}{∂α_{0}^{μ}} |^{2} χ_{n}\sqrt{g}dx ≤ C ∫_{|y|<3/8} \frac{κ_{0}^{4}|y|^{2}}{ρ^{2}|1 − a_{0}y|^{8}} dy

≤ C ∫_{|y|<3/8} \frac{1}{|1 − a_{0}y|^{4}} dy

≤ C.

For the second term, we see that

\begin{aligned}∫_{O_{n}} |\frac{∂ω_{n}^{(2)}}{∂α_{0}^{μ}} |^{2} χ_{n}\sqrt{g}dx & ≤ C ∫_{|y|<3/8} \frac{κ_{0}^{4}|\overline{e}_{μ} − 2a_{0}^{μ}\overline{y}|^{2}}{ρ^{2}|1 − a_{0}y|^{8}} dy \\ & ≤ C ∫_{|y|<3/8} \frac{1}{|1 − a_{0}y|^{4}} dy, \end{aligned}

which is again bounded by a constant C. Similarly, for the third term, we have

\begin{aligned}∫_{O_{n}} |\frac{∂ω_{n}^{(3)}}{∂α_{0}^{μ}} |^{2} χ_{n}\sqrt{g}dx & ≤ C ∫_{|y|<3/8} \frac{κ_{0}^{8}|y|^{2}}{ρ^{4}|1 − a_{0}y|^{12}} dy \\ & ≤ C ∫_{|y|<3/8} \frac{1}{|1 − a_{0}y|^{4}} dy. \end{aligned}

Considering the fourth term, we have

\begin{aligned}∫_{O_{n}} |\frac{∂ω_{n}^{(4)}}{∂α_{0}^{μ}} |^{2} χ_{n}\sqrt{g}dx & ≤ C ∫_{|y|<3/8} \frac{κ_{0}^{4}κ_{1}^{4}|y|^{2}}{ρ^{4}|1 − a_{0}y|^{6}|1 − a_{1}y|^{6}} dy \\ & ≤ C ∫_{|y|<3/8} \frac{1}{|1 − a_{1}y|^{2}|1 − a_{0}y|^{2}} dy, \end{aligned}

which is bounded by a constant C, and the same is true for the fifth term by symmetry. □

Proposition 5.5.7. For i = 0, 1, 2 and μ = 0, …, 3, we have the following upper bounds:

(i)  ∥\frac{∂ω}{∂α_{i}^{μ}} ∥^{2} ≤ C \{1 + log (\frac{1}{λ_{i}} ) + ∑_{{j≠i \atop j=0}}^{2} log (1 + \frac{1}{|a_{i} − a_{j}|})\};

(ii)  ∥\frac{∂ω}{∂a_{i}^{μ}} ∥^{2} ≤ C \{1 + log (\frac{1}{λ_{i}} ) + ∑_{{j≠i \atop j=0}}^{2} log (1 + \frac{1}{|a_{i} − a_{j}|})\} .

Proof. One makes the obvious changes in the previous estimates for the norms of the local 1-forms obtained when i = 0 to get estimates for i = 1 and 2. We then combine these inequalities to give the upper bound for the L^{2} norm of ∂ω/∂α_{i}^{μ} in (i). We then use

\begin{aligned}\frac{∂ω}{∂a_{i}^{μ}} & = \frac{∂ω}{∂κ_{i}} \frac{∂κ_{i}}{∂a_{i}^{μ}} + \frac{∂ω}{∂α_{i}^{μ}} |_{α_{i}=a_{i},β_{i}=1} \\ \frac{∂ω}{∂λ_{i}} & = \sqrt{1 + |a_{i}|^{2}} \frac{∂ω}{∂κ_{i}} , \\ \frac{∂κ_{i}}{∂a_{i}^{μ}} & = \frac{λ_{i}a_{i}^{μ}}{\sqrt{1 + |a_{i}|^{2}}} , \end{aligned}

and combine the previous estimates for the norms of ∂ω/∂λ_{i}, to give the estimate (ii). □

§5.6. Diameter and Volume of the Moduli Space

We apply our estimates for the L^{2} norms of the tangent vectors to moduli space to obtain upper bounds for the metric components of the L^{2} metric g. We then use these bounds to show that the space (M_{2}, g) has finite diameter and volume.

Theorem 5.6.1. Let M_{2} denote the moduli space of self-dual connections on a principal SU(2)-bundle P over the sphere 𝕊^{4}, where −c_{2}(P)[𝕊^{4}] = 2 and 𝕊^{4} has its standard round metric g_{0}. Let these connections be parametrized by the space T̃_{2} of unordered pairs (λ_{i}, P_{i}), i = 0, 1, 2, where λ_{0}, λ_{1}, λ_{2} are positive weight parameters satisfying the scaling condition λ_{0}^{2} + λ_{1}^{2} + λ_{2}^{2} = 1, and P_{0}, P_{1}, P_{2} are distinct points in 𝕊^{4} = ℍℙ^{1} (right projective space). Let c_{i}^{μ} denote the standard inhomogeneous coordinates of the points \underline{P}_{i} = \underline{ℍℙ}^{1} (left projective space), for i = 0, 1, 2, μ = 0, …, 3, so that c_{i} = a_{i} if \underline{P}_{i} = [a_{i}, 1], lying in the southern hemisphere, or c_{i} = b_{i} if \underline{P}_{i} = [1, b_{i}], lying in the northern hemisphere. With respect to this choice of parameters, we have the following estimates for the components of the L^{2} metric g:

g_{λ_{i}λ_{i}} = g (\frac{∂ω}{∂λ_{i}} , \frac{∂ω}{∂λ_{i}} ) ≤ C \{1 + log (\frac{1}{λ_{i}} ) + ∑_{j≠i} log (1 + \frac{1}{|c_{i} − c_{j}|})\} ,

g_{c_{i}^{μ}c_{i}^{μ}} = g (\frac{∂ω}{∂c_{i}^{μ}} , \frac{∂ω}{∂c_{i}^{μ}} ) ≤ C \{1 + log (\frac{1}{λ_{i}} ) + ∑_{j≠i} log (1 + \frac{1}{|c_{i} − c_{j}|})\} ,

for i = 0, 1, 2 and μ = 0, …, 3, where C is a universal constant independent of moduli parameters.

Proof. If we denote the moduli parameters by {t_{α}}, then we have the following estimates for the corresponding diagonal components of g:

g_{αα} = g (\frac{∂ω}{∂t_{α}} , \frac{∂ω}{∂t_{α}} ) = (h_{ω} \frac{∂ω}{∂t_{α}} , h_{ω} \frac{∂ω}{∂t_{α}} ) = ∥h_{ω} \frac{∂ω}{∂t_{α}} ∥^{2} ≤ ∥\frac{∂ω}{∂t_{α}} ∥^{2} ,

and consequently these bounds follow immediately from previous estimates. □

Remark 5.6.2. (i) If 𝕊^{4} has a metric g which is globally conformally equivalent to the standard round metric g_{0}, then the above estimates continue to hold, although the constant C will certainly depend on g.

(ii) Estimates of the non-diagonal components of g may be obtained by the Schwarz inequality.

Corollary 5.6.3. With respect to the L^{2} metric g, the moduli space M_{2} has finite diameter and volume.

Proof. It is clearly enough to show that the space has finite diameter. Fix a base point γ^{∗} = {(λ_{0}^{∗}, P_{0}^{∗}), (λ_{1}^{∗}, P_{1}^{∗}), (λ_{2}^{∗}, P_{2}^{∗})} in the parameter space T̃_{2} and let [ω^{∗}] be the corresponding basepoint in M_{2}. Let γ : (0, 1) → T̃_{2} be a smooth curve in T̃_{2} which extends continously to the boundary. In terms of our choice of parameters, we have γ(t) = (λ_{i}(t), P_{j}(t)). Assume that our curve connects the basepoint, γ(1) = γ^{∗}, with a point on the boundary, γ(0) ∈ ∂T̃_{2}. Let ω : (0, 1) → M_{2} denote the corresponding smooth curve in M_{2}, so that w(t) approaches the boundary of M_{2} as t → 0, and w(1) = ω^{∗}, the basepoint connection. Let L(ω) denote the length of the curve ω : (0, 1) → M_{2} with respect to the L^{2} metric g.

In order to show that M_{2} has finite diameter, it is enough to show that minimallength geodesics in (M_{2}, g), connecting the basepoint ω^{∗} with any point on the boundary ∂M_{2}, have lengths uniformly bounded by some constant. Since

g (\frac{dω}{dt} , \frac{dω}{dt} ) = ∥h_{ω} \frac{dω}{dt} ∥^{2} ≤ ∥\frac{dω}{dt} ∥^{2} ,

the curve length L(ω) may be estimated by

L(ω) = ∫_{0}^{1} g^{1/2} (\frac{dω}{dt} , \frac{dω}{dt} ) dt ≤ ∫_{0}^{1} ∥\frac{dω}{dt} ∥ dt.

Note that

\frac{dω}{dt} = ∑_{i=0}^{2} \frac{∂ω}{∂λ_{i}} \frac{dλ_{i}}{dt} + ∑_{i=0}^{2} ∑_{μ=0}^{3} \frac{∂ω}{∂c_{i}^{μ}} \frac{dc_{i}^{μ}}{dt} ,

and consequently

∫_{0}^{1} ∥\frac{dω}{dt} ∥ dt ≤ ∑_{i=0}^{2} ∫_{0}^{1} ∥\frac{∂ω}{∂λ_{i}} ∥ |\frac{dλ_{i}}{dt} | dt + ∑_{i=0}^{2} ∑_{μ=0}^{3} ∫_{0}^{1} ∥\frac{∂ω}{∂c_{i}^{μ}} ∥ |\frac{dc_{i}^{μ}}{dt} | dt.

Since we may choose curves of non-minimal length connecting the basepoint ω^{∗} and points on the boundary of M_{2}, we can make the following assumptions concerning the curve γ(t) = {λ_{i}(t), P_{j}(t)}, and hence ω(t):

(i) t ≤ λ_{i}(t) ≤ 1, for 0 < t < 1 and i = 0, 1, 2; (ii) t ≤ |c_{i}(t) − c_{j}(t)| ≤ 2, for 0 < t < 1, i ≠ j, and i, j = 0, 1, 2; (iii) |dλ_{i}/dt| ≤ N and |dc_{i}^{μ}/dt| ≤ N, for 0 < t < 1, i = 0, 1, 2, and μ = 0, …, 3, and

the constant N is the same for all curves γ.

Hence, we obtain

\begin{aligned}L(ω) & ≤ C \{∑_{i=0}^{2} ∫_{0}^{1} ∥\frac{∂ω}{∂λ_{i}} ∥ dt + ∑_{i=0}^{2} ∑_{μ=0}^{3} ∫_{0}^{1} ∥\frac{∂ω}{∂c_{i}^{μ}} ∥ dt\} \\ & ≤ C \{∑_{i=0}^{2} ∫_{0}^{1} ∥\frac{∂ω}{∂λ_{i}} ∥^{2} dt + ∑_{i=0}^{2} ∑_{μ=0}^{3} ∫_{0}^{1} ∥\frac{∂ω}{∂c_{i}^{μ}} ∥^{2} dt\} , \end{aligned}

for some constant C independent of the curve γ. Using our bounds for the metric components, we see that

\begin{aligned}∥\frac{∂ω}{∂λ_{i}} ∥^{2} & ≤ C \{1 + log (\frac{1}{t})\} , \\ ∥\frac{∂ω}{∂c_{i}^{μ}} ∥^{2} & ≤ C \{1 + log (\frac{1}{t})\} , \end{aligned}

for i = 0, 1, 2 and μ = 0, …, 3. Hence, the length of the curve ω is bounded by

L(ω) ≤ C ∫_{0}^{1} (1 − log t) dt
= C[2t − t log t]_{0}^{1}
= 2C,

and this completes the proof. □

BIBLIOGRAPHY

[A] Atiyah, M. F., Geometry of Yang-Mills Fields, Lezioni Fermiane, Acad. Naz. Lincei Scuola Normale Sup., Pisa 1979.

[A-B] Atiyah, M. F. and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A 308, 523-615 (1982).

[A-D-H-M] Atiyah, M. F., V. G. Drinfel’d, N. J. Hitchin, and Yu. I. Manin, Construction of instantons, Phys. Lett. 65A, 185-187 (1978).

[A-H-S] Atiyah, M. F., N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Royal Soc. London A 362, 425-461 (1978).

[A-J] Atiyah, M. F. and J. D. S. Jones, Topological aspects of Yang-Mills theory, Commun. Math. Phys. 61, 97-118 (1978).

[A-W] Atiyah, M. F. and R. S. Ward, Instantons and algebraic geometry, Commun. Math. Phys. 55, 117-124 (1977).

[Au-Do] Aupetit, H. and A. Douady, Fibrés stables de rang 2 sur ℂℙ^{3} avec c_{1} = 0, c_{2} = 2, in: Les Equations de Yang-Mills, Eds. A. Douady and J.-L. Verdier, Séminaire E.N.S. 1977-78, Astérisque 71-72 (1980), pp. 171-196.

[B-H] Barth, W. and K. Hulek, Monads and moduli of vector bundles, Manu. Math. 25, 323-347 (1978).

[Ber] Berezin, F. A., Instantons and Grassman Manifolds, Funk. Analiz. 13, 75-76 (1978).

[Bey] Beyer, W. H., Standard Mathematical Tables, 26th Ed., Chemical Rubber Co. Press, 1981.

[Bo-Tu] Bott, R. and L. Tu, Differential Forms in Algebraic Topology, Springer-Verlag: New York, 1982.

[Bo-Ma] Boyer, C. P. and B. M. Mann, Homology operations on Instantons, J. Differential Geometry 28, 423-465 (1988).

[B-H-M-M] Boyer, C. P., J. C. Hurtubise, B. M. Mann, and R. J. Milgram, The topology of instanton moduli spaces I: the Atiyah-Jones conjecture, to appear in Ann. Math.

[Br-tD] Bröcker, T. and T. tomDieck, Representations of Compact Lie Groups, Springer-Verlag: New York, 1985.

[Bu] Buchdahl, N. P., Instantons on ℂℙ^{2}, J. Differential Geometry 24, 19-52 (1986).

[C-W-S] Christ, N. H., E. J. Weinberg, and N. K. Stanton, General self-dual Yang-Mills solutions, Phys. Rev. D. 18, 2013-2025 (1978).

[C-F-G-T] Corrigan, E., D. B. Fairlie, P. Goddard, and S. Templeton, A Green function for the general self-dual gauge field, Nuc. Phys. B 140, 31-44 (1978).

[C-G-O-T] Corrigan, E., P. Goddard, H. Osborn, and S. Templeton, Zeta-function regularisation and multi-instanton determinants, Nuc. Phys. B 159, 469-496 (1979).

[D-M-M] Doi, H., Y. Matsumoto, and T. Matumoto, An explicit formula of the metric on the moduli space of BPST-instantons over 𝕊^{4}, in: A Fête of Topology, eds. Y. Matsumoto et al., Academic Press: New York, 1988.

[D-P] D’Hoker, E. and D. H. Phong, The Geometry of String Perturbation Theory, Reviews of Modern Physics 60, American Physical Society: New York, 1988.

[D1] Donaldson, S. K., Instantons and geometric invariant theory, Commun. Math. Phys. 93, 453-460 (1984).

[D2] Donaldson, S. K., Vector bundles on the flag manifold and the Ward correspondence, in: Geometry Today, ed. E. Arbarello et al., Birkhäuser: Boston, 1985.

[D3] Donaldson, S. K., Connections, cohomology and the intersection forms of four manifolds, J. Differential Geometry 24, 275-341 (1986).

[D4] Donaldson, S. K., Compactification and completion of Yang-Mills moduli spaces, in: Differential Geometry, Lecture Notes in Mathematics 1410, eds. F. J. Carreras et al., Springer-Verlag: New York, 1989.

[D5] Donaldson, S. K., Instantons in Yang-Mills theory, in: Proceedings of the IMA Conference on Geometry and Particle Physics, Oxford 1988, ed. F. Tsou, Oxford University Press: New York, 1990.

[D6] Donaldson, S. K., Polynomial invariants for smooth 4-manifolds, Topology 29, 257-315 (1990).

[D-K] Donaldson, S. K. and P. B. Kronheimer, The Geometry of Four-Manifolds, Oxford University Press: New York, 1990.

[D-M1] Drinfel’d, V. G. and Yu. I. Manin, Self-dual Yang-Mills Fields over a Sphere, Funk. Analiz. 12, 78-79 (1978).

[D-M2] Drinfel’d, V. G. and Yu. I. Manin, A description of instantons, Commun. Math. Phys. 63, 177-192 (1978).

[D-M3] Drinfel’d, V. G. and Yu. I. Manin, Instantons and bundles on ℂℙ^{3}, Funk. Analiz. 13, 59-74 (1979).

[D-M4] Drinfel’d, V. G. and Yu. I. Manin, Yang-Mills, instantons, tensor products of instantons, Soviet J. Nucl. Phys. 29, 845-849 (1979).

[Fi] Fischer, A. E., The internal symmetry group of a connection on a principal fibre bundle with applications to gauge field theories, Commun. Math. Phys. 113, 231-262 (1987).

[F-T1] Fischer, A. E. and A. J. Tromba, On a purely Riemannian proof of the structure and dimension of the unramified moduli space of a compact Riemann surface, Math. Ann. 267, 311-345 (1984).

[F-T2] Fischer, A. E. and A. J. Tromba, On the Weil-Petersson metric on Teichmüller space, Trans. A.M.S. 284, 319-335 (1984).

[F-G] Freed, D. S. and D. Groisser, The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group, Mich. Math. J. 36, 323-344 (1989).

[F-U] Freed, D. S. and K. K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer-Verlag: New York, 1991.

[F-M] Friedman, R. and J. W. Morgan, On the diffeomorphism types of certain algebraic surfaces I, II, J. Differential Geometry 27, 297-398 (1988).

[G-H] Griffiths, P. and J. Harris, Principles of Algebraic Geometry, Wiley: New York, 1978.

[G-R] Giambiagi, J. J. and K. D. Rothe, Regular N-instanton fields and singular gauge transformations, Nuclear Physics B129, 111-124 (1977).

[G-P1] Groisser, D. and T. H. Parker, The Riemannian geometry of the Yang-Mills moduli space, Commun. Math. Phys. 112, 663-689 (1987).

[G-P2] Groisser, D. and T. H. Parker, The geometry of the Yang-Mills moduli space for definite manifolds, J. Differential Geometry 29, 499-544 (1989).

[G-P3] Groisser, D. and T. H. Parker, Semi-classical Yang-Mills theory I: Instantons,

Commun. Math. Phys. 135, 101-140 (1990).

[Gu] Gunning, R. C., Lectures on Vector Bundles over Riemann Surfaces, Princeton University Press: Princeton, 1967.

[Hab] Habermann, L., On the geometry of the space of Sp(1)-instantons with Pontrjagin index 1 on the 4-sphere, Ann. Global Anal. Geom. 6, 3-29 (1988).

[Har1] Hartshorne, R., Algebraic Geometry, Springer-Verlag: New York, 1977.

[Har2] Hartshorne, R., Stable vector bundles and instantons, Commun. Math. Phys. 59, 1-15 (1978).

[Har3] Hartshorne, R., Stable vector bundles of rank 2 on ℙ^{3}, Math. Ann. 238, 229-280 (1978)

[Hat] Hattori, A., Topology of the moduli space of SU(2)-instantons with instanton

number 2, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 34, 741-761 (1987).

[H-K-L-R] Hitchin, N. J., A. Karlhede, U. Lindstrom, and M. Rǒcek, Hyperkähler metrics

and supersymmetry, Commun. Math. Phys. 108, 535-589 (1987).

[Hur] Hurtubise, J., Instantons and jumping lines, Commun. Math. Phys. 105, 107-122 (1986).

[Hus] Husemoller, D., Fibre Bundles, 2nd ed., Springer-Verlag: New York, 1966.

[I] Itoh, M., Geometry of anti-self-dual connections and Kuranishi Map, J. Math.

Soc. Japan 40, 9-33 (1988). [J-N-R] Jackiw, R., C. Nohl, and C. Rebbi, Conformal properties of pseudoparticle

configurations, Phys. Rev. D 15, 1642-1646 (1977).

[K] Kobayashi, S., Differential Geometry of Complex Vector Bundles, Princeton

University Press: Princeton, 1987. [K-N] Kobayashi, S. and K. Nomizu, Foundations of Differential Geometry, Volumes

I & II, Wiley: New York, 1963 and 1968.

[Ko] Kodaira, K., Complex Manifolds and Deformation of Complex Structures, Springer-Verlag: New York, 1986.

[La] Lawson, H. B., The Theory of Gauge Fields in Four Dimensions, American Mathematical Society: Providence, 1985.

[Le] LeBrun, C., On complete quaternionic-Kähler manifolds, Duke Math. J. 63, 723-743 (1991).

[Mac] Maciocia, A., Metrics on the moduli spaces of instantons over Euclidean 4spaces, Commun. Math. Phys. 135, 467-482 (1991).

[MC-S] Mamone Capria, M. and S. M. Salamon, Yang-Mills fields on quaternionic spaces, Nonlinearity 1, 517-530 (1988).

[M-M] Marathe, K. B. and G. Martucci, The geometry of gauge fields, J. Geometry and Physics 6, 1-105 (1989).

[Mas] Masur, H., The extension of the Weil-Petersson metric to the boundary of Teichmüller space, Duke Math. J. 43, 623-635 (1976).

[Mi-St] Milnor, J. and J. Stasheff, Characteristic Classes, Princeton University Press: Princeton, 1974.

[Mo] Morgan, J. W., The Topology of Four-Manifolds, unpublished manuscript, Columbia University: New York, 1989.

[Mu-Fo] Mumford, D. and J. Fogarty, Geometric Invariant Theory, 2nd ed., Springer-Verlag: New York, 1982.

[N-T1] Narasimhan, M. S. and G. Trautman, Compactification of M(0, 2), in: Vector Bundles on Algebraic Varieties, eds. M. F. Atiyah et al., Oxford University Press, Bombay 1987, pp. 429-443.

[N-T2] Narasimhan, M. S. and G. Trautman, Compactification of M_{ℙ_{3}}(0, 2) and Poncelet pairs of conics, Pacific J. Math. 145, 255-365 (1990).

[N-T3] Narasimhan, M. S. and G. Trautman, The Picard group of the compactification of M_{ℙ_{3}}(0, 2) J. Reine angew. Math. 422, 21-44 (1991).

[O] Osborn, H., Semi-classical functional integrals for self-dual gauge fields, Ann. Phys. 135, 373-415 (1981).

[O-S-S] Okonek, C., M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, Birkhaüser: Boston, 1980.

[R1] Rawnsley, J. H., On the Atiyah-Hitchin-Drinfeld-Manin vanishing theorem for cohomology groups of instanton bundles, Math. Ann. 241, 43-56 (1979).

[R2] Rawnsley, J. H., Self-dual Yang-Mills fields, in: Global Analysis, eds. J Marsden and M. Grmela, Lecture Notes in Mathematics 755, 295-312 (1979).

[Sa] Salamon, S., Instantons on the 4-sphere, Rend. Sem. Mat. Univers. Politec. Torino 40, 1-20 (1982).

[Sc] Schwarz, A. S., Instantons and fermions in the field of instanton, Commun. Math. Phys. 64, 223-268 (1979).

[Se] Seshadri, C. S., Theory of moduli, p. 263-304, in Algebraic Geometry, Arcata 1974, Proc. Symp. Pure Math. 29, ed. R. Hartshorne, A.M.S.: Providence, 1975.

[S-S-U] Sibner, L. M., R. J. Sibner, and K. K. Uhlenbeck, Solutions to Yang-Mills equations that are not self-dual, Proc. Natl. Acad. Sci. USA 86, 8610-8613 (1989).

[Si1] Singer, I. M., Some remarks on the Gribov ambiguity, Commun. Math. Phys. 60, 7-12 (1978).

[Si2] Singer, I. M., The geometry of the orbit space for non-abelian gauge theories, Physica Scripta 24, 817-820 (1981).

[S-T] Singhof, W. and G. Trautman, On the topology of the moduli space M(0, 2) of stable bundles of rank 2 on ℙ^{3}, Quart. J. Math. Oxford 41, 335-358, (1990).

[St] Steenrod, N., The Topology of Fibre Bundles, Princeton University Press: Princeton, 1951.

[T1] Taubes, C. H., Self-dual Yang-Mills connections on non-self-dual 4-manifolds, J. Differential Geometry 17, 139-170 (1982).

[T2] Taubes, C. H., Self-dual connections on 4-manifolds with indefinite intersection matrix, J. Differential Geometry 19, 517-560 (1984).

[T3] Taubes, C. H., Path-connected Yang-Mills moduli spaces, J. Differential Geometry 19, 337-392 (1984).

[T4] Taubes, C. H., A framework for Morse theory for the Yang-Mills functional, Invent. Math. 94, 327-402 (1988).

[T5] Taubes, C. H., The stable topology of self-dual moduli spaces, J. Differential Geometry 29, 162-230 (1989).

[Wa-We] Ward, R. S. and R. O. Wells, Twistor Geometry and Field Theory, Cambridge University Press: Cambridge, 1990.

[Wen] Wentworth, R., The asymptotics of the Arakalov-Green’s function and Falting’s delta invariant, Commun. Math. Phys. 137, 427-459 (1991).

[Wi] Witten, E., Topological quantum field theory, Commun. Math. Phys. 117, 353-386 (1988).

[Wo1] Wolpert, S. A., Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Commun. Math Phys. 112, 285-315 (1987).

[Wo2] Wolpert, S. A., Geodesic length functions and the Nielsen problem, J. Differential Geometry 25, 275-296 (1987).

[Wo3] Wolpert, S. A., Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85, 119-145 (1986).

[Wo4] Wolpert, S. A., On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107, 969-997 (1985).