Research Descriptions

Recently, I have proved a general version of the Verlinde conjecture in the framework of vertex operator algebras. This work uses results obtained during the last fifteen years by many people, including in particular essentially all the results obtained by myself (some of them jointly with Lepowsky) in a series of papers on the construction of genus-zero and genus-one chiral conformal field theories. This new theorem states: Let V be a simple vertex operator algebra satisfying the following conditions: (i) V has no nonzero elements of negative weights, any element of weight zero is proportional to the vacuum, and the contragredient module V' is isomorphic to V as a V-module. (ii) Any N-gradable weak V-module is completely reducible. (iii) V is C_2-cofinite. Then the action of the modular transformation which maps tau to -1/tau on the space of characters of irreducible V-modules diagonalizes the matrices formed by the fusion rules. The Verlinde conjecture and its well-known physical proof by Moore and Seiberg based on axioms for rational conformal field theories have played a fundamental role in the development of conformal field theory and have led to surprising mathematical results. In the special case of the Wess-Zumino-Novikov-Witten models, the Verlinde conjecture led to a surprising ``Verlinde formula'' for the dimensions of conformal blocks on Riemann surfaces, which was mathematically proved by Tsuchiya-Ueno-Yamada, Beauville-Laszlo, Faltings and Kumar-Narasimhan-Ramanathan. The proof of this theorem above is based heavily on the results establishing the duality and modular invariance properties of genus-zero and genus-one correlation functions in my recent papers. The main work in the proof has been to prove two formulas of Moore and Seiberg derived from the axioms of rational conformal field theories. Using this new theorem above, the Verlinde formula expressing the fusion rules in terms of the matrix elements of the action of the modular transformation which maps tau to -1/tau follows trivially for these vertex operator algebras.

Using this new theorem on the Verlinde conjecture, I have also proved the following theorem: For a vertex operator algebra V satisfying the conditions above, the braided tensor category structure on the category of V-modules is rigid and modular. In particular, the category of V-modules has a natural structure of a modular tensor category. This theorem has many applications. Modular tensor categories are a fundamental structure underlying the constructions of the quantum invariants of knots and three manifolds and of three-dimensional topological field theories. Motivated by the Jones and HOMFLY invariants for knots and links, Witten found the relationship between conformal field theories, Chern-Simons theory and invariants for three-manifolds with embedded links. Using what came to be called ``modular tensor categories'' constructed from representations of certain quantum groups, Reshetikhin and Turaev constructed these invariants of Jones, HOMFLY and Witten for links and three-manifolds. Assuming the existence of the structure of a modular tensor category on the category of modules for a (suitable) vertex operator algebra and the existence of conformal blocks with monodromies compatible with this modular tensor category, Felder, Frohlich, Fuchs, Runkel and Schweigert have studied open-closed conformal field theory using the theory of tensor categories and three-dimensional topological field theories. The new theorems above on the Verlinde conjecture and modular tensor categories provide foundations for these works and also provide tools for further studies.

The main goal of my research is and has been to mathematically construct conformal field theories in the sense of Kontsevich and Segal and to apply the results obtained in the construction to solve problems in algebra, geometry, topology and mathematical physics. Solving problems such as those discussed above fulfills part of that goal. In addition, I have now completely constructed genus-zero conformal field theories and, by using and considerably extending a theorem of Zhu, also completely constructed genus-one chiral rational conformal field theories, not just genus-zero and genus-one modular functors. In the process of constructing such theories, in addition to the problems discussed above, I have been able to solve a number of other problems, many of which also play important roles in the solutions of the problems above. For example, a strong version of the operator product expansion (associativity) for intertwining operators was established, the problem of constructing projective representations of the semigroup of annuli was solved, differential equations satisfied by genus-zero and genus-one conformal field theories were found and studied, and duality and modular invariance theorems for genus-zero and genus-one chiral correlation functions were proved. My early joint work with Lepowsky on the braided tensor category structure on the category of modules for a suitable vertex operator algebra is one the foundations of my recent work on the modular tensor category structure.

I also mathematically constructed and studied concrete examples in the genus-zero case, for example, the minimal models and the Z_2-orbifold theories underlying the moonshine module vertex operator algebra constructed by Frenkel, Lepowsky and Meurman. In particular, I gave a new conceptual proof that the moonshine module has a natural structure of a vertex operator algebra. Lepowsky and I constructed and studied the genus-zero Wess-Zumino-Novikov-Witten models in the framework of vertex operator algebras. Milas and I have constructed and studied N=1 and N=2 minimal or unitary models and the Gepner models. Using the reults on intertwining operator algebras, I have constructed conformal-field-theoretic analogues of binary codes and lattices, generalizing the work of Frenkel-Lepowsky-Meurman on the analogy among doubly even codes, even lattices and vertex operator algebras. This precise analogy might be useful in giving hints about possible methods to prove Frenkel-Lepowsky-Meurman's uniqueness conjecture concerning their moonshine module vertex operator algebra. Kong and I have have constructed open-string vertex algebras and established the equivalence among these algebras, associative algebras in certain tensor categories, and algebras over a certain partial operad (thus generalizing my early work in on the geometry of vertex operator algebras and my joint work with Kirillov and Lepowsky on commutative associative algebras in braided tensor categories and vertex operator algebras). Lepowsky, Zhang and I have done work on ``irrational'' conformal field theories and in particular we have solved the problem of generalizing Huang-Lepowsky's tensor product theory to a logarithmic tensor product theory. Lepowsky, Li, Zhang and I have illuminated the crucial role of Huang-Lepowsky's ``compatibility condition'' in both the early work of Huang-Lepowsky and this new work jointly with Lepowsky and Zhang.

I gave a geometric formulation of topological vertex algebra and proved an isomorphism theorem. As an application of this isomorphism theorem, I gave a new conceptual proof of the results of Lian-Zuckerman and Getzler that the cohomology of a topological vertex algebra has a natural structure of a Gerstenhaber or Batalin-Vilkovisky algebra. Zhao and I gave a geometric formulation of (strong) topological vertex operator algebra and proved an isomorphism theorem. As an application of this isomorphism theorem, we proved in the same paper a conjecture of Lian-Zuckerman and Kimura-Voronov-Zuckerman that a (strong) topological vertex operator algebra gives a (weak) homotopy Gerstenhaber algebra. Barron, Lepowsky and I proved a very general theorem giving the factorization of formal exponentials which can be thought of as an inverse of the Campbell-Baker-Hausdorff formula. Lepowsky and I proved the equivalence between the notion of vertex algebra in the sense of Borcherds and the notion of chiral algebra in the sense of Beilinson-Drinfeld.

Before I started to work on vertex operator algebras and conformal field theory, I did research on general relativity and the geometry in Minkowski spaces. Xu and I derived an analog of the Ernst equation in general relativity when there is a cosmological term in the Einstein equation. I found all types of Bäcklund transformations in the three dimensional Minkowski space. There are three types: one tranforms a space-like surface of negative constant curvature to a surface of same kind. Another transforms a time-like surface of negative constant curvature to a surface of same kind. The third transforms a time-like surface of positive constant curvature to a space-like surface of positive constant curvature and vice versa. We also discovered that the singularities of transformations of the third type occurs only when the images of the transformations go to the null infinity of the Minkowski space.