Research Interests

1. Two-dimensional orbifold conformal field theories: Two-dimensional orbifold conformal field theories are two-dimensional conformal field theories constructed from known conformal field theories and their automorphisms. The first example of two-dimensional orbifold conformal field theories, the moonshine module, was constructed by Frenkel, Lepowsky and Meurman in mathematics. The systematic study of two-dimensional orbifold conformal field theories in string theory was started by Dixon, Harvey, Vafa and Witten. Since then, two-dimensional orbifold conformal field theory has been developed in mathematics as an important part of mathematical two-dimensional conformal field theory.
   Two-dimensional orbifold conformal field theory is not just a mathematical procedure to obtain new examples of conformal field theories. More importantly, we expect that it will provide a powerful approach to solve mathematical problems and prove mathematical conjectures after it is fully developed. For example, one of the most important conjecture in the theory of vertex operator algebras and mathematical conformal field theory is the uniqueness of the moonshine module formulated by Frenkel, Lepowsky and Meurman. The moonshine module is constructed as an orbifold conformal field theory from the Leech lattice vertex operator algebra and its automorphism induced from the point reflection in the origin of the Leech lattice. From the viewpoint of orbifold conformal field theory, the uniqueness implies that every vertex operator algebra obtained as an orbifold conformal field theory satisfying the three conditions in the uniqueness conjecture must be isomorphic to the moonshine module as a vertex operator algebra. In particular, we have to study general orbifold conformal field theories satisfying the three conditions in this uniqueness conjecture. Such a study in turn means that we have to develop a general orbifold conformal field theory.
   I have formulated the main conjectures and open problems on the construction and study of orbifold conformal field theories in 2020. They are mainly about twisted intertwining operators (intertwining operators among twisted modules for vertex operator algebras), including, in particular, associativity (operator product expansion) of twisted intertwining operators, modular invariance of twisted intertwining operators and crossed braided tensor category structures on the category of twisted modules. A student of mine and I have introcduced the most general twisted intertwining operators, constructed tensor products of twisted modules and constructed the crossed braiding isomorphisms. Another student of mine has derived differential equations satisfied by products of a particular type of twisted intertwining operators.

2. The last step in the construction of rational conformal field theoires in the sence of Kontsevich and Segal: In 1987, Kontsevich and Segal independently gave a precise definition of full conformal field theory using the properties of path integrals as axioms. To explain the rich structure of chiral parts of conformal field theories, Segal further introduced the notions of modular functor and weakly conformal field theory and sketched how to obtain a conformal field theory from a suitable weakly conformal field theory. In 1988, Moore and Seiberg further formulated certain basic hypotheses for rational conformal field theories and derived some important consequences. It has been a long-standing open problem to construct chiral and full rational conformal field theories in the sense of Kontsevich and Segal.
   The genus-zero and genus-one parts of chiral rational conformal field theories were essentially constructed by me, based on results in the representation theory of vertex operator algebras obtained by myself and by many other people. A convergence problem for constructing the higher-genus chiral correlation functions was solved by Gui. The construction of genus-zero and genus-one full rational conformal field theories from chiral genus-zero and genus-one chiral conformal field theories was given by Fuchs, Runkle and Schweigert, by Fjelstad, Fuchs, Runkle and Schweigert and by Kong and me.
   To verify all the axioms of Segal for rational conformal field theories, we still need to prove the holomorphicity of the canonical isomorphisms associated to the sewing operation on the moduli spaces of higher-genus Riemann surfaces with parametrized boundaries. Here the complex structure on the moduli space of Riemann surfaces with parametrized boundaries is the one given by Eric Schippers, David Radnell and Wolfgang Staubach. Another problem needs to be worked out is to extend the results obtained for correlation functions among modules for vertex operator algebras to suitable locally convex topological completion of these modules.
3. Rigidity of the braided tensor category structure on modules of finite lengths for a C_2-cofinite vertex operator algebra : In 1988, Moore and Seiberg derived a system of polynomial equations from the axioms for rational conformal field theories. They showed that the Verlinde conjecture is a consequence of these equations. Inspired by an observation of Witten on an analogy with Mac Lane's coherence, Moore and Seiberg also demonstrated that some properties in the theory of these polynomial equations are analogous to properties in theory of tensor categories. Later, a precise notion of modular tensor category was introduced by Turaev. Then we had a conjecture that for a vertex operator algebra satisfying suitable conditions (mainly a complete reducibility condition for suitable modules and a condition called C_2-cofiniteness condition introduced first by Yongchang Zhu), the category of suitable semisimple modules for this vertex operator algebra is a modular tensor category in the sense of Turaev. This conjecture was proved by me in 2004 (with a talk announcing the result in 2004 in Osaka and with a paper posted in arXiv in 2005 and published in 2008) using the construction by Lepowsky and me of the braided tensor category structure on the category of modules for a vertex operator algebra satisfying suitable weaker condition and the results of me proving the associativity of intertwining operators, the modular invariance of intertwining operators and the proof of the Verlinde conjecture.
   Since then, an active research direction is to generalize this result to the logarithmic case in which the complete reducibility for modules and/or the C_2-cofiniteness is not assumed. There have been many results on tensor categories of suitable modules for vertex operator algebras associated to the Virasoro algebra, affine Lie algebras, W-algebras and the corresponding vertex operator superalgebras by many mathematicians.
   For a vertex operator algebra satisfying the C_2-cofiniteness condition and a minor condition on the grading of the vertex operator algebra, using a general construction of Lepowsky, Zhang and me on nonsemisimple (logarithmic) braided tensor category structure on suitable module category for a vertex operator algebra satisfying suitable conditions, I constructed a braided tensor category structure on the category of (generalized) modules of finite lengths for a vertex operator algebra satisfying suitable finiteness conditions and some other minor conditions.
   In 2009, I proposed the conjecture that this braided tensor category is rigid. It is still a main unsolved probelm in this research direction. I hope that someone will prove this conjecture in the next few years.
4. The representation theory of Virasoro vertex operator algebra and the Liouville conformal field theory: Liouville conformal field theory is a two-dimensioanl conformal field theory whose classical equation of motion is a generalization of Liouville's equation. It was first studied in physics by Dorn, Otto, A. Zamolodchikov and Al. Zamolodchikov. Mathematically it was constructed by Guillarmou, Kupiainen, Rhodes and Vargas using the approach of probability. Algebraically, physicists know that Liouville conformal field theory can be studied using the representation theory of the Virasoro algebra, especially the theory of irreducible Verma modules for the Virasoro algebra.
   Verma modules for the Virasoro algebras and their quotients are all modules for Virasoro vertex operator algebras. So in principle, Liouville conformal field theory can be studied using the representation theory of Virasoro vertex operator algebras. But the main difficulty in this approach is the convergnec of products of intertwining operators among Verma modules for the Virasoro algebra. In the case of products of intertwining operators among irreducible quotients of reducible Verma modules for the Virasoro algebra, the convergence can be proved using differential equations derived by me since these modules are C_1 cofinite. Creutzig, Jiang, Orosz Hunziker, Ridout and Yang and McRae and Yang showed that this approach also works for intertwining operators among modules of finite lengths for the Virasoro algebra with irreducible quotients of reducible Verma modules as irreducible factors. But this approach dos not work for intertwining operators among Verma modules. The construction by Guillarmou, Kupiainen, Rhodes and Vargas in principle has solved this problem. But one needs to reformulate their work using the language of vertex operator algebras. Such reformulations should be very important since we might be able to generalize the method to other two-dimensional conformal field theories which are still not constructed and to solve some open problems in the representation theory of vertex operator algebras.
5. The representtation theory of meromorphic open-string vertex operator algebras and nonlinear sigma models: Nonlinear sigma models are an important class of two-dimentional conformal field theories that play an important role in string theory and in the mathematical conjectures derived from string theory.
   String theory is a physical theory which some physicists are trying to use to unify all the fundamental interactions in the universe. The basic assumption of string theory is that the fundamental constituents of our universe are strings (one-dimesional objects), not particles (zero-dimensional objects). Up to now, as a physical theory, string theory is still a theory without experimental proof, though it is consistent with all the existing experimental results. But the mathematics underlying string theory seems to be related to all branches of mathematics. Some very abstract mathematical theories now find their use in string theory. More interestingly (at least to mathematicians), intuition of string theorists predicted surprising mathematical theorems, and techniques used by string theorists supplied powerful tools to solutions of mathematical problems. Therefore, no matter whether string theory as a physical theory is correct or not, the mathematics underlying string theory will always be interesting to mathematicians.
   Conjectures by physicists on nonlinear sigma-models are one of the most influential sources of inspirations and motivations for many recent works in geometry. Classically, a nonlinear sigma-model is just the set of all harmonic maps from a two-dimensional Riemannian manifold to a Riemannian manifold (the target). The main challenge for mathematicians is the construction of the corresponding quantum nonlinear sigma-model. The difficulties lie in the fact that in the case that the target is not flat, the nonlinear sigma-model is a quantum field theory with interaction. In physics, a quantum field theory with interaction is studied by using the methods of perturbative expansion and renormalization. Unfortunately, it does not seem to be possible to directly rigorize these physical methods to construct the correlation functions of such a quantum field theory mathematically.
   Assuming the existence of nonlinear sigma-models, as I mentioned above, physicists have obtained many surpirsing mathematical conjectures. Some of these conjectures have been proved by mathematicians using methods developed in mathematics. But there are more deep conjectures waiting to be understood and proved. Besides proving these conjectures from physics, it is also of great importance to understand mathematically what is going on underlying these deep conjectures. A mathematical construction of nonlinear sigma-models would allow us to obtain such a deep conceptual understanding and at the same time to prove these conjectures. I also hope that the construction and study of these nonlinear sigma models will provide some insight into the Clay Institute problem on the existence of Yang-Mills theory and the mass gap conjecture.
   In 2012, I started a program to construct nonlinear sigma models using the representation theory of ``noncommutative'' generalizations of vertex algebras called meromorphic open-string vertex operator algebras introduced by me in the same year. Given a Riemannian manifold, I have constructed a sheaf of meromorphic open-sring vertex operator algebras and more importantly a sheaf of left modules for this sheaf of meromorphic open-string vertex operator algebras generated from each eigenfunction on the Riemannian manifold. Though the construction of the sheaf of open-string vertex operator algebras uses the tautological method of taking parallel sections of suitable holomorphic bundles, the construction of the sheaves of left modules are completely different. Examples of these sheaves of meromorphic open-string vertex operator algebras and sheaves of left modules for them have been given explicitly by Qi. This project involves not only the representation theory of meromorphic open-string vertex algebras, but also the differential geometry of Riemmanian, Kahler and Calabi-Yau manifolds and geometric analysis on these manifolds. The Laplacian on a Riemannian manifold appear naturally in my construction. I hope that soon we will be able to construct Dirac-like operators which contain Dirac operators on spin manifolds as suitable components.
6. Cohomology theory of vertex algebras and moduli space of two-dimensional conformal field theories: Two-dimensional conformal field theory describes perturbative string theory. To understand string theory mathematically, it is necessary to understand the moduli space of two-dimensional conformal field theories. Mathematically, the mirror symmetry for Calabi-Yau manifolds is closely related to the deformations of Calabi-Yau manifolds and the deformations of two-dimensional conformal field theories.
   In 2010, I constructed a cohomology theory of grading-restricted vertex algebras. I showed that the second cohomology of a grading-restricted vertex algebra corresponds to the space of first order deformations of the vertex algebra. Recently, Qi constructed and classified the second cohomology of a class of universal type vertex algebras, including in particular the Verma module Virasoro vertex algebra, the generalized Verma module affine vertex algebras and Heisenberg vertex algebras. I also showed in uncirculated notes that the third cohomology and a certain convergence condition is the obstruction for the existence of a formal deformation started from a given first order deformation. The cohomology theory and deformation theory can be generalized to full conformal field theories. The most difficult problem is to show that formal deformations are convergent to analytic defomrations under suitable conditions. If these problems are solved, we should be able to use these cohomology theories and deformations theories as tools to study the moduli space of two-dimensional conformal field theories.
7. The representation theory of vertex operator algebras and quantum Hall states: Conformal field theory also describes many exciting phenomena in condensed matter physics. One of such phenomena is quantum Hall effects. Physicists have discovered (abelian) anyons from quantum Hall effects and predicted the existence of nonabelian anyons. If nonabelian anyons can indeed be found in experiments, it will physically realize modular tensor category structures, which will allow mathematicians, physicists and computer scientists to eventually build topological quantum computers.
   Theoretically, the wavefunctions of quantum Hall states can be described by correlation functions of suitable vertex operator algebras. If the vertex operator algebra satisfies some natural conditions, a theorem of mine mentioned above says that the category of representations of this vertex operator algebra is a modular tensor category. In view of these results, it is a very important problem to classify and study all the vertex operator algebras whose correlation functions are possible candidates of wavefunctions of quantum Hall states. This is another topic I am very interested.