Research Interests

Nonlinear sigma models are an important class of two-dimentional conformal field theories that play an important role in 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.

Recently, I have started a program to construct nonlinear sigma models using the representation theory of vertex operator algebras. For the next few years, this will be one of my main research projects. This project involves not only the representation theory of vertex operator algebras, but also the differential geometry of Riemmanian, Kahler and Calabi-Yau manifolds and geometric analysis on these manifolds.

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. I have constructed a cohomology theory of grading-restricted vertex algebras. Using this cohomology theory, I have developed a deformation theory for such grading-restricted vertex algebras. The cohomology theory and deformation theory can be generalized to full conformal field theories. I will use these cohomology theories and deformations theories as tools to study the moduli space of two-dimensional conformal field theories.

Conformal field theory also describes many exciting phenomena in condensed matter physics. One of such phenomena is quantum Hall effects. Physicists have discovered anyons from quantum Hall effects and predicted the existence of nonabelian anyons. If nonabelian anyons can indeed be found in experiments, it will realize a mathematical structure called modular tensor category, 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 vertex operator algebras. If the vertex operator algebra satisfies some natural conditions, a theorem of mine 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 also one of my main research projects in the next few years.