Research Interests

I am mainly interested in the mathematical foundation of 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.

Since I am mainly interested in using the mathematics underlying string theory to solve mathematical problems, my research must be mathematically rigorous. Though the physical intuition is very important and powerful, most of the arguments used by physicists do not solve mathematical problems at all. In most cases, these arguments are more or less the same as mathematicians' discussions on some beautiful conjectures. To prove these mathematical conjectures completely, we have to insist the mathematical standard of rigor. In physics, all the theories, no matter whether they are rigorous or not, will have to be tested eventually by experiments. Though the mathematical rigor might be useful when a theory is first proposed or when a theory cannot be tested by experiments yet, physicists will almost always ignore this issue whenever there is a hope to compare their theories with experimental results. But in mathematics, a mathematical problem is solved if and only if the proof is rigorously written down. For example, in the original paper by Wiles proving Fermat's Last Theorem, there is a gap which was filled after one year by Taylor and Wiles. Without this last step by Taylor and Wiles, Fermat's Last Theorem would not be considered as solved.

Certainly solving mathematical problems rigorously using ideas from physics does not mean filling details or reformulating the existing physical theories mathematically. Instead, it means creating and developing new mathematical theories so that the mathematical problems can be reformulated, studied and finally solved in the framework of the new theories. These new mathematical theories might also be useful in physics eventually. A good example is calculus and mathematical analysis. Newton and Leibniz invented calculus and Newton used it for the study of mechanics. But no one can deny the contributions of Cauchy, Weierstrass and many other mathematicians for developing calculus as a solid and rigorous mathematical theory. In fact these developments became the foundation of mathematics today and are applied to physics and other sciences. Another good example is the interaction between general relativity and gauge theory in physics and differential geometry in mathematics.

Right now I am mostly working on two-dimensional conformal field theory. Conformal field theory describes perturbative string theory and also critical phenomena in condensed matter physics. In mathematics, it is closely related to infinite-dimensional Lie algebras, infinite-dimensional integrable systems, the Monster (the largest finite sporadic simple group), modular functions and modular forms, Riemann surfaces, knot and three-manifold invariants, Calabi-Yau manifolds and their generalizations, and many other branches of mathematics. Many mathematical problems can be solved by constructing and studying the corresponding conformal field theories, and the study of such theories will also provide a framework for (further) unifying the branches of mathematics mentioned above. One of the main goals of my research now is to construct geometric conformal field theories from vertex operator algebras (a main ingredient of conformal field theories) and their representations and to apply the geometric conformal field theories constructed and the results obtained in the construction to solve concrete mathematical problems.

Conformal field theories discussed above actually correspond to perturbative closed string theory. To study open-string theory and D-branes, which are nonperturbative objects in string theory, one needs to study open-closed (or boundary) conformal field theories. I am now also working on the construction of open-closed conformal field theories from vertex operator algebras and their representations. I hope that such a construction will be useful in the future study of D-branes and nonperturbative string theory and will give deeper results in mathematics.

My long-term research program is to contribute to the development of a (necessarily quantum) geometric theory underlying quantum field theories and string theory. In the next few years, I intend to work on the development of quantum analysis and geometry underlying (open-closed) conformal field theory. More specifically, I intend to work on the construction of higher-genus conformal field theories, the construction of open-closed conformal field theories, the construction of superconformal field theories associated to Calabi-Yau manifolds and applications of these constructions to geometry, topology and algebra (for example, the higher-genus Verlinde formulas, mirror symmetry, quantum cohomology and the geometry of the Monster group).