From the representation theory of vertex operator algebras to modular tensor categories in conformal field theory

This is an expository article invited for the Commentary'' section of PNAS in connection with Y.-Z. Huang's article, Vertex operator algebras, the Verlinde conjecture, and modular tensor categories,‘’ appearing in the same issue of PNAS. Huang’s solution of the mathematical problem of constructing modular tensor categories from the representation theory of vertex operator algebras is very briefly discussed, along with background material. The hypotheses of the theorems entering into the solution are very general, natural and purely algebraic, and have been verified in a wide range of familiar examples, while the theory itself is heavily analytic and geometric as well as algebraic.