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

Two-dimensional conformal quantum field theory (CFT) has inspired an immense amount of mathematics and has interacted with mathematics in very rich ways, in great part through the mathematically dynamic world of string theory. One notable example of this interaction is provided by Verlinde's conjecture: E. Verlinde (1) conjectured that certain matrices formed by numbers called the “fusion rules” in a “rational” CFT are diagonalized by the matrix given by a certain natural action of a fundamental modular transformation (essentially, a certain distinguished element of the group of two-by-two matrices of determinant one with integer entries). His conjecture led him to the “Verlinde formula” for the fusion rules and, more generally, for the dimensions of spaces of “conformal blocks” on Riemann surfaces of arbitrary genera. A great deal of progress has been achieved in interpreting and proving Verlinde's (physical) conjecture and the Verlinde formula in mathematical settings, in the case of the Wess–Zumino–Novikov-Witten models in CFT, which are based on affine Lie algebras. On the other hand, Moore and Seiberg (2, 3) showed, on a physical level of rigor, that the general form of the Verlinde conjecture is a consequence of the axioms for rational CFTs, thereby providing a conceptual understanding of the conjecture. In the process, they formulated a CFT analogue, later termed “modular tensor category” (discussed in refs. 4 and 5) by I. Frenkel, of the classical …