From Three Dimensional Manifolds to Modular Tensor Categories

From Quantum Groups to Unitary Modular Tensor Categories

Modular tensor categories are generalizations of the representation theory of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFT and invariants of 3-manifolds. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to underst...

Verlinde conjecture and modular tensor categories

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0, V(0) = C1 and the contragredient module V ′ is isomorphic to V as a V -module. (ii) Every N-gradable weak V -module is completely reducible. (iii) V is C2-cofinite. We announce a proof of the Verlinde conjecture for V , that is, of the statement that the matrices formed by the fusion rules amo...

On Classification of Modular Tensor Categories

We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular S-matrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be...

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...

Hecke Algebras , Modular Categories and 3 - Manifolds Quantum Invariantschristian