Let C be a modular tensor category with a complete set of simples indexed by I. A modular invariant for C is a non-negative integer I × I-matrix that commutes with the modular data of C. In this thesis we present a novel method of associating a non-negative integer I × I-matrix to a pivotal monoidal functor M on C. This is accomplished via a construction called the tube category. The tube category shares all of its objects with C but extends the Hom-spaces. The trace of M naturally extends to a representation of the tube category that we denote TM. As irreducible representations of the tube category are indexed by pairs of elements in I, decomposing TM into irreducibles gives a non-negative integer I × I-matrix, Z(TM). For a general pivotal functor, Z(TM) will not always be a modular invariant; however, it will always commute with the T-matrix. Furthermore, under certain additional conditions on M, it is shown that TM is a haploid, symmetric, commutative Frobenius algebra. Such algebras are known to be connected to modular invariants, in particular a result of Kong and Runkel implies that Z(T M) commutes with the S-matrix if and only if the dimension of T M is equal to the dimension of C. Finally, this procedure is applied to certain pivotal monoidal functors arising from module categories over C.

- category theory
- quantum algebra
- tube category

Module Categories and Modular Invariants

Hardiman, L. (Author). 20 Nov 2019

Student thesis: Doctoral Thesis › PhD