Module Categories and Modular Invariants

  • Leonard Hardiman

Student thesis: Doctoral ThesisPhD


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.
Date of Award20 Nov 2019
Original languageEnglish
Awarding Institution
  • University of Bath
SupervisorAlastair King (Supervisor) & Gunnar Traustason (Supervisor)


  • category theory
  • quantum algebra
  • tube category

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