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Extending the Trace of a Pivotal Monoidal Functor | | Leonard Hardiman
; | | Date: |
20 Nov 2019 | | Abstract: | Let $mathcal{C}$ be a modular tensor category with a complete set of simples
$mathcal{I}$ and let $mathcal{M}colonmathcal{C} o mathcal{D}$ be a
pivotal monoidal functor. We show that the trace of $mathcal{M}$ naturally
extends to a representation of the tube category which we denote
$mathcal{TM}$. As irreducible representations of the tube category are indexed
by pairs of elements in $mathcal{I}$, decomposing $mathcal{TM}$ into
irreducibles gives a non-negative integer
$mathcal{I} imesmathcal{I}$-matrix, $Z(mathcal{TM})$. In general,
$Z(mathcal{TM})$ will not be a modular invariant, however it will always
commute with the T-matrix. Furthermore, under certain additional conditions on
$mathcal{M} $, it is shown that $mathcal{TM}$ 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
$Z(mathcal{TM})$ commutes with the S-matrix if and only if the dimension of
$mathcal{TM}$ is equal to the dimension of $mathcal{C}$. Finally, this
procedure is applied to certain pivotal monoidal functors arising from module
categories over $ ext{Rep}_k A_1^{(1)}$. | | Source: | arXiv, 1911.9024 | | Services: | Forum | Review | PDF | Favorites | |
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