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Register a new userOn invariants of Modular categories beyond modular data
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Added by
Cesar Neyit Galindo Martinez
Publication date
2018
fields
Physics
and research's language is
English
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We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the $W$-matrix--the quantum invariant of a colored framed Whitehead link from the associated TQFT of a modular category. We prove that the $W$-matrix and the set of punctured $S$-matrices are strictly beyond the modular data $(S,T)$. Whether or not the triple $(S,T,W)$ constitutes a complete invariant of modular categories remains an open question.
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In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum $mathbb{A}$ in a modular fusion category $mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum $mathbb{A}$ in $mathcal{C}$, we introduce a ribbon category $mathcal{C}_{mathbb{A}}$ and show that it is again a modular fusion category. The definition of $mathcal{C}_{mathbb{A}}$ is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when $mathbb{A}$ is given by a simple commutative $Delta$-separable Frobenius algebra $A$ in $mathcal{C}$; (ii) when $mathbb{A}$ is an orbifold datum in $mathcal{C} = operatorname{Vect}$, built from a spherical fusion category $mathcal{S}$. We show that in case (i), $mathcal{C}_{mathbb{A}}$ is ribbon-equivalent to the category of local modules of $A$, and in case (ii), to the Drinfeld centre of $mathcal{S}$. The category $mathcal{C}_{mathbb{A}}$ thus unifies these two constructions into a single algebraic setting.
We study odd-dimensional modular tensor categories and maximally non-self dual (MNSD) modular tensor categories of low rank. We give lower bounds for the ranks of modular tensor categories in terms of the rank of the adjoint subcategory and the order of the group of invertible objects. As an application of these results, we prove that MNSD modular tensor categories of ranks 13 and 15 are pointed. In addition, we show that MNSD tensor categories of ranks 17, 19, 21 and 23 are either pointed or perfect.
Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category symmetries of modular categories, which include also categorical Hopf algebras as special cases. As an application, we propose an analogue of the classification of finite simple groups to modular categories, where we define simple modular categories as the prime ones without any nontrivial normal algebras.
We use zesting and symmetry gauging of modular tensor categories to analyze some previously unrealized modular data obtained by Grossman and Izumi. In one case we find all realizations and in the other we determine the form of possible realizations; in both cases all realizations can be obtained from quantum groups at roots of unity.
The definitions of the $n^{th}$ Gauss sum and the associated $n^{th}$ central charge are introduced for premodular categories $mathcal{C}$ and $ninmathbb{Z}$. We first derive an expression of the $n^{th}$ Gauss sum of a modular category $mathcal{C}$, for any integer $n$ coprime to the order of the T-matrix of $mathcal{C}$, in terms of the first Gauss sum, the global dimension, the twist and their Galois conjugates. As a consequence, we show for these $n$, the higher Gauss sums are $d$-numbers and the associated central charges are roots of unity. In particular, if $mathcal{C}$ is the Drinfeld center of a spherical fusion category, then these higher central charges are 1. We obtain another expression of higher Gauss sums for de-equivariantization and local module constructions of appropriate premodular and modular categories. These expressions are then applied to prove the Witt invariance of higher central charges for pseudounitary modular categories.
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