Higher Gauss sums of modular categories

Abstract

The definitions of the $n^{th}$ Gauss sum and the associated $n^{th}$ central charge are introduced for premodular categories $\mathcal{C}$ and $n\in \mathbb{Z}$. 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.

arXiv comment26 pages. Typos and minor mistakes are corrected. Question 7.3 in the previous version is answered

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• Symmetry-preserving boundary of (2+1)D fractional quantum Hall states
• Modular categories with transitive Galois actions
• Gauging U(1) symmetry in (2+1)d topological phases
• Gapped boundaries of fermionic topological orders and higher central charges
• Higher Hall conductivity from a single wave function: Obstructions to symmetry-preserving gapped edge of (2+1)D topological order
• Most two-dimensional bosonic topological orders forbid sign-problem-free quantum Monte Carlo: Nonpositive Gauss sum as an indicator
• Generalization of anomaly formula for time reversal symmetry in (2+1)D abelian bosonic TQFTs