The definitions of the nth Gauss sum and the associated nth central charge are introduced for premodular categories C and n∈ Z. We first derive an expression of the nth Gauss sum of a modular category C, for any integer n coprime to the order of the T-matrix of 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 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.
Publication Source (Journal or Book title)
Selecta Mathematica, New Series
Ng, S., Schopieray, A., & Wang, Y. (2019). Higher Gauss sums of modular categories. Selecta Mathematica, New Series, 25 (4) https://doi.org/10.1007/s00029-019-0499-2