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Higher Gauss sums of modular categories

  • Siu-Hung Ng
  • Andrew Schopieray
  • Yilong WangEmail author


The definitions of the nth Gauss sum and the associated nth central charge are introduced for premodular categories \(\mathcal {C}\) and \(n\in \mathbb {Z}\). We first derive an expression of the nth 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.

Mathematics Subject Classification

18D10 57R56 



The second author would like to thank Victor Ostrik for his suggestion to explore the notion of higher Gauss sums. Both the second and third authors would like to thank MSRI (Summer Graduate School 791) for providing the opportunity to initiate this collaboration. The third author would like to thank Thomas Kerler and James Cogdell, and the first author would like to thank Ling Long for fruitful discussions.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.School of Mathematics and StatisticsUNSWSydneyAustralia

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