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 C and n ∈ Z. We first derive an expression of the n th Gauss sum of a modular category C, for any integer n coprime to the order of the Tmatrix 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.

The Witt classes of $SO(2r)_{2r}$

“…which are powerful invariants of the Witt group used in [27,25] as well as in the current paper. The Witt group W of non-degenerate braided fusion categories can be viewed as a generalization of the classical Witt group of non-degenerate quadratic forms on abelian groups (i.e., metric groups), which generate the pointed part of W denoted by W pt .…”

“…Modular data yield important numerical invariants of modular categories. For example, for any n ∈ Z, the n-th Gauss sum is defined in [27] as…”

“…Recently [25] it was shown that the 8 Witt classes of Ising categories have infinitely many independent square roots modulo the subgroup generated by pointed categories. This was achieved using the signature, a new Witt class invariant related to the higher central charge introduced in [27]. From this a verification of a conjecture of [6] was derived, namely, that the torsion subgroup sW 2 of the super-Witt group has infinite rank.…”

“…According to [27,Lem. 5.3 (c)], the dimension of any connected étale algebra A ∈ E is totally positive, i.e., all the Galois conjugates of dim(A) are positive real numbers.…”

Modular tensor categories, subcategories, and Galois orbits

“…A contains a pseudoinvertible object, and Theorem 4.2.3 implies C 0 A ≃ (C 0 A ) pt ⊠ T for some transitive modular tensor category T . Hence C 0 A is Galois conjugate to a pseudounitary modular tensor category and thus C is Galois conjugate to a pseudounitary modular tensor category by [19,Lemma 5.3(c)] and [4,Lemma 3.11]. Lastly, we may now assume…”