$SO(N)_2$ Braid group representations are Gaussian

Abstract

We give a description of the centralizer algebras for tensor powers of spin objects in the pre-modular categories $SO(N)_2$ (for $N$ odd) and $O(N)_2$ (for $N$ even) in terms of quantum $(n-1)$-tori, via non-standard deformations of $U\mathfrak{so}_N$. As a consequence we show that the corresponding braid group representations are Gaussian representations, the images of which are finite groups. This verifies special cases of a conjecture that braid group representations coming from weakly integral braided fusion categories have finite image.

Typos fixed, exposition improved, proofs and statements of Corollary 2.4 and Theorem 4.6 expanded. version 3--two typos corrected. Version 4--close to published version, typos corrected