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.