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

TL;DR

This paper investigates the Witt classes of certain modular categories derived from quantum groups, revealing infinitely many order 2 classes and providing examples that answer longstanding questions in the field.

Contribution

It introduces new Witt classes of order 2 from $SO(2r)_{2r}$ categories, including a non-pointed, anisotropic example, and explores their algebraic independence and relation to pointed classes.

Findings

Infinitely many Witt classes of order 2 are constructed.

An example of a non-pointed, anisotropic modular category with order 2 Witt class is provided.

The trivial Witt class has infinitely many square roots modulo pointed classes.

Abstract

We study the Witt classes of the modular categories $SO(2r)_{2r}$ associated with quantum groups of type $D_r$ at $4r-2$th roots of unity. From these classes we derive infinitely many Witt classes of order 2 that are linearly independent modulo the subgroup generated by the pointed modular categories. In particular we produce an example of a simple, completely anisotropic modular category that is not pointed whose Witt class has order 2, answering a question of Davydov, M\"uger, Nikshych and Ostrik. Our results show that the trivial Witt class $[Vec]$ has infinitely many square roots modulo the pointed classes, in analogy with the recent construction of infinitely many square roots of the Ising Witt classes modulo the pointed classes constructed in a similar way from certain type $B_r$ modular categories. We compare the subgroups generated by the Ising square roots and $[Vec]$ square roots and provide evidence that they also generate linearly independent subgroups.