# On finite non-degenerate braided tensor categories with a Lagrangian subcategory

## TL;DR

This paper classifies finite non-degenerate braided tensor categories containing a specific symmetric tensor subcategory, revealing a cyclic structure and distinguishing between integral and non-integral cases.

## Contribution

It establishes a classification of such categories as a torsor over Z/16Z, identifying exactly 8 integral and 8 non-integral categories.

## Key findings

- Classification as a Z/16Z torsor
- 8 integral and 8 non-integral categories identified
- Structure depends on the Lagrangian subcategory Rep(W)

## Abstract

Let $W$ be a finite dimensional purely odd supervector space over $\mathbb{C}$, and let $\sRep(W)$ be the finite symmetric tensor category of finite dimensional superrepresentations of the finite supergroup $W$. We show that the set of equivalence classes of finite non-degenerate braided tensor categories $\C$ containing $\sRep(W)$ as a Lagrangian subcategory is a torsor over the cyclic group $\mathbb{Z}/16\mathbb{Z}$. In particular, we obtain that there are $8$ non-equivalent such braided tensor categories $\C$ which are integral and $8$ which are non-integral.