Gauging the Categorical Connes' $\tilde{\chi}(M)$

TL;DR

The paper explores the categorical Connes' $ ilde{ chi}(M)$ for McDuff $ m II_1$ factors under finite group actions, providing explicit formulas and constructing examples with specific braided fusion categories.

Contribution

It generalizes Connes' short exact sequence categorically, introduces a gauging procedure, and constructs the first non-modular braided fusion category as $ ilde{ chi}(M)$.

Findings

Explicit formula for $G/K$-gauging when $L$ is trivial

Construction of a McDuff $ m II_1$ factor with $ ilde{ chi}(M)$ equivalent to $ ext{Rep}(G)$

First example of a non-modular braided fusion category as $ ilde{ chi}$

Abstract

We prove that if a finite group $G$ acts outerly on a McDuff $\rm II_1$ factor $M$, then $\mathsf{Rep}(G/KL)$ is a braided monoidal full subcategory of the categorical Connes' $\tilde{\chi}(M\rtimes G)$ defined in arXiv:2111.06378, where $K$ and $L$ are the centrally trivial and approximately inner parts in $G$ respectively. When $L$ is trivial, we give an explicit formula for the $G/K$-gauging procedure on $\tilde{\chi}(M\rtimes G)$. This is the categorical generalization of Connes' short exact sequence on $\chi(M\rtimes G)$. Using this machinery, for any finite group $G$, we construct a McDuff $\rm II_1$ factor $M$, whose $\tilde{\chi}(M)$ is braided equivalent to $\mathsf{Rep}(G)$. This is the first example of a braided fusion category which is not modular as $\tilde\chi$.