Gauging the Categorical Connes' $\tildeχ(M)$
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915952298295296 |
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| author | Chen, Quan |
| author_facet | Chen, Quan |
| contents | 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' $\tildeχ(M\rtimes G)$ 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 $\tildeχ(M\rtimes G)$. This is the categorical generalization of Connes' short exact sequence on $χ(M\rtimes G)$. Using this machinery, for any finite group $G$, we construct a McDuff $\rm II_1$ factor $M$, whose $\tildeχ(M)$ is braided equivalent to $\mathsf{Rep}(G)$. This is the first example of a braided fusion category which is not modular as $\tildeχ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21833 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gauging the Categorical Connes' $\tildeχ(M)$ Chen, Quan Operator Algebras Category Theory Quantum Algebra 46L37, 18M20, 46M15 primary, 18M15, 18M30 secondary 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' $\tildeχ(M\rtimes G)$ 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 $\tildeχ(M\rtimes G)$. This is the categorical generalization of Connes' short exact sequence on $χ(M\rtimes G)$. Using this machinery, for any finite group $G$, we construct a McDuff $\rm II_1$ factor $M$, whose $\tildeχ(M)$ is braided equivalent to $\mathsf{Rep}(G)$. This is the first example of a braided fusion category which is not modular as $\tildeχ$. |
| title | Gauging the Categorical Connes' $\tildeχ(M)$ |
| topic | Operator Algebras Category Theory Quantum Algebra 46L37, 18M20, 46M15 primary, 18M15, 18M30 secondary |
| url | https://arxiv.org/abs/2604.21833 |