Generalized Symmetries From Fusion Actions (2508.13063v1)

Abstract: Let $A$ be a condensable algebra in a modular tensor category $\EuScript{C}$. We define an action of the fusion category $\EuScript{C}A$ of $A$-modules in $\EuScript{C}$ on the morphism space $\mbox{Hom}{\EuScript{C}}(x,A)$ for any $x$ in $\EuScript{C}$, whose characters are generalized Frobenius-Schur indicators. This fusion action can be considered on $A$, and we prove a categorical generalization of Schur-Weyl duality for this action. For any fusion subcategory $\EuScript{B}$ of $\EuScript{C}_A$ containing all the local $A$-modules, we prove the invariant subobject $B=A^\EuScript{B}$ is a condensable subalgebra of $A$. The assignment of $\EuScript{B}$ to $A^\EuScript{B}$ defines a Galois correspondence between this kind of fusion subcategories of $\EuScript{C}_A$ and the condensable subalgebras of $A$. In the context of VOA, we prove for any nice VOAs $U \subset A$, $U=A^{\EuScript{C}_A}$ where $\EuScript{C}=\EuScript{M}_U$ is the $U$-module category. In particular, if $U = A^G$ for some finite automorphism group $G$ of $A,$ the fusion action of $\EuScript{C}_A$ on $A$ is equivalent to the $G$-action on $A.$