Generalized Symmetries From Fusion Actions

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