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 $\mbox{ Hom}_{ \mathcal{ C}}(x,A)$ 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 $A.$
Categories: math.QA cond-mat.str-el hep-th math.CT math.RT