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Generalized Symmetries From Fusion Actions

Chongying Dong, Siu-Hung Ng, Li Ren, Feng Xu

TL;DR

This work develops a categorical framework for fusion actions of the fusion category ${\EuScript{C}}_A$ on condensable algebras $A$ in a modular tensor category ${\EuScript{C}}$, using generalized Frobenius-Schur indicators to build and analyze the action on multiplicity spaces. It proves a categorical Schur-Weyl duality, showing an isomorphism between the relevant central-idempotent subalgebra and endomorphisms of $A$, and identifies irreducible modules $\mathcal{C}(x,A)$ under the action of $K({{\EuScript{C}}_A})$. A Galois correspondence is established between fusion subcategories of ${\EuScript{C}}_A$ containing ${\EuScript{C}}^0_A$ and condensable subalgebras of $A$, with explicit formulas for dimensions and constructions of the inverse map via ${B}=A^{\EuScript{B}}$ and $({\EuScript{C}}^0_{B})_A$. The paper further shows how these categorical results specialize to orbifold VOAs, recovering the classical Schur-Weyl duality and fixed-point correspondences for finite and group actions, and provides concrete VOA/lattice examples including coset constructions. The framework thus unifies and extends orbifold-type dualities and Galois-type correspondences within a purely categorical setting, with clear implications for VOA module categories and modular data.

Abstract

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 $A.$

Generalized Symmetries From Fusion Actions

TL;DR

This work develops a categorical framework for fusion actions of the fusion category on condensable algebras in a modular tensor category , using generalized Frobenius-Schur indicators to build and analyze the action on multiplicity spaces. It proves a categorical Schur-Weyl duality, showing an isomorphism between the relevant central-idempotent subalgebra and endomorphisms of , and identifies irreducible modules under the action of . A Galois correspondence is established between fusion subcategories of containing and condensable subalgebras of , with explicit formulas for dimensions and constructions of the inverse map via and . The paper further shows how these categorical results specialize to orbifold VOAs, recovering the classical Schur-Weyl duality and fixed-point correspondences for finite and group actions, and provides concrete VOA/lattice examples including coset constructions. The framework thus unifies and extends orbifold-type dualities and Galois-type correspondences within a purely categorical setting, with clear implications for VOA module categories and modular data.

Abstract

Let be a condensable algebra in a modular tensor category . We define an action of the fusion category of -modules in on the morphism space for any in , whose characters are generalized Frobenius-Schur indicators. This fusion action can be considered on , and we prove a categorical generalization of the Schur-Weyl duality for this action. For any fusion subcategory of containing all the local -modules, we prove the invariant subobject is a condensable subalgebra of . The assignment of to defines a Galois correspondence between this kind of fusion subcategories of and the condensable subalgebras of . In the context of VOAs, we prove for any nice VOAs , where is the category of -modules. In particular, if for some finite automorphism group of the fusion action of on is equivalent to the -action on

Paper Structure

This paper contains 17 sections, 20 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Basic terminologies on fusion categories
  3. Conventions and terminologies
  4. A-modules and their graphical conventions
  5. Fusion actions and Frobenius-Schur indicators
  6. Actions of spherical fusion categories on some hom-spaces
  7. Fusion actions on condensable subalgebras
  8. Fusion actions on condensable subalgebras
  9. Schur-Weyl duality
  10. Galois correspondence from fusion actions
  11. Galois Correspondence via relative centers
  12. Examples of fusion actions on VOAs
  13. Orbifold theory
  14. Fusion actions associated to A
  15. $l=2n$ cases
  16. ...and 2 more sections

Key Result

Proposition 3.3

Let ${\EuScript{B}}$ be a spherical fusion category. The fusion action of ${\EuScript{B}}$ on ${\EuScript{B}}(X,{\mathds{1}})$ introduced in def:action defines an $K_0({\EuScript{B}})$-module. In particular, its character value at $Y$ is the generalized Frobenius-Schur indicator $\nu_{(0,-1)}^{\math

Theorems & Definitions (48)

  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Definition 3.5
  • Proposition 3.6
  • proof
  • Proposition 3.7
  • ...and 38 more