Generalized Symmetries From Fusion Actions

Technical Summary: Generalized Symmetries from Fusion Actions

Problem Statement
The paper addresses the problem of characterizing subalgebras of a vertex operator algebra (VOA) or a condensable algebra within a modular tensor category (MTC) that cannot necessarily be realized as fixed-point subalgebras under a group action. In classical orbifold theory, if $A$ is a simple VOA and $G$ is a finite automorphism group, the fixed-point subalgebra $A^G$ and the multiplicity spaces $A^\lambda$ satisfy a Schur-Weyl duality, and there exists a Galois correspondence between subgroups of $G$ and subalgebras containing $A^G$. However, not every subalgebra $U \subset A$ arises as $A^G$ for some group $G$. The authors seek a categorical generalization of these results where the "group" is replaced by a fusion category, specifically the category of modules over a condensable algebra.

Methodology
The authors work within the framework of modular tensor categories (MTCs) and condensable algebras. Let $\mathcal{C}$ be an MTC and $A$ a condensable algebra in $\mathcal{C}$. The category $\mathcal{C}_A$ of right $A$-modules in $\mathcal{C}$ is a spherical fusion category, and its subcategory $\mathcal{C}^0_A$ of local $A$-modules is itself an MTC.

The core methodology involves defining a fusion action of the category $\mathcal{C}_A$ on the morphism spaces $\text{Hom}_{\mathcal{C}}(x, A)$ for any object $x \in \mathcal{C}$. This action is constructed using generalized Frobenius-Schur indicators. Specifically, for an object $Y \in \mathcal{C}_A$ and a morphism $f \in \text{Hom}_{\mathcal{C}}(x, A)$, the action $Y \cdot f$ is defined via diagrammatic calculus involving the half-braiding of the center $Z(\mathcal{C}_A)$ and the evaluation/coevaluation maps of the algebra $A$.

The authors utilize the structure of the Grothendieck ring $K(\mathcal{C}_A)$ and its idempotents to analyze the invariant subspaces of these morphism spaces. They define the $B$-invariant subspace of $\text{Hom}_{\mathcal{C}}(x, A)$ for a fusion subcategory $B \subseteq \mathcal{C}_A$ as the set of morphisms $f$ such that $Y \cdot f = d_A(Y)f$ for all $Y \in B$, where $d_A(Y)$ is the pivotal dimension of $Y$ in $\mathcal{C}_A$.

Key Contributions and Results

1. Categorical Schur-Weyl Duality (Theorem I):
   The paper establishes a Schur-Weyl duality for the action of the fusion category $\mathcal{C}_A$ on the algebra $A$.

   • The kernel of the action of $\mathcal{C}_A$ on $\text{Hom}_{\mathcal{C}}(A, A)$ is the ideal generated by $(1 - e_1)$, where $e_1$ is a specific primitive central idempotent in $K(\mathcal{C}_A)$ associated with the local modules $\mathcal{C}^0_A$.
   • The space $\text{Hom}_{\mathcal{C}}(x, A)$ is an irreducible module over the algebra $e_1 K(\mathcal{C}_A)$ for any simple object $x$ appearing in $A$.
   • There is an isomorphism of algebras $e_1 K(\mathcal{C}_A) \cong \text{Hom}_{\mathcal{C}}(A, A)$.
   • This result generalizes the classical Schur-Weyl duality where a group $G$ acts on a VOA, recovering the group algebra $\mathbb{C}[G]$ in the case where $A^G$ is rational and $C_2$-cofinite.

2. Galois Correspondence (Theorem II):
   The authors prove a Galois correspondence between fusion subcategories of $\mathcal{C}_A$ containing the local modules $\mathcal{C}^0_A$ and condensable subalgebras of $A$.

   • For any fusion subcategory $B \subseteq \mathcal{C}_A$ containing $\mathcal{C}^0_A$, the invariant subobject $A^B$ (defined as the image of the idempotent $e_B$ acting on $A$) is a condensable subalgebra of $A$.
   • The assignment $B \mapsto A^B$ defines an order-reversing bijection between the lattice of such fusion subcategories and the lattice of condensable subalgebras of $A$.
   • The inverse map sends a subalgebra $B \subseteq A$ to the fusion subcategory $(\mathcal{C}^0_B)_A$.
   • This provides a positive answer to the question of whether any rational, $C_2$-cofinite subalgebra $U \subset A$ can be realized as a fixed-point subalgebra under a "generalized group" action (specifically, the fusion action of $\mathcal{C}_A$).

3. Application to Vertex Operator Algebras (Theorem III):
   The results are applied to the context of VOAs. If $U \subset A$ are simple, rational, $C_2$-cofinite VOAs (with $A$ containing $U$), then the $U$-module category $\mathcal{M}_U$ is a pseudounitary MTC, and $A$ is a condensable algebra in $\mathcal{M}_U$. The fusion action of $(\mathcal{M}_U)_A$ on $A$ yields the Schur-Weyl duality and Galois correspondence for subVOAs containing $U$.

   • In the specific case where $U = A^G$ for a finite group $G$, the fusion category $(\mathcal{M}_{A^G})_A$ is a $G$-extension of the trivial category, and the fusion action recovers the standard $G$-action on $A$.

4. Examples and Extensions:
   The paper provides concrete examples involving:

   • Iterated orbifolds of group actions, showing they are invariants of actions of weakly integral fusion categories.
   • Fusion actions associated with $A_l$ lattice VOAs, where the fusion categories involved are non-abelian generalizations of Tambara-Yamagami categories (specifically, $\mathbb{Z}_2$-extensions of pointed categories associated with dihedral groups).
   • Coset constructions, linking the theory to intermediate conformal nets.

Significance
The paper claims to provide a unified categorical framework for understanding symmetries in conformal field theory and VOA theory. By replacing the group $G$ with a fusion category $\mathcal{C}_A$, the authors generalize the Schur-Weyl duality and the Galois correspondence to settings where the symmetry is not necessarily a group. This allows for the classification of subalgebras that do not arise from finite group actions, characterizing them via the invariant subobjects of the fusion action. The work bridges the gap between the theory of condensable algebras in MTCs and the representation theory of VOAs, offering a tool to analyze the structure of extensions and subalgebras in rational conformal field theories. The authors note that their approach offers an alternative to previous results based on relative centers (e.g., in [7]), utilizing the classical notion of Galois correspondence via invariant subspaces of fusion actions.