On the tensor structure of modules for compact orbifold vertex operator algebras

Abstract

Suppose $V^G$ is the fixed-point vertex operator subalgebra of a compact group $G$ acting on a simple abelian intertwining algebra $V$. We show that if all irreducible $V^G$-modules contained in $V$ live in some braided tensor category of $V^G$-modules, then they generate a tensor subcategory equivalent to the category $\mathrm{Rep}G$ of finite-dimensional representations of $G$, with associativity and braiding isomorphisms modified by the abelian $3$-cocycle defining the abelian intertwining algebra structure on $V$. Additionally, we show that if the fusion rules for the irreducible $V^G$-modules contained in $V$ agree with the dimensions of spaces of intertwiners among $G$-modules, then the irreducibles contained in $V$ already generate a braided tensor category of $V^G$-modules. These results do not require rigidity on any tensor category of $V^G$-modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known; for example they apply when $V^G$ is $C_2$-cofinite but not necessarily rational. When $V^G$ is both $C_2$-cofinite and rational and $V$ is a vertex operator algebra, we use the equivalence between $\mathrm{Rep}G$ and the corresponding subcategory of $V^G$-modules to show that $V$ is also rational. As another application, we show that a certain category of modules for the Virasoro algebra at central charge $1$ admits a braided tensor category structure equivalent to $\mathrm{Rep}SU(2)$, up to modification by an abelian $3$-cocycle.

arXiv comment32 pages, Example 4.11 added in this version, and minor correction to the formula for braiding isomorphisms in Example 4.12

References in corpus (21)

• Braided tensor categories and extensions of vertex operator algebras
• C_2-cofiniteness of cyclic-orbifold models
• Tensor categories for vertex operator superalgebra extensions
• Schur-Weyl Duality for Heisenberg Cosets
• Simple current extensions beyond semi-simplicity
• Regularity of fixed-point vertex operator subalgebras
• Logarithmic tensor category theory, II: Logarithmic formal calculus and properties of logarithmic intertwining operators
• Logarithmic tensor category theory, VI: Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms
• Logarithmic tensor category theory, VIII: Braided tensor category structure on categories of generalized modules for a conformal vertex algebra
• Logarithmic tensor category theory, V: Convergence condition for intertwining maps and the corresponding compatibility condition
• Logarithmic tensor category theory, VII: Convergence and extension properties and applications to expansion for intertwining maps
• Logarithmic tensor category theory, III: Intertwining maps and tensor product bifunctors
• Logarithmic tensor category theory, IV: Constructions of tensor product bifunctors and the compatibility conditions
• Modular categories and orbifold models
• Tensor categories and the mathematics of rational and logarithmic conformal field theory
• Twisted modules and $G$-equivariantization in logarithmic conformal field theory
• Rationality, Regularity, and C_2-cofiniteness
• Compact automorphism groups of vertex operator algebras
• A Z_2-orbifold model of the symplectic fermionic vertex operator superalgebra
• Flatness and Semi-Rigidity of Vertex Operator Algebras
• A Self-Dual Integral Form of the Moonshine Module

Cited by in corpus (26)

• Logarithmic W-algebras and Argyres-Douglas theories at higher rank
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• Gluing vertex algebras
• On ribbon categories for singlet vertex algebras
• Twisted modules and $G$-equivariantization in logarithmic conformal field theory
• Tensor categories of affine Lie algebras beyond admissible levels
• Direct limit completions of vertex tensor categories
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Q-systems and extensions of completely unitary vertex operator algebras
• Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p>1$
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• On semisimplicity of module categories for finite non-zero index vertex operator subalgebras
• Unbounded field operators in categorical extensions of conformal nets
• An ${\mathfrak{sl}}_2$-type tensor category for the Virasoro algebra at central charge $25$ and applications
• Classification of extremal vertex operator algebras with two simple modules
• A general mirror equivalence theorem for coset vertex operator algebras
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• Simplicity of higher rank triplet W-algebras
• Tensor Categories arising from the Virasoro Algebra
• On rationality for $C_2$-cofinite vertex operator algebras
• Unitarity and strong graded locality of holomorphic vertex operator superalgebras with central charge at most 24
• $N=1$ super Virasoro tensor categories
• The non-semisimple Kazhdan-Lusztig category for affine ${\mathfrak{sl}}_2$ at admissible levels
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras
• Representation theory of vertex operator algebras and orbifold conformal field theory
• Vertex Algebras and Commutative Algebras