A general mirror equivalence theorem for coset vertex operator algebras

McRae, Robert

doi:10.48550/arxiv.2107.06577

Cited by 2 publications

(3 citation statements)

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“…Part (2) was essentially proved in [CJORY,Section 4] and [CY,Section 3]; see also [McR2,Theorem 2.3].…”

Section: And Associativity Isomorphismsmentioning

confidence: 95%

“…We will show that Y M 1 ,M 2 ⊗ Y W 1 ,W 2 is a fusion product intertwining operator. To do so, we need a lemma which is a version of [DL,Proposition 13.18] and [ADL,Proposition 2.11]; it is proved the same way as [McR2,Lemma B.1]. In the statement of the lemma, X [h] denotes the generalized L V (0)-eigenspace of eigenvalue h inside a weak U ⊗ V -module X that decomposes into generalized L V (0)-eigenspaces, and P h denotes the projection onto X [h] with respect to the generalized L V (0)-eigenspace decomposition of X.…”

Section: Fusion Product Modulesmentioning

confidence: 99%

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

McRae

2019

Math. Z.

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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 Rep G of finitedimensional 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 C2-cofinite but not necessarily rational. When V G is both C2-cofinite and rational and V is a vertex operator algebra, we use the equivalence between 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 Rep SU (2), up to modification by an abelian 3-cocycle.

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“…Part (2) was essentially proved in [CJORY,Section 4] and [CY,Section 3]; see also [McR2,Theorem 2.3].…”

Section: And Associativity Isomorphismsmentioning

confidence: 95%

Section: Fusion Product Modulesmentioning

confidence: 99%

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

McRae

2019

Math. Z.

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“…Thus V is a positiveenergy vertex operator algebra. Moreover, V is simple (see for example [McR3,Theorem 2.5(4)]), so to show that V is self-contragredient, [Li1, Corollary 3.2] implies it is enough to show L V (1)V (1) = 0. In fact, because A is self-contragredient and positive-energy, we have L(1)A (1) = 0, and then for any v ∈ V (1) ⊆ A (1) , we have…”

Section: Applicationsmentioning

confidence: 99%

On rationality for $C_2$-cofinite vertex operator algebras

McRae¹

2021

Preprint

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Let V be an N-graded, simple, self-contragredient, C2-cofinite vertex operator algebra. We show that if the S-transformation of the character of V is a linear combination of characters of V -modules, then the category C of grading-restricted generalized V -modules is a rigid tensor category. We further show, without any assumption on the character of V but assuming that C is rigid, that C is a factorizable finite ribbon category, that is, a not-necessarily-semisimple modular tensor category. As a consequence, we show that if the Zhu algebra of V is semisimple, then C is semisimple and thus V is rational. The proofs of these theorems use techniques and results from tensor categories together with the method of Moore-Seiberg and Huang for deriving identities of two-point genus-one correlation functions associated to V .We give two main applications. First, we prove the conjecture of Kac-Wakimoto and Arakawa that C2-cofinite affine W -algebras obtained via quantum Drinfeld-Sokolov reduction of admissible-level affine vertex algebras are strongly rational. The proof uses the recent result of Arakawa and van Ekeren that such W -algebras have semisimple (Ramond twisted) Zhu algebras. Second, we use our rigidity results to reduce the "coset rationality problem" to the problem of C2-cofiniteness for the coset. That is, given a vertex operator algebra inclusion U ⊗ V ֒→ A with A, U strongly rational and U , V a pair of mutual commutant subalgebras in A, we show that V is also strongly rational provided it is C2-cofinite. ROBERT MCRAE 6. Applications 61 6.1. Rationality for C 2 -cofinite W -algebras 61 6.2. Rationality for C 2 -cofinite cosets 66 Appendix A. Differential equations 69 Appendix B. The assumption in Theorem 6.4 for odd nilpotent orbits 74 B.1. Classical types 74 B.2. Exceptional types 76 References 82

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A general mirror equivalence theorem for coset vertex operator algebras

Cited by 2 publications

References 23 publications

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

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

On rationality for $C_2$-cofinite vertex operator algebras

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