Mirror extensions of rational vertex operator algebras

Lin, Xingjun

doi:10.1090/tran/6749

Cited by 24 publications

(10 citation statements)

References 29 publications

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“…Our main result will be that under fairly general conditions, (sub)categories of U -and V -modules enjoy a kind of "mirror duality," that is, there is a braid-reversing tensor equivalence relating the U -modules that occur in A with the V -modules that occur in A. Such a result has obtained earlier under stronger assumptions in [Lin,CKM2]. Here, although we do assume that the representation theory of U is quite nice, we make fairly minimal assumptions on V , mainly just that V has a suitable representation category that admits the vertex algebraic braided tensor category structure of [HLZ1]- [HLZ8].…”

Section: Introductionmentioning

confidence: 76%

A general mirror equivalence theorem for coset vertex operator algebras

McRae¹

2021

Preprint

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We prove a general mirror duality theorem for a subalgebra U of a simple vertex operator algebra A and its coset V = ComA(U ), under the assumption that A is a semisimple U ⊗ V -module. More specifically, we assume that A ∼ = i∈I Ui ⊗ Vi as a U ⊗ V -module, where the U -modules Ui are simple and distinct and are objects of a semisimple braided ribbon category of U -modules, and the V -modules Vi are semisimple and contained in a (not necessarily rigid) braided tensor category of V -modules. We also assume that U and V form a dual pair in A, so that U is the coset ComA(V ). Under these conditions, we show that there is a braid-reversing tensor equivalence τ : UA → VA, where UA is the semisimple category of U -modules with simple objects Ui, i ∈ I, and VA is the category of V -modules whose objects are finite direct sums of the Vi. In particular, the V -modules Vi are simple and distinct, and VA is a rigid tensor category.

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Section: Introductionmentioning

confidence: 76%

A general mirror equivalence theorem for coset vertex operator algebras

McRae¹

2021

Preprint

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“…It is natural to consider the full subcategory C of U ⊗ V-modules whose objects are (isomorphic to) direct sums of modules X ⊗ Y where X is a module in U and Y is a module in V. For the following theorem, we make fairly minimal assumptions on the vertex operator algebras U and V; for similar results along these lines see for instance [Lin,Lemma 2.16] and [CKLinR,Proposition 3.3].…”

Section: From Tensor Categories To Vertex Operator Algebrasmentioning

confidence: 99%

“…The second part of Main Theorem 3 (in the case that A is Z-graded) has been stated in [Lin,Theorem 3.3] under the assumption that U and V are strongly rational vertex operator algebras (in particular I is finite in this setting). The proof in [Lin] uses semisimplicity of the category Rep A of left A-modules in C, citing [KO] for this result.…”

Section: Introductionmentioning

confidence: 99%

Gluing vertex algebras

Creutzig,

Kanade,

McRae

2019

Preprint

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Let U and V be vertex operator algebras with module (sub)categories U and V, respectively, satisfying suitable assumptions which hold for example if U and V are semisimple rigid braided (vertex) tensor categories with countably many inequivalent simple objects. If τ is a map from the set of inequivalent simple objects of U to the objects V with τ (U) = V, then we glue U and V along U ⊠ V via τ to obtain the objectwhere the sum is over all inequivalent simple objects of U. Assuming U and V form a commuting pair in A in the sense that the multiplicity of V is U, our main theorem is that there is a braid-reversed equivalence between U and V mapping X to τ (X) * if and only if A can be given the structure of a simple conformal vertex algebra that (conformally) extends U ⊗ V.

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“…We also initiate a study of the representation theory of these models. On general grounds, since W G and W G = W G are commutant pairs in the moonshine module, one has a Schur-Weyl like decomposition of the form [40,41]…”

Section: Introductionmentioning

confidence: 99%

Conformal Field Theories with Sporadic Group Symmetry

Bae,

Harvey,

Lee

et al. 2020

Preprint

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The monster sporadic group is the automorphism group of a central charge c = 24 vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its c = 24 stress tensor T (z), this theory contains many other conformal vectors of smaller central charge; for example, it admits 48 commuting c = 1 2 conformal vectors whose sum is T (z). Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the Goddard-Kent-Olive (GKO) coset method for affine Lie algebras. We use this procedure to produce evidence for the existence of a number of CFTs with sporadic symmetry groups and employ a variety of techniques, including Hecke operators, modular linear differential equations, and Rademacher sums, to compute the characters of these CFTs. Our examples include (extensions of) nine of the sporadic groups appearing as subquotients of the monster, as well as the simple groups 2 E 6 (2) and F 4 (2) of Lie type. Many of these examples are naturally associated to McKay's E 8 correspondence, and we use the structure of Norton's monstralizer pairs more generally to organize our presentation.

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Mirror extensions of rational vertex operator algebras

Abstract: In this paper, mirror extensions of rational vertex operator algebras are considered. The mirror extension conjecture is proved.

Cited by 24 publications

References 29 publications

A general mirror equivalence theorem for coset vertex operator algebras

A general mirror equivalence theorem for coset vertex operator algebras

Gluing vertex algebras

Conformal Field Theories with Sporadic Group Symmetry

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