On Infinite Order Simple Current Extensions of Vertex Operator Algebras

Auger, Jean; Rupert, Matt

doi:10.48550/arxiv.1711.05343

Cited by 3 publications

(4 citation statements)

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“…The up-shot of correctness of these Conjectures is the knowledge of the complete vertex tensor category of finite-dimensional weight modules of the singlet VOA and especially this is a rigid and braided tensor category so that all the results of [56] apply. We also would like to mention that the subtleties of passing to a completion of the category due to infinite order simple current extensions are discussed in both [46,62].…”

Section: The Singlet M(p)mentioning

confidence: 99%

Higgs and Coulomb branches from vertex operator algebras

Costello

Creutzig

Gaiotto

2019

J. High Energ. Phys.

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Section: The Singlet M(p)mentioning

confidence: 99%

Higgs and Coulomb branches from vertex operator algebras

Costello

Creutzig

Gaiotto

2019

J. High Energ. Phys.

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“…Simple Current Extensions. The process of extending a vertex operator algebra by (tensor powers of) modules whose tensor product is invertible (such extentions are called simple current extentions) has a long history in the conformal field theory and vertex operator algebra literature for both finite order extensions [42] and more recently also infinite ones [33,43]. At a categorical level, extensions (not necessarily the simple current type) correspond to algebra objects in a braided monoidal category [31,44,45].…”

Section: Examplementioning

confidence: 99%

“…43) Here Y( )Y( ) and Y(Y( )) denote the obvious product and iterate of intertwining operators and we have suppressed the objects labelling the natural transformations ϕ 2 , α C , α D . The left triangle commutes by the definition (see Theorem 2.12) of associativity for intertwining operators.…”

mentioning

confidence: 99%

Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson

Allen¹,

Lentner²,

Schweigert³

et al. 2021

Preprint

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have developed a module theory for vertex operator algebras that endows suitably chosen module categories with the structure of braided monoidal categories. Included in the theory is a functor which assigns to discretely strongly graded modules a contragredient module, obtained as a gradewise dual. In this paper, we show that this gradewise dual endows the module category with the structure of a ribbon Grothendieck-Verdier category. This duality structure is more general than that of a rigid monoidal category; in contrast to rigidity, it naturally accommodates the fact that a vertex operator algebra and its gradewise dual need not be isomorphic as modules and that the tensor product of modules over vertex operator algebras need not be exact.We develop criteria which allow the detection of ribbon Grothendieck-Verdier equivalences and use them to explore ribbon Grothendieck-Verdier structures in the example of the rank n Heisenberg vertex operator algebra or chiral free boson on a not necessarily full rank even lattice with arbitrary choice of conformal vector. We show that these categories are equivalent, as ribbon Grothendieck-Verdier categories, to certain categories of graded vector spaces and categories of modules over a certain Hopf algebra. R ALLEN, S LENTNER, C SCHWEIGERT, AND S WOOD where D is a contravariant equivalence of categories. In the context of vertex operator algebra module categories the dualising object K should be thought of as the gradewise dual of the vertex operator algebra, seen as a module over itself, and D as the functor which assigns to any object its gradewise dual and to any morphism its transpose. This seemingly simple definition of a Grothendieck-Verdier category has important consequences, for example, it guarantees the existence of inner Homs for all objects of C and implies that the tensor product of C is right exact, if the category is abelian.Intriguingly, every Grothendieck-Verdier category is also endowed with a second tensor product X • Y = D −1 (DY ⊗ DX) which turns out to be left exact [10,11], again, if the category is abelian. The two tensor products • and ⊗ should be considered on an equal footing. It remains to be discovered what the full implication of these two tensor products is for vertex operator algebras and conformal field theories. Rigid categories are examples of Grothendieck-Verdier categories, where the tensor unit is a dualising object, though the tensor unit being a dualising object does not imply that the category is rigid in general.The notion of a Grothendieck-Verdier category (no rigidity assumed) is nicely compatible with additional structure on the category C , for example, a braiding, and it is possible to introduce notions of a balancing and a twist. It is thus not surprising that this structure has surfaced in numerous disparate places: Grothendieck-Verdier categories are also known as * -autonomous categories [12], however, in this paper, we use the more recent terminology of Grothendieck-Verdier categories [10,11,13]. The main insight of this p...

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“…As L − 3 2 (sl 3 ) is a simple current extension in this direct limit completion [CMY20a,AR18], it now follows from [CKL20a, CKM17] that D is the category of local modules for L − 3 2 (sl 3 ), viewed as a commutative algebra object in the direct limit completion. There is therefore an induction functor that maps any module V in the completion that centralises the algebra object L − 3 2 (sl 3 ) to an object Ind V in D. Moreover, this functor is a vertex tensor functor [CKM17], meaning that it respects the fusion products of the completion and D:…”

Section: Reconstructingmentioning

confidence: 99%

A Kazhdan-Lusztig correspondence for $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$

Creutzig¹,

Ridout²,

Rupert³

2021

Preprint

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The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking. This work details and tests such a conjecture for L − 3 2 (sl3) via a logarithmic Kazhdan-Lusztig correspondence.We first investigate the representation theory of U H i (sl3), the unrolled restricted quantum group of sl3 at fourth root of unity. In particular, we analyse its finite-dimensional weight category, determining Loewy diagrams for all projective indecomposables and decomposing all tensor products of irreducibles. Our motivation is that this category is conjecturally braided tensor equivalent to a category of W 0A 2 (2)modules. Here, W 0 A 2 (2) is an orbifold of the octuplet vertex algebra WA 2 (2) of Semikhatov, the latter being the natural sl3-analogue of the well known triplet algebra. Moreover, W 0 A 2 (2) is the parafermionic coset of the affine vertex algebra LWe formulate an explicit conjecture relating the representation theory of W 0 A 2 (2) and U H i (sl3) and work out the resulting structures of the corresponding L − 3 2 (sl3)-modules. In particular, we obtain conjectural Loewy diagrams for the latter's projective indecomposables and decompositions for the fusion products of its irreducibles. These products coincide with those recently computed via Verlinde's formula. Finally, we give analogous results for WA 2 (2).

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On Infinite Order Simple Current Extensions of Vertex Operator Algebras

Cited by 3 publications

References 0 publications

Higgs and Coulomb branches from vertex operator algebras

Higgs and Coulomb branches from vertex operator algebras

Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson

A Kazhdan-Lusztig correspondence for $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$

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