Braided Tensor Categories Related to $${\mathcal {B}}_{p}$$ Vertex Algebras

Auger, Jean; Creutzig, Thomas; Kanade, Shashank; Rupert, Matthew

doi:10.1007/s00220-020-03747-8

Cited by 33 publications

(19 citation statements)

References 94 publications

(102 reference statements)

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“…In particular, the results of [40,106,107] suggest that VOAs appearing in 4d SCFTs might still satisfy the modularity properties (5.7), (5.8). Additionally, the work [48] assumes this as well (at least the weaker form of (5.7) without any insertions, i.e.…”

Section: High Temperature Limit and Modularitymentioning

confidence: 98%

“…Starting with the pioneering work of Zhu [78], there have been a number of articles proving it for C 2 -cofinite VOAs using various methods, considering both insertions of vertex operators and of more general intertwiners [100][101][102][103][104], and including the case of twisted modules into the story [98]. Yet, once we move outside the C 2 -cofinite world, the knowledge is very limited, see however [40,[105][106][107].…”

Section: High Temperature Limit and Modularitymentioning

confidence: 99%

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From VOAs to Short Star Products in SCFT

Dedushenko

2021

Commun. Math. Phys.

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We build a bridge between two algebraic structures in SCFT: a VOA in the Schur sector of 4d N = 2 theories and an associative algebra in the Higgs sector of 3d N = 4. The natural setting is a 4d N = 2 SCFT placed on S 3 × S 1 : by sending the radius of S 1 to zero, we recover the 3d N = 4 theory, and the corresponding VOA on the torus degenerates to the associative algebra on the circle. We prove that: 1) the Higgs branch operators remain in the cohomology; 2) all the Schur operators of the non-Higgs type are lifted by line operators wrapped on the S 1 ; 3) no new cohomology classes are added. We show that the algebra in 3d is given by the quotient A H = Zhu s (V )/N , where Zhu s (V ) is the non-commutative Zhu algebra of the VOA V (for s ∈ Aut(V )), and N is a certain ideal. This ideal is the null space of the (s-twisted) trace map T s : Zhu s (V ) → C determined by the torus 1-point function in the high temperature (or small complex structure) limit. It therefore equips A H with a non-degenerate (twisted) trace, leading to a short star-product according to the recent results of Etingof and Stryker. The map T s is easy to determine for unitary VOAs, but has a much subtler structure for non-unitary and non-C 2 -cofinite VOAs of our interest. We comment on relation to the Beem-Rastelli conjecture on the Higgs branch and the associated variety. A companion paper will explore further details, examples, and some applications of these ideas.

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Section: High Temperature Limit and Modularitymentioning

confidence: 98%

Section: High Temperature Limit and Modularitymentioning

confidence: 99%

From VOAs to Short Star Products in SCFT

Dedushenko

2021

Commun. Math. Phys.

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“…Next, we use our results on projective objects in O 0 c together with induction to obtain all projective covers of simple modules in C W(p) ; these modules have been constructed previously in [AM3,NT]. The following lemma is [ACKR,Theorem 17] and [CMY3,Lemma 5.0.3]; it can be proved using Frobenius reciprocity: Lemma 7.7. If P is projective in O 0 c , then F (P) is projective in C W(p) .…”

Section: On the Representation Theory Of Triplet Vertex Operator Alge...mentioning

confidence: 99%

Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p > 1$

McRae,

Yang

2020

Preprint

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Let O c be the category of finite-length central-charge-c modules for the Virasoro Lie algebra whose composition factors are irreducible quotients of reducible Verma modules. Recently, it has been shown that O c admits vertex algebraic tensor category structure for any c ∈ C. Here, we determine the structure of this tensor category when c = 13 − 6p − 6p −1 for an integer p > 1. For such c, we prove that O c is rigid, and we construct projective covers of irreducible modules in a natural tensor subcategory O 0 c . We then compute all tensor products involving irreducible modules and their projective covers. Using these tensor product formulas, we show that O c has a semisimplification which, as an abelian category, is the Deligne product of two tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig categories for affine sl 2 at levels −2 + p ±1 . Finally, as a straightforward consequence of the braided tensor category structure on O c together with the theory of vertex operator algebra extensions, we rederive known results for triplet vertex operator algebras W(p), including rigidity, fusion rules, and construction of projective covers. c 5.2. The remaining projective covers 6. Tensor product formulas and semisimplification 6.1. General fusion rules 6.2. Semisimplification 7. On the representation theory of triplet vertex operator algebras References

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“…Interpreting f as a map into X, we conclude that P is projective in Ind(O f in 1 ). Now, as in [ACKR,Thm. 17], Frobenius reciprocity implies that…”

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confidence: 95%

Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$

Creutzig,

McRae,

Yang

2020

Preprint

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We show that the Kazhdan-Lusztig category KL k of level-k finite-length modules with highest-weight composition factors for the affine Lie superalgebra gl(1|1) has vertex algebraic braided tensor supercategory structure, and that its full subcategory O f in k of objects with semisimple Cartan subalgebra actions is a tensor subcategory.We show that every simple gl(1|1)-module in KL k has a projective cover in O f in k , and we determine all fusion rules involving simple and projective objects in O f in k . Then using Knizhnik-Zamolodchikov equations, we prove that KL k and O f in k are rigid. As an application of the tensor supercategory structure on O f in k , we study certain module categories for the affine Lie superalgebra sl(2|1) at levels 1 and − 1 2 . In particular, we obtain a tensor category of sl(2|1)-modules at level − 1 2 that includes relaxed highest-weight modules and their images under spectral flow.

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Braided Tensor Categories Related to $${\mathcal {B}}_{p}$$ Vertex Algebras

Cited by 33 publications

References 94 publications

From VOAs to Short Star Products in SCFT

From VOAs to Short Star Products in SCFT

Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p > 1$

Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$

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