Weight Representations of Admissible Affine Vertex Algebras

Arakawa, Tomoyuki; Futorny, Vyacheslav; Ramirez, Luis Enrique

doi:10.1007/s00220-017-2872-3

Cited by 28 publications

(32 citation statements)

References 37 publications

(34 reference statements)

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“…However a general abstract derivation which holds also for non-Lagrangian theories is still lacking and is an open problem for future research. It would also be interesting to explore the modular properties of the resulting sums of characters such as (6.1) and (6.20) as in [19][20][21]89].…”

Section: Chiral Algebra and Line Defectsmentioning

confidence: 99%

“…The ordinary Schur index without insertions of line defects receives contributions only from operators that are annihilated by four supercharges, which can be chosen to be Q 1 − , 20 The parentheses and the square brackets denote the symmetrization and antisymmetrization of indices, respectively. That is, T (αβ) = 1 2 (T αβ + T βα ) and…”

Section: A1 Supercharges Preserved By Full Linesmentioning

confidence: 99%

“…λ • λ (C 20). there is exactly one element Λ such that the Dynkin labels of Λ + ρ are all non-negative,(Λ + ρ, α ∨ i ) ≥ 0 , (C.21)where α i 's are the simple roots of the affine Lie algebra g.Let ch M (µ) be the character of the Verma module with highest weight µ, ch M (µ) = e µ α∈∆ + (1 − e −α ) mult(α) , (C.22)…”

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confidence: 99%

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Infrared computations of defect Schur indices

Córdova

Gaiotto

Shao

2016

J. High Energ. Phys.

192

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We conjecture a formula for the Schur index of four-dimensional N = 2 theories in the presence of boundary conditions and/or line defects, in terms of the low-energy effective Seiberg-Witten description of the system together with massive BPS excitations. We test our proposal in a variety of examples for SU (2) gauge theories, either conformal or asymptotically free. We use the conjecture to compute these defect-enriched Schur indices for theories which lack a Lagrangian description, such as Argyres-Douglas theories. We demonstrate in various examples that line defect indices can be expressed as sums of characters of the associated two-dimensional chiral algebra and that for Argyres-Douglas theories the line defect OPE reduces in the index to the Verlinde algebra.

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Section: Chiral Algebra and Line Defectsmentioning

confidence: 99%

Section: A1 Supercharges Preserved By Full Linesmentioning

confidence: 99%

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Infrared computations of defect Schur indices

Córdova

Gaiotto

Shao

2016

J. High Energ. Phys.

192

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“…They have also appeared in the physics literature as integral components of the SL 2 ( ) Wess-Zumino-Witten model [8] and through requiring closure under fusion and cosets in L k (sl 2 ) conformal field theories [9][10][11][12]. More recently, relaxed highest-weight modules over L k (sl 3 ) at admissible levels have also begun to receive attention [13,14]. There are two observations relating to relaxed highest-weight modules which we find compelling as arguments for their continued study.…”

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confidence: 99%

Relaxed Highest-Weight Modules I: Rank 1 Cases

Kawasetsu

Ridout

2019

Commun. Math. Phys.

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A. Relaxed highest-weight modules play a central role in the study of many important vertex operator (super)algebras and their associated (logarithmic) conformal field theories, including the admissible-level affine models. Indeed, their structure and their (super)characters together form the crucial input data for the standard module formalism that describes the modular transformations and Grothendieck fusion rules of such theories. In this article, character formulae are proved for relaxed highestweight modules over the simple admissible-level affine vertex operator superalgebras associated to sl 2 and osp(1 |2). Moreover, the structures of these modules are specified completely. This proves several conjectural statements in the literature for sl 2 , at arbitrary admissible levels, and for osp(1 |2) at level − 5 4 . For other admissible levels, the osp(1 |2) results are believed to be new.K KAWASETSU AND D RIDOUT therefore only follow when there are no coincidences of conformal weights, modulo 1. Note that a similar character formula had been previously proven for certain critical-level relaxed highest-weight sl 2 -modules in [30].A second main input to the standard module formalism, and more widely to constructing projective covers for the highest-weight simples, is the determination of the structure of the non-simple standard modules. This structure is needed to construct the resolutions that relate the non-standard simples to standards and thereby enable the study of the modularity of the simple modules of the theory. Again, these structures were stated without proof and used extensively in [22][23][24].Our aim in this work is to rigorously prove the character formulae and structural results of [22][23][24] for all admissible levels. Instead of an explicit construction, we develop the structure theory of "relaxed Verma modules" and their simple quotients over both sl 2 and osp(1|2), the latter in both its Neveu-Schwarz and Ramond incarnations. The first main result (see below) is a means to compute the character of an arbitrary simple relaxed highest-weight module from that of an associated simple (usual) highest-weight module. When the latter character is known, for example through the Kac-Wakimoto formula for admissible-level highest-weight modules [31,32], we can thereby deduce the required relaxed characters. This is our second main result. The third settles the structures of the non-simple relaxed modules in terms of non-split short exact sequences. The key technical tools we use to prove these results are a generalisation of Mathieu's coherent families [33] to a relaxed affine setting and a study of a Shapovalov-like form on the resulting relaxed coherent families.1.1. Main results. We divide our conclusions into three main results. The first applies to general simple relaxed highestweight sl 2 -and osp(1|2)-modules of fixed level k. These sl 2 -modules are denoted by E λ;q , where λ is a coset in the quotient of the weight space of sl 2 by its root lattice and q is the eigenvalue of the quadratic Casi...

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“…These natural generalisations of the usual highest-weight modules were introduced in the conformal field theory literature in [24] for g = sl 2 , though they had already appeared in mathematics classifications such as [15], but have only recently been formalised in a general setting [25]. Since then, the role played by irreducible relaxed highest-weight modules in facilitating the study of general admissible-level WZW models has been widely appreciated and the field has been rapidly developing, see [26][27][28][29] for example.Second, one should develop techniques to reconstruct, at least in favourable circumstances, the representation theory of the algebra of interest in terms of those of subalgebras. We call this technique the (inverse) coset construction.…”

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confidence: 99%

Cosets, characters and fusion for admissible-level osp(1|2) minimal models

et al. 2019

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A. We study the minimal models associated to osp(1 |2), otherwise known as the fractional-level Wess-Zumino-Witten models of osp(1 |2). Since these minimal models are extensions of the tensor product of certain Virasoro and sl 2 minimal models, we can induce the known structures of the representations of the latter models to get a rather complete understanding of the minimal models of osp(1 |2). In particular, we classify the irreducible relaxed highest-weight modules, determine their characters and compute their Grothendieck fusion rules. We also discuss conjectures for their (genuine) fusion products and the projective covers of the irreducibles. IThis project is part of a programme to understand the admissible-level Wess-Zumino-Witten (WZW) models for a Lie algebra or superalgebra g. While the theories with non-negative integer levels and simple Lie algebras lead to rational conformal field theories and, as such, are very well understood, the situation is much more complicated and rich for other levels or when superalgebras are involved. Indeed, the non-rational admissible-level WZW models are expected to be prime examples of logarithmic conformal field theories, these being models that admit representations on which the hamiltonian acts non-diagonalisably, leading to correlation functions with logarithmic singularities.Another interesting feature of these models is that they have a continuous spectrum of modules.We view our programme as complementary to older approaches. In particular, Quella, Saleur, Schomerus et al.[1-9] approached supergroup WZW theories via free field realisations and semiclassical limits (the minisuperspace analysis), the interest being rather in features of the WZW theory of the supergroup at integer levels. Another approach employed was to learn more about the conformal field theory using the mock modular behaviour of certain irreducible characters [10][11][12]. The relatively accessible case of g = gl(1|1) has also been studied from a more algebraic perspective by two of us [13,14].Presently, we have a very good picture in the case of g = sl 2 [15][16][17][18][19][20][21][22][23]. In order to extend our understanding to more sophisticated theories, one has to develop some basic strategies. First, one has to study the general theory of relaxed highest-weight modules. These natural generalisations of the usual highest-weight modules were introduced in the conformal field theory literature in [24] for g = sl 2 , though they had already appeared in mathematics classifications such as [15], but have only recently been formalised in a general setting [25]. Since then, the role played by irreducible relaxed highest-weight modules in facilitating the study of general admissible-level WZW models has been widely appreciated and the field has been rapidly developing, see [26][27][28][29] for example.Second, one should develop techniques to reconstruct, at least in favourable circumstances, the representation theory of the algebra of interest in terms of those of subalgebras. We call this techniq...

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Weight Representations of Admissible Affine Vertex Algebras

Cited by 28 publications

References 37 publications

Infrared computations of defect Schur indices

Infrared computations of defect Schur indices

Relaxed Highest-Weight Modules I: Rank 1 Cases

Cosets, characters and fusion for admissible-level osp(1|2) minimal models

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