Modularity of logarithmic parafermion vertex algebras

Auger, Jean; Creutzig, Thomas; Ridout, David

doi:10.1007/s11005-018-1098-4

Cited by 23 publications

(14 citation statements)

References 108 publications

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“…This unfortunate observation means that the resolutions (5.27), as well as their analogues for the other atypical irreducibles, do not converge when k > 0. A similar issue was noted recently with the atypical characters of the non-unitary parafermion cosets of [52].…”

Section: Non-unitary Minimal Model Characterssupporting

confidence: 76%

“…It follows that one can determine the fusion rules of C if those of V are known, and vice versa. We have already applied this powerful realisation to the example of non-unitary (logarithmic) parafermions in [52]. A similar application involving a non-Heisenberg coset (and the vice versa direction) has also recently appeared [53,54].…”

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confidence: 99%

“…Interestingly, the resulting Fourier coefficients turn out to be mock modular forms in general. However, we expect that these computations would be quite cumbersome in our situation.In [52], the logarithmic parafermion algebras of sl(2), with k < 0, were studied, along with their infinite-order simple current extensions (which are expected to be C 2 -cofinite). There, the modularity of these extensions was analysed without resorting to contour integral machinery, despite having to deal with (negative-index) meromorphic Jacobi forms.…”

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Unitary and non-unitary N = 2 minimal models

Creutzig

Liu

Ridout

et al. 2019

J. High Energ. Phys.

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A. The unitary N = 2 superconformal minimal models have a long history in string theory and mathematical physics, while their non-unitary (and logarithmic) cousins have recently attracted interest from mathematicians. Here, we give an efficient and uniform analysis of all these models as an application of a type of Schur-Weyl duality, as it pertains to the well-known Kazama-Suzuki coset construction. The results include straightforward classifications of the irreducible modules, branching rules, (super)characters and (Grothendieck) fusion rules.1. I 1.1. Background. N = 2 supersymmetry is ubiquitous in string theory where its first appearances even predate the conception of conformal field theory as a separate discipline, see [1] for example. Upon formalising conformal invariance, physicists quickly started exploring the properties of the N = 2 superconformal algebra [2-5] and its representations, especially the unitary ones [3,[6][7][8][9][10]. The discovery [11,12] of a coset construction for the corresponding minimal models led to many generalisations, now known as Kazama-Suzuki models, and important links to the geometry of string compactifications.On the representation-theoretic side, the unitary N = 2 superconformal minimal models were studied by mathematicians and physicists interested in their characters [13][14][15][16][17], modularity [18,19] and fusion rules [20,21]. Their non-unitary cousins unfortunately attracted relatively little attention, though a new construction as a minimal quantum hamiltonian reduction [22,23] realised an important link with mock modular forms [24][25][26][27]. Moreover, their Kazama-Suzuki coset relationship with the fractional-level sl(2) Wess-Zumino-Witten models was reformulated into a number of beautiful categorical equivalences [28][29][30][31][32][33][34].With these fractional-level models now well in hand [35][36][37][38][39][40][41][42], this relationship can be exploited in both directions.Our aim here is to use this knowledge to give a uniform and direct treatment of the N = 2 superconformal minimal models, both unitary and non-unitary, with the main results being a classification of irreducible modules, explicit branching rules and characters, and (Grothendieck) fusion rules. The point is that we have established an efficient procedure to extract representation theory from coset constructions: the N = 2 superconformal minimal models provide a beautiful and important illustration of these methods. 1.2. A Schur-Weyl duality for Heisenberg cosets. Over the last few years, in a joint effort with Shashank Kanade, Robert McRae and Andrew Linshaw, two of the authors have developed a working theory of coset vertex operator algebras [43-46]. This has been strongly influenced by physics ideas, but builds on the work of many mathematicians including Kac-Radul [47], Dong-Li-Mason [48], Huang-Lepowsky-Zhang [49] and Huang-Kirillov-Lepowsky [50]. The present paper is one of a series that applies this new technology to interesting examples. The picture is the following. We have...

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Section: Non-unitary Minimal Model Characterssupporting

confidence: 76%

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confidence: 99%

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confidence: 99%

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Unitary and non-unitary N = 2 minimal models

Creutzig

Liu

Ridout

et al. 2019

J. High Energ. Phys.

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“…We call this technique the (inverse) coset construction. This formalism has recently been developed in detail and rigour in [30][31][32][33] and, as a preparatory example, we have studied the logarithmic parafermion algebras of sl 2 at (negative) admissible levels [34]. The present paper is concerned with the minimal models for g = osp(1|2), these being the admissible-level WZW models, building on the insights obtained for a particular level in [35].…”

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“…The present paper is concerned with the minimal models for g = osp(1|2), these being the admissible-level WZW models, building on the insights obtained for a particular level in [35]. In a sequel, the results of this paper will be combined with those of [34] in order to understand the minimal models of sl(2|1) at admissible levels.Recently, vertex superalgebras and their modules have appeared as invariants of four-dimensional superconformal theories. For example, those associated with sl 2 and certain subregular W-algebras at various admissible levels arise in the study of Argyres-Douglas theories [36][37][38].…”

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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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Schur–weyl Duality for Heisenberg Cosets

Creutzig

Kanade

Linshaw

et al. 2018

Transformation Groups

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Let V be a simple vertex operator algebra containing a rank n Heisenberg vertex algebra H and let C = Com (H,V) be the coset of H in V. Assuming that the representation categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of C are found and every simple C-module is shown to be contained in at least one V-module. A corollary of this is that if V is rational and C 2 -cofinite and CFT-type, and Com (C,V) is a rational lattice vertex operator algebra, then so is C. These results are illustrated with many examples and the C 1 -cofiniteness of certain interesting classes of modules is established. is exact, for M ′′ ∼ = (J ⊠ M)/M ′ = 0. But, fusion is right-exact [HLZ, Prop. 4.26], sois exact. However, M ′′ = 0 implies that J −1 ⊠ M ′′ is a non-zero quotient of M, by (1), so we must have J −1 ⊠ M ′′ ∼ = M, as M is simple. Fusing with J now gives J ⊠ M ∼ = M ′′ , so we conclude that M ′ = 0 and that J ⊠ M is simple. The simplicity of J ∼ = J ⊠ V now follows from that of V, completing the proof of (3).To prove (4), note that applying right-exactness to the short exact sequence 0where f is the induced map from J ⊠ M ′ to J ⊠ M that might not be an inclusion. Fusing with J −1 and applying (2.4), we arrive at 5)

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Modularity of logarithmic parafermion vertex algebras

Cited by 23 publications

References 108 publications

Unitary and non-unitary N = 2 minimal models

Unitary and non-unitary N = 2 minimal models

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

Schur–weyl Duality for Heisenberg Cosets

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