The Verlinde formula in logarithmic CFT

Ridout, David; Wood, Simon

doi:10.1088/1742-6596/597/1/012065

Cited by 31 publications

(59 citation statements)

References 94 publications

(134 reference statements)

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“…4.3].This lifting procedure is mathematically implemented by an induction functor. Happily, this functor is monoidal [46], meaning that the fusion product of two induced C-modules, which are V-modules, is isomorphic to the result of fusing the C-modules and then inducing [51]. It follows that one can determine the fusion rules of C if those of V are known, and vice versa.…”

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confidence: 99%

“…There are of course subtleties to overcome when dealing with the non-unitary N = 2 minimal models ( > 1). In this case, we are guided by the standard module formalism [51,55] that has worked so well in analysing similar logarithmic conformal field theories. In particular, it applies [40,41] to the fractional-level sl(2) Wess-Zumino-Witten models that appear in the (non-unitary) N = 2 coset construction.…”

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confidence: 99%

“…Here, we note that F 2n+i /2 is considered to be bosonic. Using [51,Eq. (3.3)], which has been rigorously proven in [46,Thm.…”

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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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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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Unitary and non-unitary N = 2 minimal models

Creutzig

Liu

Ridout

et al. 2019

J. High Energ. Phys.

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“…In practice, most interesting vertex operator algebras (such as the affine vertex operator algebras at admissible level) have representation categories that are not even finite, they have uncountably many inequivalent simple objects. However, jointly with David Ridout, one of us was able to nonetheless conjecture a Verlinde's formula for affine vertex operator algebras of sl 2 at admissible level by treating characters as formal distributions [CR2,CR3], see [RW1] for a review. This conjecture is completely open, except for some encouraging computations of fusion rules [Ga, Ri, AP] and a recent proof of a formula of type (1.5) for the finite subcategory of ordinary modules for all affine vertex operator algebras of simply-laced Lie algebras at admissible level [CHY,C1].…”

Section: Logarithmic Conformal Field Theory and Verlinde's Formulamentioning

confidence: 99%

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

Auger

Creutzig

Kanade

et al. 2020

Commun. Math. Phys.

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The B p -algebras are a family of vertex operator algebras parameterized by p ∈ Z ≥2 . They are important examples of logarithmic CFTs and appear as chiral algebras of type (A 1 , A 2p−3 ) Argyres-Douglas theories. The first member of this series, the B 2 -algebra, are the well-known symplectic bosons also often called the βγ vertex operator algebra.We study categories related to the B p vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of sl 2 . These categories are braided, rigid and non semisimple tensor categories. We list their simple and projective objects, their tensor products and their Hopf links. The latter are succesfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.

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“…In our opinion, it is likely that the modular story will remain under control within the so-called standard module framework [16,17]. On the other hand, brute force methods such as the NahmGaberdiel-Kausch algorithm [18,19] for fusion products are already computationally-prohibitive in all but the simplest cases.…”

Section: Fractional Level Models and Jack Symmetric Functionsmentioning

confidence: 99%

Relaxed singular vectors, Jack symmetric functions and fractional levelslˆ(2)models

Ridout

Wood

2015

Nuclear Physics B

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The fractional level models are (logarithmic) conformal field theories associated with affine Kac-Moody (super)algebras at certain levels k ∈ Q. They are particularly noteworthy because of several longstanding difficulties that have only recently been resolved. Here, Wakimoto's free field realisation is combined with the theory of Jack symmetric functions to analyse the fractional level sl (2) models. The first main results are explicit formulae for the singular vectors of minimal grade in relaxed Wakimoto modules. These are closely related to the minimal grade singular vectors in relaxed (parabolic) Verma modules. Further results include an explicit presentation of Zhu's algebra and an elegant new proof of the classification of simple relaxed highest weight modules over the corresponding vertex operator algebra. These results suggest that generalisations to higher rank fractional level models are now within reach.

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The Verlinde formula in logarithmic CFT

Cited by 31 publications

References 94 publications

Unitary and non-unitary N = 2 minimal models

Unitary and non-unitary N = 2 minimal models

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

Relaxed singular vectors, Jack symmetric functions and fractional levelslˆ(2)models

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