Unitary and non-unitary N = 2 minimal models

Abstract

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...

On fusion rules and intertwining operators for the Weyl vertex algebra

“…Verlinde formula for fusion rules was also presented by T. Creutzig and A. Milas in [12], but so far the proof was only given for the case p = 2 in [7]. We should also mention that the fusion rules and intertwining operators for some affine and superconformal vertex algebras were studied in [1], [4], [11] and [24].…”

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

“…Moreover there is a notion of local modules for a commutative algebra object and this category of local objects is equivalent as a braided tensor category to the category of modules of the vertex operator algebra A that lie in C and furthermore there is a induction functor from the category of those V -modules that centralize A to local A-modules [CKM]. In fortunate situations all local modules of interest can be realized via induction and representation categories associated to quite a few logarithmic theories have already successfully been studied in this way, see [ACR,CFK,CKLR2,CLRW]. In addition the Hopf links that play such an important role in Verlinde's formula commute with the induction functor.…”

Algebraic aspects of conformal field theory

“…More recently, there has been instrumental results on the general properties of Heisenberg algebra cosets [99]. Using these results, the authors of [100] were able to provide a concrete map between the minimal models of sl(2) at admissible level, and the minimal models of N = 2 superconformal algebras.…”

“…Following this, we introduce the algebras involved in the coset. Our presentation here generally follows the notation and structure of [100], where the authors introduced this coset for studying the (non-)unitary minimal models of the N = 2 superconformal algebras. We introduce the N = 2 superconformal algebras, their highest-weight representation theory, and their conjugation and spectral flow morphisms.…”

“…In this introduction, we focus on key points relevant to the presentation of our results. A comprehensive description of the coset construction for both unitary and non-unitary N = 2 minimal models as is given in [100].…”