Relaxed Highest-Weight Modules I: Rank 1 Cases

Abstract

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...

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].…”

Unitary and non-unitary N = 2 minimal models

“…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. In this case, the characters of the standard modules [56,57] are naturally expressed as distributions in the Jacobi variable that keeps track of the Cartan weight. They have exemplary modular properties and the standard Verlinde formula gives non-negative fusion multiplicities.…”

“…In particular, there exist reducible A 1 (u, )-modules E ± r,s , where 1 r u − 1 and 1 s − 1, whose ground states have charge equal to λ r,s mod 2 and conformal dimension ∆ aff r,s . Moreover, E ± r,s is relaxed with a submodule isomorphic to D ± r,s and the quotient by this submodule being isomorphic to D ∓ u−r, −s [57]. This is succinctly summarised in the following non-split short exact sequence:…”

“…One must be careful with (3.31) to expand the reciprocal of the theta function ϑ 1 in the correct annulus of convergence (in w), see [37,41]. The characters of the E λ, ∆ aff r , s , which must be treated as distributions in w, were originally conjectured in [41] and were subsequently proved in [56] (generically) and [57] (in complete generality).…”

Logarithmic W-algebras and Argyres-Douglas theories at higher rank

“…There are various difficult questions in the context of logarithmic vertex operator algebras, see [25] for an introduction, and it is good that they allow for gauge theory interpretations. So far the best studied logarithmic vertex operator algebras are the triplet algebras [26][27][28][29], the fractional level WZW theories of sl 2 [30][31][32], the logarithmic B(p)-algebras [33] and some progress is currently made on higher rank cases [34]. All these examples have appearances in higher dimensional super conformal field theories.…”