We introduce a notion of quasi-lisse vertex algebras, which generalizes admissible affine vertex algebras. We show that the normalized character of an ordinary module over a quasi-lisse vertex operator algebra has a modular invariance property, in the sense that it satisfies a modular linear differential equation. As an application we obtain the explicit character formulas of simple affine vertex algebras associated with the Deligne exceptional series at level −h ∨ /6 − 1, which express the homogeneous Schur indices of 4d SCFTs studied by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees, as quasi-modular forms.
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...
Abstract. A notion of intermediate vertex subalgebras of lattice vertex operator algebras is introduced, as a generalization of the notion of principal subspaces. Bases and the graded dimensions of such subalgebras are given. As an application, it is shown that the characters of some modules of an intermediate vertex subalgebra between E 7 and E 8 lattice vertex operator algebras satisfy some modular differential equations. This result is an analogue of the result concerning the "hole" of the Deligne dimension formulas and the intermediate Lie algebra between the simple Lie algebras E 7 and E 8 .
ABSTRACT. Let g be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of sl 2 inducing the minimal gradation on g. The corresponding minimal Walgebra W k (g, e −θ ) introduced by Kac and Wakimoto has strong generators in weights 1, 2, 3/2, and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra V k ′ (g ♮ ) where g ♮ ⊂ g denotes the centralizer of sl 2 . Therefore W k (g, e −θ ) has an action of a connected Lie group G ♮ 0 with Lie algebra g ♮ 0 , where g ♮ 0 denotes the even part of g ♮ . We show that for any reductive subgroup G ⊂ G ♮ 0 , and for any reductive Lie algebra g ′ ⊂ g ♮ , the orbifold O k = W k (g, e −θ ) G and the coset) are strongly finitely generated for generic values of k. Here V (g ′ ) denotes the affine vertex algebra associated to g ′ . We find explicit minimal strong generating sets for C k when g ′ = g ♮ and g is either sl n , sp 2n , sl(2|n) for n = 2, psl(2|2), or osp(1|4). Finally, we conjecture some surprising coincidences among families of cosets C k which are the simple quotients of C k , and we prove several cases of our conjecture.
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