Showing papers by "David Ridout published in 2019"

Schur–weyl duality for heisenberg cosets

Abstract: 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 module 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, C2-cofinite and CFT-type, and Com(C;V) is a rational lattice vertex operator algebra, then C is likewise rational. These results are illustrated with many examples and the C1-cofiniteness of certain interesting classes of modules is established.

Relaxed Highest-Weight Modules I: Rank 1 Cases

Abstract: 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 highest-weight modules over the simple admissible-level affine vertex operator superalgebras associated to $${\mathfrak{s}\mathfrak{l}_2}$$ and $${\mathfrak{osp} (1 \vert 2)}$$ . Moreover, the structures of these modules are specified completely. This proves several conjectural statements in the literature for $${\mathfrak{s}\mathfrak{l}_2}$$ , at arbitrary admissible levels, and for $${\mathfrak{osp} (1 \vert 2)}$$ at level $${-\frac{5}{4}}$$ . For other admissible levels, the $${\mathfrak{osp}(1 \vert 2)}$$ results are believed to be new.

Relaxed highest-weight modules II: classifications for affine vertex algebras

Abstract: This is the second of a series of articles devoted to the study of relaxed highest-weight modules over affine vertex algebras and W-algebras. The first studied the simple "rank-$1$" affine vertex superalgebras $L_k(\mathfrak{sl}_2)$ and $L_k(\mathfrak{osp}(1\vert2))$, with the main results including the first complete proofs of certain conjectured character formulae (as well as some entirely new ones). Here, we turn to the question of classifying relaxed highest-weight modules for simple affine vertex algebras of arbitrary rank. The key point is that this can be reduced to the classification of highest-weight modules by generalising Olivier Mathieu's theory of coherent families. We formulate this algorithmically and illustrate its practical implementation with several detailed examples. We also show how to use coherent family technology to establish the non-semisimplicity of category $\mathscr{O}$ in one of these examples.

Unitary and non-unitary N = 2 minimal models

Abstract: 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 straight-forward classifications of the irreducible modules, branching rules, (super)characters and (Grothendieck) fusion rules.

NGK and HLZ: Fusion for Physicists and Mathematicians

Abstract: In this expository note, we compare the fusion product of conformal field theory, as defined by Gaberdiel and used in the Nahm–Gaberdiel–Kausch (NGK) algorithm, with the P(w)-tensor product of vertex operator algebra modules, as defined by Huang, Lepowsky and Zhang (HLZ). We explain how the equality of the two “coproducts” derived by NGK is essentially dual to the P(w)-compatibility condition of HLZ and how the algorithm of NGK for computing fusion products may be adapted to the setting of HLZ. We provide explicit calculations and instructive examples to illustrate both approaches. This document does not provide precise descriptions of all statements, it is intended more as a gentle starting point for the appreciation of the depth of the theory on both sides.

Unitary and non-unitary $N=2$ minimal models

Abstract: 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.