https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

Lie Theory and Special Functions

Every four-dimensional $$ \mathcal{N}=2 $$
 superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected quantities in the same theories have yet to be completely understood. In this work, we aim to characterize the connection between the Higgs branch of the moduli space of vacua (as an algebraic geometric entity) and the associated vertex operator algebra. Ultimately our proposal is simple, but its correctness requires the existence of a number of nontrivial null vectors in the vacuum Verma module of the vertex operator algebra. Of particular interest is one such null vector whose presence suggests that the Schur index of any $$ \mathcal{N}=2 $$
 SCFT should obey a finite order modular differential equation. By way of the “high temperature” limit of the superconformal index, this allows the Weyl anomaly coefficient a to be reinterpreted in terms of the representation theory of the associated vertex operator algebra. We illustrate these ideas in a number of examples including a series of rank-one theories associated with the “Deligne-Cvitanovic exceptional series” of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class $$ \mathcal{S} $$
 theories, and $$ \mathcal{N}=2 $$
 super Yang-Mills with $$ \mathfrak{s}\mathfrak{u}(n) $$
 gauge group for small-to-moderate values of n.

/pdf/vertex-operator-algebras-higgs-branches-and-modular-3u9od3snj6.pdf

Vertex operator algebras, Higgs branches, and modular differential equations

We initiate the bootstrap program for $$ \mathcal{N}=3 $$
 superconformal field theories (SCFTs) in four dimensions. The problem is considered from two fronts: the protected subsector described by a 2d chiral algebra, and crossing symmetry for half-BPS operators whose superconformal primaries parametrize the Coulomb branch of $$ \mathcal{N}=3 $$
 theories. With the goal of describing a protected subsector of a family of $$ \mathcal{N}=3 $$
 SCFTs, we propose a new 2d chiral algebra with super Virasoro symmetry that depends on an arbitrary parameter, identified with the central charge of the theory. Turning to the crossing equations, we work out the superconformal block expansion and apply standard numerical bootstrap techniques in order to constrain the CFT data. We obtain bounds valid for any theory but also, thanks to input from the chiral algebra results, we are able to exclude solutions with $$ \mathcal{N}=4 $$
 supersymmetry,allowingustozoominonaspecific $$ \mathcal{N}=3 $$
 SCFT.

/pdf/bootstrapping-mathcal-n-3-superconformal-theories-3ebe88q80h.pdf

Bootstrapping $\mathcal{N} = 3$ superconformal theories

We investigate superconformal surface defects in four-dimensional $$ \mathcal{N}=2 $$
 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfeld-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group flows onto Higgs branches. In each case we find perfect agreement with the predicted characters.

/pdf/surface-defects-and-chiral-algebras-y52fy4dzqz.pdf

Surface Defects and Chiral Algebras

We introduce the 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 amodular 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 quasimodular forms.

Quasi-lisse Vertex Algebras and Modular Linear Differential Equations

Let $\mathfrak{g}$ be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of $\mathfrak{s} \mathfrak{l}_2$ inducing the minimal gradation on $\mathfrak{g}$ The corresponding minimal $\mathcal{W}$-algebra $\mathcal{W}^k(\mathfrak{g}, e_{-\theta})$ 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 $V^{k'}(\mathfrak{g}^{\natural})$ where $\mathfrak{g}^{\natural} \subset \mathfrak{g}$ denotes the centralizer of $\mathfrak{s} \mathfrak{l}_2$ Therefore $\mathcal{W}^k(\mathfrak{g}, e_{-\theta})$ has an action of a connected Lie group $G^{\natural}_0$ with Lie algebra $\mathfrak{g}^{\natural}_0$, where $\mathfrak{g}^{\natural}_0$ denotes the even part of $\mathfrak{g}^{\natural}$ We show that for any reductive subgroup $G \subset G^{\natural}_0$, and for any reductive Lie algebra $\mathfrak{g}' \subset \mathfrak{g}^{\natural}$, the orbifold $\mathcal{O}^k = \mathcal{W}^k(\mathfrak{g}, e_{-\theta})^{G}$ and the coset $\mathcal{C}^k = \text{Com}(V(\mathfrak{g}'),\mathcal{W}^k(\mathfrak{g}, e_{-\theta}))$ are strongly finitely generated for generic values of $k$ Here $V(\mathfrak{g}')$ denotes the affine vertex algebra associated to $\mathfrak{g}'$ We find explicit minimal strong generating sets for $\mathcal{C}^k$ when $\mathfrak{g}' = \mathfrak{g}^{\natural}$ and $\mathfrak{g}$ is either $\mathfrak{s} \mathfrak{l}_n$, $\mathfrak{s}\mathfrak{p}_{2n}$, $\mathfrak{s}\mathfrak{l}(2|n)$ for $n\neq 2$, $\mathfrak{p}\mathfrak{s}\mathfrak{l}(2|2)$, or $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|4)$ Finally, we conjecture some surprising coincidences among families of cosets $\mathcal{C}_k$ which are the simple quotients of $\mathcal{C}^k$, and we prove several cases of our conjecture

Orbifolds and cosets of minimal $\mathcal{W}$-algebras

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 I: Rank 1 Cases

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 E7 and E8 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 E7 and E8.

The Intermediate Vertex Subalgebras of the Lattice Vertex Operator Algebras

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 $\mathfrak{sl}_2$ and $\mathfrak{osp}(1|2)$. Moreover, the structures of these modules are specified completely. This proves several conjectural statements in the literature for $\mathfrak{sl}_2$, at arbitrary admissible levels, and for $\mathfrak{osp}(1|2)$ at level $-\frac{5}{4}$. For other admissible levels, the $\mathfrak{osp}(1|2)$ results are believed to be new.

Kazuya Kawasetsu

Papers

Quasi-lisse Vertex Algebras and Modular Linear Differential Equations

Orbifolds and cosets of minimal $\mathcal{W}$-algebras

Relaxed Highest-Weight Modules I: Rank 1 Cases

The Intermediate Vertex Subalgebras of the Lattice Vertex Operator Algebras

Relaxed highest-weight modules I: rank $1$ cases