arXiv subjects
Publications and source records attributed to Kazuya Kawasetsu.
We discuss a possible generalization of a result by the third-named author on the rationality of non-admissible minimal W-algebras. We then apply this generalization to finding rational non-admissible principal W-algebras.
The Rogers-Ramanujan recursions are studied from the viewpoint of free representations over free (generalized) vertex algebras. Specifically, we construct short exact sequences among the free representations over free generalized vertex algebras which lift the recursions. They naturally generalize exact sequences introduced by S. Capparelli et al. We also show that an analogue of the Rogers-Ramanujan recursions is realized as an exact sequence among finite-dimensional free representations over a certain finite-dimensional vertex algebra. As an application of exact sequences, we reveal a relation between free generalized vertex algebras and $\hat{\mathfrak{sl}}_2$ spaces of coinvariants $L_{1,0}^{N,\infty}(\mathfrak n)$ introduced by B. Feigin et al. Moreover, it is shown that generalized $q$-Fibonacci recursions may be realized as exact sequences obtained by applying several functors to exact sequences among free representations over free generalized vertex algebras. Finally, we consider a character decomposition formula of the basic $\hat{\mathrm{sl}}_2$-module $L_{1,0}$ by M. Bershtein et al. related to the Urod vertex operator algebras, from the viewpoint of free representations and exact sequences.
The Bershadsky--Polyakov algebras are the subregular quantum hamiltonian reductions of the affine vertex operator algebras associated with $\mathfrak{sl}_3$. In arXiv:2007.00396 [math.QA], we realised these algebras in terms of the regular reduction, Zamolodchikov's W$_3$-algebra, and an isotropic lattice vertex operator algebra. We also proved that a natural construction of relaxed highest-weight Bershadsky--Polyakov modules gives modules that are generically irreducible. Here, we prove that this construction, when combined with spectral flow twists, gives a complete set of irreducible weight modules whose weight spaces are finite-dimensional. This gives a simple independent proof of the main classification theorem of arXiv:2007.03917 [math.RT] for nondegenerate admissible levels and extends this classification to a category of weight modules. We also deduce the classification for the nonadmissible level $\mathsf{k}=-\frac{7}{3}$, which is new.
We study the weight modules over affine Kac-Moody algebras from the view point of vertex algebras, and determine the abelian category of weight modules for the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ at any non-integral admissible level $k$. In particular, we show that the principal block of the category of weight modules over admissible $L_k(\mathfrak{sl}_2)$ is equivalent to that of the corresponding (unrolled) small quantum group.
The coset (commutant) construction is a fundamental tool to construct vertex operator algebras from known vertex operator algebras. The aim of this paper is to provide a fundamental example of the commutants of vertex algebras ouside vertex operator algebras. Namely, the commutant $C$ of the principal subalgebra $W$ of the $A_1$ lattice vertex operator algebra $V_{A_1}$ is investigated. An explicit minimal set of generators of $C$, which consists of infinitely many elements and strongly generates $C$, is introduced. It implies that the algebra $C$ is not finitely generated. Furthermore, Zhu's Poisson algebra of $C$ is shown to be isomorphic to an infinite-dimensional algebra $\mathbb{C}[x_1,x_2,\ldots]/(x_ix_j\,|\,i,j=1,2,\ldots)$. In particular, the associated variety of $C$ consists of a point. Moreover, $W$ and $C$ are verified to form a dual pair in $V_{A_1}$.
The first part of this work uses the algorithm recently detailed in arXiv:1906.02935 to classify the irreducible weight modules of the minimal model vertex operator algebra $L_k(\mathfrak{sl}_3)$, when the level $k$ is admissible. These are naturally described in terms of families parametrised by up to two complex numbers. We also determine the action of the relevant group of automorphisms of $\hat{\mathfrak{sl}}_3$ on their isomorphism classes and compute explicitly the decomposition into irreducibles when a given family's parameters are permitted to take certain limiting values. Along with certain character formulae, previously established in arXiv:2003.10148, these results form the input data required by the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level $\mathfrak{sl}_3$ minimal models. The second part of this work applies the standard module formalism to compute these explicitly when $k=-\frac32$. We expect that the methodology developed here will apply in much greater generality.
The Nappi-Witten model is a Wess-Zumino-Witten model in which the target space is the nonreductive Heisenberg group $H_4$. We consider the representation theory underlying this conformal field theory. Specifically, we study the category of weight modules, with finite-dimensional weight spaces, over the associated affine vertex operator algebra $\mathsf{H}_4$. In particular, we classify the irreducible $\mathsf{H}_4$-modules in this category and compute their characters. We moreover observe that this category is nonsemisimple, suggesting that the Nappi-Witten model is a logarithmic conformal field theory.
We present a realisation of the universal/simple Bershadsky--Polyakov vertex algebras as subalgebras of the tensor product of the universal/simple Zamolodchikov vertex algebras and an isotropic lattice vertex algebra. This generalises the realisation of the universal/simple affine vertex algebras associated to $\mathfrak{sl}_2$ and $\mathfrak{osp}(1|2)$ given in arXiv:1711.11342. Relaxed highest-weight modules are likewise constructed, conditions for their irreducibility are established, and their characters are explicitly computed, generalising the character formulae of arXiv:1803.01989.
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.
The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $\mathfrak{sl}_3$ and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications [arxiv:1005.0185, arxiv:1910.13781]. In particular, we prove that the simple Bershadsky-Polyakov algebras with admissible nonintegral $\mathsf{k}$ are always rational in category $\mathscr{O}$, whilst they always admit nonsemisimple relaxed highest-weight modules unless $\mathsf{k}+\frac{3}{2} \in \mathbb{Z}_{\ge0}$.
Let $k$ be a field of characteristic zero. This paper studies a problem proposed by Joseph F. Ritt in 1950. Precisely, we prove that (1) If $p\geq 2$ is an integer, for every integer $i\in\mathbb{N}$, the nilpotency index of the image of $T_i$ in the ring $k\{T\}/[T^p]$ equals $(i+1)p-i$. (2) For every pair of integers $(i,j)$, the nilpotency index of the image of $T_iU_j$ in the ring $k\{T\}/[TU]$ equals $i+j+1$.
This is the third of a series of articles devoted to the study of relaxed highest weight modules over vertex operator algebras. Relaxed highest weight modules over affine vertex algebras associated to higher rank Lie algebras $A_\ell$ are extensively studied. In particular, the string functions of simple relaxed highest weight modules whose top spaces are simple cuspidal $A_\ell$-modules are shown to be the quotients by a power of the Dedekind eta series of the $q$-characters of simple ordinary modules over affine W-algebras associated with the minimal nilpotent elements of $A_\ell$.
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.
A characterization of the minimal $\mathcal{W}$-algebras associated with the Deligne exceptional series at level $-h^\vee/6$ is obtained by using one-parameter family of modular linear differential equations of order $4$. In particular, the characters of the Ramond-twisted modules of minimal $\mathcal{W}$-algebras related to the Deligne exceptional series satisfy one of these differential equations. In order to obtain the characterization, the differential equations in the one parameter family which have solutions of "CFT type" are classified, whose solutions are explicitly described.
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 $-h^{\vee}/6-1$, which express the homogeneous Schur indices of 4d SCFTs studied by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees, as quasi-modular forms.
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 $\mathcal{W}^k(\mathfrak{g}, e_{-θ})$ 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 $V^{k'}(\mathfrak{g}^{\natural})$ where $\mathfrak{g}^{\natural} \subset \mathfrak{g}$ denotes the centralizer of $\mathfrak{s} \mathfrak{l}_2$. Therefore $\mathcal{W}^k(\mathfrak{g}, e_{-θ})$ 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 $\mathcal{O}^k = \mathcal{W}^k(\mathfrak{g}, e_{-θ})^{G}$ and the coset $\mathcal{C}^k = \text{Com}(V(\mathfrak{g}'),\mathcal{W}^k(\mathfrak{g}, e_{-θ}))$ 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.
The vertex operator algebra structure of a strongly regular holomorphic vertex operator algebra $V$ of central charge $24$ is proved to be uniquely determined by the Lie algebra structure of its weight one space $V_1$ if $V_1$ is a Lie algebra of the type $A_{1,4}^{12}$, $B_{2,2}^6$, $B_{3,2}^4$, $B_{4,2}^3$, $B_{6,2}^2$, $B_{12,2}$, $D_{4,2}^2B_{2,1}^4$, $D_{8,2}B_{4,1}^2$, $A_{3,2}^4A_{1,1}^4$, $D_{5,2}^2A_{3,1}^2$, $D_{9,2}A_{7,1}$, $C_{4,1}^4$ or $D_{6,2}B_{3,1}^2C_{4,1}$.
Structure of certain simple $\mathcal{W}$-algebras assocated with the Deligne exceptional Lie algebras and non-admissible levels are described as the {\it simple current extensions} of certain vertex operator algebras. As an application, the $C_2$-cofiniteness and $\mathbb{Z}_2$-rationality of the algebras are proved.