Classifying Relaxed Highest-Weight Modules for Admissible-Level Bershadsky–Polyakov Algebras
02 Mar 2021-Communications in Mathematical Physics (Springer Berlin Heidelberg)-Vol. 385, Iss: 2, pp 859-904
TL;DR: In this article, Adamovic and Kontrec showed that the simple Bershadsky-polyakov algebras with admissible non-integral weights are always rational in the category of highest-weight modules.
Abstract: 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 (Arakawa in Commun Math Phys 323:627–633, 2013, Adamovic and Kontrec in Classification of irreducible modules for Bershadsky–Polyakov algebra at certain levels). 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 $$\mathsf {k}+\frac{3}{2} \in \mathbb {Z}_{\geqslant 0}$$
.
Citations
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TL;DR: In this paper, a realisation of the universal/simple Bershadsky-Polyakov vertex algebra as subalgebras of the tensor product of the Zamolodchikov vertex algebra and an isotropic lattice vertex algebra is presented.
Abstract: 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 $$\mathfrak {osp} (1 \vert 2)$$
given in Adamovic (Commun Math Phys 366:1025–1067, 2019). Relaxed highest-weight modules are likewise constructed, conditions for their irreducibility are established, and their characters are explicitly computed, generalising the character formulae of Kawasetsu and Ridout (Commun Math Phys 368:627–663, 2019).
18 citations
TL;DR: In this paper, a new family of simple positive energy representations for the simple affine vertex algebra V_k(sl n+1) in the minimal nilpotent orbit of sl n + 1 has been constructed in terms of Gelfand-Tsetlin tableaux.
Abstract: We explicitly construct, in terms of Gelfand--Tsetlin tableaux, a new family of simple positive energy representations for the simple affine vertex algebra V_k(sl_{n+1}) in the minimal nilpotent orbit of sl_{n+1}. These representations are quotients of induced modules over the affine Kac-Moody algebra of sl_n+1 and include in particular all admissible simple highest weight modules and all simple modules induced from sl_2. Any such simple module in the minimal nilpotent orbit has bounded weight multiplicities.
11 citations
TL;DR: In this paper, a realisation of the universal/simple Bershadsky-Polyakov vertex algebra as subalgebras of the tensor product of the Zamolodchikov vertex algebra and an isotropic lattice vertex algebra is presented.
Abstract: 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.
10 citations
TL;DR: In this article , Adamović et al. studied the modular properties of Bershadsky-Polyakov characters and deduced the associated Grothendieck fusion rules.
Abstract: Abstract The Bershadsky–Polyakov algebras are the original examples of nonregular W-algebras, obtained from the affine vertex operator algebras associated with $$\mathfrak {sl}_3$$ sl 3 by quantum Hamiltonian reduction. In Fehily et al. (Comm Math Phys 385:859–904, 2021), we explored the representation theories of the simple quotients of these algebras when the level $$\mathsf {k}$$ k is nondegenerate-admissible. Here, we combine these explorations with Adamović’s inverse quantum Hamiltonian reduction functors to study the modular properties of Bershadsky–Polyakov characters and deduce the associated Grothendieck fusion rules. The results are not dissimilar to those already known for the affine vertex operator algebras associated with $$\mathfrak {sl}_2$$ sl 2 , except that the role of the Virasoro minimal models in the latter is here played by the minimal models of Zamolodchikov’s $$\mathsf {W}_3$$ W 3 algebras.
4 citations
Posted Content•
TL;DR: In this article, the authors study the simple Bershadsky-polyakov algebra W_k = \mathcal{W}_k(sl_3,f_{\theta})$ at positive integer levels and classify their irreducible modules.
Abstract: We study the simple Bershadsky-Polyakov algebra $\mathcal W_k = \mathcal{W}_k(sl_3,f_{\theta})$ at positive integer levels and classify their irreducible modules. In this way we confirm the conjecture from arXiv:1910.13781. Next, we study the case $k=1$. We discover that this vertex algebra has a Kazama-Suzuki-type dual isomorphic to the simple afine vertex superalgebra $L_{k'} (osp(1 \vert 2))$ for $k'=-5/4$. Using the free-field realization of $L_{k'} (osp(1 \vert 2))$ from arXiv:1711.11342, we get a free-field realization of $\mathcal W_k$ and their highest weight modules. In a sequel, we plan to study fusion rules for $\mathcal W_k$.
4 citations
Cites background from "Classifying Relaxed Highest-Weight ..."
...for k+ 3/2 ∈ Z≥0, while in other cases it is a non-rational vertex algebra. More recently, for kadmissible and non-integral, irreducible Wk–modules were classified in [9] in some special cases, and in [24] in full generality. A realization of Wk, when 2k+ 3 ∈/Z≥0, and its relaxed modules is presented in [8], which gives a natural generalization of the realization of the affine vertex algebra Vk(sl(2)) fr...
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... us now mention certain problems for Bershadsky-Polyakov vertex algebras, which remain unsolved in papers listed above. A. Classification of irreducible Wk–modules for integer levels k, k+ 2 ∈ Z≥0. In [24], authors classified irreducible Wk–modules for kadmissible, nonintegral. They showed that every irreducible highest weight module for Wk is obtained as an image of the admissible modules for Lk(sl(3))...
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References
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Book•
01 Jan 1983TL;DR: The invariant bilinear form and the generalized casimir operator are integral representations of Kac-Moody algebras and the weyl group as mentioned in this paper, as well as a classification of generalized cartan matrices.
Abstract: Introduction Notational conventions 1 Basic definitions 2 The invariant bilinear form and the generalized casimir operator 3 Integrable representations of Kac-Moody algebras and the weyl group 4 A classification of generalized cartan matrices 5 Real and imaginary roots 6 Affine algebras: the normalized cartan invariant form, the root system, and the weyl group 7 Affine algebras as central extensions of loop algebras 8 Twisted affine algebras and finite order automorphisms 9 Highest-weight modules over Kac-Moody algebras 10 Integrable highest-weight modules: the character formula 11 Integrable highest-weight modules: the weight system and the unitarizability 12 Integrable highest-weight modules over affine algebras 13 Affine algebras, theta functions, and modular forms 14 The principal and homogeneous vertex operator constructions of the basic representation Index of notations and definitions References Conference proceedings and collections of paper
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TL;DR: This paper developed conformal field theory from first principles and provided a self-contained, pedagogical, and exhaustive treatment, including a great deal of background material on quantum field theory, statistical mechanics, Lie algebras and affine Lie algesas.
Abstract: Filling an important gap in the literature, this comprehensive text develops conformal field theory from first principles. The treatment is self-contained, pedagogical, and exhaustive, and includes a great deal of background material on quantum field theory, statistical mechanics, Lie algebras and affine Lie algebras. The many exercises, with a wide spectrum of difficulty and subjects, complement and in many cases extend the text. The text is thus not only an excellent tool for classroom teaching but also for individual study. Intended primarily for graduate students and researchers in theoretical high-energy physics, mathematical physics, condensed matter theory, statistical physics, the book will also be of interest in other areas of theoretical physics and mathematics. It will prepare the reader for original research in this very active field of theoretical and mathematical physics.
3,440 citations
TL;DR: In this article, it was shown that the characters of the integrable highest weight modules of affine Lie algebras and the minimal series of the Virasoro algebra give rise to conformal field theories.
Abstract: In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain representations. It is known [Fr], [KP] that for a given affine Lie algebra, the linear space spanned by the characters of the integrable highest weight modules with a fixed level is invariant under the usual action of the modular group SL2(Z). The similar result for the minimal series of the Virasoro algebra is observed in [Ca] and [IZ]. In both cases one uses the explicit character formulas to prove the modular invariance. The character formula for the affine Lie algebra is computed in [K], and the character formula for the Virasoro algebra is essentially contained in [FF]; see [R] for an explicit computation. This mysterious connection between the infinite dimensional Lie algebras and the modular group can be explained by the two dimensional conformal field theory. The highest weight modules of affine Lie algebras and the Virasoro algebra give rise to conformal field theories. In particular, the conformal field theories associated to the integrable highest modules and minimal series are rational. The characters of these modules are understood to be the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. From this point of view, the role of the modular group SL2(Z) is manifest. In the study of conformal field theory, physicists arrived at the notion of chiral algebras (see e.g. [MS]). Independently, in the attempt to realize the Monster sporadic group as a symmetry group of certain algebraic structure, an infinite dimensional graded representation of the Monster sporadic group, the so called Moonshine module, was constructed in [FLM1]. This algebraic structure was later found in [Bo] and called the vertex algebra; the first axioms of vertex operator algebras were formulated in that paper. The proof that the Moonshine module is a vertex operator algebra and the Monster group acts as its automorphism group was given in [FLM2]. Notably the character of the Moonshine module is also a modular function, namely j(τ) − 744. It turns out that the vertex operator algebra can be regarded as a rigorous mathematical definition of the chiral algebras in the physical literature. And it is expected that a pair of isomorphic vertex operator algebras and their representations (corresponding to the holomorphic and antiholomorphic sectors) are the basic objects needed to build a conformal field theory of a certain type.
1,122 citations
"Classifying Relaxed Highest-Weight ..." refers background or methods in this paper
...ndedV-module,thenMtop isnaturallyaZhu[V]-module. Infact,itmaybeidentifiedwithZhu[M]if Misalsosimple,thoughthiswillnotbetrueingeneral. Simplelower-boundedV-moduleshavethefollowingproperty. Theorem 3.7 ([40]). Zhu[−]and Ind[−]induce a bijection between the sets of isomorphism classes of simple lowerbounded V-modules and simple Zhu[]-modules. To classify the simple lower-bounded V-modules, it is therefore ...
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...dcharacters, necessary to construct partition functions in conformal field theory, we shall also require that the weight spaces of these simple relaxed highest-weight modules are finite-dimensional. By [40], it therefore suffices to classify the simple weight modules, with finite-dimensional weight spaces, of the untwisted and twisted Zhu algebras of BP(u,v). A directassaultonthis classificationseems quited...
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...treating it as a U0-module equippedwithatrivialU′ >-action,andtakingaquotientthatimposes,amongotherthings,thegeneralisedcommutation relations (Borcherds relations) of V. The details may be foundin [40,42]. Proposition 3.5 ([40]). There exists a functor, which we call the Zhu induction functor, that assigns to any Zhu[V]- module Na V-module Ind[N]such that Zhu[Ind[N]]≃N. The Zhu functor is thus a left ...
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...ols that we shall use to classify Bershadsky–Polyakovmodules are the functors induced between these modules and those of the corresponding(untwisted) Zhu algebra. Althoughoriginally introduced by Zhu [40], the idea behind this unital associative algebra was already well-known to physicists (see [41] for example). Here, we use a (slightly restricted) abstract definition that is based on the physicists’ ...
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TL;DR: In this paper, the integrable highest-weight representations of affine Lie algebras or loop algesbras were constructed by Kac i-K, inspired by the generalization of the Weyl denominator formula for affine roots systems discovered earlier by Macdonald [M].
Abstract: The first construction of the integrable highest-weight representations of affine Lie algebras or loop algebras by Kac i-K] was greatly inspired by the generalization of the Weyl denominator formula for affine roots systems discovered earlier by Macdonald [M]. Though the Macdonald identity found its natural context in representation theory, its mysterious modular invariance was not understood until the work of Witten [W-I on the geometric realization of representations of the loop groups corresponding to loop algebras. The work of Witten clearly indicated that the representations of loop groups possess a very rich structure of conformal field theory which appeared in physics literature in the work of Belavin, Polyakov, and Zamolodchikov [BPZ-I. Independently (though two years later), Borcherds, in an attempt to find a conceptual understanding of a certain algebra of vertex operators invariant under the Monster [FLM1], introduced in [B-I a new algebraic structure. We call vertex operator algebras a slightly modified version of Borcherd’s new algebras [FLM2].
940 citations
"Classifying Relaxed Highest-Weight ..." refers background in this paper
...Recall [49] that if I is an ideal of a vertex operator algebra V, then Zhu[V/I] ≃ Zhu[V]/Zhu[I]....
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...Every highest-weight module for the affine Kac–Moody algebra ŝl3 is a V (sl3)-module [49]....
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TL;DR: In this article, additional symmetries in two-dimensional conformal field theory generated by spin s = 1/2, 1,...,3 currents were investigated and the generators of the symmetry form associative algebras with quadratic determining relations.
Abstract: This paper investigates additional symmetries in two-dimensional conformal field theory generated by spin s = 1/2, 1,...,3 currents. For spins s = 5/2 and s = 3, the generators of the symmetry form associative algebras with quadratic determining relations. ''Minimal models'' of conforma field theory with such additional symmetries are considered. The space of local fields occurring in a conformal field theory with additional symmetry corresponds to a certain (in general, reducible) representation of the corresponding algebra of the symmetry.
910 citations
"Classifying Relaxed Highest-Weight ..." refers methods in this paper
...In a companion paper [28], these relaxed modules are constructed from the highest-weight modules of the Zamolodchikov algebra [29], the regular W-algebra associated to sl3, using the inverse quantum hamiltonian reduction procedure of [16, 30]....
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