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# Conformal embeddings in affine vertex superalgebras

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## Abstract

This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra Vk(g) where g=g0¯⊕g1¯ is a basic classical simple Lie superalgebra. Let Vk(g0¯) be the subalgebra of Vk(g) generated by g0¯. We first classify all levels k for which the embedding Vk(g0¯) in Vk(g) is conformal. Next we prove that, for a large family of such conformal levels, Vk(g) is a completely reducible Vk(g0¯)–module and obtain decomposition rules. Proofs are based on fusion rules arguments and on the representation theory of certain affine vertex algebras. The most interesting case is the decomposition of V−2(osp(2n+8|2n)) as a finite, non simple current extension of V−2(Dn+4)⊗V1(Cn). This decomposition uses our previous work [10] on the representation theory of V−2(Dn+4). We also study conformal embeddings gl(n|m)↪sl(n+1|m) and in most cases we obtain decomposition rules.

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... Using this result, and the fact that at level k = −5/4 there is a conformal embedding of L k (sl(2)) into L k (osp(1|2)) (cf. [7], [19,Sect. 10]), we obtain a realization of the Bershadsky-Polyakov algebra W 1 (cf. ...
... For level k = −5/4, there exists a conformal embedding of L k (sl(2)) into L k (osp(1|2)) (cf. [7]), that is, ...
... Set k = −5/4 as before. Using the following decomposition of the conformal embedding from [7], ...
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Article
We study the simple Bershadsky–Polyakov algebra 𝒲k = 𝒲k(sl3, fθ) at positive integer levels and classify their irreducible modules. In this way, we confirm the conjecture from [9]. Next, we study the case k = 1. We discover that this vertex algebra has a Kazama–Suzuki-type dual isomorphic to the simple affine vertex superalgebra Lk′ (osp(1|2)) for k′ = –5=4. Using the free-field realization of Lk′ (osp(1|2)) from [3], we get a free-field realization of 𝒲k and their highest weight modules. In a sequel, we plan to study fusion rules for 𝒲k.
... (1) The affine vertex algebras at negative levels which appeared in the decompositions of conformal embeddings in [2], [4], [6]. (2) The affine vertex algebras at collapsing levels for affine W-algebras. ...
... So KL −5/2 is semi-simple. 6. Singular vectors in V k (nω 1 ) and V k (nω 3 ) and the proof of Theorem 5.4 ...
... Straightforward calculation. Since the condition λ i (α ∨ 0 ) / ∈ Z ≥0 is satisfied for i = 1, 6,7,8,9,10,11,16, the assertion (1) gives the singular vectors in these cases. The remaining cases are given by assertion (2). ...
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We study the representation theory of non-admissible simple affine vertex algebra $L_{-5/2} (sl(4))$. We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra $V^{-5/2} (sl(4))$, and show that it generates the maximal ideal in $V^{-5/2} (sl(4))$. We classify irreducible $L_{-5/2} (sl(4))$--modules in the category ${\mathcal O}$, and determine the fusion rules between irreducible modules in the category of ordinary modules $KL_{-5/2}$. It turns out that this fusion algebra is isomorphic to the fusion algebra of $KL_{-1}$. We also prove that $KL_{-5/2}$ is a semi-simple, rigid braided tensor category. In our proofs we use the notion of collapsing level for the affine $\mathcal{W}$--algebra, and the properties of conformal embedding $gl(4) \hookrightarrow sl(5)$ at level $k=-5/2$ from arXiv:1509.06512. We show that $k=-5/2$ is a collapsing level with respect to the subregular nilpotent element $f_{subreg}$, meaning that the simple quotient of the affine $\mathcal{W}$--algebra $W^{-5/2}(sl(4), f_{subreg})$ is isomorphic to the Heisenberg vertex algebra $M_J(1)$. We prove certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction functor $H_{f_{subreg}}$. It turns out that this case is much more complicated than the case of minimal reduction.
... This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [2,3,4,5,7]. In [3] we initiated the study of conformal embeddings of affine vertex algebras in minimal affine W -algebras. ...
... In general, finding these decompositions is a hard problem, open in most cases. Some decompositions are provided in our previous papers [2,4,5,7]. In particular, the paper [5] shows decompositions of minimal affine W -algebras W k (g, x, f θ ). ...
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In this paper we prove a general result saying that under mild hypothesis an embedding of an affine vertex algebra into an affine $W$-algebra is conformal if and only if their central charges coincide. This result extends our previous result obtained in the case of minimal affine $W$-algebras. We also find a sufficient condition showing that certain conformal levels are collapsing. This new condition enables us to show collapsing of $W_k(sl(N), x, f )$ for $f$ of hook or rectangular type. Our methods can be applied to non-admissible levels. In particular, we prove Creutzig's conjecture on the conformal embedding in the hook type $W$-algebra $W_k(sl(n+m), x, f_{m,n})$ of its affine vertex subalgebra. Quite surprisingly, the problem of showing that certain conformal levels are not collapsing turns out to be very difficult. In the cases when $k$ is admissible and conformal, we prove that $W_k(sl(n+m), x, f_{m,n})$ is not collapsing. Then, by generalizing the results on semi-simplicity of conformal embeddings from our previous papers, we find many cases in which $W_k(sl(n+m), x, f_{m,n})$ is semi-simple as a module for its affine subalgebra at conformal level and we provide explicit decompositions.
... As outlined before, our present work was made possible by various quite recent results of our research teams. These are a good understanding of conformal embeddings [7,8,10,11,19], collapsing levels [9,18,26], its implications on vertex tensor categories [49] and finally a full understanding of the representation theory of the singlet algebras [40,41]. ...
Preprint
We prove that $KL_k(\mathfrak{sl}_m)$ is a semi-simple, rigid braided tensor category for all even $m\ge 4$, and $k= -\frac{m+1}{2}$ which generalizes result from arXiv:2103.02985 obtained for $m=4$. Moreover, all modules in $KL_k(\mathfrak{sl}_m)$ are simple-currents and they appear in the decomposition of conformal embeddings $\mathfrak{gl}_m \hookrightarrow \mathfrak{sl}_{m+1}$ at level $k= - \frac{m+1}{2}$ from arXiv:1509.06512. For this we inductively identify minimal affine $W$-algebra $W_{k-1} (\mathfrak{sl}_{m+2}, \theta)$ as simple current extension of $L_{k}(\mathfrak{sl}_m) \otimes \mathcal H \otimes \mathcal M$, where $\mathcal H$ is the rank one Heisenberg vertex algebra, and $\mathcal M$ the singlet vertex algebra for $c=-2$. The proof uses previously obtained results for the tensor categories of singlet algebra from arXiv:2202.05496. We also classify all irreducible ordinary modules for $W_{k-1} (\mathfrak{sl}_{m+2}, \theta)$. The semi-simple part of the category of $W_{k-1} (\mathfrak{sl}_{m+2}, \theta)$-modules comes from $KL_{k-1}(\mathfrak{sl}_{m+2})$, using quantum Hamiltonian reduction, but this $W$-algebra also contains indecomposable ordinary modules.
... 4.2); (2) Positive rational but generic levels (Sect. 4.4); (3) The levels k such that the ordinary modules for W k (g, θ) are all of finite length (Sect. 4.1). ...
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Article
We show that if V is a vertex operator algebra such that all the irreducible ordinary V-modules are $$C_1$$-cofinite and all the grading-restricted generalized Verma modules for V are of finite length, then the category of finite length generalized V-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra (resp. superalgebra) associated to a finite simple Lie algebra (resp. Lie superalgebra) $$\mathfrak {g}$$ at level k and the category $$KL_k(\mathfrak {g})$$ of its finite length generalized modules, we discover several families of $$KL_k(\mathfrak {g})$$ at non-admissible levels k, having braided tensor category structures. In particular, $$KL_k(\mathfrak {g})$$ has a braided tensor category structure if the category of ordinary modules is semisimple or more generally if the category of ordinary modules is of finite length. We also prove the rigidity and determine the fusion rules of some categories $$KL_k(\mathfrak {g})$$, including the category $$KL_{-1}(\mathfrak {sl}_n)$$. Using these results, we construct a rigid tensor category structure on a full subcategory of $$KL_1(\mathfrak {sl}(n|m))$$ consisting of objects with semisimple Cartan subalgebra actions.
... We have the following conjecture (which is also in agreement with [24]): Conjecture 8.1. For every even n ≥ 0, we have 10 . ...
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