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There is an embedding of affine vertex algebras $V^k(\mathfrak{gl}_n) \hookrightarrow V^k(\mathfrak{sl}_{n+1})$, and the coset $\mathcal{C}^k(n) = \text{Com}(V^k(\mathfrak{gl}_n), V^k(\mathfrak{sl}_{n+1}))$ is a natural generalization of the parafermion algebra of $\mathfrak{sl}_2$. It was called the algebra of generalized parafermions by the third author and was shown to arise as a one-parameter quotient of the universal two-parameter $\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,3,\dots)$. In this paper, we consider an analogous structure of orthogonal type, namely $\mathcal{D}^k(n) = \text{Com}(V^k(\mathfrak{so}_{2n}), V^k(\mathfrak{so}_{2n+1}))^{\mathbb{Z}_2}$. We realize this algebra as a one-parameter quotient of the two-parameter even spin $\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,4,\dots)$, and we classify all coincidences between its simple quotient $\mathcal{D}_k(n)$ and the algebras $\mathcal{W}_{\ell}(\mathfrak{so}_{2m+1})$ and $\mathcal{W}_{\ell}(\mathfrak{so}_{2m})^{\mathbb{Z}_2}$. As a corollary, we show that for the admissible levels $k = -(2n-2) + \frac{1}{2} (2 n + 2 m -1)$ for $\widehat{\mathfrak{so}}_{2n}$ the simple affine algebra $L_k(\mathfrak{so}_{2n})$ embeds in $L_k(\mathfrak{so}_{2n+1})$, and the coset is strongly rational. As a consequence, the category of ordinary modules of $L_k(\mathfrak{so}_{2n+1})$ at such a level is a braided fusion category., Comment: Minor corrections, final version to appear in J. Algebra

Various aspects of orbifolds and cosets of the small 𝒩 = 4 superconformal algebra are studied. First, we determine minimal strong generators for generic and specific levels. As a corollary, we obtain the vertex algebra of global sections of the chiral de Rham complex on any complex Enriques surface. We also identify orbifolds of cosets of the small 𝒩 = 4 superconformal algebra with Com(Vl(𝔰𝔩2); Vl+1(𝔰𝔩2) ⊗ 𝒲–5/2(𝔰𝔩4; frect)) and in addition at special levels with Grassmanian cosets and principal 𝒲-algebras of type A at degenerate admissible levels. These coincidences lead us to a novel level-rank duality involving Grassmannian supercosets.

We show that the Kazhdan–Lusztig category $KL_k$ of level-$k$ finite-length modules with highest-weight composition factors for the affine Lie superalgebra $\widehat{\mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure and that its full subcategory $\mathcal{O}_k^{fin}$ of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple $\widehat{\mathfrak{gl}(1|1)}$-module in $KL_k$ has a projective cover in ${\mathcal{O}}_k^{fin}$, and we determine all fusion rules involving simple and projective objects in ${\mathcal{O}}_k^{fin}$. Then using Knizhnik–Zamolodchikov equations, we prove that $KL_k$ and $\mathcal{O}_k^{fin}$ are rigid. As an application of the tensor supercategory structure on $\mathcal{O}_k^{fin}$, we study certain module categories for the affine Lie superalgebra $\widehat{\mathfrak{sl}(2|1)}$ at levels $1$ and $-\frac{1}{2}$. In particular, we obtain a tensor category of $\widehat{\mathfrak{sl}(2|1)}$-modules at level $-\frac{1}{2}$ that includes relaxed highest-weight modules and their images under spectral flow.

The vertex algebras $$\mathcal {V}^{(p)}$$ and $$\mathcal R^{(p)}$$ introduced in Adamovic (Transform Groups 21(2):299–327, 2016) are very interesting relatives of the well-known triplet algebras of logarithmic CFT. The algebra $$\mathcal {V}^{(p)}$$ (respectively, $$\mathcal {R}^{(p)}$$ ) is a large extension of the simple affine vertex algebra $$L_{k}(\mathfrak {sl}_{2})$$ (respectively, $$L_{k}(\mathfrak {sl}_{2})$$ times a Heisenberg algebra), at level $$k=-2+1/p$$ for positive integer p. Particularly, the algebra $$\mathcal {V}^{(2)}$$ is the simple small $$N=4$$ superconformal vertex algebra with $$c=-9$$ , and $$\mathcal {R}^{(2)}$$ is $$L_{-3/2}(\mathfrak {sl}_3)$$ . In this paper, we derive structural results of these algebras and prove various conjectures coming from representation theory and physics. We show that SU(2) acts as automorphisms on $$\mathcal {V}^{({p})}$$ and we decompose $$\mathcal {V}^{({p})}$$ as an $$L_{k}(\mathfrak {sl}_{2})$$ -module and $$\mathcal {R}^{({p})}$$ as an $$L_k(\mathfrak {gl}_2)$$ -module. The decomposition of $$\mathcal {V}^{({p})}$$ shows that $$\mathcal {V}^{({p})}$$ is the large level limit of a corner vertex algebra appearing in the context of S-duality. We also show that the quantum Hamiltonian reduction of $$\mathcal {V}^{({p})}$$ is the logarithmic doublet algebra $$\mathcal {A}^{({p})}$$ introduced in Adamovic and Milas (Contemp Math 602:23–38, 2013), while the reduction of $$\mathcal {R}^{({p})}$$ yields the $$\mathcal {B}^{({p})}$$ -algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014). Conversely, we realize $$\mathcal {V}^{({p})}$$ and $$\mathcal {R}^{({p})}$$ from $$\mathcal {A}^{({p})}$$ and $$\mathcal {B}^{({p})}$$ via a procedure that deserves to be called inverse quantum Hamiltonian reduction. As a corollary, we obtain that the category $$KL_{k}$$ of ordinary $$L_{k}(\mathfrak {sl}_{2})$$ -modules at level $$k=-2+1/p$$ is a rigid vertex tensor category equivalent to a twist of the category $$\text {Rep}(SU(2))$$ . This finally completes rigid braided tensor category structures for $$L_{k}(\mathfrak {sl}_{2})$$ at all complex levels k. We also establish a uniqueness result of certain vertex operator algebra extensions and use this result to prove that both $$\mathcal {R}^{({p})}$$ and $$\mathcal {B}^{({p})}$$ are certain non-principal $$\mathcal {W}$$ -algebras of type A at boundary admissible levels. The same uniqueness result also shows that $$\mathcal {R}^{({p})}$$ and $$\mathcal {B}^{({p})}$$ are the chiral algebras of Argyres-Douglas theories of type $$(A_1, D_{2p})$$ and $$(A_1, A_{2p-3})$$ .

Coset constructions of $\mathcal{W}$-algebras have many applications, and were recently given for principal $\mathcal{W}$-algebras of $A$, $D$, and $E$ types by Arakawa together with the first and third authors. In this paper, we give coset constructions of the large and small $N=4$ superconformal algebras, which are the minimal $\mathcal{W}$-algebras of $\mathfrak{d}(2,1;a)$ and $\mathfrak{psl}(2|2)$, respectively. From these realizations, one finds a remarkable connection between the large $N=4$ algebra and the diagonal coset $C^{k_1, k_2} = \text{Com}(V^{k_1+k_2}(\mathfrak{sl}_2), V^{k_1}(\mathfrak{sl}_2) \otimes V^{k_2}(\mathfrak{sl}_2))$, namely, as two-parameter vertex algebras, $C^{k_1, k_2}$ coincides with the coset of the large $N=4$ algebra by its affine subalgebra. We also show that at special points in the parameter space, the simple quotients of these cosets are isomorphic to various $\mathcal{W}$-algebras. As a corollary, we give new examples of strongly rational principal $\mathcal{W}$-algebras of type $C$ at degenerate admissible levels., 34 pages

We prove some conjectures about vertex algebras which emerge in gauge theory constructions associated to the geometric Langlands program. In particular, we present the conjectural kernel vertex algebra for the $S T^2 S$ duality transformation in $SU(2)$ gauge theory. We find a surprising coincidence, which gives a powerful hint about the nature of the corresponding duality wall. Concretely, we determine the branching rules for the small $N=4$ superconformal algebra at central charge $-9$ as well as for the generic large $N=4$ superconformal algebra at central charge $-6$. Moreover we obtain the affine vertex superalgebra of $\mathfrak{osp}(1|2)$ and the $N=1$ superconformal algebra times a free fermion as Quantum Hamiltonian reductions of the large $N=4$ superconformal algebras at $c=-6$., Comment: Relationship between large and small N=4 superconformal VOAs is clarified

In this work, we introduce Urod algebras associated to simply laced Lie algebras as well as the concept of translation of W-algebras. Both results are achieved by showing that the quantum Hamiltonian reduction commutes with tensoring with integrable representations; that is, for V and L an affine vertex algebra and an integrable affine vertex algebra associated with $\mathfrak {g}$ , we have the vertex algebra isomorphism $H_{DS,f}^{0}(V\otimes L)\cong H_{DS,f}^{0}(V)\otimes L$ , where in the left-hand-side the Drinfeld–Sokolov reduction is taken with respect to the diagonal action of $\widehat {\mathfrak {g}}$ on $V{\otimes } L$ . The proof is based on some new construction of automorphisms of vertex algebras, which may be of independent interest. As corollaries, we get fusion categories of modules of many exceptional W-algebras, and we can construct various corner vertex algebras. A major motivation for this work is that Urod algebras of type A provide a representation theoretic interpretation of the celebrated Nakajima–Yoshioka blowup equations for the moduli space of framed torsion free sheaves on $\mathbb {CP}^{2}$ of an arbitrary rank.

The question of when a vertex algebra is a quantization of the arc space of its associated scheme has recently received a lot of attention in both the mathematics and physics literature. This property was first studied by Tomoyuki Arakawa and Anne Moreau, and was given the name "classical freeness" by Jethro van Ekeren and Reimundo Heluani in their work on chiral homology. Later, it was extended to vertex superalgebras by Hao Li. In this note, we prove the classical freeness of the simple affine vertex superalgebra $L_n(\mathfrak{osp}_{m|2r})$ for all positive integers $m,n,r$ satisfying $-\frac{m}{2} + r +n+1 > 0$. In particular, it holds for the rational vertex superalgebras $L_n(\mathfrak{osp}_{1|2r})$ for all positive integers $r,n$., Minor corrections, final version to appear in Proc. Am. Math. Soc

We examine a strong/weak duality between a Heisenberg coset of a theory with $\mathfrak{sl}_n$ subregular $\mathcal{W}$-algebra symmetry and a theory with a $\mathfrak{sl}_{n|1}$-structure. In a previous work, two of the current authors provided a path integral derivation of correlator correspondences for a series of generalized Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality. In this paper, we derive correlator correspondences in a similar way but for a different series of generalized duality. This work is a part of the project to realize the duality of corner vertex operator algebras proposed by Gaiotto and Rap\v{c}\'ak and partly proven by Linshaw and one of us in terms of two dimensional conformal field theory. We also examine another type of duality involving an additional pair of fermions, which is a natural generalization of the fermionic FZZ-duality. The generalization should be important since a principal $\mathcal{W}$-superalgebra appears as its symmetry and the properties of the superalgebra are less understood than bosonic counterparts., Comment: 29 pages, final version to appear in JHEP

We study the minimal models associated to osp ( 1 | 2 ) , otherwise known as the fractional-level Wess–Zumino–Witten models of osp ( 1 | 2 ) . Since these minimal models are extensions of the tensor product of certain Virasoro and sl 2 minimal models, we can induce the known structures of the representations of the latter models to get a rather complete understanding of the minimal models of osp ( 1 | 2 ) . In particular, we classify the irreducible relaxed highest-weight modules, determine their characters and compute their Grothendieck fusion rules. We also discuss conjectures for their (genuine) fusion products and the projective covers of the irreducibles.

We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of $\mathfrak{osp}(1|2)$ and the $N=1$ super Virasoro algebra to categories of Virasoro algebra modules via certain cosets., Comment: 51 pages; in this version, Theorem 7.1 is improved by removing one of the conditions needed to apply it; corresponding improvements are made to the subsequent examples; final version to appear in Communications in Contemporary Mathematics

A W-algebra action is constructed via Hecke transformations on the equivariant Borel–Moore homology of the Hilbert scheme of points on a nonreduced plane in three-dimensional affine space. The resulting W-module is then identified to the vacuum module. The construction is based on a generalization of the ADHM construction as well as the W-action on the equivariant Borel–Moore homology of the moduli space of instantons constructed by Schiffmann and Vasserot.

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group $\overline{U}_q^H(\mfg)$ of a simple Lie algebra $\mfg$ at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the $\mfg=\mathfrak{sl}_2$ case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra $W_Q(r)$ of Feigin and Tipunin \cite{FT} and the $B_Q(r)$ algebras of \cite{C1}., 27 pages

We prove the conjecture of Gaiotto and Rap\v{c}\'ak that the $Y$-algebras $Y_{L,M,N}[\psi]$ with one of the parameters $L,M,N$ zero, are simple one-parameter quotients of the universal two-parameter $\mathcal{W}_{1+\infty}$-algebra, and satisfy a symmetry known as triality. These $Y$-algebras are defined as the cosets of certain non-principal $\mathcal{W}$-algebras and $\mathcal{W}$-superalgebras by their affine vertex subalgebras, and triality is an isomorphism between three such algebras. Special cases of our result provide new and unified proofs of many theorems and open conjectures in the literature on $\mathcal{W}$-algebras of type $A$. This includes (1) Feigin-Frenkel duality, (2) the coset realization of principal $\mathcal{W}$-algebras due to Arakawa and us, (3) Feigin and Semikhatov's conjectured triality between subregular $\mathcal{W}$-algebras, principal $\mathcal{W}$-superalgebras, and affine vertex superalgebras, (4) the rationality of subregular $\mathcal{W}$-algebras due to Arakawa and van Ekeren, (5) the identification of Heisenberg cosets of subregular $\mathcal{W}$-algebras with principal rational $\mathcal{W}$-algebras that was conjectured in the physics literature over 25 years ago. Finally, we prove the conjectures of Proch\'azka and Rap\v{c}\'ak on the explicit truncation curves realizing the simple $Y$-algebras as $\mathcal{W}_{1+\infty}$-quotients, and on their minimal strong generating types., Comment: Final version, to appear in Cambridge Journal of Mathematics

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 add more details to the proofs and also incorporate some recent progress mainly in Sec. 4 and Sec. 6, we also add Sec. 2.4 summarizing the main results on tensor category theory of vertex (super)algebra extensions that are used in this paper

We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra $\mathcal{M}(p)$, $p\geq 2$. The first category consists of all finite-length $\mathcal{M}(p)$-modules with atypical composition factors, while the second is the subcategory of modules that induce to local modules for the triplet vertex operator algebra $\mathcal{W}(p)$. We show that every irreducible module has a projective cover in the second of these categories, although not in the first, and we compute all fusion products involving atypical irreducible modules and their projective covers., Comment: 66 pages, final version incorporating referee suggestions, to appear in Comm. Math. Phys

We study relationships between the restricted unrolled quantum group U ‾ q H ( sl 2 ) at q = e π i / r , and the singlet vertex operator algebra M ( r ) , r ≥ 2 . We use deformable families of modules to efficiently compute ( 1 , 1 ) -tangle invariants colored with projective U ‾ q H ( sl 2 ) -modules. These invariants relate to the colored Alexander tangle invariants studied in [6] , [40] . It follows that the regularized asymptotic dimensions of characters of M ( r ) , studied previously by the first two authors, coincide with the corresponding modified traces of open Hopf link invariants. We also discuss various categorical properties of M ( r ) -mod in connection to braided tensor categories.

We prove Feigin-Frenkel type dualities between subregular W -algebras of type A , B and principal W -superalgebras of type sl ( 1 | n ) , osp ( 2 | 2 n ) . The type A case proves a conjecture of Feigin and Semikhatov. Let ( g 1 , g 2 ) = ( sl n + 1 , sl ( 1 | n + 1 ) ) or ( so 2 n + 1 , osp ( 2 | 2 n ) ) and let r be the lacity of g 1 . Let k be a complex number and l defined by r ( k + h 1 ∨ ) ( l + h 2 ∨ ) = 1 with h i ∨ the dual Coxeter numbers of the g i . Our first main result is that the Heisenberg cosets C k ( g 1 ) and C l ( g 2 ) of these W -algebras at these dual levels are isomorphic, i.e. C k ( g 1 ) ≃ C l ( g 2 ) for generic k. We determine the generic levels and furthermore establish analogous results for the cosets of the simple quotients of the W -algebras. Our second result is a novel Kazama-Suzuki type coset construction: We show that a diagonal Heisenberg coset of the subregular W -algebra at level k times the lattice vertex superalgebra V Z is the principal W -superalgebra at the dual level l. Conversely a diagonal Heisenberg coset of the principal W -superalgebra at level l times the lattice vertex superalgebra V − 1 Z is the subregular W -algebra at the dual level k. Again this is proven for the universal W -algebras as well as for the simple quotients. We show that a consequence of the Kazama-Suzuki type construction is that the simple principal W -superalgebra and its Heisenberg coset at level l are rational and/or C 2 -cofinite if the same is true for the simple subregular W -algebra at dual level l. This gives many new C 2 -cofiniteness and rationality results.

The Bershadsky-Polyakov algebra is the $\mathcal{W}$-algebra associated to $\mathfrak{s}\mathfrak{l}_3$ with its minimal nilpotent element $f_{\theta}$. For notational convenience we define $\mathcal{W}^{\ell} = \mathcal{W}^{\ell - 3/2} (\mathfrak{s}\mathfrak{l}_3, f_{\theta})$. The simple quotient of $\mathcal{W}^{\ell}$ is denoted by $\mathcal{W}_{\ell}$, and for $\ell$ a positive integer, $\mathcal{W}_{\ell}$ is known to be $C_2$-cofinite and rational. We prove that for all positive integers $\ell$, $\mathcal{W}_{\ell}$ contains a rank one lattice vertex algebra $V_L$, and that the coset $\mathcal{C}_{\ell} = \text{Com}(V_L, \mathcal{W}_{\ell})$ is isomorphic to the principal, rational $\mathcal{W}(\mathfrak{s}\mathfrak{l}_{2\ell})$-algebra at level $(2\ell +3)/(2\ell +1) -2\ell$. This was conjectured in the physics literature over 20 years ago. As a byproduct, we construct a new family of rational, $C_2$-cofinite vertex superalgebras from $\mathcal{W}_{\ell}$, Comment: Abstract and introduction rewritten, sections 2 and 4 expanded, minor corrections

The Schur index of the $$(A_1, X_n)$$ -Argyres–Douglas theory is conjecturally a character of a vertex operator algebra. Here such vertex algebras are found for the $$A_{\text {odd}}$$ and $$D_{\text {even}}$$ -type Argyres–Douglas theories. The vertex operator algebra corresponding to $$A_{2p-3}$$ -Argyres–Douglas theory is the logarithmic -algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014), while the one corresponding to $$D_{2p}$$ , denoted by , is realized as a non-regular quantum Hamiltonian reduction of $$L_{k}(\mathfrak {sl}_{p+1})$$ at level . For all n one observes that the quantum Hamiltonian reduction of the vertex operator algebra of $$D_n$$ -Argyres–Douglas theory is the vertex operator algebra of $$A_{n-3}$$ -Argyres–Douglas theory. As a corollary, one realizes the singlet and triplet algebras (the vertex algebras associated to the best understood logarithmic conformal field theories) as quantum Hamiltonian reductions as well. Finally, characters of certain modules of these vertex operator algebras and the modular properties of their meromorphic continuations are given.

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., Comment: Results improved substantially, references added

We study W algebras obtained by quantum Hamiltonian reduction of sl(Mn) associated to the sl(2) embedding of rectangular type. The algebra can be realized as the asymptotic symmetry of higher spin gravity with M×M matrix valued fields. In our previous work, we examined the basic properties of the W algebra and claimed that the algebra can be realized as the symmetry of Grassmannian-like coset even with finite central charge based on a proposal of holography. In this paper, we extend the analysis in the following ways. First, we compute the operator product expansions among low spin generators removing the restriction of n=2. Second, we investigate the degenerate representations in several ways, and see the relations to the coset spectrum and the conical defect geometry of the higher spin gravity. For these analyses, we mainly set M=n=2. Finally, we extend the previous analysis by introducing N=2 supersymmetry.

We examine rectangular W-algebras with $so(M)$ or $sp(2M)$ symmetry, which can be realized as the asymptotic symmetry of higher spin gravities with restricted matrix extensions. We compute the central charges of the algebras and the levels of $so(M)$ or $sp(2M)$ affine subalgebras by applying the Hamiltonian reductions of $so$ or $sp$ type Lie algebras. For simple cases with generators of spin up to two, we obtain their operator product expansions by requiring the associativity. We further claim that the W-algebras can be realized as the symmetry algebras of dual coset CFTs and provide several strong supports. The analysis can be regarded as a check of extended higher spin holographies including full quantum corrections. We also extend the analysis by introducing $\mathcal{N}=1$ supersymmetry., 49 pages, 2 tables, alternative proposals of dual coset models added, a reference added, published version

The $\mathcal{B}_p$-algebras are a family of vertex operator algebras parameterized by $p\in \mathbb Z_{\geq 2}$. They are important examples of logarithmic CFTs and appear as chiral algebras of type $(A_1, A_{2p-3})$ Argyres-Douglas theories. The first member of this series, the $\mathcal{B}_2$-algebra, are the well-known symplectic bosons also often called the $\beta\gamma$ vertex operator algebra. We study categories related to the $\mathcal{B}_p$ vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of $\mathfrak{sl}_2$. These categories are braided, rigid and non semi-simple tensor categories. We list their simple and projective objects, their tensor products and their Hopf links. The latter are successfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author., Comment: 44 pages

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for $\mathcal{C}$ a braided tensor category, we give a detailed construction of the canonical algebra in $\mathcal{C}\boxtimes\mathcal{C}^\text{rev}$: if $\mathcal{C}$ is semisimple but not necessarily finite or rigid, then $\bigoplus_{X\in\text{Irr}(\mathcal{C})}X'\boxtimes X$ is a commutative algebra, with $X'$ a representing object for $\text{Hom}_\mathcal{C}(\bullet\otimes_\mathcal{C}X,\mathbf{1}_{\mathcal{C}})$. Conversely, let $A=\bigoplus_{i\in I}U_i\boxtimes V_i$ be a simple commutative algebra in $\mathcal{U}\boxtimes\mathcal{V}$ with $\mathcal{U}$ semisimple and rigid but not necessarily finite, and $\mathcal{V}$ rigid but not necessarily semisimple. If the unit objects of $\mathcal{U}$ and $\mathcal{V}$ form a commuting pair in $A$, we show there is a braid-reversed equivalence between subcategories of $\mathcal{U}$ and $\mathcal{V}$ sending $U_i$ to $V_i^*$. When $\mathcal{U}$ and $\mathcal{V}$ are module categories for simple vertex operator algebras $U$ and $V$, we glue $U$ and $V$ along $\mathcal{U}\boxtimes\mathcal{V}$ via a map $\tau:\text{Irr}(\mathcal{U})\rightarrow\text{Obj}(\mathcal{V})$ such that $\tau(U)=V$ to create $A=\bigoplus_{X\in\text{Irr}(\mathcal{U})}X'\otimes\tau(X)$. Thus under certain conditions, $\tau$ extends to a braid-reversed equivalence between $\mathcal{U}$ and $\mathcal{V}$ if and only if $A$ is a simple conformal vertex algebra extending $U\otimes V$. As examples, we glue Kazhdan-Lusztig categories at generic levels to obtain new vertex algebras extending the tensor product of two affine vertex algebras, and we prove braid-reversed equivalences between certain module categories for affine vertex algebras and $W$-algebras at admissible levels., Comment: 58 pages, final version incorporating referee comments, abstract has been expanded

We analyze the asymptotic symmetry of higher spin gravity with $M \times M$ matrix valued fields, which is given by rectangular W-algebras with su$(M)$ symmetry. The matrix valued extension is expected to be useful for the relation between higher spin gravity and string theory. With the truncation of spin as $s=2,3,\ldots , n$, we evaluate the central charge $c$ of the algebra and the level $k$ of the affine currents with finite $c,k$. For the simplest case with $n=2$, we obtain the operator product expansions among generators by requiring their associativity. We conjecture that the symmetry is the same as that of Grassmannian-like coset based on our proposal of higher spin holography. Comparing $c,k$ from the both theories, we obtain the map of parameters. We explicitly construct low spin generators from the coset theory, and, in particular, we reproduce the operator product expansions of the rectangular W-algebra for $n=2$. We interpret the map of parameters by decomposing the algebra in the coset description., Comment: 31 pages, minor chages, a reference added, published version

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., Comment: 32 pages

We show that there is a braided tensor category structure on the category of $C_1$-cofinite modules for the (universal or simple) Virasoro vertex operator algebras of arbitrary central charge. In the generic case of central charge $c=13-6(t+t^{-1})$, with $t \notin \mathbb{Q}$, we prove semisimplicity, rigidity and non-degeneracy and also compute the fusion rules of this tensor category., Comment: 32 pages. Minor corrections and references added. To appear in Advances in Mathematics

We present a systematic study of the orbifolds of the rank $n$ symplectic fermion algebra $\mathcal{A}(n)$, which has full automorphism group $Sp(2n)$. First, we show that $\mathcal{A}(n)^{Sp(2n)}$ and $\mathcal{A}(n)^{GL(n)}$ are $\mathcal{W}$-algebras of type $\mathcal{W}(2,4,\dots, 2n)$ and $\mathcal{W}(2,3,\dots, 2n+1)$, respectively. Using these results, we find minimal strong finite generating sets for $\mathcal{A}(mn)^{Sp(2n)}$ and $\mathcal{A}(mn)^{GL(n)}$ for all $m,n\geq 1$. We compute the characters of the irreducible representations of $\mathcal{A}(mn)^{Sp(2n)\times SO(m)}$ and $\mathcal{A}(mn)^{GL(n)\times GL(m)}$ appearing inside $\mathcal{A}(mn)$, and we express these characters using partial theta functions. Finally, we give a complete solution to the Hilbert problem for $\mathcal{A}(n)$; we show that for any reductive group $G$ of automorphisms, $\mathcal{A}(n)^G$ is strongly finitely generated., Exposition streamlined, some new results added in Section 5, references added. arXiv admin note: text overlap with arXiv:1205.4469

W-algebras are constructed via quantum Hamiltonian reduction associated with a Lie algebra $\mathfrak{g}$ and an $\mathfrak{sl}(2)$-embedding into $\mathfrak{g}$. We derive correspondences among correlation functions of theories having different W-algebras as symmetry algebras. These W-algebras are associated to the same $\mathfrak{g}$ but distinct $\mathfrak{sl}(2)$-embeddings. For this purpose, we first explore different free field realizations of W-algebras and then generalize previous works on the path integral derivation of correspondences of correlation functions. For $\mathfrak{g}=\mathfrak{sl}(3)$, there is only one non-standard (non-regular) W-algebra known as the Bershadsky-Polyakov algebra. We examine its free field realizations and derive correlator correspondences involving the WZNW theory of $\mathfrak{sl}(3)$, the Bershadsky-Polyakov algebra and the principal $W_3$-algebra. There are three non-regular W-algebras associated to $\mathfrak{g}=\mathfrak{sl}(4)$. We show that the methods developed for $\mathfrak{g}=\mathfrak{sl}(3)$ can be applied straightforwardly. We briefly comment on extensions of our techniques to general $\mathfrak{g}$., Comment: 52 pages

We prove the long-standing conjecture on the coset construction of the minimal series principal $W$-algebras of $ADE$ types in full generality. We do this by first establishing Feigin's conjecture on the coset realization of the universal principal $W$-algebras, which are not necessarily simple. As consequences, the unitarity of the "discrete series" of principal $W$-algebras is established, a second coset realization of rational and unitary $W$-algebras of type $A$ and $D$ are given and the rationality of Kazama-Suzuki coset vertex superalgebras is derived., Comment: Minor corrections and typos fixed. Proposition 3.4 is strengthened, which simplifies the proofs of Lemma 5.2 and Lemma 8.1. Final version to appear in Invent. Math