arXiv subjects

Andrew R. Linshaw

Publications and source records attributed to Andrew R. Linshaw.

At least 37 records · Page 2Linked to original sources

arXiv2021

Generalized parafermions of orthogonal type

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.

math.QA↗

arXiv2020

Universal two-parameter $\mathcal{W}_{\infty}$-algebra and vertex algebras of type $\mathcal{W}(2,3,\dots, N)$

We prove the longstanding physics conjecture that there exists a unique two-parameter $\mathcal{W}_{\infty}$-algebra which is freely generated of type $\mathcal{W}(2,3,\dots)$, and generated by the weights $2$ and $3$ fields. Subject to some mild constraints, all vertex algebras of type $\mathcal{W}(2,3,\dots, N)$ for some $N$ can be obtained as quotients of this universal algebra. As an application, we show that for $n\geq 3$, the structure constants for the principal $\mathcal{W}$-algebras $\mathcal{W}^k(\mathfrak{s}\mathfrak{l}_n, f_{\text{prin}})$ are rational functions of $k$ and $n$, and we classify all coincidences among the simple quotients $\mathcal{W}^k(\mathfrak{s}\mathfrak{l}_n, f_{\text{prin}})$ for $n\geq 2$. We also obtain many new coincidences between $\mathcal{W}^k(\mathfrak{s}\mathfrak{l}_n, f_{\text{prin}})$ and other vertex algebras of type $\mathcal{W}(2,3,\dots, N)$ which arise as cosets of affine vertex algebras or nonprincipal $\mathcal{W}$-algebras

math.RT↗

arXiv2020

Invariant subalgebras of the small $\mathcal{N}=4$ superconformal algebra

Various aspects of orbifolds and cosets of the small $\mathcal{N}=4$ 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 $\mathcal{N}=4$ superconformal algebra with $\text{Com}(V^{\ell}(\mathfrak{sl}_2), V^{\ell+1}(\mathfrak{sl}_2) \otimes \mathcal{W}_{-5/2}(\mathfrak{sl}_4, f_{\text{rect}}))$ and in addition at special levels with Grassmanian cosets and principal $\mathcal{W}$-algebras of type $A$ at degenerate admissible levels. These coincidences lead us to a novel level-rank duality involving Grassmannian supercosets.

math.RT↗

arXiv2019

$N=4$ superconformal algebras and diagonal cosets

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.

math.RT↗

arXiv2019

Universal two-parameter even spin $\mathcal{W}_{\infty}$-algebra

We construct the unique two-parameter vertex algebra which is freely generated of type ${\mathcal W}(2,4,6,\dots)$, and generated by the weights $2$ and $4$ fields. Subject to some mild constraints, all vertex algebras of type ${\mathcal W}(2,4,\dots, 2N)$ for some $N$, can be obtained as quotients of this universal algebra. This includes the $B$ and $C$ type principal ${\mathcal W}$-algebras, the $\mathbb{Z}_2$-orbifolds of the $D$ type principal ${\mathcal W}$-algebras, and many others which arise as cosets of affine vertex algebras inside larger structures. As an application, we classify all coincidences among the simple quotients of the $B$ and $C$ type principal ${\mathcal W}$-algebras, as well as the $\mathbb{Z}_2$-orbifolds of the $D$ type principal ${\mathcal W}$-algebras. Finally, we use our classification to give new examples of principal ${\mathcal W}$-algebras of $B$, $C$, and $D$ types, which are lisse and rational.

math.RT↗

arXiv2019

Ito's conjecture and the coset construction for ${\mathcal W}^k(\mathfrak{sl}(3|2))$

Many ${\mathcal W}$-(super)algebras which are defined by the generalized Drinfeld-Sokolov reduction are also known or expected to have coset realizations. For example, it was conjectured by Ito that the principal ${\mathcal W}$-superalgebra ${\mathcal W}^k(\mathfrak{sl}(n+1|n))$ is isomorphic to the coset of $V^{l+1}(\mathfrak{gl}_n)$ inside $V^{l}(\mathfrak{sl}_{n+1}) \otimes {\mathcal E}(n)$ for generic values of $l$. Here ${\mathcal E}(n)$ denotes the rank $n$ $bc$-system, which carries an action of $V^1(\mathfrak{gl}_n)$, and $k$ and $l$ are related by $(k + 1) (l + n + 1) = 1$. This conjecture is known in the case $n=1$, which is somewhat degenerate, and we shall prove it in the first nontrivial case $n=2$. As a consequence, we show that the simple quotient ${\mathcal W}_k(\mathfrak{sl}(3|2))$ is lisse and rational for all positive integers $l>1$. These are new examples of rational ${\mathcal W}$-superalgebras.

math.RT↗

arXiv2019

Singular support of a vertex algebra and the arc space of its associated scheme

Attached to a vertex algebra $\mathcal{V}$ are two geometric objects. The associated scheme of $\mathcal{V}$ is the spectrum of Zhu's Poisson algebra $R_{\mathcal{V}}$. The singular support of $\mathcal{V}$ is the spectrum of the associated graded algebra $\text{gr}(\mathcal{V})$ with respect to Li's canonical decreasing filtration. There is a closed embedding from the singular support to the arc space of the associated scheme, which is an isomorphism in many interesting cases. In this note we give an example of a non-quasi-lisse vertex algebra whose associated scheme is reduced, for which the isomorphism is not true as schemes but true as varieties.

math.RT↗

arXiv2019

W-algebras as coset vertex algebras

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.

math.QA↗

arXiv2018

S-duality for the large $N=4$ superconformal algebra

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 $c=-6$.

math.QA↗

arXiv2018

Cosets of affine vertex algebras inside larger structures

Given a finite-dimensional reductive Lie algebra $\mathfrak{g}$ equipped with a nondegenerate, invariant, symmetric bilinear form $B$, let $V^k(\mathfrak{g},B)$ denote the universal affine vertex algebra associated to $\mathfrak{g}$ and $B$ at level $k$. Let $\mathcal{A}^k$ be a vertex (super)algebra admitting a homomorphism $V^k(\mathfrak{g},B)\rightarrow \mathcal{A}^k$. Under some technical conditions on $\mathcal{A}^k$, we characterize the coset $\text{Com}(V^k(\mathfrak{g},B),\mathcal{A}^k)$ for generic values of $k$. We establish the strong finite generation of this coset in full generality in the following cases: $\mathcal{A}^k = V^k(\mathfrak{g}',B')$, $\mathcal{A}^k = V^{k-l}(\mathfrak{g}',B') \otimes \mathcal{F}$, and $\mathcal{A}^k = V^{k-l}(\mathfrak{g}',B') \otimes V^{l}(\mathfrak{g}",B")$. Here $\mathfrak{g}'$ and $\mathfrak{g}"$ are finite-dimensional Lie (super)algebras containing $\mathfrak{g}$, equipped with nondegenerate, invariant, (super)symmetric bilinear forms $B'$ and $B"$ which extend $B$, $l \in \mathbb{C}$ is fixed, and $\mathcal{F}$ is a free field algebra admitting a homomorphism $V^l(\mathfrak{g},B) \rightarrow \mathcal{F}$. Our approach is essentially constructive and leads to minimal strong finite generating sets for many interesting examples. As an application, we give a new proof of the rationality of the simple $N=2$ superconformal algebra with $c=\frac{3k}{k+2}$ for all positive integers $k$.

math.RT↗

arXiv2017

Cosets of the $\mathcal{W}^k(\mathfrak{sl}_4, f_{\text{subreg}})$-algebra

Let $\mathcal {W}^k(\mathfrak{sl}_4, f_{\text {subreg}})$ be the universal $\mathcal{W}$-algebra associated to $\mathfrak{sl}_4$ with its subregular nilpotent element, and let $\mathcal {W}_k(\mathfrak{sl}_4, f_{\text {subreg}})$ be its simple quotient. There is a Heisenberg subalgebra $\mathcal{H}$, and we denote by $\mathcal{C}^k$ the coset $\text{Com}(\mathcal{H}, \mathcal {W}^k(\mathfrak{sl}_4, f_{\text {subreg}}))$, and by $\mathcal{C}_k$ its simple quotient. We show that for $k=-4+(m+4)/3$ where $m$ is an integer greater than $2$ and $m+1$ is coprime to $3$, $\mathcal{C}_k$ is isomorphic to a rational, regular $\mathcal W$-algebra $\mathcal{W}(\mathfrak{sl}_m, f_{\text{reg}})$. In particular, $\mathcal{W}_k(\mathfrak{sl}_4, f_{\text {subreg}})$ is a simple current extension of the tensor product of $\mathcal{W}(\mathfrak{sl}_m, f_{\text{reg}})$ with a rank one lattice vertex operator algebra, and hence is rational.

math.RT↗

arXiv2017

Cosets of Bershadsky-Polyakov algebras and rational $\mathcal{W}$-algebras of type $A$

The Bershadsky-Polyakov algebra is the $\mathcal{W}$-algebra associated to $\mathfrak{s}\mathfrak{l}_3$ with its minimal nilpotent element $f_θ$. For notational convenience we define $\mathcal{W}^{\ell} = \mathcal{W}^{\ell - 3/2} (\mathfrak{s}\mathfrak{l}_3, f_θ)$. 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 $\mathcal{W}_{\ell}$

math.RT↗

arXiv2017

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

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.

math.RT↗

arXiv2016

The $\mathbb{Z}_2$-orbifold of the $\mathcal{W}_3$-algebra

The Zamolodchikov $\mathcal{W}_3$-algebra $\mathcal{W}^c_3$ with central charge $c$ has full automorphism group $\mathbb{Z}_2$. It was conjectured in the physics literature over 20 years ago that the orbifold $(\mathcal{W}^c_3)^{\mathbb{Z}_2}$ is of type $\mathcal{W}(2,6,8,10,12)$ for generic values of $c$. We prove this conjecture for all $c \neq \frac{559 \pm 7 \sqrt{76657}}{95}$, and we show that for these two values, the orbifold is of type $\mathcal{W}(2,6,8,10,12,14)$. This paper is part of a larger program of studying orbifolds and cosets of vertex algebras that depend continuously on a parameter. Minimal strong generating sets for orbifolds and cosets are often easy to find for generic values of the parameter, but determining which values are generic is a difficult problem. In the example of $(\mathcal{W}^c_3)^{\mathbb{Z}_2}$, we solve this problem using tools from algebraic geometry.

math.RT↗

arXiv2016

Schur-Weyl Duality for Heisenberg Cosets

Let $V$ be a simple vertex operator algebra containing a rank $n$ Heisenberg vertex algebra $H$ and let $C=\text{Com}\left( {H}, {V}\right)$ be the coset of ${H}$ in ${V}$. Assuming that the representation categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of ${C}$ are found and every simple ${C}$-module is shown to be contained in at least one ${V}$-module. A corollary of this is that if ${V}$ is rational and $C_2$-cofinite and CFT-type, and $\text{Com}\left( {C}, {V}\right)$ is a rational lattice vertex operator algebra, then so is ${C}$. These results are illustrated with many examples and the $C_1$-cofiniteness of certain interesting classes of modules is established.

math.QA↗

arXiv2015

Simple current extensions beyond semi-simplicity

Let V be a simple VOA and consider a representation category of V that is a vertex tensor category in the sense of Huang-Lepowsky. In particular, this category is a braided tensor category. Let J be an object in this category that is a simple current of order two of either integer or half-integer conformal dimension. We prove that $V\oplus J$ is either a VOA or a super VOA. If the representation category of V is in addition ribbon, then the categorical dimension of J decides this parity question. Combining with Carnahan's work, we extend this result to simple currents of arbitrary order. Our next result is a simple sufficient criterion for lifting indecomposable objects that only depends on conformal dimensions. Several examples of simple current extensions that are $C_2$-cofinite and non-rational are then given and induced modules listed.

math.QA↗

arXiv2015

The structure of the Kac-Wang-Yan algebra

The Lie algebra $\mathcal{D}$ of regular differential operators on the circle has a universal central extension $\hat{\mathcal{D}}$. The invariant subalgebra $\hat{\mathcal{D}}^+$ under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum $\hat{\mathcal{D}}^+$-module with central charge $c\in\mathbb{C}$, and its irreducible quotient $\mathcal{V}_c$, possess vertex algebra structures, and $\mathcal{V}_c$ has a nontrivial structure if and only if $c\in \frac{1}{2}\mathbb{Z}$. We show that for each integer $n>0$, $\mathcal{V}_{n/2}$ and $\mathcal{V}_{-n}$ are $\mathcal{W}$-algebras of types $\mathcal{W}(2,4,\dots,2n)$ and $\mathcal{W}(2,4,\dots, 2n^2+4n)$, respectively. These results are formal consequences of Weyl's first and second fundamental theorems of invariant theory for the orthogonal group $\text{O}(n)$ and the symplectic group $\text{Sp}(2n)$, respectively. Based on Sergeev's theorems on the invariant theory of $\text{Osp}(1,2n)$ we conjecture that $\mathcal{V}_{-n + 1/2}$ is of type $\mathcal{W}(2,4,\dots, 4n^2+8n+2)$, and we prove this for $n=1$. As an application, we show that invariant subalgebras of $βγ$-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.

math.RT↗

arXiv2015

Jet schemes and invariant theory

Let $G$ be a complex reductive group and $V$ a $G$-module. Then the $m$th jet scheme $G_m$ acts on the $m$th jet scheme $V_m$ for all $m\geq 0$. We are interested in the invariant ring $\mathcal{O}(V_m)^{G_m}$ and whether the map $p_m^*\colon\mathcal{O}((V//G)_m) \rightarrow \mathcal{O}(V_m)^{G_m}$ induced by the categorical quotient map $p\colon V\rightarrow V//G$ is an isomorphism, surjective, or neither. Using Luna's slice theorem, we give criteria for $p_m^*$ to be an isomorphism for all $m$, and we prove this when $G=SL_n$, $GL_n$, $SO_n$, or $Sp_{2n}$ and $V$ is a sum of copies of the standard representation and its dual, such that $V//G$ is smooth or a complete intersection. We classify all representations of $\mathbb{C}^*$ for which $p^*_{\infty}$ is surjective or an isomorphism. Finally, we give examples where $p^*_m$ is surjective for $m=\infty$ but not for finite $m$, and where it is surjective but not injective.

math.AG↗