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

Abstract

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.

arXiv commentFinal version. New examples of lisse, rational W-algebras of B and C type are given, some details added in Section 9. Note: our main construction is similar to the one in arXiv:1710.02275 but there are some additional subtleties

References in corpus (28)

• Vertex Algebras and Algebraic Curves
• W-symmetry in Conformal Field Theory
• Quantum Reduction for Affine Superalgebras
• W_{1+\infty} and W(gl_N) with central charge N
• Triality in Minimal Model Holography
• Finite vs. affine W-algebras
• W-symmetry, topological vertex and affine Yangian
• Coset Realization of Unifying W-Algebras
• W-algebras as coset vertex algebras
• Rationality of W-algebras: principal nilpotent cases
• Cosets of affine vertex algebras inside larger structures
• Joseph ideals and lisse minimal W-algebras
• Webs of W-algebras
• Exploring ${\mathcal{W}}_{\infty }$ in the quadratic basis
• Poisson-Lie group of pseudodifferential symbols
• Freely generated vertex algebras and non-linear Lie conformal algebras
• Orbifolds and cosets of minimal $\mathcal{W}$-algebras
• Rationality of Bershadsky-Polyakov vertex algebras
• Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size
• A Class of W-Algebras with Infinitely Generated Classical Limit
• The super W_{1+\infty} algebra with integral central charge
• Orbifolds of symplectic fermion algebras
• Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras
• Classification of Structure Constants for W-algebras from Highest Weights
• The structure of the Kac-Wang-Yan algebra
• Logarithmic conformal field theories of type $B_n,\ell =4$ and symplectic fermions
• $\mathcal{W}$-algebras with non-admissible levels and the Deligne exceptional series
• Representations of ${\mathbb{Z}}_2$-orbifold of the parafermion vertex operator algebra $K(s{l}_2,k)$

Cited by in corpus (16)

• Instanton R-matrix and W-symmetry
• Rectangular W-algebras, extended higher spin gravity and dual coset CFTs
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• A geometric free field realisation for the genus-two class $\mathcal{S}$ theory of type ${\mathfrak{a}}_1$
• $N=4$ superconformal algebras and diagonal cosets
• Rationality of the exceptional W-algebras ${\mathcal{W}}_k({\mathfrak{sp}}_4,f_{subreg})$ associated with subregular nilpotent elements of ${\mathfrak{sp}}_4$
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• Generalized parafermions of orthogonal type
• On even spin $W_{\infty }$
• Fusion rules for ${\mathbb{Z}}_2$-orbifolds of affine and parafermion vertex operator algebras
• Orbifolds of Gaiotto-Rapčák $Y$-algebras
• The Grassmannian VOA
• Orthosymplectic Feigin-Semikhatov duality
• The $\mathcal{W}$-algebra bootstrap of 6d $\mathcal{N}=(2,0)$ theories
• Vertex Algebras and Commutative Algebras
• Duality via convolution of W-algebras