Schur-Weyl Duality for Heisenberg Cosets

Abstract

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.

41 pages

References in corpus (21)

• Braided tensor categories and extensions of vertex operator algebras
• Braided tensor categories of admissible modules for affine Lie algebras
• Logarithmic conformal field theory, log-modular tensor categories and modular forms
• Fusion in Fractional Level sl^(2)-Theories with k=-1/2
• Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models
• The GL(1|1)-symplectic fermion correspondence
• Orbifolds and cosets of minimal $\mathcal{W}$-algebras
• Screening operators for W-algebras
• The Verlinde formula in logarithmic CFT
• On the triplet vertex algebra W(p)
• Logarithmic intertwining operators and vertex operators
• Modularity of logarithmic parafermion vertex algebras
• Weight representations of admissible affine vertex algebras
• Cosets of the ${\mathcal{W}}^k({\mathfrak{sl}}_4,f_{\text{subreg}})$-algebra
• Zhu's algebra, C_2-algebra and C_2-cofiniteness of parafermion vertex operator algebras
• Flatness and Semi-Rigidity of Vertex Operator Algebras
• Logarithmic tensor product theory for generalized modules for a conformal vertex algebra
• Logarithmic Link Invariants of ${\overline{U}}_q^H({\mathfrak{sl}}_2)$ and Asymptotic Dimensions of Singlet Vertex Algebras
• Some applications and constructions of intertwining operators in LCFT
• On C_2-cofiniteness of parafermion vertex operator algebras
• Equivalences between weight modules via $\mathcal{N}=2$ coset constructions

Cited by in corpus (52)

• Tensor categories for vertex operator superalgebra extensions
• Braided tensor categories of admissible modules for affine Lie algebras
• Logarithmic conformal field theory, log-modular tensor categories and modular forms
• Duality of subregular W-algebras and principal W-superalgebras
• On the tensor structure of modules for compact orbifold vertex operator algebras
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• Gluing vertex algebras
• Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1|2)$ minimal models
• On ribbon categories for singlet vertex algebras
• Universal two-parameter ${\mathcal{W}}_{\infty }$-algebra and vertex algebras of type $\mathcal{W}(2,3,\dots ,N)$
• Tensor categories of affine Lie algebras beyond admissible levels
• Conformal Field Theories with Sporadic Group Symmetry
• Rectangular W-algebras, extended higher spin gravity and dual coset CFTs
• Braided Tensor Categories related to ${\mathcal{B}}_p$ Vertex Algebras
• A realisation of the Bershadsky--Polyakov algebras and their relaxed modules
• An admissible level $\widehat{\mathfrak{osp}}\left(1|2\right)$-model: modular transformations and the Verlinde formula
• Modularity of logarithmic parafermion vertex algebras
• S-duality for the large $N=4$ superconformal algebra
• Systematic Orbifold Constructions of Schellekens' Vertex Operator Algebras from Niemeier Lattices
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
• Cosets of the ${\mathcal{W}}^k({\mathfrak{sl}}_4,f_{\text{subreg}})$-algebra
• Unitary and non-unitary $N=2$ minimal models
• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• Modularity of Bershadsky-Polyakov minimal models
• 3d Mirror Symmetry and the $\beta \gamma$ VOA
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p>1$
• Generalized parafermions of orthogonal type
• On the semisimplicity of the category $K{L}_k$ for affine Lie superalgebras
• Free field realisation of boundary vertex algebras for Abelian gauge theories in three dimensions
• A general mirror equivalence theorem for coset vertex operator algebras
• Automorphism groups and uniqueness of holomorphic vertex operator algebras of central charge $24$
• Equivalences between weight modules via $\mathcal{N}=2$ coset constructions
• Cosets of free field algebras via arc spaces
• Kazama-Suzuki coset construction and its inverse
• Representation theory of $L_k\left(\mathfrak{osp}(1|2)\right)$ from vertex tensor categories and Jacobi forms
• From vertex operator superalgebras to graded-local conformal nets and back
• On the structure of W-algebras in type A
• On parafermion vertex algebras of $\mathfrak{sl}(2{)}_{-3/2}$ and $\mathfrak{sl}(3{)}_{-3/2}$
• Representations of ${\mathbb{Z}}_2$-orbifold of the parafermion vertex operator algebra $K(s{l}_2,k)$
• Unitarity and strong graded locality of holomorphic vertex operator superalgebras with central charge at most 24
• Staggered modules of $N=2$ superconformal minimal models
• Invariant subalgebras of the small $\mathcal{N}=4$ superconformal algebra
• Orthosymplectic Feigin-Semikhatov duality
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras
• Conformal nets from minimal W-algebras
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras
• Vertex Algebras and Commutative Algebras
• On some vertex algebras related to $V_{-1}(\mathfrak{sl}(n))$ and their characters