Duality of subregular W-algebras and principal W-superalgebras

Abstract

We prove Feigin-Frenkel type dualities between subregular W-algebras of type A, B and principal W-superalgebras of type $\mathfrak{sl}(1|n),\mathfrak{osp}(2|2n)$. The type A case proves a conjecture of Feigin and Semikhatov. Let $({\mathfrak{g}}_1,{\mathfrak{g}}_2)=({\mathfrak{sl}}_{n+1},\mathfrak{sl}(1|n+1))$ or $({\mathfrak{so}}_{2n+1},\mathfrak{osp}(2|2n))$ and let $r$ be the lacity of ${\mathfrak{g}}_1$. Let k be a complex number and $\ell$ defined by $r(k+h_1^{\vee })(\ell +h_2^{\vee })=1$ with $h_i^{\vee }$ the dual Coxeter numbers of the ${\mathfrak{g}}_i$. Our first main result is that the Heisenberg cosets ${\mathcal{C}}^k({\mathfrak{g}}_1)$ and ${\mathcal{C}}^{\ell }({\mathfrak{g}}_2)$ of these W-algebras at these dual levels are isomorphic, i.e. ${\mathcal{C}}^k({\mathfrak{g}}_1)\simeq {\mathcal{C}}^{\ell }({\mathfrak{g}}_2)$ 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 $V_{\mathbb{Z}}$ is the principal W-superalgebra at the dual level $\ell$. Conversely a diagonal Heisenberg coset of the principal W-superalgebra at level $\ell$ times the lattice vertex superalgebra $V_{\sqrt{-1}\mathbb{Z}}$ 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 $\ell$ are rational and/or C_2-cofinite if the same is true for the simple subregular W-algebra at dual level $\ell$. This gives many new C_2-cofiniteness and rationality results.

arXiv comment42 pages, accepted in Advances in Mathematics

References in corpus (48)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Vertex Algebras and Algebraic Curves
• Quantum Reduction for Affine Superalgebras
• TFT construction of RCFT correlators III: Simple currents
• W Symmetry in six dimensions
• Finite vs. affine W-algebras
• C_2-cofiniteness of cyclic-orbifold models
• Equivalence between Chain Categories of Representations of Affine sl(2) and N=2 Superconformal Algebras
• W-algebras as coset vertex algebras
• Tensor categories for vertex operator superalgebra extensions
• Schur-Weyl Duality for Heisenberg Cosets
• Rationality of W-algebras: principal nilpotent cases
• Coset Constructions of Logarithmic (1,p)-Models
• $W_n^{(2)}$ algebras
• Cosets of affine vertex algebras inside larger structures
• Cohomological Hall algebras, vertex algebras and instantons
• Simple current extensions beyond semi-simplicity
• Regularity of fixed-point vertex operator subalgebras
• Plane partitions with a "pit": generating functions and representation theory
• Trialities of $\mathcal{W}$-algebras
• Free Field Realizations from the Higgs Branch
• The GL(1|1)-symplectic fermion correspondence
• Rationality of Bershadsky-Polyakov vertex algebras
• Free Field Realizations of Affine Current Superalgebras, Screening Currents and Primary Fields
• $\mathcal{W}$-algebra Modules, Free Fields, and Gukov-Witten Defects
• Screening operators for W-algebras
• On fusion rules and intertwining operators for the Weyl vertex algebra
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• Rationality, Regularity, and C_2-cofiniteness
• Braided Tensor Categories related to ${\mathcal{B}}_p$ Vertex Algebras
• Affine sl(2|1) and D(2|1;alpha) as Vertex Operator Extensions of Dual Affine sl(2) Algebras
• Orbifolds of symplectic fermion algebras
• Rationality and Fusion Rules of Exceptional W-Algebras
• Cosets of Bershadsky-Polyakov algebras and rational $\mathcal{W}$-algebras of type $A$
• Free Field Representations of Extended Superconformal Algebras
• Cosets of the ${\mathcal{W}}^k({\mathfrak{sl}}_4,f_{\text{subreg}})$-algebra
• Correspondences among CFTs with different W-algebra symmetry
• Extended Superconformal Algebras and Free Field Realizations from Hamiltonian Reduction
• $N=4$ superconformal algebras and diagonal cosets
• A commutant realization of W^(2)_n at critical level
• VOA[M4]
• Equivalences between weight modules via $\mathcal{N}=2$ coset constructions
• Kazama-Suzuki coset construction and its inverse
• Strong generators of the subregular W-algebra and combinatorial description at critical level
• Representations of N=2 superconformal vertex algebra
• Fermionic Coset, Critical Level W^(2)_4-Algebra and Higher Spins
• On resolution of highest weight modules over the $\mathcal{N}=2$ superconformal algebra
• Ito's conjecture and the coset construction for ${\mathcal{W}}^k(\mathfrak{sl}(3|2))$

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• Trialities of orthosymplectic $\mathcal{W}$-algebras
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• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
• Correspondences among CFTs with different W-algebra symmetry
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• Subregular W-algebras of type A
• 3d Mirror Symmetry and the $\beta \gamma$ VOA
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• Generalized parafermions of orthogonal type
• Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator
• On Miura maps for W-superalgebras
• Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$
• FZZ-triality and large $\mathcal{N}=4$ super Liouville theory
• Cosets from equivariant W-algebras
• Orthosymplectic Feigin-Semikhatov duality
• Relaxed and logarithmic modules of $\widehat{{\mathfrak{sl}}_3}$
• Vertex Algebras and Commutative Algebras
• Feigin-Semikhatov conjecture and related topics
• Correlator correspondences for subregular $\mathcal{W}$-algebras and principal $\mathcal{W}$-superalgebras
• A duality between vertex superalgebras $L_{-3/2}(\mathfrak{osp}(1|2))$ and ${\mathcal{V}}^{(2)}$ and generalizations to logarithmic vertex algebras
• Duality via convolution of W-algebras