Feigin-Semikhatov conjecture and related topics

Abstract

Feigin-Semikhatov conjecture, now established, states algebraic isomorphisms between the cosets of the subregular $\mathcal{W}$-algebras and the principal $\mathcal{W}$-superalgebras of type A by their full Heisenberg subalgebras. It can be seen as a variant of Feigin-Frenkel duality between the ${\mathcal{W}}_n$-algebras and also as a generalization of the connection between the $\mathcal{N}=2$ superconformal algebra and the affine algebra ${\widehat{\mathfrak{sl}}}_{2,k}$.We review the recent developments on the correspondence of the subregular W-algebras and the principal W-superalgebras of type A at the level of algebras, modules and intertwining operators, including fusion rules.

arXiv commentThis is a review paper for Int. J. Mod. Phys. A. 25 pages. Comments welcome

References in corpus (31)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Quantum Reduction for Affine Superalgebras
• W Symmetry in six dimensions
• Finite vs. affine W-algebras
• The affine Yangian of ${\mathfrak{gl}}_1$ revisited
• Equivalence between Chain Categories of Representations of Affine sl(2) and N=2 Superconformal Algebras
• W-algebras as coset vertex algebras
• Rationality of W-algebras: principal nilpotent cases
• $W_n^{(2)}$ algebras
• Cosets of affine vertex algebras inside larger structures
• Cohomological Hall algebras, vertex algebras and instantons
• Webs of W-algebras
• Plane partitions with a "pit": generating functions and representation theory
• Resolutions and Characters of Irreducible Representations of the N=2 Superconformal Algebra
• Trialities of $\mathcal{W}$-algebras
• Relaxed highest-weight modules I: rank $1$ cases
• Duality of subregular W-algebras and principal W-superalgebras
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• The supersymmetric affine Yangian
• Twin-plane-partitions and $\mathcal{N}=2$ affine Yangian
• Universal two-parameter ${\mathcal{W}}_{\infty }$-algebra and vertex algebras of type $\mathcal{W}(2,3,\dots ,N)$
• Braided Tensor Categories related to ${\mathcal{B}}_p$ Vertex Algebras
• A realisation of the Bershadsky--Polyakov algebras and their relaxed modules
• Quantum Langlands duality of representations of W-algebras
• Affine Super Yangians and Rectangular $W$-superalgebras
• Level-rank duality via tensor categories
• Cosets of Bershadsky-Polyakov algebras and rational $\mathcal{W}$-algebras of type $A$
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• Subregular W-algebras of type A
• Infinitesimal Cherednik algebras as W-algebras