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Orthosymplectic Feigin-Semikhatov duality

Justine Fasquel, Shigenori Nakatsuka

TL;DR

The paper develops an orthosymplectic analogue of Feigin–Semikhatov duality, relating the subregular $\\mathcal{W}^k(\\mathfrak{so}_{2n+1},f_{sub})$ and the principal $\\mathcal{W}^\\ell(\\mathfrak{osp}_{2|2n})$ via inverse Hamiltonian reduction and coset constructions. It proves blockwise equivalences of weight-module categories using relative semi-infinite cohomology and spectral flow twists, and establishes a Wakimoto free-field correspondence between the two sides. In the exceptional rational case, it classifies simple modules and derives their characters, showing consistency with Feigin–Frenkel duality and yielding modular tensor category structures in favorable settings. The work connects dualities, free-field realizations, and BRST reductions to provide a coherent framework for the representation theory of orthosymplectic W-algebras, with potential applications to fusion rules and quantum geometric Langlands. Overall, the results illuminate the structural parallels between W-algebras of type B and W-superalgebras of orthosymplectic type, revealing a rich interaction between cohomological, categorical, and free-field methods in vertex algebra theory.

Abstract

We study the representation theory of the subregular W-algebra $\mathcal{W}^k(\mathfrak{so}_{2n+1},f_{sub})$ of type B and the principal W-superalgebra $\mathcal{W}^\ell(\mathfrak{osp}_{2|2n})$, which are related by an orthosymplectic analogue of Feigin-Semikhatov duality in type A. We establish a block-wise equivalence of weight modules over the W-superalgebras by using the relative semi-infinite cohomology functor and spectral flow twists, which generalizes the result of Feigin-Semikhatov-Tipunin for the N=2 superconformal algebra. In particular, the correspondence of Wakimoto type free field representations is obtained. When the level of the subregular W-algebra is exceptional, we classify the simple modules over the simple quotients $\mathcal{W}_k(\mathfrak{so}_{2n+1},f_{sub})$ and $\mathcal{W}_\ell(\mathfrak{osp}_{2|2n})$ and derive the character formulae.

Orthosymplectic Feigin-Semikhatov duality

TL;DR

The paper develops an orthosymplectic analogue of Feigin–Semikhatov duality, relating the subregular and the principal via inverse Hamiltonian reduction and coset constructions. It proves blockwise equivalences of weight-module categories using relative semi-infinite cohomology and spectral flow twists, and establishes a Wakimoto free-field correspondence between the two sides. In the exceptional rational case, it classifies simple modules and derives their characters, showing consistency with Feigin–Frenkel duality and yielding modular tensor category structures in favorable settings. The work connects dualities, free-field realizations, and BRST reductions to provide a coherent framework for the representation theory of orthosymplectic W-algebras, with potential applications to fusion rules and quantum geometric Langlands. Overall, the results illuminate the structural parallels between W-algebras of type B and W-superalgebras of orthosymplectic type, revealing a rich interaction between cohomological, categorical, and free-field methods in vertex algebra theory.

Abstract

We study the representation theory of the subregular W-algebra of type B and the principal W-superalgebra , which are related by an orthosymplectic analogue of Feigin-Semikhatov duality in type A. We establish a block-wise equivalence of weight modules over the W-superalgebras by using the relative semi-infinite cohomology functor and spectral flow twists, which generalizes the result of Feigin-Semikhatov-Tipunin for the N=2 superconformal algebra. In particular, the correspondence of Wakimoto type free field representations is obtained. When the level of the subregular W-algebra is exceptional, we classify the simple modules over the simple quotients and and derive the character formulae.
Paper Structure (37 sections, 42 theorems, 286 equations, 6 figures, 1 table)

This paper contains 37 sections, 42 theorems, 286 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Results
  3. Outlook
  4. Plan of the paper
  5. Feigin--Semikhatov duality of orthosymplectic type
  6. $\mathcal{W}$-superalgebras
  7. Duality
  8. Some special levels
  9. Case (i): $\mathscr{C}_{D^+}^k$ degenerates.
  10. Case (ii): $\pi^{J^+}$ degenerates.
  11. Feigin--Frenkel duality
  12. Inverse Hamiltonian reduction
  13. Feigin--Frenkel duality
  14. Spectral flow twists
  15. Spectral flow twist
  16. ...and 22 more sections

Key Result

Theorem 2.1

For the levels $(k,\ell)$ satisfying level condition for FS, we have the following isomorphisms of vertex superalgebras: where $\pi^{J^\pm_\Delta}$ denote the Heisenberg vertex algebras generated by the fields Moreover, the isomorphisms FST and KS still hold by replacing $\mathcal{W}^k_{D^+}(n,1)$ and $\mathcal{W}^{\ell}_{D^-}(n,1)$ by their simple quotients $\mathcal{W}_k^{D^+}(n,1)$ and $\math

Figures (6)

  • Figure 1: $\mathfrak{g}=\mathfrak{so}_{2n+1}$
  • Figure 2: $\mathfrak{g}=\mathfrak{osp}_{2|2n}$
  • Figure 3:
  • Figure 4:
  • Figure 5: $\tilde{B}_n$
  • ...and 1 more figures

Theorems & Definitions (56)

  • Theorem 2.1: CGN
  • Corollary 2.2
  • Proposition 2.3
  • Conjecture 2.4
  • Lemma 3.1
  • Theorem 3.2
  • Example 3.3: cf. BM21
  • Lemma 3.4
  • Corollary 3.5
  • Proposition 4.1: Li2
  • ...and 46 more