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Representation theory of W-algebras, II: Ramond twisted representations

Tomoyuki Arakawa

Abstract

We study the Ramond twisted representations of the affine W-algebra W^k(g,f) in the case that f admits a good even grading. We establish the vanishing and the almost irreducibility of the corresponding BRST cohomology. This confirms some of the recent conjectures of Kac and Wakimoto. In type A, our results give the characters of all irreducible ordinary Ramond twisted representations of W^k(sl_n,f) for all nilpotent elements f and all non-critical k, and prove the existence of modular invariant representations conjectured by Kac and Wakimoto.

Representation theory of W-algebras, II: Ramond twisted representations

Abstract

We study the Ramond twisted representations of the affine W-algebra W^k(g,f) in the case that f admits a good even grading. We establish the vanishing and the almost irreducibility of the corresponding BRST cohomology. This confirms some of the recent conjectures of Kac and Wakimoto. In type A, our results give the characters of all irreducible ordinary Ramond twisted representations of W^k(sl_n,f) for all nilpotent elements f and all non-critical k, and prove the existence of modular invariant representations conjectured by Kac and Wakimoto.

Paper Structure

This paper contains 34 sections, 28 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Vertex Algebras and their Twisted Representations
  3. Fields
  4. Vertex Algebras
  5. Twisted Representations of Vertex Algebras
  6. $H$-Twisted Zhu Algebras
  7. Affine $W$-algebras
  8. The Setting
  9. Root Data
  10. Kac-Moody Lie Algebras
  11. Universal Affine Vertex Algebras
  12. Clifford Vertex Algebras
  13. The $W$-Algebra $\mathscr{W}^k(\bar{\mathfrak{g}}, f)$
  14. The Hamiltonian of $\mathscr{W}^k(\bar{\mathfrak{g}}, f)$
  15. Generating Fields of $\mathscr{W}^k(\bar{\mathfrak{g}}, f)$
  16. ...and 19 more sections

Key Result

Theorem 2.4.1

The map $M\mapsto M_{{\mathrm{top}}}$ gives a bijective correspondence between simple objects of $V \text{-}\mathfrak{Mod}_{\sigma_H}$ and irreducible $\mathrm{Z}\mathrm{h}_{H}V$-modules.

Theorems & Definitions (43)

  • Theorem 2.4.1: Zhu96De-Kac06
  • Lemma 3.3.1
  • Definition 3.6.1
  • Lemma 3.7.1
  • proof
  • Theorem 3.8.1: KacWak04
  • Remark 3.8.2
  • Remark 4.1.1
  • Proposition 4.1.2
  • proof
  • ...and 33 more