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Whittaker vectors for $\mathcal{W}$-algebras from topological recursion

Gaëtan Borot, Vincent Bouchard, Nitin Kumar Chidambaram, Thomas Creutzig

Abstract

We identify Whittaker vectors for $\mathcal{W}_k(\mathfrak{g})$-modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable compactification of the moduli space of $G$-bundles over $\mathbb{P}^2$ for $G$ a complex simple Lie group, can be computed by a non-commutative version of the Chekhov-Eynard-Orantin topological recursion. We formulate the connection to higher Airy structures for Gaiotto vectors of type A, B, C, and D, and explicitly construct the topological recursion for type A (at arbitrary level) and type B (at self-dual level). On the physics side, it means that the Nekrasov partition function for pure $\mathcal{N} = 2$ four-dimensional supersymmetric gauge theories can be accessed by topological recursion methods.

Whittaker vectors for $\mathcal{W}$-algebras from topological recursion

Abstract

We identify Whittaker vectors for -modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable compactification of the moduli space of -bundles over for a complex simple Lie group, can be computed by a non-commutative version of the Chekhov-Eynard-Orantin topological recursion. We formulate the connection to higher Airy structures for Gaiotto vectors of type A, B, C, and D, and explicitly construct the topological recursion for type A (at arbitrary level) and type B (at self-dual level). On the physics side, it means that the Nekrasov partition function for pure four-dimensional supersymmetric gauge theories can be accessed by topological recursion methods.

Paper Structure

This paper contains 55 sections, 35 theorems, 290 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results and outline
  4. Notational conventions
  5. Preliminaries
  6. Instanton moduli spaces and W-algebras
  7. Uhlenbeck spaces and intersection cohomology
  8. W-algebra action
  9. Instanton partition function
  10. Airy structures and topological recursion
  11. Airy structures
  12. Topological recursion
  13. Whittaker modules
  14. Background on vertex algebras
  15. Vertex algebras
  16. ...and 40 more sections

Key Result

Theorem 2.1

The space $M_{\mathbb{F}_G}(Q)$ can be equipped with the structure of a $\mathcal{W}^{\mathsf{k}}(\mathfrak g)$-module with shifted level $\kappa = - \frac{\epsilon_2}{\epsilon_1}$, satisfying:

Figures (1)

  • Figure 1: A two-level graph with $b_1 = 5$.

Theorems & Definitions (108)

  • Theorem 2.1: Braverman:2014xca
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6: KS
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • ...and 98 more