Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher Airy structures and topological recursion for singular spectral curves

Gaëtan Borot, Reinier Kramer, Yannik Schüler

Abstract

We give elements towards the classification of quantum Airy structures based on the $W(\mathfrak{gl}_r)$-algebras at self-dual level based on twisted modules of the Heisenberg VOA of $\mathfrak{gl}_r$ for twists by arbitrary elements of the Weyl group $\mathfrak{S}_{r}$. In particular, we construct a large class of such quantum Airy structures. We show that the system of linear ODEs forming the quantum Airy structure and determining uniquely its partition function is equivalent to a topological recursion à la Chekhov-Eynard-Orantin on singular spectral curves. In particular, our work extends the definition of the Bouchard-Eynard topological recursion (valid for smooth curves) to a large class of singular curves, and indicates impossibilities to extend naively the definition to other types of singularities. We also discuss relations to intersection theory on moduli spaces of curves, giving a general ELSV-type representation for the topological recursion amplitudes on smooth curves, and formulate precise conjectures for application in open $r$-spin intersection theory.

Higher Airy structures and topological recursion for singular spectral curves

Abstract

We give elements towards the classification of quantum Airy structures based on the -algebras at self-dual level based on twisted modules of the Heisenberg VOA of for twists by arbitrary elements of the Weyl group . In particular, we construct a large class of such quantum Airy structures. We show that the system of linear ODEs forming the quantum Airy structure and determining uniquely its partition function is equivalent to a topological recursion à la Chekhov-Eynard-Orantin on singular spectral curves. In particular, our work extends the definition of the Bouchard-Eynard topological recursion (valid for smooth curves) to a large class of singular curves, and indicates impossibilities to extend naively the definition to other types of singularities. We also discuss relations to intersection theory on moduli spaces of curves, giving a general ELSV-type representation for the topological recursion amplitudes on smooth curves, and formulate precise conjectures for application in open -spin intersection theory.

Paper Structure

This paper contains 68 sections, 59 theorems, 566 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Classification of W(glr)-Airy structures
  3. Constructing Airy structures
  4. Preliminaries on Airy structures
  5. Finite dimension
  6. Infinite dimension
  7. Preliminaries on the W(glr)-algebra and its twisted modules
  8. The Heisenberg VOA and W(glr)-algebras
  9. The mode algebra and its subalgebras
  10. Twisted modules
  11. Airy structures from twisted W(glr) modules
  12. From a twist with a transitive element
  13. A generalisation to arbitrary twists
  14. Arbitrary dilaton shifts and changes of polarisation
  15. Necessary conditions
  16. ...and 53 more sections

Key Result

Theorem 2.2

KSTR, BBCCN18 If $(H_i)_{i\in I}$ is an Airy structure on $E$, then the system of linear differential equations admits a unique solution $Z$ of the form $Z= \exp(F)$ with

Figures (1)

  • Figure 1: Left panel: contours in the $\zeta$-plane for $r = 6$. The striped regions correspond to ${\rm Re}\,x > M$. Right panel: Hankel contour in the $x$ plane.

Theorems & Definitions (169)

  • Remark 1.1
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Fateev:1988AM17
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8
  • Lemma 2.9
  • ...and 159 more