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The hypergeometric functions of the Faber-Zagier and Pixton relations

A. Buryak, F. Janda, R. Pandharipande

Abstract

The relations in the tautological ring of the moduli space M_g of nonsingular curves conjectured by Faber-Zagier in 2000 and extended to the moduli space of stable pointed curves by Pixton in 2012 are based upon two hypergeometric series A and B. The question of the geometric origins of these series has been solved in at least two ways (via the Frobenius structures associated to 3-spin curves and to CP1). The series A and B also appear in the study of descendent integration on the moduli spaces of open and closed curves. We survey here the various occurrences of A and B starting from their appearance in the asymptotic expansion of the Airy function (calculated by Stokes in the 19th century). Several open questions are proposed.

The hypergeometric functions of the Faber-Zagier and Pixton relations

Abstract

The relations in the tautological ring of the moduli space M_g of nonsingular curves conjectured by Faber-Zagier in 2000 and extended to the moduli space of stable pointed curves by Pixton in 2012 are based upon two hypergeometric series A and B. The question of the geometric origins of these series has been solved in at least two ways (via the Frobenius structures associated to 3-spin curves and to CP1). The series A and B also appear in the study of descendent integration on the moduli spaces of open and closed curves. We survey here the various occurrences of A and B starting from their appearance in the asymptotic expansion of the Airy function (calculated by Stokes in the 19th century). Several open questions are proposed.

Paper Structure

This paper contains 42 sections, 4 theorems, 174 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Tautological classes
  3. Relations
  4. Differential equations
  5. Origins of ${\mathsf{A}}$ and ${\mathsf{B}}$
  6. Acknowledgements
  7. Asymptotic expansion of the Airy function
  8. Moduli of stable curves and the infinite Grassmanian
  9. Overview
  10. Witten's conjecture
  11. Kontsevich's matrix integral
  12. Matrix integrals
  13. Determinantal formulas
  14. Case $N=1$
  15. The Infinite Grassmanian and tau-functions of the KP hierarchy
  16. ...and 27 more sections

Key Result

Theorem 1

In $R^r({\mathcal{M}}_g)$, the Faber-Zagier relation holds when $g-1+|\sigma|< 3r$ and $g\equiv r+|\sigma|+1 \mod 2$.

Figures (2)

  • Figure 1: Shift of the integration contour
  • Figure 2: Lefschetz thimble for $i(z^3 + 3z)$ through critical point $i$.

Theorems & Definitions (4)

  • Theorem 1: Pandharipande-Pixton
  • Theorem 2: Buryak
  • Theorem 3: Janda
  • Theorem 4: Janda