Pure and Applied Mathematics Quarterly

Volume 11 (2015)

Number 4

Special Issue: In Honor of Eduard Looijenga, Part 1 of 3

Guest Editor: Gerard van der Geer

The hypergeometric functions of the Faber–Zagier and Pixton relations

Pages: 591 – 631

DOI: http://dx.doi.org/10.4310/PAMQ.2015.v11.n4.a3

Authors

A. Buryak (Department of Mathematics, ETH Zürich, Switzerland)

F. Janda (Department of Mathematics, ETH Zürich, Switzerland)

R. Pandharipande (Department of Mathematics, ETH Zürich, Switzerland)

Abstract

The relations in the tautological ring of the moduli space $\mathcal{M}_g$ of nonsingular curves conjectured by Faber–Zagier in 2000 and extended to the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves by Pixton in 2012 are based upon two hypergeometric series $\mathsf{A}$ and $\mathsf{B}$. 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 $\mathbb{P}^1$). The series $\mathsf{A}$ and $\mathsf{B}$ also appear in the study of descendent integration on the moduli spaces of open and closed curves. We survey here the various occurrences of $\mathsf{A}$ and $\mathsf{B}$ starting from their appearance in the asymptotic expansion of the Airy function (calculated by Stokes in the 19th century). Several open questions are proposed.

Full Text (PDF format)