The hypergeometric functions of the Faber–Zagier and Pixton relations

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.

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Published 15 February 2017