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.
Keywords
Cite
@article{arxiv.1502.05150,
title = {The hypergeometric functions of the Faber-Zagier and Pixton relations},
author = {A. Buryak and F. Janda and R. Pandharipande},
journal= {arXiv preprint arXiv:1502.05150},
year = {2024}
}
Comments
25 pages, typos corrected