The hypergeometric functions of the faber-zagier and pixton relations

Abstract

The relations in the tautological ring of the moduli space Mg of nonsingular curves conjectured by Faber-Zagier in 2000 and extended to the moduli space Mg,n of stable 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 P1). 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.