Logarithmic series and Hodge integrals in the tautological ring (with an appendix by D. Zagier)

Abstract

We describe a new perspective on the intersection theory of the moduli space of curves involving both Virasoro constraints and Gorenstein conditions. The main result of the paper is the computation of a basic 1-point Hodge integral series occurring in the tautological ring of the moduli space of nonsingular curves. In the appendix by D. Zagier, "Polynomials arising from the tautological ring", a detailed study is made of certain polynomials whose coefficients are intersection numbers on moduli space. The paper and the appendix together provide proofs of all previously conjectured formulas for 1-point integrals in the tautological ring, and of natural extensions of these formulas as well.