A new cohomology class on the moduli space of curves

Abstract

We define a collection of cohomology classes $\Theta_{g,n}\in H^{4g-4+2n}(\overline{\cal M}_{g,n})$ for $2g-2+n>0$ that restrict naturally to boundary divisors. We prove that a generating function for the intersection numbers $\int_{\overline{\cal M}_{g,n}}\Theta_{g,n}\prod_{i=1}^n\psi_i^{m_i}$ is a tau function of the KdV hierarchy. This is analogous to the theorem conjectured by Witten and proven by Kontsevich that a generating function for the intersection numbers $\int_{\overline{\cal M}_{g,n}}\prod_{i=1}^n\psi_i^{m_i}$ is a tau function of the KdV hierarchy.