A new cohomology class on the moduli space of curves

Abstract

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