A new cohomology class on the moduli space of curves

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 ${\int }_{{\bar{\mathcal{M}}}_{g,n}}{\mathrm{\Theta }}_{g,n}{\prod }_{i=1}^n{\psi }_i^{m_i}$ 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.