A new cohomology class on the moduli space of curves

We define a collection Theta(g,n) is an element of H4g-4 ((M) over bar (g,n), Q) for 2g - 2 + n> 0 of cohomology classes that restrict naturally to boundary divisors. We prove that the intersection numbers integral(M) over bar (g,n), Theta(g,n) Pi(n)(i=1) psi(mi)(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 integral(M) over bar (g,n), Theta(g,n) Pi(n)(i=1) psi(mi)(i) is a tau function of the KdV hierarchy.