A new cohomology class on the moduli space of curves

We define a collection Θg,n∈ H⁴ᵍ⁻⁴⁺²ⁿ( M̄g,n,Q) for $2g-2+n>0$ of cohomology classes that restrict naturally to boundary divisors. We prove that the intersection numbers ∫_ M̄g,nΘg,n∏ᵢ₌₁ⁿψᵢmᵢ 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 ∫_ M̄g,n∏ᵢ₌₁ⁿψᵢmᵢ is a tau function of the KdV hierarchy.