A new cohomology class on the moduli space of curves

Abstract

We define a collection ${\Theta }_{g,n}\in H^{4g-4+2n}({\overline{\mathcal{M}}}_{g,n},\mathbb{Q})$ for $2g-2+n>0$ of cohomology classes that restrict naturally to boundary divisors. We prove that the intersection numbers ${\int }_{{\overline{\mathcal{M}}}_{g,n}}{\Theta }_{g,n}{\prod }_{i=1}^n{\psi }_i^{m_i}$ 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 ${\int }_{{\overline{\mathcal{M}}}_{g,n}}{\prod }_{i=1}^n{\psi }_i^{m_i}$ is a tau function of the KdV hierarchy.

53 pages, added rigidity result which uniquely determines initial value of classes, KdV relation is now a conjecture true up to genus 7

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• The factorial growth of topological recursion
• On tau-functions for the KdV hierarchy
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• Asymptotics of Weil-Petersson volumes and two-dimensional quantum gravities