A new cohomology class on the moduli space of curves

Norbury, Paul¹

¹ School of Mathematics and Statistics, University of Melbourne, Melbourne VIC, Australia

Geometry & topology, Tome 27 (2023) no. 7, pp. 2695-2761.

Résumé

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.

Détail

DOI : 10.2140/gt.2023.27.2695

Keywords: moduli space, cohomology, spin structure

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