A new cohomology class on the moduli space of curves

Norbury, Paul

doi:10.2140/gt.2023.27.2695

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A new cohomology class on the moduli space of curves

Paul Norbury

Geometry & Topology 27 (2023) 2695–2761

DOI: 10.2140/gt.2023.27.2695

Abstract

We define a collection $Θ_{g, n} \in H^{4 g - 4 + 2 n} ({\bar{ℳ}}_{g, n}, ℚ)$ for $2 g - 2 + n > 0$ of cohomology classes that restrict naturally to boundary divisors. We prove that the intersection numbers $\int_{{\bar{ℳ}}_{g, n}} Θ_{g, n} \prod_{i = 1}^{n} ψ_{i}^{m_{i}}$ 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 $\int_{{\bar{ℳ}}_{g, n}} \prod_{i = 1}^{n} ψ_{i}^{m_{i}}$ is a tau function of the KdV hierarchy.

Keywords

moduli space, cohomology, spin structure

Mathematical Subject Classification

Primary: 14D23, 32G15, 53D45

References

Publication

Received: 2 April 2020

Revised: 22 September 2021

Accepted: 16 October 2021

Published: 19 September 2023

Proposed: Haynes R Miller

Seconded: Jim Bryan, Mark Gross

Authors

Paul Norbury
School of Mathematics and Statistics University of Melbourne Melbourne VIC Australia

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Volume 27, issue 7 (2023)