Intersection numbers with Pixton's class and the noncommutative KdV hierarchy

TL;DR

This paper proves that the generating series of intersection numbers involving Pixton's class on moduli spaces solves the noncommutative KdV hierarchy, linking geometric intersection theory with integrable systems.

Contribution

It confirms a conjecture that connects Pixton's class intersection numbers with solutions to the noncommutative KdV hierarchy, advancing understanding of moduli space geometry and integrable hierarchies.

Findings

Generated series satisfy the noncommutative KdV equations

Established link between Pixton's class and integrable systems

Confirmed conjecture from previous work

Abstract

The Pixton class is a nonhomogeneous cohomology class on the moduli space of stable curves $\overline{\mathcal{M}}_{g,n}$, with nontrivial terms in degree $0,2,4,\ldots,2g$, whose top degree part coincides with the double ramification cycle. In this paper, we prove our conjecture from a previous work, claiming that the generating series of intersection numbers of the Pixton class with monomials in the psi-classes gives a solution of the noncommutative KdV hierarchy.