Quadratic double ramification integrals and the noncommutative KdV hierarchy

TL;DR

This paper computes intersection numbers of double ramification cycles on moduli spaces of curves and shows their connection to a noncommutative KdV hierarchy, revealing new links between algebraic geometry and integrable systems.

Contribution

It introduces quadratic double ramification integrals and demonstrates their role in defining a noncommutative KdV hierarchy related to cohomological field theories.

Findings

Computed intersection numbers of double ramification cycles.

Established the equivalence of the hierarchy with a noncommutative KdV system.

Connected algebraic geometry with noncommutative integrable systems.

Abstract

In this paper we compute the intersection number of two double ramification cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic double ramification integrals are the main ingredient for the computation of the double ramification hierarchy associated to the infinite dimensional partial cohomological field theory given by $\exp(\mu^2 \Theta)$ where $\mu$ is a parameter and $\Theta$ is Hain's theta class, appearing in Hain's formula for the double ramification cycle on the moduli space of curves of compact type. This infinite rank double ramification hierarchy can be seen as a rank $1$ integrable system in two space and one time dimensions. We prove that it coincides with a natural analogue of the KdV hierarchy on a noncommutative Moyal torus.