Moduli spaces of residueless meromorphic differentials and the KP hierarchy

TL;DR

This paper establishes a connection between the geometry of moduli spaces of residueless meromorphic differentials and integrable systems, specifically showing that a certain Hamiltonian PDE system reduces to the KP hierarchy.

Contribution

It constructs a partial cohomological field theory from moduli spaces of residueless meromorphic differentials and links it to the KP hierarchy via the double ramification hierarchy.

Findings

The cohomology classes form a partial CohFT of infinite rank.

Application of the double ramification hierarchy yields a Hamiltonian system.

Reduction of the system matches the KP hierarchy after variable change.

Abstract

We prove that the cohomology classes of the moduli spaces of residueless meromorphic differentials, i.e., the closures, in the moduli space of stable curves, of the loci of smooth curves whose marked points are the zeros and poles of prescribed orders of a meromorphic differential with vanishing residues, form a partial cohomological field theory (CohFT) of infinite rank. To this partial CohFT we apply the double ramification hierarchy construction to produce a Hamiltonian system of evolutionary PDEs. We prove that its reduction to the case of differentials with exactly two zeros and any number of poles coincides with the KP hierarchy up to a change of variables.