Spin refinement of moduli spaces of residueless meromorphic differentials and the BKP hierarchy

TL;DR

This paper links the geometry of moduli spaces of residueless meromorphic differentials with spin structures to integrable systems, showing that a certain hierarchy of PDEs aligns with the BKP hierarchy after specific reductions and transformations.

Contribution

It introduces a spin refinement of the cohomological field theory related to residueless meromorphic differentials and connects it to the BKP hierarchy through the DR hierarchy construction.

Findings

The partial CohFT of these strata forms a rich algebraic structure.

The reduced PDE system matches the BKP hierarchy after a coordinate change.

A new result on reconstructing the BKP hierarchy from Lax formalism is established.

Abstract

We consider strata of curves carrying a residueless meromorphic differential inducing a spin structure on the curve. The cohomology classes of the closures of these strata, weighted by the parity of the spin structures, form a partial cohomological field theory (CohFT) of infinite rank. After applying the DR hierarchy construction to this partial CohFT and reducing to differentials with two zeros and arbitrarily many poles, we show that the resulting system of evolutionary PDEs coincides with the BKP hierarchy up to a coordinate transformation. This is a spin refinement of an analogous result from arXiv:2110.01419. Our proof relies on a new result regarding the reconstruction of the BKP hierarchy from a limited amount of information in the Lax formalism.