The Spin Gromov-Witten/Hurwitz correspondence for $\mathbb{P}^1$

TL;DR

This paper establishes a spin Gromov-Witten/Hurwitz correspondence for $ ext{P}^1$, demonstrating its connection to integrable hierarchies and confirming conjectures about spin invariants and their degeneration formulas.

Contribution

It proves the spin GW/Hurwitz correspondence for $ ext{P}^1$ and links it to the 2-BKP hierarchy, extending known results to the spin setting.

Findings

Equivariant potential expressed via operator formalism

Spin GW/Hurwitz correspondence proven for $ ext{P}^1$

Connection to conjectural degeneration formula for spin curves

Abstract

We study the spin Gromov-Witten (GW) theory of $\mathbb{P}^1$. Using the standard torus action on $\mathbb{P}^1$, we prove that the associated equivariant potential can be expressed by means of operator formalism and satisfies the 2-BKP hierarchy. As a consequence of this result, we prove the spin analogue of the GW/Hurwitz correspondence of Okounkov-Pandharipande for $\mathbb{P}^1$, which was conjectured by J. Lee. Finally, we prove that this correspondence for a general target spin curve follows from a conjectural degeneration formula for spin GW invariants that holds in virtual dimension 0.