The spin Gromov–Witten/Hurwitz correspondence for ${\mathbb{P}}^1$

Abstract

We study the spin Gromov–Witten 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 Gromov–Witten/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 Gromov–Witten invariants that holds in virtual dimension zero.