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

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.