Equivariant Open Gromov-Witten Theory of $\mathbb{R}\mathbb{P}^{2m} \hookrightarrow \mathbb{C}\mathbb{P}^{2m}$

Abstract

We define equivariant open Gromov-Witten invariants for $\mathbb{R}\mathbb{P}^{2m} \hookrightarrow \mathbb{C}\mathbb{P}^{2m}$ as sums of integrals of equivariant forms over resolution spaces, which are blowups of products of moduli spaces of stable disc-maps modeled on trees. These invariants encode the quantum deformation of the equivariant cohomology of $\mathbb{R}\mathbb{P}^{2m}$ by holomorphic discs in $\mathbb{C}\mathbb{P}^{2m}$ and, for $m=1$, specialize to give Welschinger's signed count of real rational planar curves in the non-equivariant limit.