Fixed-point Localization for $\mathbb{RP}^{2m} \subset \mathbb{CP}^{2m}$

Abstract

We derive a fixed-point formula for integrals on moduli spaces of stable maps to projective spaces of even dimension. This gives a formula for the equivariant open Gromov-Witten invariants of (RP^{2m},CP^{2m}) and the structure constants of the equivariant Fukaya A-infinity algebra of RP^2m in CP^{2m} with bulk deformations. The formula involves contributions from Givental's correlators for the closed theory and the descendent integrals of discs, and specializes to give a new expression for the Welschinger count of real rational curves in the plane passing through some real and conjugation invariant pairs of points.