Abstract:
In this pair of talks I will discuss how to obtain fixed-point expressions
for open Gromov-Witten invariants. The talks will be self-contained,
and the second talk will only require a small part of the first talk,
which we will review.
The Atiyah-Bott localization formula has become a valuable tool for
computation of symplectic invariants given in terms of integrals on
the moduli spaces of closed stable maps. In contrast, the moduli spaces
of open stable maps have boundary which must be taken into account
in order to apply fixed-point localization. Homological perturbation
for twisted $A_{\infty}$ algebras allows one to write down an integral
sum which effectively eliminates the boundary. For genus zero maps
to $\left(\mathbb{CP}^{2m},\mathbb{RP}^{2m}\right)$ we'll explain how one can
define numerical equivariant invariants using this idea, and then
flow to a fixed-point limit which can be computed explicitly as a
sum over certain even-odd diagrams. These invariants specialize to
open Gromov-Witten invariants, and in particular produce new expressions
for Welschinger's signed counts of real rational plane curves. We'll
also mention the two-sided information flow with the intersection
theory of Riemann surfaces with boundary, which provides evidence
to a conjectural generalization of the localization formula to higher
genus.
This is joint work with Jake Solomon.