Hebrew University Topology and Geometry Seminar

March 23, 2016
Ross building, Seminar Room 70A

Amitai Zernik

Hebrew University

Fixed-point Expressions for Open Gromov-Witten Invariants - overview and $A_{\infty}$ perspective

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.