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.

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.

## Date:

Wed, 23/03/2016 - 11:00 to 12:45

## Location:

Ross building, Hebrew University (Seminar Room 70A)