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# Algebra/Topology seminar

Amitai Zernik (IAS)

Title: From Massey products to a fixed-point formula for open Gromov-Witten Theory

Abstract: For any topological space L, there's an A_\infty algebra extending the cup product, which is closely related to the Massey products. We'll see that if L is a closed smooth manifold this A_\infty algebra can be represented by a sum over trees where the edges carry a certain ``heat propagator''. If L is equipped with an S^1-action and has cohomology concentrated in even degrees, this construction admits an equivariant extension, defined over the equivariant cohomology of a point.

Now assume, in addition, that L is equipped with a Lagrangian embedding into a symplectic manifold X. Then we can also introduce a ``quantum'' deformation of the A_\infty algebra, coming from (pseudo-)holomorphic disks inside X whose boundary

lie on L. If X is equipped with a compatible S^1 action, this also admits an equivariant extension.

For the special case where L=RP^{2m} inside X=CP^{2m}, it turns out the combined equivariant and quantum deformation A_\infty algebra can be computed explicitly, by a fixed point formula.

For motivation, you can see this formula in action (without having to install anything) by clicking ``play'' here.

I'll try to make the talk as self-contained as possible. In particular, I'll assume no previous experience with symplectic geometry.