TQFT structure of Gromov-Witten theory and the Seidel representation

Mar 2025

Summary:

First, I describe the TQFT structure of the topological sigma model with target a symplectic manifold. To a surface, the TQFT assigns gromov witten invariants. To a circle, it assigns quantum cohomology. To a point, it assigns the fukaya category. I describe the origin of quantum cohomology from the top down (by neck-stretching curves and using gromov-witten invariants) and bottom up (as the Hochschild cohomology of the Fukaya category).

Next, I discuss the Seidel representation. Given a loop $\Psi_t$ of hamiltonain diffeomorphisms, Seidel constructs a unit in the quantum cohomology ring, the “seidel element” $S(\Psi_t)$. An element of quantum cohomology is acted on by $\Psi_t$through multiplication by $S(\Psi_t)$. Seidel constructs $S(\Psi_t)$ by building a symplectic bundle over $\mathbb{P}^1$ with clutching function $\Psi_t$, then counting $J$-holomorphic sections. This is the first taste of the categorical action of Hamiltonian diffeomorphisms on the fukaya category.

Presented at:

• gauged TQFTs seminar , Spring 2025

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