Quantum Cohomology and the Seidel Representation

We define the quantum cohomology ring $QH^*(X)$ of a symplectic manifold, which is a deformation of the ordinary cohomology ring $H^*(X)$ by "higher-order terms", or more concretely, using Gromov-Witten invariants.

The Seidel representation is a map $\pi_0(\widetilde G) \to QH^*(X)$, where $\widetilde G$ is a covering space of the free loop space on $Ham(X)$. To define this, we will also define Hamiltonian Floer (co)homology, and study $S^1$-actions on symplectic manifolds.

To conclude, we present two applications of the theory. The first is using the Seidel representation to find elements in $\pi_1(Ham(X))$ of infinite order. The second is to use $S^1$-actions and the Seidel representation to compute the quantum cohomology ring of toric manifolds.