Quantum Cohomology and Morse Theory on the Loop Space of Toric Varieties

Abstract

On a symplectic manifold $M$, the quantum product defines a complex, one parameter family of flat connections called the A-model or Dubrovin connections. Let $\hbar$ denote the parameter. Associated to them is the quantum $\mathcal{D}$ - module ${\mathcal{D}}/I$ over the Heisenberg algebra of first order differential operators on a complex torus. An element of $I$ gives a relation in the quantum cohomology of $M$ by taking the limit as $\hbar\to 0$. Givental (HomGeom), discovered that there should be a structure of a $\mathcal{D}$ - module on the (as yet not rigorously defined) ${S^1}$ equivariant Floer cohomology of the loop space of $M$ and conjectured that the two modules should be equal. Based on that, we formulate a conjecture about how to compute the quantum cohomology $\mathcal{D}$ - module in terms of Morse theoretic data for the symplectic action functional. The conjecture is proven in the case of toric manifolds with $\int_d{c_1}> 0$ for all nonzero classes $d$ of rational curves in $M$.