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

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 ~ denote the parameter. Associated to them is the quantum D module 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 ~ → 0. Givental [10], discovered that there should be a structure of a D module on the (as yet not rigorously defined) S 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 D module in terms of Morse theoretic data for the symplectic action functional. The conjecture is proven in the case of toric manifolds with ∫ d c1 > 0 for all nonzero classes d of rational curves in M .