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

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