Quantum D-modules and Equivariant Floer Theory for Free Loop Spaces
نویسنده
چکیده
The objective of this paper is to clarify the relationships between quantum Dmodule and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper “Homological Geometry”. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohomology for the universal cover of the free loop space. First, motivated by the work of Guest, we formulate the abstract notion of quantum D-module which generalizes the D-module defined by the small quantum cohomology algebra. By using this framework, we show that quantum D-module completely determines the small quantum products. Second, we define the equivariant Floer cohomology of toric complete intersections explicitly, using Givental’s model. This is shown to satisfy the axioms of abstract quantum D-module. Then we prove that quantum D-module and equivariant Floer cohomology are isomorphic for nef toric complete intersections. Mathematics Subject Classification 2000. Primary 53D45; Secondary 14N35, 53D40.
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