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Quantum $$D$$-modules and equivariant Floer theory for free loop spaces. (English) Zbl 1121.53062
Summary: The objective of this paper is to clarify the relationships between the quantum $$D$$-module and equivariant Floer theory. Equivariant Floer theory was introduced by A. Givental in his paper “Homological Geometry” [Sel. Math., New Ser. 1, No. 2, 325–345 (1995; Zbl 0920.14028)]. 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 notion of abstract quantum $$D$$-module which generalizes the $$D$$-module defined by the small quantum cohomology algebra. Second, we define the equivariant Floer cohomology of toric complete intersections rigorously as a $$D$$-module, using Givental’s model. This is shown to satisfy the axioms of abstract quantum $$D$$-module. By Givental’s mirror theorem A. Givental [A mirror theorem for toric complete intersections. Prog. Math. 160, 141–175 (1998; Zbl 0936.14031)], it follows that the equivariant Floer cohomology defined here is isomorphic to the quantum cohomology $$D$$-module.

##### MSC:
 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 53D40 Symplectic aspects of Floer homology and cohomology
##### Keywords:
symplectic manifold
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