Quantum D-modules and equivariant Floer theory for free loop spaces

Details Quantum D-modules and equivariant Floer theory for free loop spaces Quantum D-modules and equivariant Floer theory for free loop spaces

2004-10-22

arxiv.org/abs/math/0410487v3

Math. Z. 252 (3) (2006), pp.577--622

Mathematics

Differential Geometry

37 pages, the original version was written in June,2003, v2 added e-mail address, v3 final version

Scientific paper

10.1007/s00209-005-0867-9

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 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 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, it follows that equivariant Floer cohomology defined here is isomorphic to the quantum cohomology D-module.

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