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

Abstract

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.

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

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• Quantum periods - I. Semi-infinite variations of Hodge structures
• Quantum D-modules and generalized mirror transformations
• Mirror Principle II
• Quantum Cohomology and Morse Theory on the Loop Space of Toric Varieties
• Quantum cohomology via D-modules
• Introduction to homological geometry: part I

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• Lagrangian Floer theory on compact toric manifolds I
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• Loop structure on equivariant $K$-theory of semi-infinite flag manifolds
• Orbifold quantum D-modules associated to weighted projective spaces
• Vortex invariants and toric manifolds
• The Quantum Orbifold Cohomology of Weighted Projective Spaces
• Logarithmic Frobenius manifolds, hypergeometric systems and quantum D-modules
• A Fock Sheaf For Givental Quantization
• Reconstructing GKZ via topological recursion