Adiabatic Fredholm Theory

TL;DR

This paper introduces a comprehensive functional analytic framework for adiabatic limits, defining adiabatic Fredholm families and their properties, with applications to Floer theory and related areas.

Contribution

It develops the concept of adiabatic Fredholm families, providing a unified approach to regularization and finite dimensional reductions in adiabatic limits.

Findings

Finite dimensional reductions inherit regularity properties.

Framework applies to Atiyah-Floer conjecture and quilted Floer theory.

Explicit construction of adiabatic Fredholm families.

Abstract

We develop a robust functional analytic framework for adiabatic limits. This framework consist of a notion of adiabatic Fredholm family, several possible regularity properties, and an explicit construction that provides finite dimensional reductions that fit into all common regularization theories. We show that thhese finite dimensional reductions inherit global continuity and differentiability properties from the adiabatic Fredholm family. Moreover, we indicate how to construct adiabatic Fredholm families that describe the adiabatic limits for the nondegenerate Atiyah-Floer conjecture and strip-shrinking in quilted Floer theory.