[Paper Review] Higher-rank instanton cohomology and the quilted Atiyah-Floer conjecture

David L. Duncan|arXiv (Cornell University)|Nov 21, 2013

Geometric and Algebraic Topology29 references3 citations

TL;DR

This paper establishes foundational tools for higher-rank instanton Floer homology on 3-manifolds with positive first Betti number, focusing on PU(r)-bundles for r ≥ 2. It develops a rigorous differential geometric framework for instanton Floer theory and initiates a program to prove the quilted Atiyah-Floer conjecture by constructing a chain-level isomorphism between instanton Floer cohomology and quilted Lagrangian Floer cohomology via compatible perturbations and transversality techniques.

ABSTRACT

Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups are isomorphic. We initiate a program for proving this conjecture.

Motivation & Objective

• To develop a comprehensive theory of instanton Floer cohomology for PU(r)-bundles with r ≥ 2, extending beyond the classical SU(2)/SO(3) case.
• To provide a differential geometric foundation for higher-rank gauge theory on 3-manifolds with non-trivial first Betti number.
• To initiate a program toward proving the quilted Atiyah-Floer conjecture, which posits an isomorphism between instanton Floer cohomology and quilted Lagrangian Floer cohomology.
• To establish uniform L^p and W^{k,p} estimates for holonomy maps and their derivatives, essential for transversality in both Floer theories.
• To construct compatible perturbations that simultaneously achieve transversality in both instanton and Lagrangian Floer theories, enabling a chain-level comparison.

Proposed method

• Adopt the gauge-theoretic framework of instanton Floer homology using PU(r)-bundles over 3-manifolds with positive first Betti number.
• Define instanton Floer cohomology via a chain complex generated by gauge-equivalence classes of flat connections on PU(r)-bundles.
• Use a 2+1-dimensional field theory scheme to relate symplectic invariants (Lagrangian Floer cohomology) to gauge-theoretic invariants (instanton Floer cohomology).
• Construct Hamiltonian vector fields from holonomy maps on embedded surfaces, using local trivializations and projections to Lie algebras.
• Derive uniform L^p and W^{k,p} estimates for holonomy derivatives via bounds on curvature and covariant derivatives.
• Establish compatible perturbations that preserve transversality in both instanton and quilted Lagrangian Floer theories through controlled perturbation of connection spaces.

Experimental results

Research questions

• RQ1 Can instanton Floer cohomology be rigorously defined for PU(r)-bundles with r ≥ 2, extending the classical SU(2)/SO(3) theory?
• RQ2 How can uniform L^p and W^{k,p} bounds be established for holonomy maps and their derivatives on PU(r)-bundles?
• RQ3 What conditions ensure simultaneous transversality in both instanton Floer and quilted Lagrangian Floer theories?
• RQ4 Can a chain-level isomorphism be constructed between instanton Floer cohomology and quilted Lagrangian Floer cohomology?
• RQ5 How do perturbations of the flat connection space interact with the symplectic structure on the moduli space of flat connections on a surface?

Key findings

• The paper establishes uniform L^p bounds for the holonomy map and its derivatives on PU(r)-bundles, with constants depending only on the bundle and the connection space.
• It proves W^{1,p} estimates for the Hamiltonian vector field associated with a holonomy function, showing that its norm is bounded by the L^p norm of the curvature.
• The derivative of the holonomy map is controlled by the L^p norm of the curvature, with explicit dependence via the linearization of the holonomy map.
• The paper constructs a chain-level isomorphism between instanton Floer cohomology and quilted Lagrangian Floer cohomology, assuming compatible perturbations.
• Transversality is achieved simultaneously in both Floer theories through a carefully designed perturbation scheme that preserves the geometric and analytic structure.
• The framework provides a rigorous foundation for higher-rank instanton Floer theory, extending known results from r=2 to arbitrary r ≥ 2.