[Paper Review] Seiberg-Witten Equations on Asymptotically-Flat Three-Manifolds

Yi-Jen Lee|arXiv (Cornell University)|Jun 24, 1997

Geometric and Algebraic Topology10 references1 citations

TL;DR

This paper establishes Seiberg-Witten theory on asymptotically flat three-manifolds by constructing a Banach manifold structure for the quotient of the configuration space modulo gauge group actions and developing a proper Fredholm theory for the unperturbed and perturbed Seiberg-Witten equations. The key contribution is the foundational framework enabling applications in 3-manifold topology, including the equivalence of the Seiberg-Witten invariant to other invariants.

ABSTRACT

We construct the Seiberg-Witten theory on asymptotically flat three manifolds and describe the structure of the moduli space. The analysis should serve as the basis for many applications in 3-manifold topology, including a proof of the equivalence of the Seiberg-Witten invariant of In this paper we investigate the perturbed and unperturbed version of Seiberg-Witten theory on asymptotically flat 3-manifolds. We prove the Banach manifold structure of the quotient space (of the configuration space by the gauge group action), establish a proper Fredholm theory, which in turn leads

Motivation & Objective

• To extend Seiberg-Witten theory to the setting of asymptotically flat three-manifolds, where standard compactness assumptions do not hold.
• To establish a Banach manifold structure for the quotient space of configurations modulo gauge group actions.
• To develop a proper Fredholm theory for the Seiberg-Witten equations on non-compact 3-manifolds.
• To lay the groundwork for topological applications, including the equivalence of the Seiberg-Witten invariant to other invariants in 3-manifold theory.

Proposed method

• Construct the configuration space of spin-c connections and sections of the spinor bundle on an asymptotically flat 3-manifold.
• Define the action of the gauge group on the configuration space and quotient by this action to form a Banach manifold.
• Introduce perturbations to the Seiberg-Witten equations to ensure regularity and avoid reducible solutions.
• Apply elliptic regularity and Fredholm theory in weighted Sobolev spaces to analyze the linearized operator.
• Establish the proper Fredholm index theory for the Seiberg-Witten operator in the asymptotically flat setting.
• Use the Banach implicit function theorem to prove the smoothness of the moduli space as a Banach manifold.

Experimental results

Research questions

• RQ1 How can Seiberg-Witten theory be extended to asymptotically flat three-manifolds where the standard compactness tools fail?
• RQ2 What is the correct geometric and analytic framework for the configuration space and its quotient under gauge transformations in this setting?
• RQ3 Can a Fredholm theory be developed for the Seiberg-Witten equations on non-compact 3-manifolds with controlled decay at infinity?
• RQ4 What is the structure of the moduli space of solutions to the unperturbed and perturbed Seiberg-Witten equations on such manifolds?
• RQ5 How does this framework support topological applications, such as proving invariance or equivalence of invariants?

Key findings

• The quotient space of the configuration space modulo the gauge group action is shown to carry a natural Banach manifold structure.
• A proper Fredholm theory is established for the Seiberg-Witten operator in the asymptotically flat setting using weighted Sobolev spaces.
• The moduli space of solutions to the unperturbed and perturbed Seiberg-Witten equations is shown to be a smooth Banach manifold of finite dimension.
• The linearized operator of the Seiberg-Witten equations is proven to be Fredholm with a well-defined index in the weighted Sobolev setting.
• The framework enables the definition of a Seiberg-Witten invariant for asymptotically flat 3-manifolds, paving the way for topological applications.
• The analysis provides the necessary foundation for proving the equivalence of the Seiberg-Witten invariant to other invariants in 3-manifold topology.