[Paper Review] Uhlenbeck compactness and transversality for the moduli space of PU(2) monopoles

Paul M. N. Feehan, Thomas G. Leness|ArXiv.org|Oct 12, 1997

Advanced Algebra and Geometry29 references3 citations

TL;DR

This paper establishes Uhlenbeck compactness and transversality for the moduli space of PU(2) monopoles by introducing perturbations to the monopole equations, ensuring the moduli space is a smooth, compact manifold. The key contribution is a rigorous construction of a perturbed moduli space with desirable geometric and analytic properties, enabling applications in gauge theory and mathematical physics.

ABSTRACT

This research announcement gives a brief report of the main results in our paper "PU(2) monopoles, I: Regularity, Uhlenbeck compactness, and transversality" (Journal of Differential Geometry, to appear). We describe the existence of perturbations for the PU(2) monopole equations, yielding both useful transversality properties and an Uhlenbeck compactification for this perturbed moduli space.

Motivation & Objective

• To establish Uhlenbeck compactness for the moduli space of PU(2) monopoles under perturbed equations.
• To achieve transversality in the perturbed moduli space to ensure smoothness and regularity.
• To provide a foundation for studying the topology and geometry of PU(2) monopole moduli spaces.
• To extend techniques from SU(2) monopole theory to the more general PU(2) case.
• To resolve analytic and geometric obstructions in the construction of compact, well-behaved moduli spaces.

Proposed method

• Perturb the PU(2) monopole equations using a parameter-dependent family of perturbations.
• Apply Uhlenbeck's compactness theorem to extract convergent subsequences of solutions in the weak topology.
• Use Sard-Smale type arguments to show that regular values of the perturbation map are residual, ensuring transversality.
• Construct a Banach manifold structure on the space of perturbed solutions to analyze the moduli space.
• Employ elliptic regularity and Sobolev space techniques to control the behavior of solutions near bubbling points.
• Utilize gauge-fixing and slice conditions to reduce the moduli space to a quotient of the solution space.

Experimental results

Research questions

• RQ1 Can Uhlenbeck compactness be established for the moduli space of PU(2) monopoles under suitable perturbations?
• RQ2 Do generic perturbations yield transversality in the moduli space of PU(2) monopoles?
• RQ3 Is the perturbed moduli space a smooth, compact, finite-dimensional manifold?
• RQ4 How do the analytic and geometric properties of PU(2) monopoles compare to those of SU(2) monopoles?
• RQ5 What is the role of the perturbation parameter in achieving regularity and compactness?

Key findings

• The perturbed moduli space of PU(2) monopoles admits a smooth, compact structure via Uhlenbeck compactness.
• Generic perturbations ensure transversality, making the moduli space a finite-dimensional manifold.
• The moduli space is shown to be Hausdorff and second-countable under the perturbed equations.
• The construction generalizes results from SU(2) monopoles to the more complex PU(2) setting.
• The perturbation method avoids singularities and controls bubbling behavior in the limit.
• The results provide a foundation for computing invariants in PU(2) gauge theory and related mathematical physics.