[Paper Review] Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory

Tobias Diez|arXiv (Cornell University)|Sep 2, 2019

Black Holes and Theoretical Physics106 references4 citations

TL;DR

This paper establishes a normal form theorem for equivariant maps between Fréchet manifolds and develops an infinite-dimensional theory of singular symplectic reduction, extending the classical Marle-Guillemin-Sternberg normal form to infinite-dimensional settings. By combining slice theorems and Kuranishi-type constructions, it proves that reduced phase spaces decompose into symplectic manifolds with refined strata—'seams'—in cotangent bundle settings, and applies the framework to gauge theories including Yang-Mills and Yang-Mills-Higgs models.

ABSTRACT

A local normal form theorem for smooth equivariant maps between Fréchet manifolds is established. Moreover, an elliptic version of this theorem is obtained. The proof these normal form results is inspired by the Lyapunov-Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces, and uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence of this equivariant normal form theorem, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space. Moreover, the theory of singular symplectic reduction is developed in the infinite-dimensional Fréchet setting. By refining the above construction, a normal form for momentum maps similar to the classical Marle-Guillemin-Sternberg normal form is established. Analogous to the reasoning in finite dimensions, this normal form result is then used to show that the reduced phase space decomposes into smooth manifolds each carrying a natural symplectic structure. Finally, the singular symplectic reduction scheme is further investigated in the situation where the original phase space is an infinite-dimensional cotangent bundle. The fibered structure of the cotangent bundle yields a refinement of the usual orbit-momentum type strata into so-called seams. Using a suitable normal form theorem, it is shown that these seams are manifolds. Taking the harmonic oscillator as an example, the influence of the seams on dynamics is illustrated. The general results stated above are applied to various gauge theory models. The moduli spaces of anti-self-dual connections in four dimensions and of Yang-Mills connections in two dimensions is studied. Moreover, the stratified structure of the reduced phase space of the Yang-Mills-Higgs theory is investigated in a Hamiltonian formulation.

Motivation & Objective

• To extend the classical normal form theorems for momentum maps to infinite-dimensional Fréchet manifolds.
• To develop a rigorous framework for singular symplectic reduction in infinite dimensions, particularly for gauge-theoretic phase spaces.
• To refine the stratification of reduced phase spaces by identifying 'seams'—submanifolds arising from the fibered structure of cotangent bundles.
• To establish that the reduced spaces decompose into smooth symplectic manifolds, even in singular settings.
• To apply the abstract framework to concrete gauge field theories, including anti-self-dual and Yang-Mills connections, and Yang-Mills-Higgs theory.

Proposed method

• Adapts the Lyapunov-Schmidt reduction and Kuranishi method to construct a normal form for smooth equivariant maps between Fréchet manifolds.
• Employs a slice theorem for Fréchet manifolds as the central technical tool to achieve local normal forms.
• Constructs the abstract moduli space as a quotient of a level set by the group action, showing it inherits the structure of a Kuranishi space.
• Refines the momentum map normal form to an infinite-dimensional analogue of the Marle-Guillemin-Sternberg theorem.
• Analyzes the cotangent bundle structure of the phase space to identify 'seams'—refined strata beyond standard orbit-momentum type strata.
• Uses the refined normal form to prove that seams are smooth submanifolds, enabling a stratified symplectic structure on the reduced space.

Experimental results

Research questions

• RQ1 Can a normal form theorem for equivariant maps be established in the infinite-dimensional Fréchet setting?
• RQ2 How can singular symplectic reduction be generalized to infinite-dimensional symplectic manifolds with group actions?
• RQ3 What is the role of the fibered structure of cotangent bundles in refining the stratification of reduced phase spaces?
• RQ4 Are the refined strata—'seams'—in the reduced space of gauge theories smooth manifolds with natural symplectic structures?
• RQ5 How do the abstract results on normal forms and reduction apply to concrete gauge field theories like Yang-Mills and Yang-Mills-Higgs?

Key findings

• A local normal form theorem for smooth equivariant maps between Fréchet manifolds is established, generalizing finite-dimensional results.
• The abstract moduli space obtained by quotienting a level set under group action is shown to carry the structure of a Kuranishi space.
• An infinite-dimensional version of the Marle-Guillemin-Sternberg normal form is derived for momentum maps in Fréchet settings.
• The reduced phase space decomposes into smooth symplectic manifolds, even in singular cases, under the refined reduction scheme.
• In cotangent bundle settings, the standard orbit-momentum strata are refined into 'seams', which are proven to be smooth submanifolds via the normal form.
• The harmonic oscillator model illustrates that seams influence dynamics, confirming their geometric and physical relevance in the reduced system.