[Paper Review] Group-valued momentum maps for actions of automorphism groups

Tobias Diez, Tudor S. Raţiu|arXiv (Cornell University)|Feb 4, 2020

Geometry and complex manifolds52 references4 citations

TL;DR

This paper introduces a novel framework for group-valued momentum maps in infinite-dimensional symplectic geometry, extending classical momentum map theory to actions of automorphism groups on spaces of sections of symplectic fiber bundles. It constructs momentum maps that assign principal circle bundles with connection to each section, resolving obstructions in classical momentum map theory and enabling applications to fluid dynamics, gauge theory, and Teichmüller theory.

ABSTRACT

The space of smooth sections of a symplectic fiber bundle carries a natural symplectic structure. We provide a general framework to determine the momentum map for the action of the group of bundle automorphism on this space. Since, in general, this action does not admit a classical momentum map, we introduce the more general class of group-valued momentum maps which is inspired by the Poisson Lie setting. In this approach, the group-valued momentum map assigns to every section of the symplectic fiber bundle a principal circle-bundle with connection. The power of this general framework is illustrated in many examples: we construct generalized Clebsch variables for fluids with integral helicity; the anti-canonical bundle turns out to be the momentum map for the action of the group of symplectomorphisms on the space of compatible complex structures; the Teichmüller moduli space is realized as a symplectic orbit reduced space associated to a coadjoint orbit of $\mathrm{SL}(2,\mathbb{R})$ and spaces related to the other coadjoint orbits are identified and studied. Moreover, we show that the momentum map for the group of bundle automorphisms on the space of connections over a Riemann surface encodes, besides the curvature, also topological information of the bundle.

Motivation & Objective

• To extend classical momentum map theory to infinite-dimensional symplectic geometry where standard momentum maps fail due to topological obstructions.
• To develop a general framework for momentum maps in the context of symplectic fiber bundles with structure group actions.
• To resolve the lack of classical momentum maps for actions of diffeomorphism and symplectomorphism groups on spaces of sections.
• To unify and generalize existing momentum map constructions in gauge theory and Kähler geometry under a single formalism.
• To demonstrate the framework's utility in geometric mechanics, moduli space theory, and mathematical physics through concrete examples.

Proposed method

• Introduce group-valued momentum maps as a generalization of classical momentum maps, inspired by Poisson-Lie structures.
• Construct the momentum map for the action of the automorphism group on the space of sections of a symplectic fiber bundle.
• Use fiber integration and differential characters (Cheeger-Simons classes) to encode curvature and topological data in the momentum map.
• Apply the framework to derive momentum maps for gauge groups, diffeomorphism groups, and symplectomorphisms on sections.
• Utilize the hat product of differential characters to relate curvature and characteristic classes in the momentum map construction.
• Leverage the coadjoint orbit picture and reduction techniques to identify moduli spaces such as Teichmüller space as symplectic quotients.

Experimental results

Research questions

• RQ1 Can a momentum map be defined for the action of the automorphism group on the space of sections of a symplectic fiber bundle when the classical momentum map fails?
• RQ2 How can group-valued momentum maps encode both geometric and topological data in infinite-dimensional symplectic geometry?
• RQ3 What is the role of differential characters and prequantum bundles in constructing momentum maps for symplectic fiber bundles?
• RQ4 How does the momentum map for symplectomorphisms on sections relate to known structures in Kähler geometry and Teichmüller theory?
• RQ5 Can the momentum map framework unify constructions in gauge theory, fluid dynamics, and moduli space geometry?

Key findings

• The momentum map for the action of the group of bundle automorphisms on the space of sections is constructed as a group-valued map assigning a principal circle bundle with connection to each section.
• The momentum map for the group of symplectomorphisms on sections is shown to exist only when certain cohomological obstructions vanish, and is otherwise replaced by a group-valued momentum map.
• The generalized Clebsch variables for fluids with integral helicity are constructed via the momentum map framework, linking helicity to prequantum line bundles.
• The anti-canonical bundle is identified as the momentum map for the action of the symplectomorphism group on the space of compatible complex structures.
• Teichmüller moduli space is realized as a symplectic reduction of a coadjoint orbit of SL(2, R), with other related moduli spaces identified through the same framework.
• The momentum map for the quantomorphism group on connections over a Riemann surface encodes both curvature and topological invariants of the bundle, via differential characters.