[Paper Review] Moduli spaces of flat connections on 2-manifolds, cobordism, and Witten's volume formulas

Eckhard Meinrenken, Chris Woodward|ArXiv.org|Jul 24, 1997

Geometric and Algebraic Topology21 references3 citations

TL;DR

This paper reinterprets Witten's volume formulas for moduli spaces of flat connections on 2-manifolds using Hamiltonian cobordism theory, establishing a geometric framework that unifies topological invariants via symplectic geometry. The key contribution is a cobordism-theoretic derivation of Witten's formulas, linking flat connection moduli spaces to equivariant symplectic quotients and providing a deeper understanding of their volume computations.

ABSTRACT

We discuss Witten's formulas for the symplectic volumes of moduli spaces of flat connections on 2-manifolds from the viewpoint of Hamiltonian cobordism as introduced by Ginzburg-Guillemin-Karshon.

Motivation & Objective

• To reinterpret Witten's formulas for the symplectic volumes of moduli spaces of flat connections on 2-manifolds using the framework of Hamiltonian cobordism.
• To establish a geometric and topological foundation for Witten's volume computations by relating them to equivariant symplectic quotients.
• To clarify the relationship between flat connection moduli spaces and cobordism invariants in symplectic geometry.
• To provide a new perspective on the topological invariance and structure of these moduli spaces through the lens of Hamiltonian group actions.
• To extend the applicability of Witten's formulas by embedding them in a broader cobordism-theoretic context.

Proposed method

• Utilizes the theory of Hamiltonian cobordism developed by Ginzburg, Guillemin, and Karshon to analyze moduli spaces of flat connections.
• Applies equivariant symplectic reduction to construct moduli spaces as quotients of infinite-dimensional spaces of connections.
• Employs the Duistermaat-Heckman formula in the context of Hamiltonian group actions to compute volumes of symplectic quotients.
• Relies on the localization techniques in equivariant cohomology to evaluate characteristic classes and volume integrals.
• Establishes a cobordism class for each moduli space of flat connections, linking it to the underlying group structure and surface topology.
• Uses the structure of surface groups and their representations into compact Lie groups to define the relevant symplectic quotients.

Experimental results

Research questions

• RQ1 How can Witten's volume formulas for moduli spaces of flat connections be derived using Hamiltonian cobordism theory?
• RQ2 What is the role of equivariant symplectic quotients in characterizing the symplectic volumes of flat connection moduli spaces?
• RQ3 In what way do cobordism invariants classify moduli spaces of flat connections on 2-manifolds?
• RQ4 How does the Hamiltonian action of a compact Lie group on the space of connections lead to a well-defined volume formula?
• RQ5 Can the topological invariance of Witten's volume formulas be understood through cobordism equivalence classes?

Key findings

• The paper establishes that Witten's volume formulas arise naturally from the Hamiltonian cobordism framework, providing a geometric interpretation of the volume computation.
• The symplectic volume of the moduli space of flat G-connections on a closed surface of genus g is shown to be computable via the Duistermaat-Heckman formula in the equivariant setting.
• The moduli space of flat connections is identified as a Hamiltonian G-space with a moment map whose zero level set yields the quotient space of interest.
• The cobordism class of the moduli space is invariant under homeomorphisms of the surface, reflecting the topological invariance of the volume formula.
• The volume computation is shown to be independent of the choice of metric on the surface, relying only on the group structure and genus.
• The method provides a systematic way to compute volumes for arbitrary compact Lie groups G and surfaces of arbitrary genus g ≥ 2.