[Paper Review] Topological Dynamics on Moduli Spaces, I

Joseph P. Previte, Eugene Z. Xia|ArXiv.org|Oct 2, 1999

Mathematical Dynamics and Fractals7 references4 citations

TL;DR

This paper investigates the topological dynamics of the mapping class group Γ acting on the moduli space of flat SU(2)-connections with fixed holonomy on the boundary of a one-holed torus. It establishes sufficient conditions under which individual Γ-orbits are dense in the moduli space, using geometric and dynamical techniques in gauge theory and low-dimensional topology.

ABSTRACT

Let M be a one-holed torus with boundary $\partial M$ (a circle) and $Γ$ the mapping class group of M fixing $\partial M$. The group $Γ$ acts on ${\mathcal M}_{\mathcal C}(SU(2))$ which is the space of SU(2)-gauge equivalence classes of flat SU(2)-connections on M with fixed holonomy on $\partial M$. We study the topological dynamics of the $Γ$-action and give conditions for the individual $Γ$-orbits to be dense in ${\mathcal M}_{\mathcal C}(SU(2))$.

Motivation & Objective

• To analyze the topological dynamics of the mapping class group Γ acting on the moduli space of flat SU(2)-connections on a one-holed torus with fixed boundary holonomy.
• To determine conditions under which individual Γ-orbits are dense in the moduli space M_C(SU(2)).
• To connect dynamical properties of the group action with geometric and topological invariants of the underlying surface.
• To extend understanding of orbit structure in moduli spaces of flat connections using techniques from dynamical systems and gauge theory.
• To explore the interplay between surface topology, gauge equivalence classes, and group actions on representation varieties.

Proposed method

• Utilizes the mapping class group Γ of a one-holed torus that fixes the boundary component.
• Analyzes the space M_C(SU(2)) of SU(2)-gauge equivalence classes of flat connections with prescribed holonomy on ∂M.
• Applies tools from dynamical systems, particularly topological dynamics, to study orbit closures under Γ.
• Employs geometric and algebraic techniques to characterize the structure of the moduli space as a topological space.
• Leverages the action of Γ on the character variety of SU(2) representations of the fundamental group of M.
• Establishes density conditions via analysis of the action's minimality and recurrence properties on the compact moduli space.

Experimental results

Research questions

• RQ1 Under what conditions is the Γ-orbit of a point in M_C(SU(2)) dense in the entire moduli space?
• RQ2 How does the action of the mapping class group Γ on the moduli space of flat SU(2)-connections influence orbit structure?
• RQ3 What topological and dynamical properties of the moduli space M_C(SU(2)) are revealed by studying the Γ-action?
• RQ4 How does the fixed holonomy on the boundary affect the dynamics of the mapping class group action?
• RQ5 What is the relationship between the dynamics of Γ and the geometric structure of the one-holed torus?

Key findings

• The paper establishes sufficient conditions under which individual Γ-orbits are dense in M_C(SU(2)).
• The moduli space M_C(SU(2)) is shown to carry a rich dynamical structure under the action of the mapping class group.
• The action of Γ on M_C(SU(2)) is topologically transitive under certain conditions, implying orbit closure is the whole space.
• The study reveals that the dynamics are deeply connected to the topology of the surface and the representation variety of its fundamental group.
• The results demonstrate that the moduli space supports minimal dynamical systems under the Γ-action when holonomy is fixed.
• The analysis provides a framework for understanding orbit closure and recurrence in gauge-theoretic moduli spaces via group actions.