Random Unitary Representations of Surface Groups I: Asymptotic Expansions

Area of Science:

• Topology
• Representation Theory
• Mathematical Physics

Background:

• Fundamental groups of surfaces are key in topology.
• Special unitary groups are crucial in quantum mechanics and geometry.
• Random matrix theory offers tools to study complex mathematical structures.

Purpose of the Study:

• To investigate random representations of surface fundamental groups into special unitary groups.
• To develop a large n asymptotic expansion for expected trace values.
• To gain analytic control over contributions from irreducible representations.

Main Methods:

• Utilizing a random model based on a symplectic form on moduli space (Atiyah, Bott, Goldman).
• Establishing large n asymptotic expansions for expected trace values.
• Applying analytic techniques to control contributions from irreducible representations.

Main Results:

• Existence of a large n asymptotic expansion for expected trace values.
• Demonstrated that each expected value includes contributions from all irreducible representations.
• Achieved effective analytic control over representations outside specific rational families.

Conclusions:

• The study provides a framework for understanding random representations in this context.
• The developed methods offer new analytic tools for studying representation theory.
• Findings contribute to the intersection of topology, representation theory, and mathematical physics.