[Paper Review] Geometry of representation spaces in SU(2)

Julien Marché|arXiv (Cornell University)|Jan 14, 2010

Homotopy and Cohomology in Algebraic Topology17 references3 citations

TL;DR

This paper develops the differential and symplectic geometry of SU(2) representation spaces for surfaces and 3-manifolds using twisted cohomology, gauge theory, and Chern-Simons theory. It constructs a geometric quantization of the moduli space of flat SU(2) connections on a surface via Bohr-Sommerfeld orbits with metaplectic correction, deriving quantum Clebsch-Gordan conditions that parametrize the quantized Hilbert space and linking it to conformal field theory and TQFT.

ABSTRACT

These notes of a course given at IRMA in April 2009 cover some aspects of the representation theory of fundamental groups of manifolds of dimension at most 3 in compact Lie groups, mainly $\\su$. We give detailed examples, develop the techniques of twisted cohomology and gauge theory. We review Chern-Simons theory and describe an integrable system for the representation space of a surface. Finally, we explain some basic ideas on geometric quantization. We apply them to the case of representation spaces by computing Bohr-Sommerfeld orbits with metaplectic correction.

Motivation & Objective

• To develop a geometric framework for SU(2) representation spaces of surfaces and 3-manifolds using twisted cohomology and gauge theory.
• To describe the symplectic structure of the moduli space of flat SU(2) connections on surfaces via Chern-Simons theory.
• To construct a geometric quantization of the moduli space using Lagrangian fibrations and metaplectic correction.
• To identify Bohr-Sommerfeld orbits in the moduli space and relate them to quantum invariants via Clebsch-Gordan conditions.
• To lay the foundation for a topological quantum field theory (TQFT) by connecting geometric quantization to conformal blocks and link polynomials.

Proposed method

• Use twisted cohomology to analyze the tangent space of the representation space via the cohomology of the fundamental group with coefficients in the adjoint representation.
• Apply gauge theory to interpret flat connections on principal SU(2)-bundles as critical points of the Chern-Simons functional.
• Construct a prequantum line bundle over the moduli space using the Chern-Simons functional and its holonomy properties.
• Define a Lagrangian fibration on the moduli space via trace functions associated to a pants decomposition of the surface.
• Implement geometric quantization with metaplectic correction by identifying Bohr-Sommerfeld leaves and computing the half-form bundle.
• Compute the holonomy of a path in the frame bundle to determine the metaplectic structure and classify quantized states.

Experimental results

Research questions

• RQ1 How can the moduli space of flat SU(2) connections on a surface be described using twisted cohomology and gauge-theoretic methods?
• RQ2 What is the symplectic structure of the representation space for surfaces, and how does it arise from Chern-Simons theory?
• RQ3 Which Lagrangian submanifolds in the moduli space correspond to Bohr-Sommerfeld orbits under geometric quantization with metaplectic correction?
• RQ4 How do the quantum Clebsch-Gordan conditions arise from the topological and geometric data of the surface?
• RQ5 Can the geometric quantization of the moduli space be related to a TQFT and to conformal field theory via the Verlinde formula and BKS pairing?

Key findings

• The moduli space of flat SU(2) connections on a surface of genus g ≥ 2 admits a Lagrangian fibration via trace functions associated to a pants decomposition.
• Bohr-Sommerfeld orbits in the moduli space are parametrized by tuples (σiπ/K)i∈I where σi ∈ [1, K−1] satisfy quantum Clebsch-Gordan conditions: σi ≤ σj + σk, σi + σj + σk odd, and σi + σj + σk ≤ 2K.
• The geometric quantization of the moduli space yields a finite-dimensional Hilbert space whose dimension matches the Verlinde formula, confirming consistency with conformal field theory.
• The metaplectic correction is essential for correctly identifying the quantized states, and the holonomy computation shows the path in GL(C1) is nontrivial, confirming the nontriviality of the metalinear structure.
• The construction of the prequantum bundle via the Chern-Simons functional provides a natural connection with holonomy related to Reidemeister torsion and flat geometry.
• The BKS pairing between quantizations from different pants decompositions is expected to be a unitary isomorphism, suggesting a consistent TQFT structure underlying the geometric quantization.