[Paper Review] Quantum Moduli Spaces of Flat Connections

Anton Alekseev, Volker Schomerus|ArXiv.org|Dec 31, 1996

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper constructs quantum moduli algebras as quantizations of the moduli spaces of flat connections on punctured Riemann surfaces using discrete quantum group gauge theory. It establishes a closed formula for mapping class group representations on conformal blocks, providing a quantum algebraic framework for Chern-Simons theory with explicit topological symmetry actions.

ABSTRACT

Using the formalism of discrete quantum group gauge theory, one can construct the quantum algebras of observables for the Hamiltonian Chern-Simons model. The resulting moduli algebras provide quantizations of the algebra of functions on the moduli spaces of flat connections on a punctured 2-dimensional surface. In this note we describe some features of these moduli algebras with special emphasis on the natural action of mapping class groups. This leads, in particular, to a closed formula for representations of the mapping class groups on conformal blocks.

Motivation & Objective

• To develop a quantum algebraic framework for the moduli spaces of flat connections on punctured 2D surfaces.
• To apply discrete quantum group gauge theory to construct quantum algebras of observables in the Hamiltonian Chern-Simons model.
• To describe the natural action of the mapping class group on these quantum moduli algebras.
• To derive a closed formula for the representation of the mapping class group on conformal blocks.
• To provide a quantization of the classical algebra of functions on moduli spaces of flat connections.

Proposed method

• Utilizes discrete quantum group gauge theory to construct quantum algebras of observables for the Chern-Simons model.
• Applies the formalism to surfaces with punctures, encoding the punctures as Wilson lines or defects.
• Identifies the quantum moduli algebra as a quantization of the classical algebra of functions on the moduli space of flat connections.
• Derives the action of the mapping class group on the quantum moduli algebra via braided tensor category structures.
• Establishes a closed-form expression for the representation of the mapping class group on conformal blocks using the quantum algebra structure.
• Relies on the algebraic structure of quantum groups and their representations to encode topological invariance and symmetry.

Experimental results

Research questions

• RQ1 How can the moduli space of flat connections on a punctured Riemann surface be quantized using gauge-theoretic methods?
• RQ2 What is the structure of the quantum algebra of observables in the Hamiltonian Chern-Simons theory on such surfaces?
• RQ3 How does the mapping class group act on the quantum moduli algebra and on conformal blocks?
• RQ4 Can a closed formula be derived for the representation of the mapping class group on conformal blocks in this setting?
• RQ5 What is the precise relationship between the quantum moduli algebra and the classical algebra of functions on the moduli space?

Key findings

• The quantum moduli algebra is constructed as a quantization of the classical algebra of functions on the moduli space of flat connections on a punctured surface.
• The mapping class group acts naturally on the quantum moduli algebra, preserving its algebraic structure.
• A closed formula is derived for the representation of the mapping class group on conformal blocks, expressed in terms of the quantum group structure.
• The construction uses discrete quantum group gauge theory to systematically realize the quantum algebra of observables.
• The resulting quantum algebra encodes the topological and geometric data of the surface and its punctures.
• The formalism provides a rigorous algebraic realization of the quantum geometry of flat connections in 2+1D topological field theory.