[Paper Review] SU(2) WZNW Model at Higher Genera from Gauge Field Functional Integral

Krzysztof Gawędzki|ArXiv.org|Dec 7, 1993

Physics of Superconductivity and Magnetism14 references4 citations

TL;DR

This paper computes the gauge field functional integral for SU(2) Chern-Simons theory on Riemann surfaces of genus greater than one, deriving explicit finite-dimensional integrals that express the higher-genus partition functions of the SU(2) WZNW conformal field theory. The method provides a direct link between Chern-Simons theory and WZNW model amplitudes, offering new insights into the functional integral structure of Liouville theory.

ABSTRACT

We compute the gauge field functional integral giving the scalar product of the SU(2) Chern-Simons theory states on a Riemann surface of genus > 1. The result allows to express the higher genera partition functions of the SU(2) WZNW conformal field theory by explicit finite dimensional integrals. Our calculation may also shed new light on the functional integral of the Liouville theory.

Motivation & Objective

• To derive the scalar product of SU(2) Chern-Simons theory states on Riemann surfaces of genus > 1.
• To express the higher-genus partition functions of the SU(2) WZNW conformal field theory in terms of finite-dimensional integrals.
• To provide a functional integral formulation that may clarify the structure of Liouville theory's path integral.
• To establish a concrete computational framework for topological and conformal field theory amplitudes on higher-genus Riemann surfaces.

Proposed method

• Uses the gauge field functional integral formulation to compute the scalar product in SU(2) Chern-Simons theory on genus g > 1 Riemann surfaces.
• Applies the path integral approach to relate the quantum states of Chern-Simons theory to the correlation functions of the WZNW model.
• Derives explicit finite-dimensional integrals that encode the partition functions of the SU(2) WZNW model at higher genera.
• Corrects a missing factor in a key equation (Eq. (1.4)) in the revised version, ensuring consistency with known results.
• Employs techniques from topological quantum field theory and conformal field theory to connect geometric data on Riemann surfaces to quantum amplitudes.
• Relies on the correspondence between Chern-Simons theory and WZNW model, leveraging known results in 2+1D topological field theory.

Experimental results

Research questions

• RQ1 How can the scalar product of SU(2) Chern-Simons theory states be computed on Riemann surfaces of genus greater than one?
• RQ2 What is the explicit functional integral representation of the higher-genus partition functions in the SU(2) WZNW model?
• RQ3 How does the gauge field functional integral formulation relate to the conformal field theory amplitudes of the WZNW model?
• RQ4 Can this approach shed light on the structure of the Liouville field theory path integral?
• RQ5 What finite-dimensional integral formulae emerge from the functional integral computation in the SU(2) WZNW model at higher genera?

Key findings

• The scalar product of SU(2) Chern-Simons theory states on genus g > 1 Riemann surfaces is computed via a gauge field functional integral.
• The higher-genus partition functions of the SU(2) WZNW conformal field theory are expressed as explicit finite-dimensional integrals.
• The corrected version of the paper (v2) fixes a missing factor in Eq. (1.4), ensuring consistency with known normalization in Chern-Simons theory.
• The result provides a new computational tool for studying correlation functions in the SU(2) WZNW model on higher-genus Riemann surfaces.
• The method may offer new insights into the functional integral formulation of Liouville theory, suggesting deeper connections between 2D conformal field theories and topological field theories.