Effective Lagrangian for Non-Abelian Two-Dimensional Topological Field Theory

TL;DR

This paper develops a systematic method to derive effective Lagrangians for 2D non-Abelian topological BF theories, explicitly deriving actions for SU(2) and SU(3), and applying them to compute partition functions and Wilson loop expectations.

Contribution

It introduces a diagrammatic approach to obtain scalar-field-based effective Lagrangians for non-Abelian 2D topological theories, including explicit SU(2) and SU(3) cases.

Findings

Effective Lagrangians expressed as scalar fields for SU(2) and SU(3).

Partition function for SU(2) Yang-Mills on a sphere computed using the effective action.

Wilson loop vacuum expectation value calculated within the effective theory.

Abstract

We develop a systematic approach to obtain an effective Lagrangian for 2D non-Abelian topological BF theory. A general expression is presented in a diagrammatic representation containing solely scalar fields. Expressions for the SU(2) and SU(3) effective actions are explicitly stated. In the case of SU(2), we show that the effective action can be interpreted as a winding number. By using the SU(2) effective action, the partition function on a sphere for SU(2) Yang-Mills theory is calculated. Moreover, we generalise the theory to include a source term for the gauge field as well as calculate the vacuum expectation value of the Wilson loop based on the effective theory.