Lectures on 2d Gauge Theories: Topological Aspects and Path Integral Techniques

TL;DR

Blau and Thompson develop a path-integral framework for extracting the topological content of two-dimensional Yang-Mills and G/G models. By applying a Weyl integral formula and an Abelianization procedure, they reduce the theories to tractable Abelian systems and derive exact partition functions that reproduce Verlinde numbers and CS–BF correspondences, including level shifts via determinants. They show how YM at ε → 0 yields the symplectic volume of moduli spaces of flat connections and how G/G computes CS Hilbert-space dimensions (and their marked-point generalizations) through the Verlinde formula, with a clear link to BF theory in appropriate limits. The approach provides a gauge-theoretic, first-principles route to topological invariants on closed surfaces, with potential extensions to broader classes of topological field theories via Abelianization.

Abstract

These are lecture notes of lectures presented at the 1993 Trieste Summer School, dealing with two classes of two-dimensional field theories, (topological) Yang-Mills theory and the G/G gauged WZW model. The aim of these lectures is to exhibit and extract the topological information contained in these theories, and to present a technique (a Weyl integral formula for path integrals) which allows one to calculate directly their partition function and topological correlation functions on arbitrary closed surfaces. Topics dealt with are (among others): solution of Yang-Mills theory on arbitrary surfaces; calculation of intersection numbers of moduli spaces of flat connections; coupling of Yang-Mills theory to coadjoint orbits and intersection numbers of moduli spaces of parabolic bundles; derivation of the Verlinde formula from the G/G model; derivation of the shift k to k+h in the G/G model via the index of the twisted Dolbeault complex.