Topology Change and theta-Vacua in 2D Yang-Mills Theory

TL;DR

This paper explores the existence and classification of theta-vacua in 2D Yang-Mills theory, linking different quantum theories to irreducible unitary representations of fundamental groups based on manifold properties.

Contribution

It provides a systematic procedure to classify quantum Yang-Mills theories in 2D by their topological and gauge group characteristics, including boundary conditions.

Findings

Classifies theta-vacua in 2D Yang-Mills for various gauge groups and manifolds.

Establishes correspondence between quantum theories and irreducible unitary representations.

Differentiates cases based on orientability of the manifold.

Abstract

We discuss the existence of $\theta$-vacua in pure Yang-Mills theory in two space-time dimensions. More precisely, a procedure is given which allows one to classify the distinct quantum theories possessing the same classical limit for an arbitrary connected gauge group G and compact space-time manifold M (possibly with boundary) possessing a special basepoint. For any such G and M it is shown that the above quantizations are in one-to-one correspondence with the irreducible unitary representations (IUR's) of $\pi_1(G)$ if M is orientable, and with the IUR's of $\pi_1(G)/2\pi_1(G)$ if M is nonorientable.