Two Dimensional Gauge Theories Revisited

TL;DR

This paper reveals a direct mapping between topological and physical Yang-Mills theories in two dimensions, clarifying their relationship and enabling exact computations of the topological theory.

Contribution

It demonstrates a direct path integral mapping between two-dimensional topological and physical Yang-Mills theories, explaining their connection and computational properties.

Findings

Partition function expressed as a sum over classical solutions

Topological theory made fully computable

Clarification of the relation between topological and physical theories

Abstract

Topological gravity is equivalent to physical gravity in two dimensions in a way that is still mysterious, though by now it has been proved by Kontsevich. In this paper it is shown that a similar relation between topological and physical Yang-Mills theory holds in two dimensions; in this case, however, the relation can be explained by a direct mapping between the two path integrals. This (1) explains many strange facts about two dimensional Yang-Mills theory, like the way the partition function can be expressed exactly as a sum over classical solutions, including unstable ones; (2) makes the corresponding topological theory completely computable.