$n$-point functions of $2d$ Yang-Mills theories on Riemann surfaces

TL;DR

This paper computes $n$-point functions of field strength in 2D Yang-Mills theories on Riemann surfaces, revealing free field behavior and decomposing correlators into distinct parts for both abelian and non-abelian cases.

Contribution

It introduces a simple path integral approach to calculate correlators on arbitrary Riemann surfaces, identifying free and almost $x$-independent components in both abelian and non-abelian theories.

Findings

Correlators in $U(1)$ are composed of free and $x$-independent parts.

Non-abelian correlators in Schwinger-Fock gauge split into free and almost $x$-independent parts.

Gauge-invariant Green functions correspond to a free field theory.

Abstract

Using the simple path integral method we calculate the $n$-point functions of field strength of Yang-Mills theories on arbitrary two-dimensional Riemann surfaces. In $U(1)$ case we show that the correlators consist of two parts , a free and an $x$-independent part. In the case of non-abelian semisimple compact gauge groups we find the non-gauge invariant correlators in Schwinger-Fock gauge and show that it is also divided to a free and an almost $x$-independent part. We also find the gauge-invariant Green functions and show that they correspond to a free field theory.