Prepotential of $N=2$ Supersymmetric Yang-Mills Theories in the Weak Coupling Region

TL;DR

This paper derives the explicit low-energy effective action for N=2 supersymmetric Yang-Mills theories in the weak coupling regime using hyperelliptic Riemann surfaces, confirming instanton correction results for various gauge groups.

Contribution

It provides an explicit integral-based method to compute the effective action and instanton corrections for N=2 SYM theories, aligning with recent direct instanton calculations.

Findings

Explicit form of the low-energy effective action derived.

Leading instanton corrections calculated for multiple gauge groups.

One-instanton contributions match recent direct calculations.

Abstract

We show how to obtain the explicite form of the low energy quantum effective action for $N=2$ supersymmetric Yang-Mills theory in the weak coupling region from the underlying hyperelliptic Riemann surface. This is achieved by evaluating the integral representation of the fields explicitly. We calculate the leading instanton corrections for the group $SU(\nc), SO(N)$ and $SP(2N)$ and find that the one-instanton contribution of the prepotentials for the these group coincide with the one obtained recently by using the direct instanton caluculation.