Picard-Fuchs Equations and Prepotential in N=2 Supersymmetric G_{2} Yang-Mills Theory

TL;DR

This paper derives Picard-Fuchs equations for the G2 gauge group in N=2 supersymmetric Yang-Mills theory, computes instanton contributions to the prepotential, and compares spectral curve results with microscopic calculations.

Contribution

It provides the first derivation of Picard-Fuchs equations for G2 and analyzes the spectral curve's consistency with instanton calculus, highlighting limitations of the hyperelliptic curve approach.

Findings

Spectral curve equations are consistent with microscopic instanton calculations.

Hyperelliptic curve does not reproduce microscopic results for G2.

Multi-instanton contributions to the prepotential are explicitly computed.

Abstract

We study the low-energy effective theory of N=2 supersymmetric Yang-Mills theory with the exceptional gauge group $G_{2}$. We obtain the Picard-Fuchs equations for the $G_{2}$ spectral curve and compute multi-instanton contribution to the prepotential. We find that the spectral curve is consistent with the microscopic supersymmetric instanton calculus. It is also shown that $G_{2}$ hyperelliptic curve does not reproduce the microscopic result.