Picard-Fuchs Equation and Prepotential of Five Dimensional SUSY Gauge Theory Compactified on a Circle

TL;DR

This paper derives the Picard-Fuchs equation for 5D SUSY gauge theory compactified on a circle, connecting it with integrable systems and obtaining the prepotential including perturbative and non-perturbative effects.

Contribution

It introduces a novel derivation of the Picard-Fuchs equation for 5D SUSY gauge theories using the relativistic Toda chain, linking it to spectral curves and supersymmetric cycles.

Findings

Derived the Picard-Fuchs equation for 5D SUSY gauge theory on a circle.

Connected the equation with spectral curves of XXZ spin chain and M theory cycles.

Obtained the prepotential including Kaluza-Klein and instanton effects.

Abstract

Five dimensional supersymmetric gauge theory compactified on a circle defines an effective N=2 supersymmetric theory for massless fields in four dimensions. Based on the relativistic Toda chain Hamiltonian proposed by Nekrasov, we derive the Picard-Fuchs equation on the moduli space of the Coulomb branch of SU(2) gauge theory. Our Picard-Fuchs equation agrees with those from other approaches; the spectral curve of XXZ spin chain and supersymmetric cycle in compactified M theory. By making use of a relation to the Picard-Fuchs equation of SU(2) Seiberg-Witten theory, we obtain the prepotential and the effective coupling constant that incorporate both a perturbative effect of Kaluza-Klein modes and a non-perturbative one of four dimensional instantons. In the weak coupling regime we check that the prepotential exhibits a consistent behavior in large and small radius limits of the circle.