Picard-Fuchs equation and prepotential of five-dimensional SUSY gauge theory compactified on a circle

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 the XXZ spin chain and the 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.