Extended superspace, higher derivatives and SL(2, Z) duality

TL;DR

This paper analyzes the low-energy effective action of an N=2 supersymmetric gauge theory, emphasizing its invariance under SL(2,Z) duality and the modular properties of its terms.

Contribution

It demonstrates the natural SL(2,Z) invariance of the N=2 superspace formalism and characterizes the modular invariance of the real analytic function in the effective action.

Findings

The effective action's leading term is the holomorphic prepotential.

The next-to-leading term is a real analytic, SL(2,Z)-invariant function.

The superspace formalism maintains duality invariance regardless of the action form.

Abstract

We consider the low-energy effective action for the Coulomb phase of an $N = 2$ supersymmetric gauge theory with a rank one gauge group. The $N = 2$ superspace formalism is naturally invariant under an $SL(2, {\bf Z})$ group of duality transformations, regardless of the form of the action. The leading and next to leading terms in the long distance expansion of the action are given by the holomorphic prepotential and a real analytic function respectively. The latter is shown to be modular invariant with respect to $SL(2, {\bf Z})$.