On the Geometry of Moduli Space of Vacua in N=2 Supersymmetric Yang-Mills Theory

TL;DR

This paper explores the geometric structure of the moduli space in N=2 supersymmetric Yang-Mills theory, revealing differential equations and duality properties that differ from higher supersymmetry cases, especially when coupled to gravity.

Contribution

It identifies Picard-Fuchs type differential equations governing the moduli space and analyzes how electric-magnetic duality is altered in the presence of gravity.

Findings

Derivation of second-order differential equations for the moduli space.

Demonstration that electric-magnetic duality differs from the SL(2,Z) symmetry in N=2 with gravity.

Insights into the geometric and duality structure of N=2 supersymmetric Yang-Mills theory.

Abstract

We consider generic properties of the moduli space of vacua in $N=2$ supersymmetric Yang--Mills theory recently studied by Seiberg and Witten. We find, on general grounds, Picard--Fuchs type of differential equations expressing the existence of a flat holomorphic connection, which for one parameter (i.e. for gauge group $G=SU(2)$), are second order equations. In the case of coupling to gravity (as in string theory), where also ``gravitational'' electric and magnetic monopoles are present, the electric--magnetic S duality, due to quantum corrections, does not seem any longer to be related to $Sl(2,\mathbb{Z})$ as for $N=4$ supersymmetric theory.