On the geometry of moduli space of vacua in N = 2 supersymmetric Yang-Mills theory

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,\IZ) as for N=4 supersymmetric theory.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 $Sl(2,\IZ)$ as for $N=4$ supersymmetric theory.We consider generic properties of the moduli space of vacua in N = 2 supersymmetric Yang-Mills theory recently studied 4by 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” dyons are present, the electric-magnetic S duality, due to quantum corrections, does not seem any longer to be directly related to Sl(2, Z ) as for N = 4 supersymmetric theory.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