Ivancevic_Applied-Diff-Geom

720 Applied Differential Geometry: A Modern Introduction(such as confinement of the color electric charge of quarks) are mappedby duality into problems of weak magnetic charge. It was conjecturedby [Montonen and Olive (1977)] that the N = 4 supersymmetric YMtheory for any gauge algebra g is mapped under the interchange ofelectric and magnetic charges, i.e., under e ↔ 1/e into the theory withdual gauge algebra g ∨ . When combined with the shift–invariance ofthe instanton angle θ this symmetry is augmented to the duality groupSL(2, Z), or a subgroup thereof.Of central interest to many of these exciting developments is the 4Dsupersymmetric YM theory with maximal supersymmetry, N = 4, andwith arbitrary gauge algebra g. As an N = 2 supersymmetric theory,the theory has a g−gauge multiplet, and a hypermultiplet in the adjointrepresentation of g with mass m.This generalized theory enjoys many of the same properties as theN = 4 theory: it has the same field contents; it is ultra–violet finite; it hasvanishing renormalization group β−function, and it is expected to haveMontonen–Olive duality symmetry. For vanishing hypermultiplet massm = 0, the N = 4 theory is recovered. For m −→ ∞, it is possible tochoose dependences of the gauge coupling and of the gauge scalar expectationvalues so that the limiting theory is one of many interesting N = 2supersymmetric YM theories. Amongst these possibilities for g = SU(N)for example, are the theories with any number of hypermultiplets in thefundamental representation of SU(N), or with product gauge algebrasSU(N 1 ) × SU(N 2 ) × · · · × SU(N p ), and hypermultiplets in fundamentaland bi-fundamental representations of these product algebras.An outstanding problem in non–Abelian gauge theory has been to makereliable predictions about the (non–perturbative) strong coupling region.N. Seiberg and E. Witten in [Seiberg and Witten (1994a); Seiberg andWitten (1994b)] studied N = 2 supersymmetric gauge theories in 4D withmatter multiplets. For all such models for which the gauge group is SU(2),they derived the exact metric on the moduli space of quantum vacua andthe exact spectrum of the stable massive states. Seiberg and Witten haveshown that the local part of the effective action is governed by a singleanalytic function F of a complex variable; they made an Ansatz for the Fthat satisfies all the physical criteria and embodies electromagnetic duality,thus directly connecting the weak to the strong coupling regions. A numberof new physical phenomena occurred, such as chiral symmetry breakingthat was driven by the condensation of magnetic monopoles that carried