What is the physical motivation of Seiberg-Witten theory? Seiberg-Witten equations ushered in a new era in gauge theory and enabled mathematicians to find simpler proofs of some of the deepest results of Donaldson and others. The mathematical formalism of the Seiberg-Witten equations is straightforward but I was wondering about its physical motivation. Can this be formulated in terms accessible to a nonexpert?
 A: As my PhD adviser Tom Banks (who was very helpful in the development of these SW papers and is thanked in the acknowledgements) has stressed, the actual physical intuition that Seiberg and Witten exploited to find the right answer was tightly connected with string theory. 
In the final form of the papers, they eliminated all dependence on string theory – possibly in order to increase the number of readers of the papers (not to discourage people who haven't learned or dislike string theory). But this editing turned out to be temporary – the stringy realization of these physical systems and formulae is so useful and natural that it became widespread, anyway.
In the papers, Seiberg and Witten have analyzed the mathematical functions (generalizing the form of the potential energy) that fully describe the behavior of electrically and magnetically charged objects in a theory with supersymmetry. These functions, like the "prepotential", receive "quantum corrections" which is the difference between the exact answer and the answer in the classical limit.
The quantum corrections are "difficult" and the functions could a priori be the most difficult general functions one could imagine. However, supersymmetry guarantees that the functions are holomorphic (a mathematical property but one that physicists in this kind of business "feel" almost intuitively, as if it were very physical). And when this fact is connected with some rules about the "duality" between electric and magnetic charges (the generalization of the permutation symmetry between electricity and magnetism that we know even from Maxwell's equations) and with the behavior around special points of the configuration space for the scalar fields, one may determine all the functions exactly.
One may solve these theories and the holomorphy; right monodromies (transformation of magnetic charges into mixtures of magnetic and electric charges under certain "cyclic" operations); and right limits at some points are enough to uniquely determine these solutions.
A special importance of the string-theoretical thinking was that quantum field theory seemingly "totally distinguishes" the elementary particles – which are often electrically but not magnetically charged – from magnetic monopoles – which are not elementary because they have to be described by an extended solution of the equations of motion. No symmetry between them seems to be possible. However, string theory blurs the differences between such elementary and composite objects, allows lots of transformations that seem forbidden in quantum field theory. And these transformations turned out to be relevant not just in string theory but also in quantum field theory.
The functions defining the Seiberg-Witten solutions may be visualized as certain geometric shapes or manifolds. In string theory, such shapes may be literally interpreted as shapes of some objects or manifolds in the higher-dimensional spacetime. String theory geometrizes lots of physical laws and phenomena that seemingly didn't involve any interesting geometry.
