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Analytic tools to brane technology in N=2 gauge theories with matter

Exact solutions to the low-energy effective action (LEEA) of the four-dimensional N=2 supersymmetric gauge field theories are known to be obtained either by quantum field theory methods from S-duality in the Seiberg-Witten approach, or by the Type-IIA superstring/M-Theory methods of brane technology. After a brief review of the standard field-theoretical results for the N=2 gauge (Seiberg-Witten) LEEA, we consider a field-theoretical derivation of the exact hypermultiplet LEEA by using the N=2 harmonic superspace methods. We illustrate our techniques on a number of explicit examples. Our main purpose, however, is to discuss the existing analytical (calculational) support for the alternative methods of brane technology. We summarize known exact solutions to the eleven-dimensional and ten-dimensional type-IIA supergravities, which describe classical configurations of intersecting BPS branes with eight supercharges relevant to the non-perturbative N=2 gauge field theory with fundamental hypermultiplet matter. The crucial role of M-Theory in providing a classical resolution of singularities in the ten-dimensional (Type-IIA superstring) brane picture, as well as the N=2 extended supersymmetry in four dimensions, are made manifest. The two approaches to a derivation of the exact N=2 gauge theory LEEA are thus seen to be complementary to each other and mutually dependent.