Seiberg-Witten Theory et Riemann Surfaces
In this thesis we study Riemann surfaces with a view to understanding Seiberg Written theory. In their seminal work, Seiberg and Witten derived the low energy approximation of the supersymmetric gauge theory having for gauge group SU(2) and N = 2 supersymmetries by relating the problem to the data of an elliptic curve. The supersymmetric gauge theory is completely determined by a holomorphic function of the fields, the prepotential, that contains both perturbative and non-perturbative corrections of the theory. The identification with the elliptic curve allowed Seiberg and Witten to compute the prepotential of the theory exactly, including the non-perturbative corrections which are extremely difficult to obtain by traditional techniques. Seiberg and Witten identified the elliptic curve by studying the singularity structure of the moduli space of the theory, and comparing the monodromy data of the physical problem with the monodromy data coming from the elliptic curve. In this thesis we review the arguments of Seiberg and Witten to arrive at the monodromy data of the physical problem. We then derive this same data from the elliptic curve in detail thus showing the correspondence. Using this correspondence, we derive the asymptotic expressions for the variables (aD, a) determined in the physical theory by computing some contour integrals on the elliptic curve in some detail. Finally, using hypergeometric representations of the contour integrals, we also reproduce the calculation of the non-perturbative corrections and determine some of the so called instanton numbers which arise from them.
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