Studies in non-anticommutative gauge theories, geometric dualities, and twistor strings

In this Dissertation we consider three different topics. The first one is the study of instantons in U(2) super Yang-Mills theories defined on non-anticommutative superspace. We extend the ordinary instanton calculus to this class of theories by solving the appropriate equations of motion iteratively in the deformation parameter C. In the case without matter, we solve the equations exactly. We find that the SU(2) part of the instanton is the same as in ordinary SU (2) N = 1 super Yang-Mills, but acquires in addition a non-trivial U(1) part which depends on the fermionic collective coordinates and the deformation parameter C. In the case with matter we solve the equations of motion to leading order in the coupling constant. We find that also the profile of the matter fields is deformed through linear and quadratic corrections in C. The instanton effective action for pure gluodynamics is unaffected by C, but gets a contribution of order C2 in addition to the usual 't Hooft term when the matter is included. Surprisingly, this extra term is compensated by a deformation in the moduli space integration measure and the usual non-perturbative ADS superpotential remains the same as in N = 1 super Yang-Mills.; The second topic is the analysis of a general class of symmetries of hyperkahler quotients which can be interpreted as analogues of Seiberg duality for arbitrary N = 2 supersymmetric quiver gauge theories in the Higgs branch. We also find a new classical duality for Kahler potentials of N = 1 quivers that generalizes the trivial duality Gr( N, N+M) ≡ Gr(M, N +M) for Grassmannians. A limit application of the N = 2 duality yields some new exotic realizations of ALE spaces as hyperkahler quotients.; The third topic is the extension of a recent construction by Witten which reinterprets the perturbative expansion of N = 4 super Yang-Mills as a D-instanton expansion of the topological B-model on the supermanifold C P¾. We generalize the concept of ordinary orbifolds by considering fermionic orbifolds of C P¾ in order to obtain superconformal quiver theories with reduced supersymmetry from Witten's framework. We consider specific examples of N = 1 and N = 2 orbifolds, and explicitely compute some amplitudes for the corresponding theories. We finally consider possible extensions of Witten's construction to gravity amplitudes.