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Non-perturbative aspects of gauge and string theories and their holographic relations

In this thesis we discuss non-perturbative phenomena emerging in gauge and in string/supergravity theories. We compute the partition function of 5D minimal supersymmetric U(1) gauge theory with extra adjoint matter in general Omega-background. These N=2^* theories are massive deformations of N=4 SYM and can be thought of as a minimal supersymmetric five dimensional theory compactified on a circle. Such partition functions encode very rich topological information. We show that this partition function at some special values of the parameters directly reproduces the generating function for the Poincare` polynomial of the moduli space of instantons. We discuss the basic aspects of worldsheet and penta-brane instantons as well as (unoriented) D-brane instantons and threshold corrections to BPS-saturated couplings in superstring theories. We consider non-perturbative superpotentials generated by `gauge' and `exotic' instantons living on D3-branes at orientifold singularities. We also discuss the interplay between worldsheet and D-string instantons on T^4/Z_2. We focus on a 4-fermi amplitude and offer a multi D-string instanton interpretation. We discuss also AdS/CFT correspondence and the role of instantons there. We concentrate on AdS4/CFT3 correspondence. We revisit Kaluza-Klein compactification of 11-d supergravity on S^7/Z_k using group theory techniques that may find application in other flux vacua with internal coset spaces. Among the SO(2) neutral states, we identify marginal deformations and fields that couple to the recently discussed worldsheet instanton of Type IIA on CP^3. We also discuss charged states, dual to monopole operators, and the Z_k projection of the Osp(8|4) singleton and its tensor products. We show that the doubleton spectrum may account for N =6 higher spin symmetry enhancement in the limit of vanishing 't Hooft coupling in the boundary CS theory.