**Abstract:**

In the first part of the thesis we exploit supersymmetric localization to study aspects of supersymmetric gauge theories relevant to holography.

In chapter 2 we study the 1/2-BPS circular Wilson loop in the totally antisymmetric representation of the gauge group in N = 4 supersymmetric Yang-Mills. We compute the first 1/N correction at leading order in ’t Hooft coupling by means of the matrix model loop equations for comparison with the 1-loop effective action of the holographically dual D5-brane. Our result suggests the need to account for gravitational backreaction on the string theory side.

In chapter 3 we solve the planar N = 2^∗ super-Yang-Mills theory at large ’t Hooft coupling again using localization on S^4. The solution permits detailed investigation of the resonance phenomena responsible for quantum phase transitions in infinite volume, and leads to quantitative predictions for the semiclassical string dual of the N = 2^∗ theory.

The second part of the thesis deals with the Schwinger effect in scalar quantum electrodynamics and in bosonic string theory. Chapter 4 presents a detailed study of the semiclassical expansion of the world line path integral for a charged relativistic particle in a constant external electric field. It is demonstrated that the Schwinger formula for charged particle pair production is reproduced exactly by the semiclassical expansion around classical instanton solutions when the leading order of fluctuations is taken into account. By a localization argument we prove that all corrections to this leading approximation vanish and that the WKB approximation to the world line path integral is exact.

Finally, in chapter 5 we analyse the problem of charged string pair creation in a constant external electric field. We find the instantons in the worldsheet sigma model which are responsible for the tunneling events, and evaluate the sigma model partition function in the multi-instanton sector in the WKB approximation. We further identify a fermionic symmetry associated with collective coordinates, which we use to localize the worldsheet functional integral onto its WKB limit, proving that our result is exact.