On squashed spheres and warm strings : Applications of supersymmetric localization and integrability in gauge and string theory

Non-perturbative aspects of quantum field theories are notoriously hard to explore. In this thesis we study applications of two different techniques that give non-perturbative results for supersymmetric quantum field theories. The first exact technique we use is supersymmetric localization which allows for the exact computation of partition functions on compact manifolds and squashed spheres of various dimensions are the manifolds of our choice. In three dimensions we use the squashed sphere partition functions to test dualities of N=4 gauge theories. For a squashed sphere preserving six supercharges we lift analytic results from the round sphere. On a squashed sphere preserving 4 supercharges we numerically evaluate the ABJM and N=8 super Yang-Mills (SYM) partition functions at low rank and find equality within estimated error margins. In four dimensions we present a framework to obtain partially integrated correlators of 4d N=2 gauge theories from their localized partition functions. Moreover we discuss the general form of the free energy of N=2 superconformal field theories on deformed four-dimensional spheres and use localization in N=4 SYM for an explicit example. In seven dimensions we study the super Yang-Mills on a sphere and propose a contribution of three-dimensional membrane instantons to its localized partition function. We then outline an approach to study the weak negative coupling limit of the SYM theory on the seven-sphere. As the second approach to exact results we use the integrability of N=4 SYM and ABJM theory in the planar limit. Using the quantum spectral curve we compute the Hagedorn temperature for finite coupling both in N=4 SYM and ABJM theory. On the dual AdS side we use an effective model to compute subleading terms in the curvature expansion of the Hagedorn temperature. We use the numeric CFT calculation to conjecture the analytic form of an unfixed coefficient in the effective model.