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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.

Using the supergravity dual and the plane-wave limit as a guide, we conjecture the asymptotic large coupling form of the Hagedorn temperature for planar N=4 super Yang-Mills to order 1/root lambda . This is two orders beyond the presently known behavior. Using the quantum spectral curve procedure of Harmark and Wilhelm, we show that our conjectured form is in excellent agreement with the numerical results.

We consider N = 4 supersymmetric gauge theories on the squashed three-sphere with six preserved supercharges. We first discuss how Wilson and vortex loops preserve up to four of the supercharges and we find squashing independence for the expectation values 2 of these 3-BPS loops. We then show how the additional supersymmetries facilitate the analytic matching of partition functions and loop operator expectation values to those in the mirror dual theory, allowing one to lift all the results that were previously established on the round sphere to the squashed sphere. Additionally, on the squashed sphere with four preserved supercharges, we numerically evaluate the partition functions of ABJM and its dual super-Yang-Mills at low ranks of the gauge group. We find matching values of their partition functions, prompting us to conjecture the general equality on the squashed sphere. From the numerics we also observe the squashing dependence of the Lee-Yang zeros and of the non-perturbative corrections to the all order large N expression for the ABJM partition function.

We consider the partition function for Euclidean SU(N) super Yang-Mills on a squashed seven-sphere. We show that the localization locus of the partition function has instanton membrane solutions wrapping the six "fixed" three-spheres on the S7. The ADHM vari-ables of these instantons are fields living on the membrane world volume. We compute their contribution by localizing the resulting three-dimensional supersymmetric field theory. In the round-sphere limit the individual instanton contributions are singular, but the singularities cancel when adding the contributions of all six three-spheres. The full partition function on the S7 is well-defined even when the square of the effective Yang -Mills coupling is negative. We show for an SU(2) gauge theory in this regime that the bare negative tension of the instanton membranes is canceled off by contributions from the instanton partition function, indicating the existence of tensionless membranes. We provide evidence that this phase is distinct from the usual weakly coupled super Yang -Mills and, in fact, is gravitational.

We consider mass-deformed theories with N >= 2 supersymmetry on round and squashed three-spheres. By embedding the supersymmetric backgrounds in extended supergravity we show that at special values of mass deformations the supersymmetry is enhanced on the squashed spheres. When the 3d partition function can be obtained by a limit of a 4d index we also show that for these special mass deformations only the states annihilated by extra supercharges contribute to the index. By using an equivalence between partition functions on squashed spheres and ellipsoids, we explain the recently observed squashing independence of the partition function of mass-deformed ABJ(M) theory on the ellipsoid. We provide further examples of such simplification for various 3d supersymmetric theories.

We study N = 2 theories on four-dimensional manifolds that admit a Killing vector v with isolated fixed points. It is possible to deform these theories by coupling position-dependent background fields to the flavor current multiplet. The partition function of the deformed theory only depends on the value of the background scalar fields at the fixed points of v. For a single adjoint hypermultiplet, the partition function becomes independent of the supergravity as well as the flavor background if the scalars attain special values at the fixed points. For these special values, supersymmetry at the fixed points enhances from the Donaldson-Witten twist to the Marcus twist or the Vafa-Witten twist of N = 4 SYM. Our results explain the recently observed squashing independence of N = 2* theory on the squashed sphere and provide a new squashing independent point. Interpreted through the AGT-correspondence, this implies the b-independence of torus one-point functions of certain local operators in Liouville/Toda CFT. The position-dependent deformations imply relations between correlators of partially integrated operators in any N = 2 SCFT with flavor symmetries.

We study the free energy of four-dimensional CFTs on deformed spheres. For generic nonsupersymmetric CFTs only the coefficient of the logarithmic divergence in the free energy is physical, which is an extremum for the round sphere. We then specialize to N = 2 SCFTs where one can preserve some supersymmetry on a compact manifold by turning on appropriate background fields. For deformations of the round sphere the c anomaly receives corrections proportional to the supersymmetric completion of the (Weyl)(2) term, which we determine up to one constant by analyzing the scale dependence of various correlators in the stress-tensor multiplet. We further show that the double derivative of the free energy with respect to the marginal couplings is proportional to the two-point function of the bottom components of the marginal chiral multiplet placed at the two poles of the deformed sphere. We then use anomaly considerations and counter-terms to parametrize the finite part of the free energy which makes manifest its dependence on the Kahler potential. We demonstrate these results for a theory with a vector multiplet and a massless adjoint hypermultiplet using results from localization. Finally, by choosing a special value of the hypermultiplet mass where the free energy is independent of the deformation, we derive an infinite number of constraints between various integrated correlators in N = 4 super Yang-Mills with any gauge group and at all values of the coupling, extending previous results.