Resurgence in Supersymmetric Localisable Quantum Field Theories

In this thesis we consider the application of resurgence and Picard-Lefschetz theory to supersymmetric localisable quantum field theories in 2, 3 and 4 dimensions. We consider two problems. First, in the theories we study, observables can be calculated exactly using localization methods, and written in the form of a transseries. However in each non-perturbative sector, the associated perturbation series is not asymptotic, seemingly rendering the application of resurgence theory impossible. This problem is solved by deploying a Cheshire Cat analysis; we slightly deform the theory rendering the series asymptotic, perform a resurgence analysis in the deformed theory, and analytically continue the deformation back to 0, returning the non-perturbative data in the undeformed theories. This is achieved in N=(2,2) theories in 2 dimensions, and N=2 theories in 3 dimensions. Comments are made about how we might generalize this to 4 dimensional theories. The second problem is the disappearance of the resurgence triangle structure in N = 2 theories on a 3-sphere. This structure is recovered by means of introducing a complex squashing parameter, uncovering a hidden topological angle present in the theory. Finally, in the two above mentioned theories and N = 2 theories in 4 dimensions, a method is given for how to combine a resurgence analysis with additional nonperturbative structures present in these theories to compute non-perturbative contributions with different topological charge from the perturbative data.