Studies of five dimensional superconformal field theories, localization and geometry

In this thesis, we study the U-plane of rank-one 5d N = 1 superconformal field theories (SCFTs), which is the one-dimensional Coulomb branch (CB) of such theories on $R^4 \times S^1$. This circle compactification gives us 4d N = 2 supersymmetric field theories of ‘Kaluza-Klein (KK) type’, and thus the Coulomb branch physics can be described by its Seiberg-Witten (SW) geometry. The total space of the SW geometry, which consists of the SW curve fibered over the CB, can be modelled as a rational elliptic surface. As such, a classification of all possible Coulomb branch configurations, for the 5d field theories and their 4d descendants, is given by Persson’s classification of rational elliptic surfaces. This formalism trivializes renormalization group flows, while also containing information about the global form of the flavour symmetry group and the one-form symmetry, through the Mordell-Weil group of the SW geometry. Moreover, in a number of important instances, the U-plane is a modular curve, and we use its beautiful modular properties to investigate aspects of the low-energy physics, such as the spectrum of light particles at strong coupling and the associated BPS quivers. A related approach to studying the strong-coupling dynamics of these theories is to consider them on curved backgrounds M5. Our focus is on closed five-manifolds M5 which are principal circle bundles over simply-connected K\"ahler four-manifolds, $M_4$. Starting with the topologically twisted 4d N = 2 KK theory on $M_4$, we propose a new approach to compute the supersymmetric partition function on $M_5$ through the insertion of a fibering operator in the 4d theory, which introduces a non-trivial fibration over $M_4$. We determine the so-called Coulomb branch partition function on any such $M_5$, which is conjectured to be the holomorphic ‘integrand’ of the full partition function. We precisely match the low-energy effective field theory approach to explicit one-loop computations, and we discuss the effect of non-perturbative 5d BPS particles ...