Mathematical Methods for 4d N=2 QFTs

In this work we study different aspects of 4d N = 2 superconformal field theories. Not only we accurately define what we mean by a 4d N = 2 superconformal field theory, but we also invent and apply new mathematical methods to classify these theories and to study their physical content. Therefore, although the origin of the subject is physical, our methods and approach are rigorous mathematical theorems: the physical picture is useful to guide the intuition, but the full mathematical rigor is needed to get deep and precise results. No familiarity with the physical concept of Supersymmetry (SUSY) is need to understand the content of this thesis: everything will be explained in due time. The reader shall keep in mind that the driving force of this whole work are the consequences of SUSY at a mathematical level. Indeed, as it will be detailed in part II, a mathematician can understand a 4d N = 2 superconformal field theory as a complexified algebraic integrable system. The geometric properties are very constrained: we deal with special K ̈ahler geometries with a few other additional structures (see part II for details). Thanks to the rigidity of these structures, we can compute explicitly many interesing quantities: in the end, we are able to give a coarse classification of the space of "action" variables of the integrable system, as well as a fine classification -- only in the case of rank k = 1 -- of the spaces of "angle" variables. We were able to classify conical special K ̈ahler geometries via a number of deep facts of algebraic number theory, diophantine geometry and class field theory: the perfect overlap between mathematical theorems and physical intuition was astonishing. And we believe we have only scratched the surface of a much deeper theory: we can probably hope to get much more information than what we already discovered; of course, a deeper study of the subject -- as well as its generalizations -- is required. A 4d N = 2 superconformal field theory can thus be defined by its geometric structure: its scaling dimensions, its singular fibers, the monodromy around them and so on. But giving a proper and detailed definition is only the beginning: one may be interested in exploring its physical content. In particular, we are interested in supersymmetric quantities such as BPS states, framed BPS states and UV line operators. These quantities, thanks to SUSY, can be computed independently of many parameters of the theory: this peculiarity makes it possible to use the language of category theory to analyze the aforementioned aspects. As it will be proven in part V, to each 4d N = 2 superconformal field theory we can associate a web of categories, all connected by functors, that describe the BPS states, the framed BPS states (IR) and the UV line operators. Hence, following the old ideas of ‘t Hooft, it is possible to describe the phase space of gauge theories via categories, since the vacuum expectation values of such line operators are the order parameters of the confinement/deconfinement phase transitions. Mathematically, the (quantum) cluster algebra of Fomin and Zelevinski is the structure needed. Moreover, the analysis of BPS objects led us to a deep understanding of generalized S-dualities. Not only were we able to precisely define -- abstractly and generally -- what the S-duality group of a 4d N = 2 superconformal field theory should be, but we were also able to write a computer algorithm to obtain these groups in many examples (with very high accuracy).