N=2 dualities, quivers and boundaries

Since the seminal work by Pestun on the supersymmetric localization of $\mathcal{N}=2$ super-Yang-Mills (SYM) on round $S^4$, supersymmetric gauge theories on curved backgrounds have become an intensely active area of research. In particular, much effort has been put into understanding general curved manifolds, different numbers of supersymmetries and various observables. A powerful tool, localization has inspired the remarkable work by Alday, Gaiotto and Tachikawa on the relation between Liouville theory and $\mathcal{N}=2$ SYM, the so-called AGT correspondence. This beautiful relation sheds light on a curious web of dualities between various theories in diverse dimensions, connecting their underlying geometrical structure to the deep language of string theory and M-theory. \\ \noindent In this work we study $\mathcal{N}=2^* \ SU(N)$ SYM on the hemisphere $HS^4$ coupled to a 3d boundary quiver theory that engineers Nahm pole boundary conditions for the bulk theory, effectively leaving us with a 4d $\mathcal{N}=1^*$ gauge theory. We begin with an in-depth review of supersymmetric localization for $\mathcal{N}=2$ gauge theories and discuss the construction of such theories on curved backgrounds. We then move on to a quick survey of the AGT correspondence and its M-theoretical setting, describing also the more general setup of the 3d-3d correspondence which provides the proper geometrical background of our work. After that, we construct the 4d gauge theory coupled to a 3d Nahm pole boundary quiver and obtain its matrix model for the $SU(2)$ case.