MODERN TOPICS IN SUPERSYMMETRY

Quantum field theory (QFT) stands as one of the most remarkable mathematical frameworks for understanding fundamental particles and their interactions. Its predictive power has delivered profound insights into the behavior of particles, culminating in the Standard Model of particle physics. However, the theory encounters significant challenges when confronted with the complexities of strong coupling regimes, such as those governing phenomena like confinement and the nature of strongly interacting quarks and gluons. These intriguing phenomena remain pivotal open questions in theoretical and mathematical physics. In this context, supersymmetry emerges as a powerful and elegant tool for exploring the dynamics of strongly coupled gauge theories. Its application to gauge theories has led to groundbreaking insights into the non-perturbative aspects of QFT, especially through the concept of dualities. Supersymmetric dualities, in fact, usually provide weakly coupled descriptions to asymptotically free gauge theories. These dualities have been checked in various fashions: 't Hooft anomaly matching, supersymmetric localization for partition functions on various manifolds, good behavior under renormalization group flows, matching of global symmetries and, as a more recent development, matching of generalized symmetries by the SymTFT approach. String theory and its higher-dimensional counterpart, M-theory, provide an elegant and unified approach to these aspects. The dynamics of D-brane systems, as well as their embedding in geometric engineering within M-theory, usually give a more fundamental description of such dualities. Particularly interesting are non-Lagrangian theories that can be built in string/M-theory. Likewise, the AdS/CFT correspondence also provides the tools to probe the quantum dynamics of exotic gravitational systems such as black holes (BH) by using QFT tools like the superconformal index. Here, the BH Hawking-Page (HP) phase transition could be exploited to probe the confinement/deconfinement transition in four-dimensional maximally supersymmetric YM and beyond. Motivated by this, in this thesis we will mainly analyze supersymmetric QFTs in various dimensions. Part of the thesis will be devoted to the study of supersymmetric dualities, where we will initially discuss the fundamental result of Seiberg, and then apply such reasoning to theories in three dimensions in the presence of adjoint matter. These dualities will be checked by means of matching the 3d partition functions on both the electric and magnetic sides. Then we will focus on the topic of generalized symmetries in a special class of theories which are non-Lagrangian and arise in string theory by stacking D3 branes on particular orbifold geometries. Next, we will delve into the realm of AdS/CFT by using its powerful tools to, on one hand, compute observables of two-dimensional SCFTs arising from compactifications of M5 branes on a special geometry known as a Spindle, and on the other hand to compute the microstates of black holes in the presence of probe defects from the dual field theory.