Aspects of symmetry structures in quantum field theory via brane dynamics

This thesis aims to explore the correspondence between operators implementing symmetries in a quantum field theory (QFT) and branes (extended dynamical objects present in string theory) that embeds in the holographic dual backgrounds of string theory. In particular the thesis will focus on d-dimensional QFTs and their full set of (discrete) generalized symmetries (mathematically described by fusion categories). Certain theories can be realized holographically via the AdS/CFT correspondence or in string theory via geometric engineering. If such construction exists, we are allowed to explore the set of probe branes extending in this background in order to study the symmetry operators and the mathematical category that they form. The thesis will specifically aim to compute the lowest-energy configurations that these branes form in the string theory background that engineers the studied QFT. It consists of studying the world-volume effective field theory on the brane and its fluctuations at a classical level. The final brane configuration will be given by solving the variation calculus of the brane effective action. In the string theory geometric background this configuration will describe a specific aspect of the symmetry of the engineered (or dual) QFT. This will finally allow us to access the sometimes-complicated mathematical structure of the underlying generalized symmetries in a more physical way.