Symmetries and their Holographic Aspects in Quantum Field Theory

In this thesis we study generalized global symmetries in Quantum Field Theory. In the modern definition, symmetries are viewed as extended topological operators. On the one hand this new paradigm efficiently encodes all the information a symmetry usually comes with (e.g. 't Hooft anomalies) and on the other it allows for many interesting generalizations. After the Introduction, the first chapter is a brief review of basic concepts used throughout the thesis. The rest of this work is conceptually divided into two parts. In the first one, we consider conventional symmetries in exotic theories characterized by randomly distributed interaction couplings as well as non-invertible symmetries in well-known $2$ dimensional theories corresponding to Calabi-Yau non-linear sigma models. In the second part instead we showcase two applications of the holographic approach to symmetries, the SymTFT. In particular we show how this tool can be used efficiently to discuss anomalies (defined as obstructions to gauging) of non-invertible symmetries in higher dimensions, and also how it can be used to establish holographic correspondences relating a Topological Quantum Field Theories and the universal effective theories describing the spontaneous breaking phases of continuous symmetries.