Geometric and Category-Theoretic Structures in Generalized Symmetries: Gauging and Decomposition

Abstract

The concept of symmetry has been a crucial guiding principle in the development of modern physics. Recently, its mathematical characterization has been significantly expanded under the framework of "generalized symmetries," leading to conceptual, theoretical, and practical advancements in quantum field theory and related fields. Simultaneously, this has generated a fruitful collaboration with contemporary mathematics. This thesis explores several aspects of generalized symmetries: on the one hand, their relation with several branches of mathe- matics, notably category theory, geometry, and algebra; on the one hand, their implications for physical phenomena such as gauging and decomposition. The present work is divided into two main parts, following the two-fold generalization of symmetries in the generalized sym- metries framework: that of higher-form symmetries, and that of non-invertible symmetries. The first part delves into higher-form symmetries and their relation with their mathematical counterpart, higher geometry. We provide rigorous mathematical constructions making this connection concrete, using the theory of higher principal bundles with adjusted connections in cohesive ∞-topos theory. Then, we concentrate on a specific example, that of three- dimensional σ-models involving quotients by higher finite groups. In this case study, we derive decomposition statements and work out the role higher cohomology groups play. The second part concentrates on non-invertible symmetries, specializing in two-dimensional theo- ries. Following the same model, we first develop a general mathematical construction, in this case concerning the calculation of partition functions of theories with gauged non-invertible symmetries. Then, we turn to applying this formalism to describe decomposition. Along the way, we constantly highlight the natural connections with non-commutative algebra, chiefly with Hopf algebras and their representation categories.