Generalized Global Symmetries, Quantum Field Theory, and Geometry

Over the past ten years tremendous progress has been achieved in quantum field theory thanks to the realization that symmetries can be interpreted in terms of the action of topological defects on the space of operators. This gives a generalization of the notion of symmetries, explicitly realized in many instances exploiting Lagrangian field theory techniques, uncovering a rich and unexpected landscape of physical effects. The product rules for symmetry defects are organized by higher structures, and moreover examples of non-invertible symmetry defects have been found, indicating that global symmetries are categorical. Thanks to geometric field theory methods in string theory, an understanding of generalized global symmetries for field theories without a conventional Lagrangian formulation is at reach in terms of appropriate defect groups, determined exploiting the dynamics of membranes in the string theory realization.