Gauge It or Break It: Lectures on Global Symmetries and the Cobordism Conjecture

Abstract

These lecture notes introduce the No Global Symmetries Conjecture in the context of the Swampland Program, with particular emphasis on the Cobordism Conjecture. We begin with a review of ordinary and higher-form symmetries, their realization by extended operators, and mechanisms for gauging or breaking them. We then discuss the no global symmetries principle, including evidence from black hole physics, string perturbation theory, and AdS/CFT, and its relation to other Swampland constraints. A central theme is the reformulation of symmetry constraints in topological terms: we introduce bordism groups, with special focus on spin bordism, and explain how they provide a natural framework for organizing would-be global symmetries in gravitational theories. With this at hand, we discuss the Cobordism Conjecture, which proposes that nontrivial bordism classes are eliminated by dynamical processes in quantum gravity. We conclude with examples and a discussion of dynamical aspects involving defects, boundaries, and related open questions. These notes are based on lectures delivered by Matilda Delgado at the 21st Modave Summer School on Mathematical Physics. A similar series of lectures was also presented at the KMPB School on D-branes in Physics and Mathematics.