Symmetries, charges and thermodynamics at the corners of spacetime

In the presence of boundaries, the physics of systems with gauge symmetries becomes significantly richer: new degrees of freedom emerge, and would-be redundancies are promoted to true symmetries. In recent years, the study of boundaries has rapidly grown, especially in the context of gravity, where it is expected to shed light on foundational problems such as the definition of observables and the origin of gravitational entropy. This thesis covers some aspects of this wide area. After a brief introduction to the necessary technical tools in classical mechanics, we present the covariant phase space formalism, which enables us to apply these tools to study the symplectic structure of gravity in a covariant way. In this setting, we can formulate a refined definition of symmetry that allows us to distinguish between physical symmetries and gauge redundancies, and identify the associated new degrees of freedom. Next, we discuss the relevance of boundaries in the formulation of a variational principle for field theories and how it is related to ambiguity resolution. At timelike boundaries, imposing suitable conditions provides a way to unambiguously define charges associated with symmetries and to ensure their conservation, in accordance with the spirit of Noether's work. We investigate the compatibility of this construction with renormalization, showing that the running of couplings, induced by quantum effects, can jeopardize the variational principle. Afterward, we apply covariant phase space techniques to the thermodynamics of black holes in Lanczos--Lovelock theories. These are the simplest extensions of General Relativity, as they introduce new interactions but no additional degrees of freedom. Our results demonstrate that the covariant phase space formalism is suited to deriving both the first law of black hole thermodynamics and the Smarr formula, of which we present a natural generalization. We also find a background-independent prescription for the thermodynamic potentials associated with the theory's couplings. Finally, we turn to the problem of identifying the degrees of freedom that exhibit the newly uncovered symmetries. We present the universal form of the symmetry group and study its representation theory, paying attention to the complications that may appear in a quantum context. As a first step of a larger program, we construct the unitary irreducible representations of the quantum corner symmetry group in two-dimensional gravity using the orbit method. We find that a dressed version of the generators of boosts (in the plane normal to the corner) plays a key role in this analysis. This research adheres to the paradigm that much can be learned about gravity by exploiting its classical symmetry structure. In the spirit of Wigner’s program, identifying the correct symmetry group --- which is possible only after a careful construction of the charges --- is a crucial step in understanding gravity at the quantum level.