Abstract :
[en] Since the early 20th century, physicists have pursued a quantum theory of gravitation, with notable breakthroughs such as 1970 Bekenstein's proposal linking black hole entropy to the event horizon surface area. By leveraging upon this observation, 't Hooft and Susskind then introduced the holographic principle, proposing that quantum gravity degrees of freedom might be encoded in lower-dimensional surfaces. Maldacena's work extended this idea with the AdS/CFT correspondence, connecting gravitational theories on Anti de Sitter (AdS) spacetime to conformal field theories (CFT) defined on the boundary of AdS. However, this duality involves a negative cosmological constant, conflicting with its observed positive value. This PhD thesis aims to explore selected aspects of the AdS/CFT correspondence and their generalization in the limit of vanishing cosmological constant.
In particular, this manuscript focuses on asymptotic symmetries and corner or, equivalently, surface charges through the Lagrangian approach to general relativity and covariant phase space. This framework offers insights into observables in gravity and dual gauge theories. Identifying physical asymptotic symmetries allows indeed one to identify the global symmetries of the dual conformal field theory and thus sets up crucial constraints allowing to identify the latter. In their turn, the relevant symmetries are selected by non-trivial surface charges. However, determining the surface charges faces challenges due to divergences as one approaches the asymptotic boundary. To tackle this, we confront variational and symplectic structure ``renormalization schemes'', opting for the latter for a systematic study.
To illustrate these techniques, we analyze asymptotic symmetries of Maxwell theory in both Anti de Sitter and flat backgrounds, aiming to recover the flat space results from AdS. This leads to studying the relaxation of the standard Fefferman-Graham gauge within Einstein gravity, resulting in the Weyl-Fefferman-Graham gauge, which restores the broken boundary Weyl covariance and introduces new charges associated with the underlying Weyl geometry. This raises questions about new charges related to different available choices for the underlying symplectic structure. These issues are also linked with the current efforts in the literature to transition towards gauge-free analyses. As a general guideline, one could argue that the more physical charges the better, as this would lead to larger symmetry algebras that are more powerful to organize the observables of the theory. While the Fefferman-Graham gauge is suited to AdS/CFT, it falls short for asymptotically flat spaces. In contrast, the Bondi gauge, designed for flat spacetimes and gravitational waves, is universally applicable. Introducing a relaxation, the covariant Bondi gauge combines advantages of all aforementioned gauges, providing insights into boundary anomalies through a fluid/gravity representation and deepening the understanding of the holographic duality through new finite corner physical charges.
