Notes from the bulk

In most cases Quantum Field Theories (QFTs) are considered without boundaries and have been successful in providing descriptions of fundamental interactions, including gravity and cosmology. This is because one is generally interested in bulk effects, where the boundary can be neglected. Nevertheless, boundaries do exist, and in some cases, their effects are self-evident and dominant. Important phenomena pertaining to condensed matter physics, like the Fractional Quantum Hall Effect and the behavior of Topological Insulators have been explained in terms of topological QFTs with boundaries. This is rather counterintuitive : topological QFTs, when considered without boundaries, have a vanishing Hamiltonian and no energy-momentum tensor. They might appear as the least physical theories one can imagine. Despite this, when a boundary is introduced, an extremely rich physics emerges, which can be observed experimentally. The scope of this Ph.D thesis is to study the effects of the presence of a boundary from a Quantum Field Theoretical perspective, searching for new physics and explanations of observed phenomena. In particular, thanks to the formal QFT setting, the issue of the existence of local, accelerated, edge modes in Hall systems is analyzed and understood in terms of the bulk-to-boundary approach as related to a curved background in topological QFTs with boundary. Within this formalism the induced metric on the boundary can be associated to the ad hoc potential introduced in the phenomenological models in order to obtain such non-constant edge velocities. This also leads to the prediction of local modes for Topological Insulators, and Quantum Spin Hall systems in general. The paradigm for which only topological QFTs have a physical content on the boundary is broken, and also non-Topological Quantum Field Theories such as fracton models and Linearized Gravity are shown to have non-trivial boundary dynamics. Indeed due to the breaking of their defining symmetry both models have a current algebra of the Kac-Moody type on the boundary. In the case of fractons this algebra is in a generalized form, which also appears in some kinds of higher order Topological Insulators, a sign of a possible relation between these materials and edge states of fracton quasiparticles. Concerning the theory of Linearized Gravity, instead, the algebra is a standard Kac-Moody one, whose presence was suspected, but never proved before. Physical results on the boundary range between condensed matter, elasticity and (massive) gravity models. A collateral result, which enrich this Thesis, is the building of a new covariant QFT for fractons with a peculiar gauge structure. This new model better highlight the properties of these quasiparticles.