Edge Modes: from classical to quantum and from discrete to continuum

Abstract

In this talk, I will describe recent developments in our understanding of edge modes associated to subregions in gauge theories, leveraging their realization as reference frames. I will begin with the picture in classical Maxwell theory, where I will introduce the concept of subregional Goldstone mode as a relational observable parametrizing the corner symmetry group. With this as a guiding principle, I will then move on to non-Abelian lattice gauge theory, where we can carry out the construction directly at the quantum level. I will characterize a novel hierarchy of relational subregional algebras, which encompasses the so-called electric and magnetic center algebras usually considered in the literature, for which we provide a new general definition. This leads to corresponding entropy hierarchies. Interestingly, some of the relational algebras can be factors, and so the physical Hilbert space factorizes. Except in the Abelian case, the subregional Goldstone mode is generically only defined on a subspace of the Hilbert space, stemming from the incompleteness of certain edge mode frames. I will conclude with on-going efforts to have a quantum description of edge modes in the continuum, employing algebraic QFT methods. An overarching theme will be the relation between the subregional Goldstone mode and the asymptotic soft sector of the theory.