Edge modes and Surface-Preserving Symmetries in Einstein-Maxwell Theory

TL;DR

This paper explores edge modes and surface-preserving symmetries in Einstein-Maxwell theory, revealing a semi-direct sum structure of symmetries and linking the Casimir of SL(2,R) to the area element.

Contribution

It introduces a combined diffeomorphism and gauge transformation, extends phase space with new fields, and characterizes the symmetry group structure in Einstein-Maxwell theory.

Findings

Symplectic potential is not invariant under combined transformations.

New fields produce edge modes in the theory.

Surface-preserving symmetry group is a semi-direct sum involving diffeomorphisms, SL(2,R), and U(1).

Abstract

Einstein-Maxwell theory is not only covariant under diffeomorphisms but also is under $U(1)$ gauge transformations. We introduce a combined transformation constructed out of diffeomorphism and $U(1)$ gauge transformation. We show that symplectic potential, which is defined in covariant phase space method, is not invariant under combined transformations. In order to deal with that problem, following Donnelly and Freidel proposal \cite{1}, we introduce new fields. In this way, phase space and consequently symplectic potential will be extended. We show that new fields produce edge modes. We consider surface-preserving symmetries and we show that the group of surface-preserving symmetries is semi-direct sum of 2-dimensional diffeomorphism group on a spacelike codimension two surface with $SL(2,\mathbb{R})$ and $U(1)$. Eventually, we deduce that the Casimir of $SL(2,\mathbb{R})$ is the area element, similar to the pure gravity case \cite{1}.