# Gauge from Holography and Holographic Gravitational Observables

Zapata, José A.  (Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Morelia, MICH, Mexico)

08 February 2019

Abstract: In a spacetime divided into two regions ${U}_{\mathrm{1}}$ and ${U}_{\mathrm{2}}$ by a hypersurface $\mathrm{\Sigma }$ , a perturbation of the field in ${U}_{\mathrm{1}}$ is coupled to perturbations in ${U}_{\mathrm{2}}$ by means of the holographic imprint that it leaves on $\mathrm{\Sigma }$ . The linearized gluing field equation constrains perturbations on the two sides of a dividing hypersurface, and this linear operator may have a nontrivial null space. A nontrivial perturbation of the field leaving a holographic imprint on a dividing hypersurface which does not affect perturbations on the other side should be considered physically irrelevant. This consideration, together with a locality requirement, leads to the notion of gauge equivalence in Lagrangian field theory over confined spacetime domains. Physical observables in a spacetime domain $U$ can be calculated integrating (possibly nonlocal) gauge invariant conserved currents on hypersurfaces such that $\partial \mathrm{\Sigma }\subset \partial U$ . The set of observables of this type is sufficient to distinguish gauge inequivalent solutions. The integral of a conserved current on a hypersurface is sensitive only to its homology class $\left[\mathrm{\Sigma }\right]$ , and if $U$ is homeomorphic to a four ball the homology class is determined by its boundary $S=\partial \mathrm{\Sigma }$ . We will see that a result of Anderson and Torre implies that for a class of theories including vacuum general relativity all local observables are holographic in the sense that they can be written as integrals of over the two-dimensional surface $S$ . However, nonholographic observables are needed to distinguish between gauge inequivalent solutions.

Published in: Advances in High Energy Physics 2019 (2019) 9781620