In this paper, we find the boundary dual of the symplectic form for the bulk fields in any entanglement wedge. The key ingredient is Uhlmann holonomy, which is a notion of parallel transport of purifications of density matrices based on a maximisation of transition probabilities. Using a replica trick, we compute this holonomy for curves of reduced states in boundary subregions of holographic QFTs at large N , subject to changes of operator insertions on the boundary. It is shown that the Berry phase along Uhlmann parallel paths may be written as the integral of an abelian connection whose curvature is the symplectic form of the entanglement wedge. This generalises previous work on holographic Berry curvature.

The covariant phase space technique is a powerful formalism for understanding the Hamiltonian description of covariant field theories. However, applications of this technique to problems involving subregions, such as the exterior of a black hole, have heretofore been plagued by ambiguities arising at the boundary. We provide a resolution of these ambiguities by directly computing the symplectic structure from the path integral, showing that it may be written as a contour integral around a partial Cauchy surface. We comment on the implications for gauge symmetry and entanglement.

We describe a completely general and fully non-perturbative framework for constructing dynamical reference frames in generally covariant theories, and for understanding the gauge-invariant observables that they yield. Our approach makes use of a 'universal dressing space', which contains as a subset every possible dynamical frame. We describe examples of such frames, including matter frames, a popular construction based on boundary-anchored geodesics and one using minimal surfaces -but our formalism does not depend on the existence of a boundary. The class of observables we construct generalises and unifies the dressed and relational approaches to constructing gravitational observables, including single-integral and canonical power-series constructions. All these (possibly gravitationally charged) relational observables describe physics in a precise sense relative to the dynamical frame and respect a notion of 'relational' locality based on the relationships between fields. By using 'relational atlases', i.e. collections of dynamical frames glued together by field-dependent maps (which are relational observables too), we can construct relationally local observables throughout spacetime. This further establishes a framework for dynamical frame covariance that permits us to change between arbitrary relational frame perspectives. Relational locality obeys many desirable properties: we prove that it satisfies microcausality in the bulk (in tension with previous work done mainly in a perturbative setting which we comment on), and show that it permits a relational version of local bulk dynamics. Relational locality is therefore arguably more physically meaningful than the ordinary notion of locality. Thus, our formalism -which we argue to be an updated, gauge-invariant version of general covariance -refutes the commonly claimed non-existence of local gravitational bulk physics.

Large gauge transformations (LGT) in asymptotically flat space are generated by charges defined at asymptotic infinity. No method for unambiguously localising these charges into the interior of spacetime has previously been established. We determine what this method must be, and use it to find localised expressions for the LGT charges. Applying the same principle to the case of a charged black hole spacetime leads to angle-dependent generalisations of the Smarr formula and the first law of black hole mechanics, both of which have important thermodynamical implications. In particular, the presence of a heat current intrinsic to the event horizon is observed.jjvk2@cam.ac.uk

We describe an explicit mechanism for the emergence of a dynamical holographic bulk from the structure of entanglement in a quantum state. We start with a generic system in complete isolation, assuming it has a classical limit involving coherent states. Then we entangle it with another system of that kind, and subject the pair to a decohering process. We make a number of broadly applicable and physically reasonable assumptions about this setup. First, we assume that the states selected by the decoherence (called pointer states) have the same local symmetries as the isolated systems, in a sense which is made precise. We also assume that the modular Hamiltonians of pointer states scale inversely with Planck's constant, so that the pointer states are highly entangled in the classical limit. Finally, we require the timescale of decoherence to scale in a certain way with Planck's constant, so that decoherence happens very frequently in the classical limit, but not too frequently. Given these assumptions, we demonstrate that the semiclassical evolution of the system is dominated by a certain dynamical generalisation of Uhlmann holonomy. We construct a coherent state path integral for this evolution, showing that the semiclassical fields evolve in a spacetime with one more dimension than the isolated case. The additional dimension is generated by modular flow. Contents 1 Introduction 2 Highly entangled decohering systems 2.1 Polar decomposition of pointer states 2.2 Frequent decoherence (∆t → 0) 2.3 Dynamical Uhlmann holonomy 2.4 Bures metric 2.5 Classical limit ( → 0) 2.6 Semiclassical correlators