Abstract
We relate the Riemann curvature of a holographic spacetime to an entangle- ment property of the dual CFT state: the Berry curvature of its modular Hamiltonians. The modular Berry connection encodes the relative bases of nearby CFT subregions while its bulk dual, restricted to the code subspace, relates the edge-mode frames of the cor- responding entanglement wedges. At leading order in 1/N and for sufficiently smooth HRRT surfaces, the modular Berry connection simply sews together the orthonormal co- ordinate systems covering neighborhoods of HRRT surfaces. This geometric perspective on entanglement is a promising new tool for connecting the dynamics of entanglement and gravitation.
Highlights
Central to this task is the spacetime connection that relates the Lorentz frames of nearby tangent spaces and endows spacetime with its curvature
We explain how holographic spacetimes are assembled by the set of entanglement wedges by means of a geometric connection, which we propose is determined microscopically by the entanglement structure of the dual CFT state
At leading order in GN, bulk modular zero modes consist of large diffeomorphisms that do not vanish at the HRRT surface
Summary
2.1 A toy example The central idea underlying this work is that entanglement plays the role of a connection for subsystems of a quantum state [12]. The reduced density matrix of each qubit is maximally mixed Both ρA and ρB are invariant under unitary transformations on the respective Hilbert space, which translates to a symmetry of expectation values for operators localized in A or B: σAi = UA† σAi UA σBi = UB† σBi UB. The operators σB are a simple example of the mirror operators of σA as discussed in [17], with |ψ AB a cyclic and separating vector for the algebra of operators acting on subsystem A It follows from definition (2.3) that the map between the two Hilbert spaces transforms under the action of a local SU(2) symmetry on each qubit as: Wij → UA† , ikWkl UB, lj. A reminder of the relevant concepts from differential geometry, as well as a description of the fiber bundle at hand, is given in figure 2
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