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If a holographic bulk spacetime is built out of quantum entanglement in the boundary theory, how do we understand the bulk connection? To inspect the entanglement structure of a boundary state, we dissect it into components and look at their quantum correlations. Each boundary component reconstructs a region of the bulk called entanglement wedge. The entanglement wedge (and its corresponding component subregion of the boundary) has an internal symmetry called modular flow, which has two properties that will be useful for our purposes. First, modular flow is a gauge symmetry because it relates to one another different ways of presenting the same physical system--the entanglement wedge. Second, modular flow is a generalization of choosing the phase of a pure quantum state in a Hilbert space. When we glue together two overlapping entanglement wedges to build a larger spacetime, we must specify how to map the observables in the first wedge (presented in some modular frame--in some gauge) to observables in the second wedge (also presented in some gauge). Thus, gluing together two component subregions of the boundary--as well as two entanglement wedges--requires a connection that relates their respective modular frames. This connection is analogous to specifying the phase of a quantum state that evolves under a time-dependent Hamiltonian, that is the Berry phase. I argue that the modular Berry connection is the boundary origin of the usual, geometric connection in the bulk. I will sketch some subtleties in the formal construction of the modular Berry connection, give examples and list key questions for the future. |
