Large N von Neumann Algebras and the Renormalization of Newton’s Constant

I derive a family of Ryu–Takayanagi formulae that are valid in the large N limit of holographic quantum error-correcting codes, and parameterized by a choice of UV cutoff in the bulk. The bulk entropy terms are matched with a family of von Neumann factors nested inside the large N von Neumann algebra describing the bulk effective field theory. These factors are mapped onto one another by a family of conditional expectations, which are interpreted as a renormalization group flow for the code subspace. Under this flow, I show that the renormalizations of the area term and the bulk entropy term exactly compensate each other. This result provides a concrete realization of the ER=EPR paradigm, as well as an explicit proof of a conjecture due to Susskind and Uglum.