Ryu-Takayanagi Area as an Entanglement Edge Term

TL;DR

The paper argues that the Ryu-Takayanagi area term in AdS/CFT can be understood as an entanglement edge term arising from gauge constraints at entangling surfaces. By analyzing extended Hilbert space constructions in lattice gauge theories and mapping UV-finite emergent gauge theories to their IR gauge-theory representatives, it shows that the UV-exact EE decomposes into a boundary Shannon term plus a $\log \dim R$ edge term and interior entanglement, up to a state-independent constant. Using the entanglement wedge reconstruction framework, the RT area $A/4G_N$ is identified with a $\log \dim R$-type edge contribution in the bulk, connecting microscopic gauge-edge physics to holographic geometry. The work clarifies the role of edge modes in gauge theories and discusses speculative string-theoretic interpretations of the gravitational entanglement edge, highlighting future directions to formalize the duality between density matrices, edge degrees of freedom, and bulk geometry.

Abstract

By comparing entanglement in emergent gauge theories to the bulk in AdS/CFT, I suggest that the Ryu-Takayanagi area term is an entanglement edge term related to a natural measure on the gauge group. The main technical result in this paper is an argument why the "extended Hilbert space" definition of entanglement entropy in a lattice gauge theory is applicable to an emergent gauge theory.

Figures (5)

• Figure 1: EE across an interval on ${\bf S}^1$.
• Figure 2: The left-hand side illustrates the extended Hilbert space definition of EE in a lattice gauge theory. We take the entangling region $A$ (in blue) to go through a set of links, and extend the Hilbert space at each intersection of $\partial A$ with a link (in red). The middle picture depicts the situation in section \ref{['s221']}, where we specify an entangling region by a collection of links. The right-hand side illustrates the generic situation in an emergent gauge theory, where the UV Hilbert space might be the tensor product of microscopic Hilbert spaces at the sites of a UV lattice.
• Figure 3: Wilson loop in an emergent gauge theory cut by surface charges on the lattice; closed strings cut by the entangling surface in AdS?
• Figure 4: Left: The analogy between the Wilson loop and closed string suggests that the string dual of $\rho_A$ is the entanglement wedge with open strings in a mixed state of Chan-Paton factors on a stack of branes at the entangling surface. Right: A naive Euclidean continuation of the left-hand picture is that the perturbative string background whose partition sum reproduces $\log \hbox{Tr}_A\rho_A$ is the Euclidean cigar with a $T \sim 1/g_s^2$ defect at the tip.
• Figure 5: The electric center choice of Casini:2013rba defines the EE for the collection of links marked in black above as the algebraic EE of the maximal subalgebra supported on those links. The magnetic center choice defines the EE for the collection of links marked in black to be the algebraic EE of the maximal subalgebra supported on the red links.