Holographic Entanglement Entropy of the Coulomb Branch

TL;DR

This paper calculates the entanglement entropy in a holographic model of $ ext{SU}(N)$ Yang-Mills theory on the Coulomb branch, revealing monotonic behavior consistent with the $a$-theorem and exploring properties of Wilson lines and solitons.

Contribution

It introduces a method to compute entanglement entropy from probe branes without back-reaction and applies it to analyze Coulomb branch states and related objects in holography.

Findings

Entanglement entropy decreases monotonically with sphere radius, consistent with the $a$-theorem.

The EE of a screened Wilson line also decreases monotonically.

The EE of a spherical soliton does not scale with surface area at the soliton radius.

Abstract

We compute entanglement entropy (EE) of a spherical region in $(3+1)$-dimensional $\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory in states described holographically by probe D3-branes in $AdS_5 \times S^5$. We do so by generalising methods for computing EE from a probe brane action without having to determine the probe's back-reaction. On the Coulomb branch with $SU(N)$ broken to $SU(N-1)\times U(1)$, we find the EE monotonically decreases as the sphere's radius increases, consistent with the $a$-theorem. The EE of a symmetric-representation Wilson line screened in $SU(N-1)$ also monotonically decreases, although no known physical principle requires this. A spherical soliton separating $SU(N)$ inside from $SU(N-1)\times U(1)$ outside had been proposed to model an extremal black hole. However, we find the EE of a sphere at the soliton's radius does not scale with the surface area. For both the screened Wilson line and soliton, the EE at large radius is described by a position-dependent W-boson mass as a short-distance cutoff. Our holographic results for EE and one-point functions of the Lagrangian and stress-energy tensor show that at large distance the soliton looks like a Wilson line in a direct product of fundamental representations.