Entanglement Entropy in Internal Spaces and Ryu-Takayanagi Surfaces

TL;DR

This paper investigates minimal area surfaces in warped product geometries involving AdS spaces and internal spaces, relating their areas to entanglement entropy of internal degrees of freedom and exploring their applications in holography.

Contribution

It demonstrates the existence of RT surfaces in warped geometries, extends previous results, and explores their implications for entanglement and bulk reconstruction.

Findings

RT surfaces can exist in warped internal spaces, avoiding previous limitations.

Such surfaces relate to entanglement entropy of internal degrees of freedom.

They serve as probes for correlations, thermodynamics, and boundary observables.

Abstract

We study minimum area surfaces associated with a region, $R$, of an internal space. For example, for a warped product involving an asymptotically $AdS$ space and an internal space $K$, the region $R$ lies in $K$ and the surface ends on $\partial R$. We find that the result of Graham and Karch can be avoided in the presence of warping, and such surfaces can sometimes exist for a general region $R$. When such a warped product geometry arises in the IR from a higher dimensional asymptotic AdS, we argue that the area of the surface can be related to the entropy arising from entanglement of internal degrees of freedom of the boundary theory. We study several examples, including warped or direct products involving $AdS_2$, or higher dimensional $AdS$ spaces, with the internal space, $K=R^m, S^m$; $Dp$ brane geometries and their near horizon limits; and several geometries with a UV cut-off. We find that such RT surfaces often exist and can be useful probes of the system, revealing information about finite length correlations, thermodynamics and entanglement. We also make some preliminary observations about the role such surfaces can play in bulk reconstruction, and their relation to subalgebras of observables in the boundary theory.