Holographic entanglement entropy and the internal space

TL;DR

This paper explores how extremal surfaces in the internal space of AdS/CFT can quantify entanglement between subsectors of the theory, refining the understanding of entanglement entropy in holographic duals, especially for N=4 SYM.

Contribution

It proposes a refined method to interpret extremal surfaces in the internal space as measures of entanglement between subsectors, addressing gauge invariance issues and applying to N=4 SYM on the Coulomb branch.

Findings

Extremal surfaces in internal space relate to subsector entanglement.

A modified proposal links internal space to R-symmetry for entanglement interpretation.

Application to Coulomb branch mitigates gauge invariance issues.

Abstract

We elaborate on the role of extremal surfaces probing the internal space in AdS/CFT. Extremal surfaces in AdS quantify the "geometric" entanglement between different regions in physical space for the dual CFT. This, however, is just one of many ways to split a given system into subsectors, and extremal surfaces in the internal space should similarly quantify entanglement between subsectors of the theory. For the case of AdS$_5\times$S$^5$, their area was interpreted as entanglement entropy between U(n) and U(m) subsectors of U(n+m) N=4 SYM. Making this proposal precise is subtle for a number of reasons, the most obvious being that from the bulk one usually has access to gauge-invariant quantities only, while a split into subgroups is inherently gauge variant. We study N=4 SYM on the Coulomb branch, where some of the issues can be mitigated and the proposal can be sharpened. Continuing back to the original AdS$_5\times$S$^5$ geometry, we obtain a modified proposal, based on the relation of the internal space to the R-symmetry group.