# Holographic calculation of entanglement entropy in the presence of boundaries

## TL;DR

This paper investigates boundary contributions to entanglement entropy in spacetimes with boundaries, comparing weak coupling calculations with holographic methods, and finds agreement under specific boundary conditions and minimal hypersurface extensions.

## Contribution

It provides a detailed holographic calculation of boundary entanglement entropy contributions and compares them with weak coupling results, highlighting conditions for their agreement.

## Key findings

- Boundary contributions to entanglement entropy are computed in 3D and 4D.
- Holographic and weak coupling calculations agree under supersymmetry-preserving boundary conditions.
- The boundary extension into AdS must be a minimal hypersurface for agreement.

## Abstract

When a spacetime has boundaries, the entangling surface does not have to be necessarily compact and it may have boundaries as well. Then there appear a new, boundary, contribution to the entanglement entropy due to the intersection of the entangling surface with the boundary of the spacetime. We study the boundary contribution to the logarithmic term in the entanglement entropy in dimensions $d=3$ and $d=4$ when the entangling surface is orthogonal to the boundary. In particular, we compute a boundary term in the entropy of ${\mathcal{N}}=4$ super-gauge multiplet at weak coupling. This result is compared with the holographic calculation of the entropy based on the Ryu-Takayanagi proposal adapted appropriately to the present situation. We find a complete agreement between these two calculations provided the boundary conditions imposed on the gauge multiplet preserve $1/2$ of the supersymmetry and the extension of the boundary into the AdS bulk is a minimal hypersurface.