Complex Entangling Surfaces for AdS and Lifshitz Black Holes?

TL;DR

This paper explores the role of complex extremal surfaces in holographic entanglement entropy, proposing that their real parts may correspond to CFT entropy, especially in black hole spacetimes like AdS and Lifshitz.

Contribution

It identifies complex extremal surfaces in various black hole backgrounds and suggests their real parts could be relevant for holographic entanglement entropy calculations.

Findings

Real parts of complex extremal surfaces match physical entropy expectations.

Complex surfaces have smaller real areas than real extremal surfaces.

Proposed relation between complex surface areas and CFT entropy is plausible.

Abstract

We discuss the possible relevance of complex codimension-two extremal surfaces to the the Ryu-Takayanagi holographic entanglement proposal and its covariant Hubeny-Rangamani-Takayanagi (HRT) generalization. Such surfaces live in a complexified bulk spacetime defined by analytic continuation. We identify surfaces of this type for BTZ, Schwarzschild-AdS, and Schwarzschild-Lifshitz planar black holes. Since the dual CFT interpretation for the imaginary part of their areas is unclear, we focus on a straw man proposal relating CFT entropy to the real part of the area alone. For Schwarzschild-AdS and Schwarzschild-Lifshitz, we identify families where the real part of the area agrees with qualitative physical expectations for the appropriate CFT entropy and, in addition, where it is smaller than the area of corresponding real extremal surfaces. It is thus plausible that the CFT entropy is controlled by these complex extremal surfaces.