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Modifications to Holographic Entanglement Entropy in Warped CFT

Wei Song, Qiang Wen, Jianfei Xu

TL;DR

This work shows that holographic entanglement entropy in AdS$_3$/WCFT and WAdS$_3$/WCFT setups is sensitive to asymptotic boundary conditions, necessitating a WCFT-aware Rindler method rather than the standard Ryu–Takayanagi prescription. It develops a bulk quotient construction to generate WAdS$_3$ black strings whose thermal entropy reproduces the entanglement entropy of WCFTs, and it extends the analysis to Rényi entropy with matching bulk and boundary results. The results establish a consistent holographic dictionary for WCFTs under Dirichlet–Neumann (CSS) boundary conditions, connecting warped conformal mappings, vacuum charges, and the warped Cardy formula. This advances our understanding of spacetime–entanglement interrelations beyond AdS/CFT and demonstrates concrete, calculable tests of WAdS$_3$/WCFT and AdS$_3$/WCFT dualities. The work highlights how boundary conditions shape holographic entanglement, offering a framework to explore quantum gravity in non-AdS spacetimes.

Abstract

In arXiv:1601.02634 it was observed that asymptotic boundary conditions play an important role in the study of holographic entanglement beyond AdS/CFT. In particular, the Ryu-Takayanagi proposal must be modified for warped AdS$_3$ (WAdS$_3$) with Dirichlet boundary conditions. In this paper, we consider AdS$_3$ and WAdS$_3$ with Dirichlet-Neumann boundary conditions. The conjectured holographic duals are warped conformal field theories (WCFTs), featuring a Virasoro-Kac-Moody algebra. We provide a holographic calculation of the entanglement entropy and Rényi entropy using AdS$_3$/WCFT and WAdS$_3$/WCFT dualities. Our bulk results are consistent with the WCFT results derived by Castro-Hofman-Iqbal using the Rindler method. Comparing with arXiv:1601.02634, we explicitly show that the holographic entanglement entropy is indeed affected by boundary conditions. Both results differ from the Ryu-Takayanagi proposal, indicating new relations between spacetime geometry and quantum entanglement for holographic dualities beyond AdS/CFT.

Modifications to Holographic Entanglement Entropy in Warped CFT

TL;DR

This work shows that holographic entanglement entropy in AdS/WCFT and WAdS/WCFT setups is sensitive to asymptotic boundary conditions, necessitating a WCFT-aware Rindler method rather than the standard Ryu–Takayanagi prescription. It develops a bulk quotient construction to generate WAdS black strings whose thermal entropy reproduces the entanglement entropy of WCFTs, and it extends the analysis to Rényi entropy with matching bulk and boundary results. The results establish a consistent holographic dictionary for WCFTs under Dirichlet–Neumann (CSS) boundary conditions, connecting warped conformal mappings, vacuum charges, and the warped Cardy formula. This advances our understanding of spacetime–entanglement interrelations beyond AdS/CFT and demonstrates concrete, calculable tests of WAdS/WCFT and AdS/WCFT dualities. The work highlights how boundary conditions shape holographic entanglement, offering a framework to explore quantum gravity in non-AdS spacetimes.

Abstract

In arXiv:1601.02634 it was observed that asymptotic boundary conditions play an important role in the study of holographic entanglement beyond AdS/CFT. In particular, the Ryu-Takayanagi proposal must be modified for warped AdS (WAdS) with Dirichlet boundary conditions. In this paper, we consider AdS and WAdS with Dirichlet-Neumann boundary conditions. The conjectured holographic duals are warped conformal field theories (WCFTs), featuring a Virasoro-Kac-Moody algebra. We provide a holographic calculation of the entanglement entropy and Rényi entropy using AdS/WCFT and WAdS/WCFT dualities. Our bulk results are consistent with the WCFT results derived by Castro-Hofman-Iqbal using the Rindler method. Comparing with arXiv:1601.02634, we explicitly show that the holographic entanglement entropy is indeed affected by boundary conditions. Both results differ from the Ryu-Takayanagi proposal, indicating new relations between spacetime geometry and quantum entanglement for holographic dualities beyond AdS/CFT.

Paper Structure

This paper contains 30 sections, 151 equations, 3 figures.

Table of Contents

  1. Introduction
  2. AdS$_3$/WCFT and WAdS$_3$/WCFT
  3. WCFT
  4. AdS$_3$/WCFT
  5. AdS$_3$/WCFT$_{({\hat{u}},{\hat{v}})}$
  6. From WCFT$_{({\hat{u}},{\hat{v}})}$ to WCFT$_{(\hat{x},\hat{t})}$
  7. From WCFT$_{(u,v)}$ to WCFT$_{( {\hat{u}},{\hat{v}})}$
  8. WAdS$_3$/WCFT
  9. Entanglement entropy for WCFT
  10. The gravity side story
  11. The strategy
  12. The story with $T_{V}=1,\, T_{U}=0$
  13. The quotient: from WAdS$_{(U,V,\rho)}$ to WAdS$_{(u,v,r)}$
  14. A bulk calculation of the entanglement entropy
  15. The geometric quantity in WAdS$_{(U,V,\rho)}$
  16. ...and 15 more sections

Figures (3)

  • Figure 1: Diagram that depicts the region of (X,T) covered by the (x,t) space, which is the shaded strip. The solid line segment is the interval (\ref{['intervalXT']}), and the identification of the dashed line gives the thermal circle.
  • Figure 2: The solid line segment in the bulk is the geometric quantity $\gamma_{\mathcal{A}}$ (\ref{['image1']}) whose length is proportional to the entanglement entropy of the interval on the boundary (\ref{["intervalUV'1"]}).
  • Figure 3: Diagram that make a conclusion about all the spacetimes and field theories we have discussed, and their relationships.