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Timelike entanglement entropy with gravitational anomalies

Chong-Sun Chu, Himanshu Parihar

TL;DR

The authors investigate timelike entanglement entropy (TEE) in two-dimensional CFTs with gravitational anomalies (c_L ≠ c_R) and develop a holographic description using topologically massive gravity in AdS$_3$. They show that the imaginary part of TEE exhibits an asymmetric dependence, proportional to c_R, while the real part depends on c_L+c_R, enabling a potential probe of gravitational anomalies. Using analytic continuation from spacelike EE and a bulk computation combining spacelike and timelike geodesics with a Chern–Simons twist, they obtain results for zero temperature, finite temperature with angular potential, and zero temperature with finite angular potential, all of which match precisely between field theory and holography. The findings reinforce the duality between anomalous CFTs and TMG and suggest TEE as a diagnostic for gravitational anomalies in holographic contexts and beyond.

Abstract

We study the timelike entanglement entropy (TEE) in two dimensional conformal field theories (CFT) with gravitational anomalies. We employ analytical continuation to compute the timelike entanglement entropy for a pure timelike interval in such CFTs. We find that, unlike the real part, the imaginary part of the TEE displays an asymmetric dependence on the central charges of the left and right moving modes. We propose that the asymmetric dependence on central charges of the imaginary part of the TEE can be used to probe the presence of gravitational anomalies in chiral CFT. Furthermore, we propose a holographic construction to obtain the timelike entanglement entropy from the bulk dual geometries involving topologically massive gravity in AdS$_3$. The holographic results obtained match exactly with the dual field theory results.

Timelike entanglement entropy with gravitational anomalies

TL;DR

The authors investigate timelike entanglement entropy (TEE) in two-dimensional CFTs with gravitational anomalies (c_L ≠ c_R) and develop a holographic description using topologically massive gravity in AdS. They show that the imaginary part of TEE exhibits an asymmetric dependence, proportional to c_R, while the real part depends on c_L+c_R, enabling a potential probe of gravitational anomalies. Using analytic continuation from spacelike EE and a bulk computation combining spacelike and timelike geodesics with a Chern–Simons twist, they obtain results for zero temperature, finite temperature with angular potential, and zero temperature with finite angular potential, all of which match precisely between field theory and holography. The findings reinforce the duality between anomalous CFTs and TMG and suggest TEE as a diagnostic for gravitational anomalies in holographic contexts and beyond.

Abstract

We study the timelike entanglement entropy (TEE) in two dimensional conformal field theories (CFT) with gravitational anomalies. We employ analytical continuation to compute the timelike entanglement entropy for a pure timelike interval in such CFTs. We find that, unlike the real part, the imaginary part of the TEE displays an asymmetric dependence on the central charges of the left and right moving modes. We propose that the asymmetric dependence on central charges of the imaginary part of the TEE can be used to probe the presence of gravitational anomalies in chiral CFT. Furthermore, we propose a holographic construction to obtain the timelike entanglement entropy from the bulk dual geometries involving topologically massive gravity in AdS. The holographic results obtained match exactly with the dual field theory results.
Paper Structure (16 sections, 80 equations, 3 figures)

This paper contains 16 sections, 80 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Entanglement entropy in CFT$_2$ with gravitational anomalies
  3. Zero temperature
  4. Finite temperature and angular potential
  5. Finite angular potential and zero temperature
  6. Timelike entanglement entropy in CFT$_2$ with gravitational anomaly
  7. Zero temperature
  8. Finite temperature and angular potential
  9. Zero temperature and finite angular potential
  10. Holographic Timelike entanglement entropy in AdS$_3$/CFT$_2$ with gravitational anomaly
  11. Review of earlier works
  12. Anomalous contribution from timelike geodesic
  13. Holographic TEE at zero temperature
  14. Holographic TEE at finite temperature and angular potential
  15. Holographic TEE at zero temperature and finite angular potential
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Configuration for a single boosted interval on a complex plane.
  • Figure 2: RT surfaces (geodesics) for a pure timelike interval in Poincaré coordinates are labelled by red and green curves. The normal vectors $n_i$ and $n_f$ are timelike for spacelike geodesics.
  • Figure 3: Schematic for the boundary value of normal vectors $n_1$ and $n_2$ which are spacelike for timelike geodesics.