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Universal Entanglement and Boundary Geometry in Conformal Field Theory

Christopher P. Herzog, Kuo-Wei Huang, Kristan Jensen

TL;DR

The paper addresses the universal logarithmic term in entanglement entropy for even-dimensional CFTs by focusing on boundary contributions to the trace anomaly. It develops boundary-inclusive anomaly actions, derives explicit boundary terms for the Euler (A-type) anomaly in $d=4$ and $d=6$, and demonstrates that, under maps to hyperbolic space or de Sitter, the sphere EE is governed by a purely boundary effect encoded in the dimensionally regulated action $\widetilde{W}[\delta_{\mu\nu}]$. The authors provide a complete Wess-Zumino–consistent framework for the boundary anomalies, identify boundary central charges in four dimensions, and present explicit dilaton effective actions with boundary terms that yield the correct universal EE for spheres. The results unify higher-dimensional EE calculations with anomaly-based methods and offer holographic cross-checks, clarifying the boundary-origin of universal entanglement terms and their geometric interpretation through Euler characteristics.

Abstract

Employing a conformal map to hyperbolic space cross a circle, we compute the universal contribution to the vacuum entanglement entropy (EE) across a sphere in even-dimensional conformal field theory. Previous attempts to derive the EE in this way were hindered by a lack of knowledge of the appropriate boundary terms in the trace anomaly. In this paper we show that the universal part of the EE can be treated as a purely boundary effect. As a byproduct of our computation, we derive an explicit form for the A-type anomaly contribution to the Wess-Zumino term for the trace anomaly, now including boundary terms. In d=4 and 6, these boundary terms generalize earlier bulk actions derived in the literature.

Universal Entanglement and Boundary Geometry in Conformal Field Theory

TL;DR

The paper addresses the universal logarithmic term in entanglement entropy for even-dimensional CFTs by focusing on boundary contributions to the trace anomaly. It develops boundary-inclusive anomaly actions, derives explicit boundary terms for the Euler (A-type) anomaly in and , and demonstrates that, under maps to hyperbolic space or de Sitter, the sphere EE is governed by a purely boundary effect encoded in the dimensionally regulated action . The authors provide a complete Wess-Zumino–consistent framework for the boundary anomalies, identify boundary central charges in four dimensions, and present explicit dilaton effective actions with boundary terms that yield the correct universal EE for spheres. The results unify higher-dimensional EE calculations with anomaly-based methods and offer holographic cross-checks, clarifying the boundary-origin of universal entanglement terms and their geometric interpretation through Euler characteristics.

Abstract

Employing a conformal map to hyperbolic space cross a circle, we compute the universal contribution to the vacuum entanglement entropy (EE) across a sphere in even-dimensional conformal field theory. Previous attempts to derive the EE in this way were hindered by a lack of knowledge of the appropriate boundary terms in the trace anomaly. In this paper we show that the universal part of the EE can be treated as a purely boundary effect. As a byproduct of our computation, we derive an explicit form for the A-type anomaly contribution to the Wess-Zumino term for the trace anomaly, now including boundary terms. In d=4 and 6, these boundary terms generalize earlier bulk actions derived in the literature.

Paper Structure

This paper contains 29 sections, 229 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Two Dimensional Case and Rényi entropy
  3. Anomaly Action with Boundary and Entanglement Entropy
  4. The Schwarzian Derivative
  5. Entanglement Entropy from the Plane and Cylinder
  6. Rényi Entropies from the Annulus
  7. Anomaly Actions in More than Two Dimensions
  8. Boundary Term of the Euler Characteristic
  9. An Explicit Expression For The Boundary Term
  10. Wess-Zumino Consistency
  11. A Complete Classification in $d=4$ and Boundary Central Charges
  12. Dimensional Regularization
  13. Dilaton Effective Actions and Boundary Terms
  14. The Dilaton Effective Action in $d=4$
  15. The Dilaton Effective Action in $d=6$
  16. ...and 14 more sections

Figures (2)

  • Figure 1: The causal development of an interval of length $L$. The dots indicate the endpoints of the interval.
  • Figure 2: a) Blue dashed curves are constant $u$ contours. Red curves are constant $\tau$ contours. b) Blue dashed curves are constant $u$ contours. Red curves are constant $\tau_E$ contours. Note that we have plotted negative values for $r$ and $u$ even though both technically are restricted to be positive.