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Renormalized pseudoentropy in dS/CFT

Giorgos Anastasiou, Ignacio J. Araya, Avijit Das, Javier Moreno

Abstract

We study holographic pseudoentropy for subregions in non-unitary Euclidean conformal field theories (CFTs) within the framework of the de Sitter/conformal field theory (dS/CFT) correspondence. Pseudoentropy, defined as the von Neumann entropy of a transition matrix, is computed holographically from codimension-two extremal surfaces in dS space and is divergent due to the asymptotic bulk volume at future infinity. We show that a finite and regulator-independent definition follows from the on-shell action of conformal gravity in four and six dimensions, implemented through the replica construction. We illustrate the formalism for spherical entangling surfaces and small shape deformations thereof. The renormalized pseudoentropy isolates the universal contribution, which for a spherical entangling surface is proportional to the complex-valued central charge $a^\star$ of the non-unitary CFT. On an equal footing, for infinitesimal deformations away from the sphere, we recover, at quadratic order in the deformation parameter, an analytic continuation of the Mezei-like formula in its anti-de Sitter counterpart.

Renormalized pseudoentropy in dS/CFT

Abstract

We study holographic pseudoentropy for subregions in non-unitary Euclidean conformal field theories (CFTs) within the framework of the de Sitter/conformal field theory (dS/CFT) correspondence. Pseudoentropy, defined as the von Neumann entropy of a transition matrix, is computed holographically from codimension-two extremal surfaces in dS space and is divergent due to the asymptotic bulk volume at future infinity. We show that a finite and regulator-independent definition follows from the on-shell action of conformal gravity in four and six dimensions, implemented through the replica construction. We illustrate the formalism for spherical entangling surfaces and small shape deformations thereof. The renormalized pseudoentropy isolates the universal contribution, which for a spherical entangling surface is proportional to the complex-valued central charge of the non-unitary CFT. On an equal footing, for infinitesimal deformations away from the sphere, we recover, at quadratic order in the deformation parameter, an analytic continuation of the Mezei-like formula in its anti-de Sitter counterpart.
Paper Structure (18 sections, 129 equations, 5 figures, 1 table)

This paper contains 18 sections, 129 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Pseudoentropy from conformal renormalization: four-dimensional case
  3. Codimension-two local conformal invariant in four dimensions
  4. Renormalized pseudoentropy for dS$_{4}$/CFT$_{3}$
  5. Explicit examples
  6. Spherical entangling surface
  7. Small deformations of the sphere
  8. Pseudoentropy from conformal renormalization: six-dimensional case
  9. Codimension-two local conformal invariants in six dimensions
  10. Renormalized pseudoentropy for dS$_6$/CFT$_5$
  11. Explicit examples
  12. Spherical entangling surface
  13. Small deformations of the sphere
  14. Conclusions
  15. Notation and conventions
  16. ...and 3 more sections

Figures (5)

  • Figure 1: A pictorial representation of the dS/CFT correspondence. The HH geometry is prepared by Wick-rotating the past half of dS space to a Euclidean section.
  • Figure 2: Transition matrix $\tau_A$ for a subsystem $A$ on the equator of $\mathbb S^d$, where the Euclidean CFT dual to dS is defined.
  • Figure 3: (Left) We show pictorially the disk entangling region $A=\mathbb B^2$ of unit radius defined in $\mathbb S^2$, which is in the equator---this is, at $\theta=0$ following \ref{['eq:GdS1']}---of $\mathbb S^3$. (Right) We show the same disk-like entangling region of unit radius perturbed harmonically, $\mathbb B_\epsilon^2$, following \ref{['eq:tET']}.
  • Figure 4: The RT surface associated with $\mathbb B^2$ subregion. The timelike (Lorentzian) section is shown in orange, while the spacelike (Euclidean) section is in blue. The curves are smoothly matched at $\tau = 0$, representing the junction between Lorentzian and Euclidean parts of the extremal surface. The plot is schematic and not drawn to scale. (Left) The RT surface front view (Right) The RT surface top view.
  • Figure 5: The RT surface for a small deformation of a ball-shaped subregion. The timelike (Lorentzian) section is shown in orange, while the spacelike (Euclidean) section is in blue. The curves are smoothly matched at $\tau = 0$, representing the junction between Lorentzian and Euclidean parts of the extremal surface. The plot is schematic and not drawn to scale. (Left) The RT surface front view (Right) The RT surface top view.