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Topological shadows and complexity of islands in multiboundary wormholes

Aranya Bhattacharya, Anindya Chanda, Sabyasachi Maulik, Christian Northe, Shibaji Roy

TL;DR

The paper studies the evolution of subregion complexity for Hawking radiation in evaporating black holes by modeling evaporation with three- and multi-boundary wormholes in AdS$_3$. It connects the Page curve to topological volume changes below HRT surfaces, using Gauss-Bonnet and hyperbolic-polygon methods, and encodes time through throat lengths and island inclusions. A central thread is the use of kinematic space to recast bulk volumes as integrals over geodesics, yielding integral identities for causal shadows and islands, and a tensor-network perspective via Coxeter tessellations to visualize complexity changes. The findings show a universal discrete-volume jump at Page time tied to island formation, imply a nonlocal boundary data requirement for bulk volumes, and suggest that these MbW models capture qualitative features of radiation complexity while leaving open questions about exponential growth and higher-dimensional generalization.

Abstract

Recently, remarkable progress in recovering the Page curve of an evaporating black hole (BH) in Jackiw-Teitelboim gravity has been achieved through use of Quantum Extremal surfaces (QES). Multi-boundary Wormhole (MbW) models have been crucial in parallel model building in three dimensions. Motivated by this we here use the latter models to compute the subregion complexity of the Hawking quanta of the evaporating BH in AdS$_{3}$ and obtain the Page curve associated with this information theoretic measure. We use three- and $n$-boundary wormhole constructions to elucidate our computations of volumes below the Hubeny-Rangamani-Takayanagi (HRT) surfaces at different times. Time is represented by the growing length of the throat horizons corresponding to smaller exits of the multi-boundary wormhole and the evaporating bigger exit shrinks with evolving time. We track the change in choice of HRT surfaces with time and plot the volume with time. The smooth transition of Page curve is realized by a discontinuous jump at Page time in volume subregion complexity plots and the usual Page transition is realized as a phase transition due to the inclusion of the island in this context. We discuss mathematical intricacies and physical insights regarding the inclusion of the extra volume at Page time. The analysis is backed by calculations and lessons from kinematic space and tensor networks.

Topological shadows and complexity of islands in multiboundary wormholes

TL;DR

The paper studies the evolution of subregion complexity for Hawking radiation in evaporating black holes by modeling evaporation with three- and multi-boundary wormholes in AdS. It connects the Page curve to topological volume changes below HRT surfaces, using Gauss-Bonnet and hyperbolic-polygon methods, and encodes time through throat lengths and island inclusions. A central thread is the use of kinematic space to recast bulk volumes as integrals over geodesics, yielding integral identities for causal shadows and islands, and a tensor-network perspective via Coxeter tessellations to visualize complexity changes. The findings show a universal discrete-volume jump at Page time tied to island formation, imply a nonlocal boundary data requirement for bulk volumes, and suggest that these MbW models capture qualitative features of radiation complexity while leaving open questions about exponential growth and higher-dimensional generalization.

Abstract

Recently, remarkable progress in recovering the Page curve of an evaporating black hole (BH) in Jackiw-Teitelboim gravity has been achieved through use of Quantum Extremal surfaces (QES). Multi-boundary Wormhole (MbW) models have been crucial in parallel model building in three dimensions. Motivated by this we here use the latter models to compute the subregion complexity of the Hawking quanta of the evaporating BH in AdS and obtain the Page curve associated with this information theoretic measure. We use three- and -boundary wormhole constructions to elucidate our computations of volumes below the Hubeny-Rangamani-Takayanagi (HRT) surfaces at different times. Time is represented by the growing length of the throat horizons corresponding to smaller exits of the multi-boundary wormhole and the evaporating bigger exit shrinks with evolving time. We track the change in choice of HRT surfaces with time and plot the volume with time. The smooth transition of Page curve is realized by a discontinuous jump at Page time in volume subregion complexity plots and the usual Page transition is realized as a phase transition due to the inclusion of the island in this context. We discuss mathematical intricacies and physical insights regarding the inclusion of the extra volume at Page time. The analysis is backed by calculations and lessons from kinematic space and tensor networks.

Paper Structure

This paper contains 18 sections, 14 theorems, 46 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Volumes in AdS$_{3}$ and multi-boundary models:
  3. Mathematical preliminaries on pairs of pants
  4. A Short Review of The Hyperbolic Plane:
  5. Hyperbolic Surfaces and their Construction
  6. Introducing the MbW models of black hole evaporation
  7. Volumes in AdS$_{3}$
  8. Volumes in Multiboundary Wormholes:
  9. Gauss-Bonnet & hyperbolic polygons
  10. Islands and Kinematic Space:
  11. A crash course on kinematic space
  12. Volumes of AdS subspaces as integrals in ${\cal K}$
  13. Causal Shadow in $\text{AdS}_3$ from Kinematic Space
  14. Kinematic Space and Islands
  15. Causal shadow & complexity in tensor networks
  16. ...and 3 more sections

Key Result

Theorem 2.3

The isometry group of $\mathbb{H}^{2}$, $Isom(\mathbb{H}^{2})$, consists of the maps of the form $z \rightarrow \frac{az+b}{cz+d}$ and $z\rightarrow \frac{-a\overline{z}+b}{-c\overline{z}+d}$ where $a,b,c,d\in \mathbb{R}$ and $ad-cb=1$. The maps, $z \rightarrow \frac{az+b}{cz+d}$, are the orientatio

Figures (18)

  • Figure 1: Pair of Pants and Polygonal Representation. In the figure, concentric geodesic edges are denoted by $C_{1}$ and $C_{2}$, whereas $C_{a}$ and $C_{b}$ stand for geodesics that are not concentric. $c_{1}$ and $c_{2}$ denote the centres of the non-concentric geodesics on the horizontal plane.
  • Figure 2: Describing higher number of boundaries.
  • Figure 3: Four boundary case in upper half space model.
  • Figure 4: The $t = 0$ slice of the two-sided BTZ obtained by quotienting the upper half plane by dilatation.
  • Figure 5: Change of preferred HRT in $3$ boundary model. Left: Before the Page time the HRT surface separates the $R_i$ from the remainder of the pair of pants. Right: After the Page time $L_0$ has shrunk to $L_0'$ and the HRT surface has jumped to include the island $I$.
  • ...and 13 more figures

Theorems & Definitions (24)

  • Theorem 2.3
  • Definition 1
  • Theorem 2.4
  • Lemma 2.5
  • Lemma 2.8
  • Definition 2
  • Lemma 2.9
  • Definition 3
  • Lemma 2.10: schultens2014introduction
  • Definition 4
  • ...and 14 more