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Towards a holographic description of closed universes

Hong Liu

TL;DR

This work advances an algebraic holographic framework in which all spacetime asymptotics, including closed universes like AS$^2$ PETS and Maldacena-Maoz, are encoded by boundary operator algebras and states via GNS constructions. It shows how a baby universe emerges as the commutant of the single-trace algebra in the large-$N$ limit when $N$-dependent phases are allowed in heavy operator matrix elements, and argues for an averaged large-$N$ limit as part of the AdS/CFT dictionary to capture semiclassical bulk physics. The paper also demonstrates the absence of a geometric entanglement wedge at $O(N^0)$ entanglement, the necessity of cross-boundary operators for bulk BU observables, and provides a holographic MM description through invariant boundary algebras; it connects Euclidean path integrals to averaged boundary correlators, suggesting a unified algebraic treatment of spacetimes with and without boundaries. Overall, the approach offers a mathematically robust route to quantum gravity in closed cosmologies and motivates further exploration of averaged large-$N$ holography and the precise structure of the emergent bulk from boundary algebras.

Abstract

We study baby closed universes in AdS, focusing on the Antonini-Sasieta-Swingle (AS$^2$) cosmology, which arises in the gravitational description of partially entangled thermal states (PETS), as well as the classical example of Maldacena-Maoz (MM). We show that the algebraic formulation of AdS/CFT--and in particular the subregion-subalgebra duality--provides a natural framework for describing such universes within the standard AdS/CFT setting, phrased in terms of an operator algebra ${\cal M}$ and a state $ω$ on that algebra, with no need to introduce observers. The algebra encodes all physical operations in the closed universe, and, in principle, all physical observables are obtainable from the pair $({\cal M},ω)$. Along the way, we propose a resolution to a puzzle raised by Antonini and Rath (AR) and reinforced by recent arguments of Engelhardt and Gesteau and a no-go theorem of Gesteau: that a semiclassical baby universe in the AS$^2$ cosmology cannot be understood from the boundary in the usual manner. Our analysis motivates an averaged large-$N$ limit as part of the AdS/CFT dictionary and points toward a unified treatment of spacetimes of all asymptotics in terms of operator algebras. Finally, our boundary descriptions of closed universes indicate that for small but finite $G_N$ there should exist a semiclassical description of a macroscopic closed universe, rather than a one-dimensional Hilbert space, and we discuss weaknesses in arguments favoring the latter.

Towards a holographic description of closed universes

TL;DR

This work advances an algebraic holographic framework in which all spacetime asymptotics, including closed universes like AS PETS and Maldacena-Maoz, are encoded by boundary operator algebras and states via GNS constructions. It shows how a baby universe emerges as the commutant of the single-trace algebra in the large- limit when -dependent phases are allowed in heavy operator matrix elements, and argues for an averaged large- limit as part of the AdS/CFT dictionary to capture semiclassical bulk physics. The paper also demonstrates the absence of a geometric entanglement wedge at entanglement, the necessity of cross-boundary operators for bulk BU observables, and provides a holographic MM description through invariant boundary algebras; it connects Euclidean path integrals to averaged boundary correlators, suggesting a unified algebraic treatment of spacetimes with and without boundaries. Overall, the approach offers a mathematically robust route to quantum gravity in closed cosmologies and motivates further exploration of averaged large- holography and the precise structure of the emergent bulk from boundary algebras.

Abstract

We study baby closed universes in AdS, focusing on the Antonini-Sasieta-Swingle (AS) cosmology, which arises in the gravitational description of partially entangled thermal states (PETS), as well as the classical example of Maldacena-Maoz (MM). We show that the algebraic formulation of AdS/CFT--and in particular the subregion-subalgebra duality--provides a natural framework for describing such universes within the standard AdS/CFT setting, phrased in terms of an operator algebra and a state on that algebra, with no need to introduce observers. The algebra encodes all physical operations in the closed universe, and, in principle, all physical observables are obtainable from the pair . Along the way, we propose a resolution to a puzzle raised by Antonini and Rath (AR) and reinforced by recent arguments of Engelhardt and Gesteau and a no-go theorem of Gesteau: that a semiclassical baby universe in the AS cosmology cannot be understood from the boundary in the usual manner. Our analysis motivates an averaged large- limit as part of the AdS/CFT dictionary and points toward a unified treatment of spacetimes of all asymptotics in terms of operator algebras. Finally, our boundary descriptions of closed universes indicate that for small but finite there should exist a semiclassical description of a macroscopic closed universe, rather than a one-dimensional Hilbert space, and we discuss weaknesses in arguments favoring the latter.

Paper Structure

This paper contains 21 sections, 74 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Algebraic formulation of the AdS/CFT duality in the large-$N$ limit
  3. Holographic Description of Baby Universes in AS$^2$ Cosmology
  4. Review of the setup
  5. Algebraic description of the baby universe: the basic picture
  6. Existence of commutant of the single-trace algebra and the AR puzzle
  7. The nature of the large $N$ limit
  8. Finite $N$ origin of operators in the baby universe
  9. Absence of a geometric entanglement wedge and failure of QES at $O(N^0)$
  10. Comments on large finite $N$
  11. Comments on the Maldacena-Maoz Closed Universe
  12. Review of the setup
  13. Holographic description
  14. Connection with Euclidean path integrals
  15. Conclusions and discussion
  16. ...and 6 more sections

Figures (3)

  • Figure 1: (a) Global AdS. (b) A closed universe. (c) A causal diamond region (shaded region) in global AdS that does not touch the boundary. The algebra for (a) contains "local" operators that arise from the boundary limit of bulk operators (represented by a dot in the plot), which are absent in those for (b) and (c). The algebras for (a) and (b) are type I, while that for (c) is type III$_1$.
  • Figure 2: Gravity description of PETS \ref{['PETS']}. The insertion of the heavy operator ${\mathbb{{O}}}$ generates a matter shell in the bulk geometry, represented in the plots by thick vertical lines. (a) In the TAdS-BU phase (for sufficiently low temperature), the geometry consists of two copies of global AdS entangled with a baby universe. The baby universe is constructed by gluing together two copies of \ref{['einn0']} along $\xi_c(t)$, the location of the shell. The center $\xi = 0$ of each patch is shown as a dashed line. The shell position $\xi_0$ on the $t=0$ slice (shown as a dotted blue line) diverges as $\mu_{\mathbb{{O}}} \to \infty$, effectively covering an entire slice of global AdS. Dotted red lines are schematic cartoons for entanglement. (b) At sufficiently high temperature, the system is described by a long two-sided black hole.
  • Figure 3: (a) The part of AdS covered by the coordinates \ref{['frws']} lies inside the red diamond in the plot. Each point in figure represents a $S_{d-1}$. Various constant $T$ surfaces are show in the figure. Note that $T=0$ hypersurface coincides with $t={0}$ surface. (b) The closed universe \ref{['euw']} resulted from the quotient. (c) The corresponding Euclidean manifold \ref{['euw1']} resulted from the quotient. Inserting boundary operators (e.g. $X$) at $\tau =-\infty$ can create "excited" states in the Fock space of closed universe.