Vacuum Transitions in Two-Dimensions and their Holographic Interpretation

Technical Summary: Vacuum Transitions in Two-Dimensions and their Holographic Interpretation

Problem Statement
The study of vacuum transitions provides a critical arena for exploring quantum aspects of gravity, particularly regarding the stability of vacua with different cosmological constants and the emergence of unitarity-violating behaviors (analogous to the black hole information paradox). While four-dimensional (4D) vacuum transitions have been extensively studied using three primary formalisms—Coleman-De Luccia (CDL), Brown-Teitelboim (BT), and Fischler-Morgan-Polchinski (FMP)—their application to two-dimensional (2D) gravity presents unique challenges. Specifically, previous analyses in 4D revealed that transitions involving Minkowski spacetimes often lead to divergent actions or forbid "up-tunnelling" (transitions to higher energy vacua) in the absence of black holes. Furthermore, the interpretation of transition amplitudes in terms of entropy and their holographic embedding in the context of islands and quantum extremal surfaces (QES) remains an open question in lower dimensions. This work aims to calculate 2D vacuum transition amplitudes using all three formalisms, reconcile their differences, and provide a holographic interpretation involving islands and entanglement.

Methodology
The authors employ three distinct approaches to calculate transition amplitudes ($\Gamma$) in 2D Jackiw-Teitelboim (JT) gravity coupled to a scalar field:

1. Euclidean Methods (CDL and BT):

   • CDL: The authors extend the CDL formalism to 2D using the Almheiri-Polchinski (AP) action. They analyze the system both with and without the scalar kinetic term. Removing the kinetic term via rescaling allows for a calculation without gravity, while removing the Ricci scalar term (with appropriate boundary terms) recovers the BT action.
   • BT: The Brown-Teitelboim approach is derived from the AP setup by restoring the kinetic term and adding boundary terms to account for the ADM mass, ensuring energy conservation during brane nucleation.
   • The bounce ($B$) is calculated as the difference between the Euclidean action on the instanton and the background ($B = S_E|_{inst} - S_E|_{bckgr}$).

2. Lorentzian Method (FMP):

   • The Fischler-Morgan-Polchinski formalism is applied to 2D JT gravity using a Hamiltonian approach. The transition amplitude is defined as a conditional probability ratio between nucleation from "nothing" to a spacetime with a wall versus nucleation to the background alone.
   • The authors introduce a constant energy term ($B$) in the Lagrangian density, which acts as a parameter for the black hole mass. This allows for the existence of non-extremal black holes and distinct physical turning points, which are crucial for defining transitions involving Minkowski limits.

3. Holographic Interpretation:

   • The authors utilize the AdS$_2$/CFT$_1$ $\subset$ AdS$_3$/CFT$_2$ correspondence and the concept of "islands" (regions contributing to the generalized entropy).
   • They map the bulk vacuum transitions to boundary phenomena, specifically relating the transition rates to entropies of $T\bar{T}$-deformed CFTs, Boundary CFTs (BCFT), and internal CFTs.

Key Contributions and Results

• Unification of Formalisms: The paper demonstrates that while CDL, BT, and FMP yield different intermediate expressions, they are consistent under specific parametric redefinitions.

  • CDL (without kinetic term): Describes transitions in the absence of gravity. The total bounce is expressed as the product of relative entropies of an internal CFT$_2$.
  • BT and FMP (without black holes): These methods, which include gravity, yield transition rates expressible as the difference of generalized entropies (specifically entropies of $T\bar{T}$-deformed CFTs).
  • FMP (with black holes): By introducing the black hole mass parameter ($B$), the authors show that the total action can be written as the difference of generalized entropies ($S_{gen}$) involving an island.

• Resolution of Flat Limit Divergences:

  • In the absence of black holes, the authors confirm that up-tunnelling from Minkowski space and transitions to Minkowski space from de Sitter (dS) lead to divergent actions, effectively forbidding these processes in pure 2D dS/AdS setups without black holes.
  • Crucially, the introduction of a black hole (via the parameter $B$) renders the flat limit finite. The black hole mass acts as a regulator, allowing for well-defined transition amplitudes and enabling up-tunnelling.

• Islands and Unitarity:

  • A central finding is that the emergence of an "island" (a region contributing to the entropy via a quantum extremal surface) is contingent on the presence of non-extremal black holes.
  • When an island is present (i.e., when the black hole mass lies within a specific range defined by the theory's parameters), up-tunnelling becomes possible. The transition rate is then governed by the difference of generalized entropies, $S_{gen} = \text{Area}(\partial I)/4G_N + S_{EE}$.
  • In the absence of black holes, no island emerges, and the transition corresponds to an AdS$_2$/CFT$_1$ $\subset$ AdS$_3$/CFT$_2$ embedding where the process is local. With black holes, the embedding becomes AdS$_2$/CFT$_1$ $\not\subset$ AdS$_3$/CFT$_2$.

• Holographic Interpretation:

  • CDL: The bounce is interpreted as an entropy product of an internal CFT$_2$, geometrizing the RG-flow between different dilaton values.
  • BT/FMP (No Black Holes): The transition rate corresponds to the difference of entropies of $T\bar{T}$-deformed CFTs, illustrating the locality of the nucleation process.
  • BT/FMP (With Black Holes): The action corresponds to the difference of generalized entropies. The black hole mass parameter $B$ plays a role analogous to time in Hawking evaporation, driving a phase transition in the brane action (analogous to the Page transition) that signals the emergence of the island.
  • The authors propose that vacuum transitions in the bulk correspond to confinement/deconfinement phase transitions on the boundary CFT side.

Significance
The paper claims that the total action (or bounce) associated with direct vacuum transition processes carries an internal entropic interpretation. Specifically:

1. Entropic Hierarchy: The results establish a hierarchy where CDL (no gravity) relates to relative entropy, while BT and FMP (with gravity) relate to generalized entropy differences.
2. Necessity of Islands: The work provides evidence that for up-tunnelling to be possible and for the flat limit to remain finite in 2D, an island must be present, which requires the presence of non-extremal black holes.
3. Holographic Embedding: The study successfully embeds 2D vacuum transitions into a holographic framework involving wedge holography and BCFTs, showing that the transition amplitudes can be understood as mutual approximations of boundary states belonging to different CFTs.
4. Unitarity: By resolving the divergence issues in the flat limit through the introduction of black holes and islands, the paper suggests a mechanism to preserve unitarity in vacuum transitions, drawing a parallel to the resolution of the black hole information paradox.

The authors conclude that while the three formalisms (CDL, BT, FMP) appear distinct, they offer complementary holographic perspectives on 2D vacuum transitions, with the FMP formalism providing the necessary setup (via the black hole mass parameter) to describe transitions involving islands and up-tunnelling.