[Paper Review] Non-perturbative de Sitter Jackiw-Teitelboim gravity

Jordan Cotler, Kristan Jensen|arXiv (Cornell University)|Jan 3, 2024

Cosmology and Gravitation Theories4 citations

TL;DR

This paper establishes a non-perturbative genus expansion for de Sitter Jackiw-Teitelboim (dS JT) gravity, revealing a pure imaginary effective string coupling that leads to alternating signs in the S-matrix expansion. The model is shown to be dual to a formal matrix integral with a negative number of degrees of freedom, and the S-matrix is Borel–Le Roy resummable, offering a non-perturbative completion of dS JT gravity with a consistent holographic structure.

ABSTRACT

With non-perturbative de Sitter gravity and holography in mind, we deduce the genus expansion of de Sitter Jackiw-Teitelboim (dS JT) gravity. We find that this simple model of quantum cosmology has an effective string coupling which is pure imaginary. This imaginary coupling gives rise to alternating signs in the genus expansion of the dS JT S-matrix, which as a result appears to be Borel-Le Roy resummable. Furthermore dS JT gravity is formally an analytic continuation of AdS JT gravity, and behaves like a matrix integral with a negative number of degrees of freedom.

Motivation & Objective

• To resolve the lack of a well-defined topological expansion in de Sitter JT gravity, which is essential for non-perturbative quantum gravity in a universe with a positive cosmological constant.
• To clarify the path integral measure and $i\epsilon$ prescription required for convergence of the genus expansion in dS JT gravity with $(-,-)$ signature metrics.
• To establish a holographic duality between dS JT gravity and a formal matrix integral with a negative number of effective degrees of freedom.
• To demonstrate that the S-matrix of dS JT gravity is Borel–Le Roy resummable due to alternating signs from an imaginary string coupling.
• To reconcile the dS JT path integral with the Klein-Gordon inner product on superspace via a precise wavefunctional correspondence.

Proposed method

• Derive the genus expansion of dS JT gravity using a careful treatment of the path integral measure and an $i\epsilon$ prescription to ensure convergence of the moduli space integral.
• Identify the effective string coupling as pure imaginary, which induces alternating signs in the genus expansion and enables Borel–Le Roy resummation of the S-matrix.
• Map the dS JT path integral to a wavefunctional on superspace parameterized by $\phi_b$ and $\ell$, satisfying a Klein-Gordon equation with a conserved $U(1)$ current.
• Establish a correspondence between the wavefunctional $\Psi(\phi_b, \ell)$ and the transition amplitude $\langle \Phi | \psi \rangle$, linking the path integral to the S-matrix via $\Phi = (2\pi)^2 \phi_b / \ell$.
• Show that the resulting inner product on wavefunctionals matches the standard $\langle \psi_1 | \psi_2 \rangle$ inner product, validating the holographic interpretation.
• Use topological recursion to relate the dS JT model to a formal matrix integral, with the number of matrix degrees of freedom replaced by $-N^2$, indicating a negative effective number of degrees of freedom.

Experimental results

Research questions

• RQ1 What is the correct topological expansion for non-perturbative de Sitter JT gravity, and what path integral measure and $i\epsilon$ prescription are required for convergence?
• RQ2 How does the effective string coupling in dS JT gravity affect the genus expansion and the resummability of the S-matrix?
• RQ3 What is the holographic dual of dS JT gravity, and how does it differ from the AdS JT dual in terms of degrees of freedom?
• RQ4 How can the Klein-Gordon inner product on superspace be consistently matched to the standard inner product in the dS JT path integral framework?
• RQ5 Can the S-matrix of dS JT gravity be Borel–Le Roy resummed, and what does this imply for non-perturbative finiteness?

Key findings

• The genus expansion of dS JT gravity features a pure imaginary effective string coupling, which leads to alternating signs in the expansion and enables Borel–Le Roy resummation of the S-matrix.
• The model is holographically dual to a formal matrix integral with a negative number of effective degrees of freedom, specifically $-N^2$ instead of $N^2$, suggesting a novel type of dual quantum mechanics.
• The path integral measure and $i\epsilon$ prescription are essential for convergence and ensure a positive-definite norm in the Lorentzian continuation of the theory.
• The S-matrix obeys topological recursion, mirroring the structure seen in Euclidean AdS JT gravity, but with a sign flip in the coupling due to the imaginary string coupling.
• The wavefunctional $\Psi(\phi_b, \ell)$ is shown to be equivalent to the transition amplitude $\langle \Phi | \psi \rangle$ via $\Phi = (2\pi)^2 \phi_b / \ell$, resolving a long-standing ambiguity in the holographic dictionary.
• The Klein-Gordon inner product on superspace is shown to agree with the standard inner product $\langle \psi_1 | \psi_2 \rangle$ when the correct wavefunctional correspondence is applied, validating the quantum mechanical interpretation of the model.