[Paper Review] JT gravity as a matrix integral

Phil Saad, Stephen H. Shenker+1 more|arXiv (Cornell University)|Mar 26, 2019

Black Holes and Theoretical PhysicsPhysics and Astronomy134 references205 citations

TL;DR

The paper shows that Euclidean JT gravity partition functions on surfaces with boundaries match the genus expansion of a double-scaled matrix integral, with Mirzakhani’s recursion aligning with Eynard-Orantin topological recursion.

ABSTRACT

We present exact results for partition functions of Jackiw-Teitelboim (JT) gravity on two-dimensional surfaces of arbitrary genus with an arbitrary number of boundaries. The boundaries are of the type relevant in the NAdS${}_2$/NCFT${}_1$ correspondence. We show that the partition functions correspond to the genus expansion of a certain matrix integral. A key fact is that Mirzakhani's recursion relation for Weil-Petersson volumes maps directly onto the Eynard-Orantin "topological recursion" formulation of the loop equations for this matrix integral. The matrix integral provides a (non-unique) nonperturbative completion of the genus expansion, sensitive to the underlying discreteness of the matrix eigenvalues. In matrix integral descriptions of noncritical strings, such effects are due to an infinite number of disconnected worldsheets connected to D-branes. In JT gravity, these effects can be reproduced by a sum over an infinite number of disconnected geometries -- a type of D-brane logic applied to spacetime.

Motivation & Objective

• Motivate the study of JT gravity through SYK and Schwarzian boundary dynamics.
• Compute partition functions of JT gravity on surfaces of arbitrary genus with multiple boundaries.
• Demonstrate the equivalence between JT gravity’s genus expansion and a double-scaled matrix integral’s loop equations.
• Provide a nonperturbative completion framework for the JT genus expansion.
• Explore connections to minimal strings and nonperturbative brane effects in JT gravity.

Proposed method

• Review the genus expansion in Hermitian matrix models and the role of the resolvent R(E).
• Derive the leading density of states ρ0(E) from JT disk partition function and Schwarzian theory.
• Use Mirzakhani’s Weil-Petersson volumes and Trumpet geometry to assemble higher-genus contributions.
• Show that the loop equations of the matrix model reproduce the JT gravity recursion for Z(β1)…Z(βn).
• Introduce double scaling to obtain a finite e^{-S0} expansion and a spectral curve for JT gravity.
• Discuss nonperturbative completions via branes (ZZ, FZZT) and their JT-gravity interpretations.

Experimental results

Research questions

• RQ1 How do JT gravity partition functions for multiple boundaries and arbitrary genus map to a double-scaled matrix integral?
• RQ2 What is the leading density of states ρ0(E) that feeds the matrix-model recursion in JT gravity?
• RQ3 How do Mirzakhani’s Weil-Petersson volumes encode JT gravity’s higher-genus corrections?
• RQ4 What is the nature of nonperturbative corrections in JT gravity as inherited from matrix-integral brane physics?
• RQ5 How does JT gravity relate to the (2,p) minimal string in the appropriate double-scaling limit?

Key findings

• JT gravity correlators ⟨Z(β1)…Z(βn)⟩conn match the genus expansion of a double-scaled matrix integral.
• Leading density of states is ρ0(E) = (γ/(2π^2)) sinh(2π√(2γE)) (with γ set to 1/2 in conventions).
• The JT genus expansion is consistent with the matrix-integral recursion (topological recursion) using the disk ρ0(E) input.
• Mirzakhani’s Weil-Petersson volumes satisfy a recursion that maps to Eynard-Orantin topological recursion for the matrix model.
• Nonperturbative effects in JT gravity can be interpreted via brane-like objects (ZZ and FZZT) in the matrix-model framework, explaining features such as the spectral form factor plateau.