JT gravity with matter, generalized ETH, and Random Matrices

TL;DR

This paper builds a rigorous bridge between Jackiw-Teitelboim gravity with propagating matter and single-trace, two-matrix models, establishing disk-level matches for O insertions and exploring how cylinder/annulus observables depend on the chosen double-scaling regulator. By embedding ETH into a two-matrix ensemble, the authors derive two complementary approaches: (i) an operator-equation constraint between O and the Hamiltonian H that yields a generalized free-field structure in the disk, and (ii) a constructive program to reconstruct a regulated matrix potential V(H,O) from gravity data using epsilon-regulators (Selberg and q-deformed). They then show that cylinder amplitudes, including the double-trumpet, can be computed directly from disk correlators, while UV divergences and wormhole amplitudes map to Hessian instabilities of the matrix potential, signaling a perturbative breakdown that may require non-perturbative completion. The work also connects to double-scaled SYK through the q-deformed regulator and to Berkooz et al.’s results, suggesting a broader link between random-matrix ensembles, gravitational path integrals, and chaotic quantum dynamics. Overall, the paper provides concrete frameworks to study holographic ensembles via matrix models and highlights both the power and the limits (UV divergences and non-perturbative issues) of the effective theories involved.

Abstract

We present evidence for a duality between Jackiw-Teitelboim gravity minimally coupled to a free massive scalar field and a single-trace two-matrix model. One matrix is the Hamiltonian $H$ of a holographic disorder-averaged quantum mechanics, while the other matrix is the light operator $\cal O$ dual to the bulk scalar field. The single-boundary observables of interest are thermal correlation functions of $\cal O$. We study the matching of the genus zero one- and two-boundary expectation values in the matrix model to the disk and cylinder Euclidean path integrals. The non-Gaussian statistics of the matrix elements of $\cal O$ correspond to a generalization of the ETH ansatz. We describe multiple ways to construct double-scaled matrix models that reproduce the gravitational disk correlators. One method involves imposing an operator equation obeyed by $H$ and $\cal O$ as a constraint on the two matrices. Separately, we design a model that reproduces certain double-scaled SYK correlators that may be scaled once more to obtain the disk correlators. We show that in any single-trace, two-matrix model, the genus zero two-boundary expectation value, with up to one $\cal O$ insertion on each boundary, can be computed directly from all of the genus zero one-boundary correlators. Applied to the models of interest, we find that these cylinder observables depend on the details of the double-scaling limit. To the extent we have checked, it is possible to reproduce the gravitational double-trumpet, which is UV divergent, from a systematic classification of matrix model `t Hooft diagrams. The UV divergence indicates that the matrix integral saddle of interest is perturbatively unstable. A non-perturbative treatment of the matrix models discussed in this work is left for future investigations.

Figures (32)

• Figure 1: An example of a Feynman diagram that contributes to the thermal six-point $\mathcal{O}$ correlator. Each blue line is a Wick-contraction through the bulk of two boundary operators with scaling dimension $\Delta$. The correlator is a sum over all ways of contracting the operators. Each disk-shaped region is labeled by an $s$ parameter that is integrated in the range $s \in (0,\infty)$ to obtain the value of the correlator. The Euclidean-time separation between the external operators is indicated by the $\beta_1,\cdots,\beta_6$ parameters. We adopt a convention where a blue bulk line always has scaling dimension $\Delta$.
• Figure 2: The blue geodesic divides the pair of pants into three regions with cylinder topology. The length of the blue geodesic $B$ is given in \ref{['eq:Bstar']}.
• Figure 3: Top row: The left disk represents the disk computation of $\braket{\text{Tr } e^{- \beta H}}$ in the SSS model Saad:2019lba. One should imagine filling in the disk with all possible planar 't Hooft diagrams of $H$. The middle disk represents a correction from a single insertion of the double-trace term in $\tilde{V}(H)$ (in general, there could be arbitrarily many insertions, which are all summed over). One should imagine filling in the regions inside and outside the red double-line loop with the t' Hooft diagrams of $H$ in the SSS model of disk and cylinder topology, respectively. In our terminology, the red double-line loop in the center is an example of an "$\mathcal{O}$ bubble diagram." The right disk represents a correction from the single-trace counterterm potential that is designed to cancel the contribution from the $\mathcal{O}$ bubble diagram. The counterterm is a single-trace term in the potential and hence is represented by a single loop. Bottom row: we provide an example of one of the infinitely many ways the diagrams in the top row can be filled in with planar 't Hooft diagrams involving the $H$ matrix. A black double-line represents the propagator of the $H$ matrix.
• Figure 4: Two adjacent red line loops $X(b_j)$ represents the double-trace term in \ref{['VXrel']}. These double-lines separate the double-trumpet into $n+1$ regions, and one should imagine filling in these regions with diagrams in the SSS model with cylinder topology. Any diagrams that contain a contractible red double-line loop are canceled because $\braket{X(b)}_\text{disk} = 0$.
• Figure 5: To compute the correct gravitational observables to order $\epsilon$, it suffices to let the quartic $\mathcal{O}$ coupling be the only interaction to order $\epsilon$. (a) $\mathcal{O}$ bubble diagrams that contribute to the disk partition function at order $\epsilon$. As in Figure \ref{['fig:threedisks']}, these are cancelled by counterterms in $V_{c.t.}$. (b) The 't Hooft diagrams that contribute to the two-point function. The order $\epsilon$ correction to the quadratic $\mathcal{O}$ term is chosen to ensure that these diagrams sum to the known gravitational answer for the two-point function. (c) The value of the quartic $\mathcal{O}$ coupling is determined by matching the four-point tree diagram to the known gravitational answer for the connected four-point function.