Matrix Models and Deformations of JT Gravity

TL;DR

The paper demonstrates that deformations of Jackiw–Teitelboim gravity by a potential W(φ) admit a dual description as Hermitian matrix models, enabling the extraction of the eigenvalue density ρ(E) from the disc path integral. A controlled perturbative approach maps exponential perturbations to conical singularities and leverages Weil–Petersson volumes to compute corrections, yielding an explicit ρ(E) when U(0)=0 and a principled framework for determining a shifted threshold E0(U) when U(0)≠0. Exact results for ρ(E) and E0(U) are obtained under the matrix-model assumption, complemented by checks on higher-topology observables (double/triple trumpets, one-holed torus) that agree with matrix-model expectations. The work connects deformed JT gravity to topological gravity and Mirzakhani-type volumes, providing a coherent, testable bridge between gravitational theories and random matrix ensembles with broad implications for holography in low dimensions.

Abstract

Recently, it has been found that JT gravity, which is a two-dimensional theory with bulk action $ -\frac{1}{2}\int {\mathrm d}^2x \sqrt gφ(R+2)$, is dual to a matrix model, that is, a random ensemble of quantum systems rather than a specific quantum mechanical system. In this article, we argue that a deformation of JT gravity with bulk action $ -\frac{1}{2}\int {\mathrm d}^2x \sqrt g(φR+W(φ))$ is likewise dual to a matrix model. With a specific procedure for defining the path integral of the theory, we determine the density of eigenvalues of the dual matrix model. There is a simple answer if $W(0)=0$, and otherwise a rather complicated answer.

Figures (9)

• Figure 1: By a "trumpet," we mean a hyperbolic two-manifold that is topologically an annulus. One "inner" boundary is a geodesic, here of length $b$. The other "outer" boundary is exceedingly large; it represents an approximation to the asymptotic boundary of ${\mathrm{AdS}}_2$.
• Figure 2: A three-holed sphere with a hyperbolic metric and with geodesic boundaries of lengths $b_1,b_2,b_3$. By gluing together three-holed spheres and/or trumpets along their geodesic boundaries, one can make more complicated hyperbolic two-manifolds. A simple example involving gluing of a trumpet to a three-holed sphere is in fig. \ref{['FigOne']}(a).
• Figure 3: (a) A Riemann surface $\Sigma$ of genus 0 with an asymptotically AdS boundary $C$, and two geodesic boundaries $B_1$, $B_2$. We consider loops homologous to $C$. Minimizing the length of such a loop in its homology class, we arrive at the closed geodesic $\gamma$. Once we find $\gamma$, $\Sigma$ can be reconstructed by gluing a trumpet and a three-holed sphere along $\gamma$. (b) The geodesic boundaries of $\Sigma$ have been replaced by conical singularities $p_1$, $p_2$, with deficit angles $\alpha_1$, $\alpha_2$. Again we consider loops homologous to $C$. As long as $\alpha_1,\alpha_2>\pi$, minimizing the length of such a loop leads to a picture rather similar to that in (a), again with a closed geodesic $\gamma$. For the reason for the constraint $\alpha_1,\alpha_2>\pi$, see fig. \ref{['FigTwo']}. $\Sigma$ can be reconstructed by gluing two simple-building blocks along $\gamma$. One building block is a trumpet, and the other is similar to a three-holed sphere, but with two of the holes replaced by conical singularities.
• Figure 4: In the presence of conical singularities, hyperbolic Riemann surfaces can still be built by gluing together elementary building blocks along their geodesic boundaries, but one has to include two additional elementary building blocks. The new building blocks are a sphere with two geodesic boundaries and one conical singularity or with one geodesic boundary and two conical singularities, as pictured here. The geodesic boundaries are labeled by their circumferences and the conical singularities are labeled by their deficit angles.
• Figure 5: A Riemann surface $\Sigma$ with a conical singularity at a point $p$ has been, locally, "unwrapped" to a wedge-shaped region of the plane. (The black rays emanating from $p$ should be identified to build $\Sigma$.) The curvature of $\Sigma$ does not affect the issue that will be discussed here, because it is unimportant in a small neighborhood of $p$; it is neglected in this figure. (a) The deficit angle at $p$ is greater than $\pi$. Consider a curve passing between specified endpoints $s_1, s_2$ on a given side of $p$. In minimizing the length of a path from $s_1$ to $s_2$, the path is "repelled" from p. There is always a geodesic between $s_1$ and $s_2$ on any pre-chosen side of $p$. (As an exercise, the reader can try to describe the geodesic from $s_1$ to $s_2$ that goes around $p$ on the other side; one has to use the fact that the two black rays from $p$ are identified.) (b) The deficit angle at $p$ is less than $\pi$. A geodesic with specified endpoints does not necessarily exist on a given side of $p$. For example, in the figure, to shorten the indicated path from $s_1$ to $s_2$ to a geodesic, one would have to pull it across the point $p$.