JT Supergravity, Minimal Strings, and Matrix Models

TL;DR

This work builds a non-perturbative definition of JT supergravity by assembling type 0A minimal string theories labeled by Γ within the (2Γ+1,2) Altland–Zirnbauer class. Central to the construction are string equations and the resolvent/Gelfand–Dikii framework, which yield disc and non-perturbative spectral data consistent with JT/SJT, and reveal special, tractable cases at half-integer Γ. By interpolating among minimal models with carefully chosen t_k, the authors reproduce the disc spectral density and provide a complete, stable non-perturbative completion, while exposing rich perturbative structures and intriguing features at half-integer Γ. The approach offers a complementary perspective to recursive JT gravity results and opens avenues to analyze broader JT-like theories and observables within a controlled minimal-string setting.

Abstract

It is proposed that a family of Jackiw-Teitelboim supergravites, recently discussed in connection with matrix models by Stanford and Witten, can be given a complete definition, to all orders in the topological expansion and beyond, in terms of a specific combination of minimal string theories. This construction defines non-perturbative physics for the supergravity that is well-defined and stable. The minimal models come from double-scaled complex matrix models and correspond to the cases $(2Γ{+}1,2)$ in the Altland-Zirnbauer $(\boldsymbolα,\boldsymbolβ)$ classification of random matrix ensembles, where $Γ$ is a parameter. A central role is played by a non-linear `string equation' that naturally incorporates $Γ$, usually taken to be an integer, counting e.g., D-branes in the minimal models. Here, half-integer $Γ$ also has an interpretation. In fact, $Γ{=}{\pm}\frac12$ yields the cases $(0,2)$ and $(2,2)$ that were shown by Stanford and Witten to have very special properties. These features are manifest in this definition because the relevant solutions of the string equation have special properties for $Γ{=}{\pm}\frac12$. Additional special features for other half-integer $Γ$ suggest new surprises in the supergravity models.

Figures (9)

• Figure 1: Examples of perturbative contributions. A cross represents a crosscap insertion making a non--orientable surface.
• Figure 2: Features of the leading potential for the minimal models. For $x{<}\mu$, $u_0(x){=}(-x)^{1/k}$ ($k{=}2$ plotted here). This is common to both the bosonic and the type 0A minimal models. For $x{>}\mu$, $u_0(x){=}0$, a key feature of the type 0A minimal models. (Figure \ref{['fig:potential2']} shows the full $u(x)$ for $k{=}2$ type 0A.)
• Figure 3: A schematic diagram of the interpolating flow as a pattern of operator deformations. A circle with $i$ in it is the $i$th minimal model. Here, $k$ should be understood to be taken to infinity. There are $k{-}1$ operators in the model, labelled ${\cal O}_l$, and deformation with coefficient $t_l$ is equivalent to turning on the $l$th model. The length of the bonds/arrows signifies the differing strengths of the $t_l$.
• Figure 4: The special Bessel spectral densities for $\Gamma{=}\frac{1}{2}$ (solid) and $\Gamma{=}{-}\frac{1}{2}$ (dashed).
• Figure 5: The potential $u(x)$ that is supplied by equation (\ref{['eq:string-equation-2']}) for the case $k{=}2$. c.f., figure \ref{['fig:leading-potential']} for the leading part, $u_0(x)$, in this case.