Solving Puzzles in Deformed JT Gravity: Phase Transitions and Non-Perturbative Effects

TL;DR

The paper analyzes deformations of Jackiw–Teitelboim gravity through double‑scaled matrix models and shows that negative disc spectral densities ρ0(E) arise from multi‑valued leading Genus string equations u0(x). It introduces a non‑perturbative completion based on a non‑Hermitian/string‑equation framework for u(x), and applies it to two explicit deformations (Models A and B), revealing a rich phase structure with semiclassical black‑hole transitions and (in some cases) perturbative/non‑perturbative inconsistencies. Positive‑λ sectors can be made consistent via phase transitions that move the spectral edge E0, whereas negative‑λ sectors often exhibit a first‑order transition and potential non‑perturbative instability, with the multi‑valuedness of u0(x) identifying the root cause. Across the models, the semiclassical analysis partially agrees with non‑perturbative matrix‑model results, and the work clarifies when deformations yield well‑defined theories and when they do not, guiding future explorations of non‑perturbative gravity in two dimensions.

Abstract

Recent work has shown that certain deformations of the scalar potential in Jackiw-Teitelboim gravity can be written as double-scaled matrix models. However, some of the deformations exhibit an apparent breakdown of unitarity in the form of a negative spectral density at disc order. We show here that the source of the problem is the presence of a multi-valued solution of the leading order matrix model string equation. While for a class of deformations we fix the problem by identifying a first order phase transition, for others we show that the theory is both perturbatively and non-perturbatively inconsistent. Aspects of the phase structure of the deformations are mapped out, using methods known to supply a non-perturbative definition of undeformed JT gravity. Some features are in qualitative agreement with a semi-classical analysis of the phase structure of two-dimensional black holes in these deformed theories.

Figures (12)

• Figure 1: The non-perturbative potential (left) and spectral density $\rho(E)$ (right) for JT gravity computed in ref. Johnson:2020exp using equation (\ref{['eq:13']}), truncated to $k=7$. The dashed line is the disc level result for $\rho_0(E)$ given in equation (\ref{['eq:undeformed']}). The inset shows the region near the origin, displaying a non-zero $\rho(E=0)$. Here, $\hbar$ is set to unity.
• Figure 2: Deformed JT gravity dilaton potential $W(\phi)=2\phi+U(\phi)$ in equation (\ref{['eq:70']}) for the values $(\alpha_1,\alpha_2)=(0.9,0.95)$ and several values of $\lambda$. Segments of the curve in red, blue and green indicate whether the black hole solution with $\phi_h=\phi$ is nonexistent (violates (\ref{['eq:35']})), unstable (negative specific heat) or stable (positive specific heat).
• Figure 3: The left diagram shows the phase transition at $\lambda=\lambda_c$ in equation (\ref{['eq:51']}), where we plot ${\phi_h=\phi_0(\lambda)}$ obtained by numerically solving $W(\phi_0)=0$. In the right diagram is plotted the black hole energy as a function of the temperature for $\lambda=-40$, taking into account the phase transition (left diagram in figure \ref{['fig:5']}). We observe a gap in the spectrum generated by the phase transition at $\lambda$ below $\lambda_c'$ in (\ref{['eq:72']}).
• Figure 4: Examples of the curve $x=x(u_0)$ obtained from (\ref{['eq:25']}) in the $(x,u_0)$ plane for several values of $\lambda$ and $(\alpha_1,\alpha_2)=(1/4,1/3)$. The green curve is the unique choice that yields a single valued function defined in the range $x<0$. While for $\lambda\le \lambda_c$ the green curve always satisfy $u(0)=0$, for $\lambda>\lambda_c$ we observe $u(0)=E_0>0$.
• Figure 5: Threshold energy $E_0$ as a function of $\lambda$ for $(\alpha_1,\alpha_2)=(1/4,1/3)$, obtained by numerically solving (\ref{['eq:30']}). For $\lambda=\lambda_c$ in (\ref{['eq:54']}) we observe a first order phase transition.