Discrete Bulk Spectrum in Jackiw-Teitelboim Theory

TL;DR

The paper addresses reconciling Lorentzian JT gravity with Euclidean SSS duality by positing an extra left confining potential that makes the bulk spectrum discrete with random statistics for the renormalized geodesic length $\ell_{\gamma}$. This potential is fixed by matching the Euclidean density of states $\rho_{JT}(E)=\frac{e^{S_0}}{4\pi^2}\sinh(2\pi\sqrt{E})$ via an Abel's integral equation, yielding a function $W(X)$ with $X=-e^{-S_0} q$ and a relation to the string equation. In the SSS duality, the same $W$ arises from the matrix-model string equation, though the bulk and boundary pictures differ; the spectrum becomes discrete with level spacing $O(e^{S_0})$, and Krylov complexity evolves through ramp, peak, slope, and plateau phases. The results are supported by a semiclassical Schrödinger analysis and a numerical demonstration of Krylov complexity behavior, and the work offers a concrete mechanism to connect Euclidean and Lorentzian JT gravity via a confining potential.

Abstract

We argue that a discrete bulk spectrum with random statistics appears naturally in the Lorentzian description of Jackiw-Teitelboim (JT) gravity if an extra confining potential is introduced in the region where the renormalized geodesic length becomes of order $e^{S_0}$. The existence of such an extra confining potential may be inferred from the late behavior of complexity and also from the Saad-Shenker-Stanford (SSS) duality between JT gravity and the matrix model. We derive the explicit form of the extra confining potential from the well-established density of states obtained in the Euclidean approach to JT gravity. This extra potential is implicitly determined by the solution of the Abel's integral equation which turns out to be identical to the string equation of the matrix model in the SSS duality formulation of JT gravity. Thanks to the extra confining potential and the random nature of the spectrum, the time evolution of the Krylov complexity, which is identified with the renormalized geodesic length, naturally exhibits four phases, namely a ramp, a peak, a slope, and a plateau.

Figures (3)

• Figure 1: On the left we draw the left and right cutoff trajectories as purple curves near the AdS$_2$ boundaries, respectively. On the right we illustrate the genus expansion with one boundary.
• Figure 2: A schematic form of the potential $V(q)=e^q+W(q)$, where the left confining potential $W(q)$ becomes $O(1)$ only when $q$ becomes of $O(-e^{S_0})$. With the left confining potential, the spectrum becomes discrete.
• Figure 3: Complexity $\langle \ell_\gamma(t) \rangle = - \langle q(t) \rangle$ as a function of time. This figure is generated with the parameters $S_0 = 4.5$, $v_0=4.4$, and $k_{max}=1000$ for definiteness, but the generic behavior is not changed with other values. The random number $\alpha$ is from the interval $[0,1]$.