Explorations of nonperturbative Jackiw-Teitelboim gravity and supergravity

Some recently proposed definitions of Jackiw-Teitelboim (JT) gravity and supergravities in terms of combinations of minimal string models are explored, with a focus on physics beyond the perturbative expansion in spacetime topology. While this formally involves solving infinite-order nonlinear differential equations, it is shown that the physics can be extracted to arbitrarily high accuracy in a simple controlled truncation scheme, using a combination of analytical and numerical methods. The nonperturbative spectral densities are explicitly computed and exhibited. The full spectral form factors, involving crucial nonperturbative contributions from wormhole geometries, are also computed and displayed, showing the nonperturbative details of the characteristic “slope,” “dip,” “ramp,” and “plateau” features. It is emphasized that results of this kind can most likely be readily extracted for other types of JT gravity using the same methods.

Figure 1

The full spectral form factor, showing the classic (saxophone) shape made up of a slope, dip, ramp, and plateau. This is computed using the methods of this paper for the (2,2) model of JT supergravity. Here, $\beta =35$, $\hslash =1/5$. (See text.)

Figure 2

The full spectral density, computed using the methods of this paper, for the (2,2) model of JT supergravity. The dashed blue line is the disc level result of Eq. (5). Here $\hslash =1$.

Figure 3

The “nearly ${\mathrm{AdS}}_2$” geometry, presented in two equivalent ways.

Figure 4

Black holes vs wormholes.

Figure 5

The complete classical potential for JT supergravity (solid line). The uppermost dotted line is a truncation up to $t_2$, while the lower dotted line is a truncation up to $t_4$.

Figure 6

The $\mathrm{\Gamma }=+\frac{1}{2}$ solution (solid line) of the string equation for truncation up to $t_6$. The inset shows a closeup of the smooth transitional region in the interior. For comparison, the full classical solution line is shown too (dotted line).

Figure 7

The $\mathrm{\Gamma }=-\frac{1}{2}$ solution (solid line) of the string equation for truncation up to $t_6$. The inset shows the smooth well that developed in the interior. For comparison, the full classical solution is shown too (dotted line).

Figure 8

The full spectral density (made out of red dots), computed using the methods of this paper, for the (0,2) model of JT supergravity. The dashed blue line is the disc-level result of Eq. (5). Here, $\hslash =1$. The companion result for the (2,2) model is in Fig. 2.

Figure 9

Comparison of the full spectral density for the (2,2) model of JT supergravity with Eq. (25) for the leading form (dark blue dots). The dashed line is the disc-level result [Eq. (23)]. The cases $\mu =1$, 5, and 10 are shown (lower to upper). Here, $\hslash =1$.

Figure 10

Comparison of the full spectral density for the (0,2) model of JT supergravity with Eq. (25) for the leading form (dark blue dots). The dashed line is the disc-level result [Eq. (23)]. The cases $\mu =1$, 5, and 10 are shown (lower to upper). Here, $\hslash =1$.

Figure 11

The solid lines show close-ups of the $\mathrm{\Gamma }=\frac{1}{2}$ or (2,2) solution (left) and the $\mathrm{\Gamma }=-\frac{1}{2}$ or (0,2) solution (right) of the string equation for truncation up to $t_6$, for $\hslash =\frac{1}{5}$. For comparison, the full classical solution is shown too (dotted lines).

Figure 12

Two examples ($\mu =1$, lower; $\mu =5$, upper) of the full spectral density for the (2,2) model of JT supergravity, for $\hslash$ reduced to $\frac{1}{5}$. The (blue) dots are points from Eq. (25). In each case, the dashed line is the disc result of Eq. (23). Cf. the $\hslash =1$ cases of Fig. 9.

Figure 13

The disconnected (starting higher at the left) vs the connected (lower) two-point function of the partition function as a function of $\beta$ at $\hslash =1$, showing a phase transition at ${\beta }_{\mathrm{cr}}\simeq 13.56$.

Figure 14

The disconnected part of the (2,2) JT supergravity spectral density function vs $t$, at $\beta =50$ and $\hslash =1$, showing the classic slope feature.

Figure 15

The disconnected part of the (0,2) JT supergravity spectral density function vs $t$, at $\beta =50$ and $\hslash =1$.

Figure 16

The connected part of the (2,2) JT supergravity spectral density function vs $t$, at $\beta =50$ and $\hslash =1$, showing the classic ramp and plateau features.

Figure 17

The connected part of the (0,2) JT supergravity spectral density function vs $t$, at $\beta =50$ and $\hslash =1$.

Figure 18

The full spectral form factor for the (2,2) model of JT supergravity at $\beta =50$, $\hslash =1$. (The case of $\hslash =1/5$ is in Fig. 1).

Figure 19

The full spectral form factor for (0,2) JT supergravity. Here, $\beta =50$ and $\hslash =1$. Cf. the (2,2) case in Fig. 18.

Figure 20

A series showing the evolution, with $\beta$, of the disconnected part of the spectral form factor, for (2,2) JT supergravity. Here $\hslash =1$.

Figure 21

A series showing the evolution, with $\beta$, of the full spectral form factor, for (2,2) JT supergravity. Here, $\hslash =1$. For comparison, the curves have been rescaled to start out at the same height as the highest-temperature curve.

Figure 22

The full spectral form factor at $\beta =14$, $\hslash =1$, for (2,2) JT supergravity. A (relatively) high-temperature feature appears near the crossover from ramp to plateau. See text for discussion.

Figure 23

Disconnected part of the spectral form factor at $\beta =9$, $\hslash =1$, for (2,2) JT supergravity. The growing dips are consistent with a pattern of zeros developing at infinite temperature.

Figure 24

The solution (solid line) of the string equation for truncation up to $t_7$. The inset shows the well that develops in the interior. The full classical solution is shown too (dotted line).

Figure 25

The full spectral density, for the nonperturbative completion of JT gravity in Ref. [3]. The dashed blue line is the disc-level result of Eq. (4).

Figure 26

The disconnected part of the JT gravity spectral form factor vs $t$, at $\beta =50$ and $\hslash =1$, for the nonperturbative scheme of Ref. [3].

Figure 27

The connected part of the JT gravity spectral form factor vs $t$, at $\beta =50$ and $\hslash =1$, for the nonperturbative scheme of Ref. [3].

Figure 28

The full JT gravity spectral form factor vs $t$, at $\beta =50$ and $\hslash =1$, for the nonperturbative scheme of Ref. [3].

Figure 29

The full spectral density for the exactly solvable Airy model, at $\hslash =1$. The dashed line is the disc-level result.

Figure 30

The disconnected (starting higher at the left) vs the connected (lower) two-point function of the Airy model’s partition function as a function of $\beta$, showing a phase transition at $\beta ={\beta }_{\mathrm{cr}}\simeq 0.92$. A dashed line shows the exact result; the dots are for a numerical truncation.

Figure 31

The disconnected part of the Airy model’s spectral form factor vs $t$, at $\beta =1/2$, showing the classic slope feature. A dashed line shows the exact result; dots are for a numerical truncation. (The undulations at the end are numerical errors at ultrasmall values, and so should be ignored.)

Figure 32

The connected part of the Airy model’s spectral form factor vs $t$, at $\beta =1/2$. A dashed line shows the exact result; dots are for a numerical truncation.

Figure 33

The full spectral form factor of the Airy model vs $t$, at $\beta =1/2$.