Degenerate operators in JT and Liouville (super)gravity

TL;DR

This work identifies and analyzes a highly tractable integrable subsector of boundary bilocal operators in JT gravity and its N=1 supersymmetric extension, corresponding to degenerate Virasoro representations with weights $h\in-\mathbb{N}/2$. On the disk, these degenerate bilocals yield explicit, finite-structure expressions for the correlators, illuminate the asymptotic nature of the 1/C Schwarzian expansion (which is convergent only at the degenerate values), and enable controlled exploration of higher-genus corrections by embedding in minimal string and Liouville supergravity frameworks. The authors extend the construction to $\mathcal{N}=1$ JT supergravity, deriving disk super-Schwarzian bilocals, one-loop self-energies, and chaotic behavior that saturates the chaos bound at leading order in $1/C$, while also formulating fixed-length Liouville supergravity amplitudes and their JT limits, including boundary tachyon correlators in the minimal superstring and their matrix-model interpretation. A central theme is that for degenerate $h$, higher-topology corrections mirror those of the partition function and do not generate cross-handles across the bilocal line, in contrast to generic $h$; this links JT gravity with minimal-string/matrix-model structure and provides a robust, nonperturbative handle on correlators in gravity with boundaries. The results have potential implications for bulk observables, non-perturbative effects in quantum gravity, and holographic ensembles in supersymmetric contexts. Overall, the paper advances a coherent, multi-faceted program to understand degenerate sectors in JT and Liouville supergravity through explicit disk amplitudes, perturbative expansions, and a minimal-string/matrix-model perspective that clarifies higher-genus behavior and JT limits.

Abstract

We derive explicit expressions for a specific subclass of Jackiw-Teitelboim (JT) gravity bilocal correlators, corresponding to degenerate Virasoro representations. On the disk, these degenerate correlators are structurally simple, and they allow us to shed light on the 1/C Schwarzian bilocal perturbation series. In particular, we prove that the series is asymptotic for generic weight $h\notin - \mathbb{N}/2$. Inspired by its minimal string ancestor, we propose an expression for higher genus corrections to the degenerate correlators. We discuss the extension to the $\mathcal{N}=1$ super JT model. On the disk, we similarly derive properties of the 1/C super-Schwarzian perturbation series, which we independently develop as well. As a byproduct, it is shown that JT supergravity saturates the chaos bound $λ_L = 2π/β$ at first order in 1/C. We develop the fixed-length amplitudes of Liouville supergravity at the level of the disk partition function, the bulk one-point function and the boundary two-point functions. In particular we compute the minimal superstring fixed length boundary two-point functions, which limit to the super JT degenerate correlators. We give some comments on higher topology at the end.

Figures (12)

• Figure 2: Left: degenerate two-point correlator for $j=1/2$ and $\beta=1$ and $C=1/2$. Blue (bottom): genus zero result \ref{['halfexact']}. Red (top): genus zero \ref{['halfexact']} plus genus one \ref{['halfg1']} result, for a suitable choice of genus counting parameter $e^{-S_0}$. Right: $\tau \to 0$ limit can be non-trivial when wrapping around higher topology.
• Figure 3: Two Feynman graphs contributing at order $1/C$ in the supersymmetric theory.
• Figure 4: Lowest graviton (left) and gravitino (right) contribution to the four-point function. The gravitino contribution only starts at $1/C^2$.
• Figure 5: Integration contour in the $\mu_B$ plane in the supersymmetric case. The initial contour is in green, and the final one in blue. The red lines denote the branch cuts of the integrand. Left: $\eta=+1$. Right: $\eta=-1$.
• Figure 6: FZZT brane segments between $n$ marking operators $M_2(x_i)$\ref{['marki2']} leads upon transforming to the fixed length basis with length $\ell \equiv \sum_j \ell_j$. In the figure we show an example with $n=3$.