$N=2$ JT Supergravity and Matrix Models

TL;DR

This work establishes a precise holographic duality between ${ t N}=2$ JT supergravity on asymptotically AdS${}_2$ spaces and a family of independent ${ t N}=2$ random-matrix ensembles, one for each $R$-charge multiplet. By extending Mirzakhani’s recursion to ${ t N}=2$ hyperbolic surfaces and deriving the corresponding measure via torsion, the authors show that gravity loop equations match random-matrix loop equations for an AZ$(1,2)$ ensemble, validating the duality to all orders in the topological expansion. They analyze the disk, cylinder, and three-holed sphere, compute the leading black-hole spectrum, and identify how BPS states contribute discretely while non-BPS sectors yield continuous densities governed by the spectral curve $y_q(x)$. Time-reversal symmetry (both ${ t T}$ and ${ t CT}$) is incorporated, with detailed matrix-model predictions for crosscaps and unorientable geometries, including anomalies and possible deformations via defects. The paper also sketches extensions to ${ t N}=4$ JT gravity, where the multiplets remain statistically independent and the dual random-matrix description persists, providing a comprehensive framework linking Jackiw–Teitelboim supergravity, topological recursion, and random-matrix theory in a supersymmetric setting.

Abstract

Generalizing previous results for $N=0$ and $N=1$, we analyze $N=2$ JT supergravity on asymptotically AdS${}_2$ spaces with arbitrary topology and show that this theory of gravity is dual, in a holographic sense, to a certain random matrix ensemble in which supermultiplets of different $R$-charge are statistically independent and each is described by its own $N=2$ random matrix ensemble. We also analyze the case with a time-reversal symmetry, either commuting or anticommuting with the $R$-charge. In order to compare supergravity to random matrix theory, we develop an $N=2$ analog of the recursion relations for Weil-Petersson volumes originally discovered by Mirzakhani in the bosonic case.

Figures (13)

• Figure 1: Spectral properties of JT gravity in the large $S_0$ approximation, for different number of supercharges. The theory is unique for $\mathcal{N}=0,1,4$ but multiple theories exist for $\mathcal{N}=2$ depending on an odd integer $\widehat{q}$, the $R$-charge of the supercharge, and a number $\delta~{\rm mod}~\mathbb{Z}$, a non-integer background $R$-charge. For most values of these parameters, there is an order $e^{S_0}$ number of BPS states but the index vanishes due to cancellations. (A refinement of the index exists which does not vanish, and predicts the presence of BPS states.) One can deform $\mathcal{N}=1$ JT gravity with Ramond punctures, which produces a finite number of BPS states and a non-vanishing index, which does not necessarily grow with $S_0$. (This case is not shown in the table.)
• Figure 2: Summary of $\mathcal{N}=2$ random matrix ensembles. One of our main results is that different supermultiplets are statistically independent and can therefore be described by their own ensemble. We label them by their symmetry group, with $L$ and $L'$ being the appropriate dimensions for each supermultiplet of $R$-charges $(k,k+\widehat{q}\,)$ with $k\in \mathbb{Z}+\delta$. ${\sf T}$ is a time-reversal symmetry that commutes with all bosonic generators (can exist for any $\delta$) while ${\sf CT}$ anticommutes with the $R$-charge generator (can only exist for $\delta=0$ or $1/2$) and is defined such that ${\sf CT} Q^\dagger {\sf CT}^{-1} = Q$. For this reason, in the presence of ${\sf CT}$ symmetry the multiplets $(k,k+\widehat{q}\,)$ and $(-k-\widehat{q},-k)$ are identified. The final two exceptions only are relevant when $\delta= 1/2$. In section \ref{['timereversal']} we explain in what sense these are the only cases to consider.
• Figure 3: Level spacing statistics for ${\cal N}=2$ SYK with $N=10$ fermions, averaging over $400$ instances of the couplings. The plots show the probability distribution of $r=(\lambda_{j+2}-\lambda_{j+1})/(\lambda_{j+1}-\lambda_j)$ for the multiplets of charges $(-4,-1)$, $(-3,0)$, and $(-2,1)$. We omit multiplets related to these by ${\sf {CT}}$ symmetry or with only one state. The smooth curves denote the surmise $P_\upbeta(r)$ with $\upbeta=1$ (dashed blue), $\upbeta=2$ (black) and $\upbeta=4$ (blue). The distribution of all multiplets is consistent with $\upbeta=2$. The case $(-4,-1)$ has larger deviations since the Hilbert space dimension is smaller.
• Figure 4: Level spacing statistics for ${\cal N}=2$ SYK with $N=11$ fermions, averaging over $200$ instances of the couplings. The plots show the probability distribution of $r$ for multiplets with the indicated $R$-charges. The smooth curves denote the surmise $P_\upbeta(r)$ with $\upbeta=1$ (dashed blue), $\upbeta=2$ (black) and $\upbeta=4$ (blue). The distribution of all multiplets is consistent with $\upbeta=2$, except for the case of charges $(-3/2,3/2)$, which is matched by $\upbeta=1$.
• Figure 5: A cylinder with two NAdS boundaries -- the leading connected two-boundary wormhole. We label each NAdS boundary by their inverse temperature $\beta$ and ${\mathrm U}(1)$$R$-symmetry chemical potential $\alpha$. Each ${\mathcal{N}}=2$ hyperbolic surface has an interior geodesic, indicated by the dashed line, with length $b$ and ${\mathrm U}(1)$ holonomy $\phi$, that should be integrated over.