Exploring supersymmetric wormholes in $\cal{N} = 2$ SYK with chords

TL;DR

This work probes zero-temperature supersymmetric wormholes in the $\mathcal{N}=2$ double-scaled SYK model using chord techniques, establishing a precise bulk–boundary link: the probability that a SUSY wormhole has vanishing length equals the fraction of extremal ground states in a fixed $U(1)_R$ sector. It develops a refined two-sided, $\mathcal{N}=4$-symmetric chord Hilbert space, solves for the Hartle–Hawking SUSY ground state via $q$-Hermite recursions, and derives nonperturbative ground-state degeneracies that match refined index computations. The paper then computes zero-temperature two-point functions and the wormhole length, including horizontal matter chords and the structure of the superchord algebra and coproduct, thereby extending the analysis to SUSY wormholes with matter. A detailed discussion of the bulk-to-boundary map and boundary-sign issues clarifies how UV completion affects nonperturbative corrections and suggests generalizations to other extremal black holes, highlighting the UV-sensitive interplay between gravity and microstate counting. Overall, the results showcase how the UV-complete super chord theory captures nonperturbative microstate data beyond the super-Schwarzian, and provide a robust framework for studying SUSY wormholes and their informational properties in holographic systems.

Abstract

A feature the $\mathcal{N}=2$ supersymmetric Sachdev-Ye-Kitaev (SYK) model shares with extremal black holes is an exponentially large number of ground states that preserve supersymmetry. In fact, the dimension of the ground state subsector is a finite fraction of the total dimension of the SYK Hilbert space. This fraction has a remarkably simple bulk interpretation as the probability that the zero-temperature wormhole -- a supersymmetric Einstein-Rosen bridge -- has vanishing length. Using chord techniques, we compute the zero-temperature Hartle-Hawking wavefunction; the results reproduce the ground state count obtained from boundary index computations, including non-perturbative corrections. Along the way, we improve the construction [arXiv:2003.04405] of the super-chord Hilbert space and show that the transfer matrix of the empty wormhole enjoys an enhanced $\mathcal{N} = 4 $ supersymmetry. We also obtain expressions for various two point functions at zero temperature. Finally, we find the expressions for the supercharges acting on more general wormholes with matter and present the superchord algebra.

Figures (4)

• Figure 1: Wavefunction of the $j=0$ ground state. We plot the probability that the wormhole has some fixed chord number. As $\lambda$ decreases, the wavefunction recedes to larger values of $n$. The probability that the wormhole has zero length $|\Psi(0)|^2$ gives the fraction of states in the Hilbert space that are supersymmetric. In the small $\lambda$ regime this agrees with the super-Schwarzian prediction.
• Figure 2: Fraction of ground states as a function of charge $j_R$ for different values of $\lambda$. For small $\lambda$, the number of ground states is a tiny fraction of the total number of states of charge $j_R$.
• Figure 3: Zero temperature 2-pt function in double scaled SYK as a function of $\Delta$. We are considering the fixed charge $j_R = 0$ ensemble. As $\lambda = 2p^2/N$ tends to zero, we recover the $\mathcal{N} = 2$ super-Schwarzian predictions.
• Figure 4: Length as computed by the chords and the Schwarzian prediction. As $q\to 1$ we recover the $\mathcal{N} = 2$ super-Schwarzian predictions.