Hartle-Hawking wavefunction in double scaled SYK

TL;DR

This work computes the Hartle-Hawking wavefunction $\langle \ell|e^{-\beta T}|0\rangle$ in the double-scaled SYK model, identifying $\ell$ with a discretized bulk geodesic length and establishing the HH wavefunction as a fundamental bulk building block. It shows that un-crossed matter correlators arise by gluing HH wavefunctions with appropriate weights, and provides explicit closed-form expressions and a propagator $\langle\ell_1|e^{-\beta T}|\ell_2\rangle$, linking boundary data to a chord-based bulk picture. The paper also analyzes the probability distribution $P_{\ell}(\beta,q)$ and its dependence on temperature and the deformation parameter $q$, illustrating a transition from a peak at $\ell=0$ to nonzero optimal $\ell$ at low temperature. Overall, the results suggest a coherent bulk spacetime interpretation for DSSYK built from chords, and open avenues toward crossing correlators, $1/N$ corrections, and higher-genus extensions.

Abstract

We compute the transition amplitude between the chord number $0$ and $\ell$ states in the double scaled SYK model and interpret it as a Hartle-Hawking wavefunction of the bulk gravitational theory. We observe that the so-called un-crossed matter correlators of double scaled SYK model are obtained by gluing the Hartle-Hawking wavefunctions with an appropriate weight.

Figures (2)

• Figure 1: Plot of $P_\ell(\beta,q)$ against $\ell$ for various values of $\beta$ and $q$.
• Figure 2: Plot of $\overline{\ell}(\beta,q)$ against $\beta$ with fixed $q$.