End of the world brane in double scaled SYK

TL;DR

This paper investigates the end of the world brane in double scaled SYK, revealing its boundary state as a coherent state of q-deformed oscillators, connecting it to JT gravity, and analyzing the half-wormhole amplitude.

Contribution

It introduces a novel description of the EOW brane in DSSYK using q-deformed oscillators and connects it to known gravity wavefunctions, providing new insights into holographic dualities.

Findings

Boundary state is a coherent state of q-deformed oscillators.

Associated orthogonal polynomial is the continuous big q-Hermite polynomial.

In a certain limit, the polynomial reduces to the Whittaker function, matching JT gravity wavefunction.

Abstract

We study the end of the world (EOW) brane in double scaled SYK (DSSYK) model. We find that the boundary state of EOW brane is a coherent state of the $q$-deformed oscillators and the associated orthogonal polynomial is the continuous big $q$-Hermite polynomial. In a certain scaling limit, the big $q$-Hermite polynomial reduces to the Whittaker function, which reproduces the wavefunction of JT gravity with an EOW brane. We also compute the half-wormhole amplitude in DSSYK and show that the amplitude is decomposed into the trumpet and the factor coming from the EOW brane.