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End of the world brane in double scaled SYK

Kazumi Okuyama

TL;DR

This work analyzes the end-of-the-world (EOW) brane in the double-scaled SYK (DSSYK) model, showing the boundary state is a coherent state of $q$-deformed oscillators and the EOW wavefunction is described by the continuous big $q$-Hermite polynomial. In a triple scaling limit, this wavefunction reproduces the JT gravity Whittaker function, establishing a concrete bridge between DSSYK and JT gravity with EOW branes. The half-wormhole amplitude in DSSYK factorizes into a trumpet component and a brane-dependent factor, mirroring the JT result and providing a precise holographic interpretation of EOW brane effects in the DSSYK ensemble. The findings reveal quantized bulk lengths controlled by the DSSYK parameter $\\lambda$, and suggest avenues for generalizations to multiple branes and matter couplings, offering a pathway toward a richer DSSYK-based gravity dictionary.

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.

End of the world brane in double scaled SYK

TL;DR

This work analyzes the end-of-the-world (EOW) brane in the double-scaled SYK (DSSYK) model, showing the boundary state is a coherent state of -deformed oscillators and the EOW wavefunction is described by the continuous big -Hermite polynomial. In a triple scaling limit, this wavefunction reproduces the JT gravity Whittaker function, establishing a concrete bridge between DSSYK and JT gravity with EOW branes. The half-wormhole amplitude in DSSYK factorizes into a trumpet component and a brane-dependent factor, mirroring the JT result and providing a precise holographic interpretation of EOW brane effects in the DSSYK ensemble. The findings reveal quantized bulk lengths controlled by the DSSYK parameter , and suggest avenues for generalizations to multiple branes and matter couplings, offering a pathway toward a richer DSSYK-based gravity dictionary.

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 -deformed oscillators and the associated orthogonal polynomial is the continuous big -Hermite polynomial. In a certain scaling limit, the big -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.
Paper Structure (12 sections, 118 equations, 1 figure)

This paper contains 12 sections, 118 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of DSSYK
  3. EOW brane in JT gravity
  4. EOW brane in DSSYK
  5. Triple scaling limit
  6. Boundary state of EOW brane
  7. Representation of the $q$-oscillator algebra
  8. Half-wormhole in DSSYK
  9. Trumpet and cylinder in DSSYK
  10. Conclusions and outlook
  11. Useful formulae
  12. Continuous big $q$-Hermite polynomials

Figures (1)

  • Figure 1: Plot of the spectral form factor $Z_{\text{cylinder}}(\beta+\mathrm{i} t,\beta-\mathrm{i} t,\Delta)$ as a function of $t$. We set $u_0=1,\beta=1,q=1/2$ in \ref{['eq:cyl-matter']} and evaluate it numerically by truncating the summation up to $\mathfrak{b}_{\text{cut}}=100$.