Degenerate chords in double-scaled SYK

TL;DR

This work studies degenerate chords in the double-scaled SYK (DSSYK) model, focusing on a degenerate matter operator D = M_{-1/2}. By leveraging a null vector equation for D and a two-sided chord Hilbert space, the authors derive recursion relations for the matter two-point function G(Delta|theta1,theta2) that bypass the need to sum infinite chord diagrams. They demonstrate delta-functional matrix elements at Delta = -1/2 and show how composites DM_Delta generate shifts Delta -> Delta ± 1/2, enabling OPE-like relations and a structured representation-theoretic analysis of degenerate sectors. The approach parallels boundary Liouville CFT methods, revealing algebraic structures tied to q-deformed oscillators and suggesting extensions to higher-point functions and connections to bulk Liouville-type theories and quantum groups.

Abstract

The matter operator in the double-scaled SYK model exhibits special properties when its dimension is analytically continued to -1/2. At this dimension, the operator is in a degenerate representation of the q-deformed oscillator algebra and satisfies a null vector equation. Its peculiar fusion property gives rise to recursion relations among matter correlation functions. We find that these relations allow us to determine the two-point function without having to sum over infinitely many chord diagrams.

Figures (7)

• Figure 1: (left) a chord diagram contributing $q^2$ to $m_8$, and (right) a quantum mechanics interpretation of the same diagram.
• Figure 2: An uncrossed four-point function (\ref{['4ptfn2']}) and the parameters $\beta_i,\theta_i$.
• Figure 3: OPE of boundary operators.
• Figure 4: A disk correlator that can be expressed in terms of $c_\pm$ and $d$ coefficients.
• Figure 5: A half-disk with two matter chords and $3+4+2$$H$-chords, corresponding to a basis vector $|3,4,2\rangle$ in the two-sided chord Hilbert space $[\Delta_1\times\Delta_2]$.