$p$-Chords, Wee-Chords, and de Sitter Space

TL;DR

This paper investigates the algebraic structures underlying a proposed holographic duality between the double-scaled SYK model and de Sitter space, resolving previous disagreements by introducing the concept of wee-chords.

Contribution

It identifies a third q-deformed algebra, the wee-chords, which clarifies the relation between DSSYK and Chern-Simons line operators, resolving prior inconsistencies.

Findings

Identification of wee-chords as a key algebraic structure

Resolution of the duality discrepancy between DSSYK and de Sitter space

Restoration of the scale separation in the duality framework

Abstract

One of us (L.S.) and H. Verlinde independently conjectured a holographic duality between the double-scaled SYK model at infinite temperature and dimensionally reduced $(2+1)$-dimensional de Sitter space [1]-[8]. Beyond the statement that such a duality exists there was deep disagreement between the two proposals [9]. In this note, we trace the origin of the disagreement to a superficial similarity between two q-deformed algebraic structures: the algebra of "chords" in DSSYK, and the algebra of line operators in the Chern-Simons formulation of 3D de Sitter gravity. Assuming that these two structures are the same requires an identification of parameters [7][10] which leads to a collapse of the separation of scales [9] -- the separation being required by the semiclassical limit [3][9]. Dropping that assumption restores the separation of scales but leaves unexplained the relation between chords and Chern-Simons line operators. In this note we point out the existence of a third q-deformed algebra that appears in DSSYK: the algebra of ``wee-chords." Identifying the Chern-Simons line operators with wee-chords removes the discrepancy and leads to a satisfying relation between the two sides of the duality.