Towards a full solution of the large N double-scaled SYK model

TL;DR

The paper provides an exact all-energy solution to the large-N double-scaled SYK model by mapping correlation functions to chord diagrams and developing a robust bi-local operator framework. It derives non-perturbative diagrammatic rules, obtains the exact 4-point function, and analyzes chaos exponents via a q-deformed SL(2) perspective, with the R-matrix connected to quantum-group 6j-symbols. A central result is the identification of a quantum-group structure (U_q(su(1,1))) governing the full model, including a finite-energy spectrum and Schwarzian-like behavior in the appropriate limit. The work also clarifies the interplay between large-N and large-p limits, offering a unified view through chord-counting and q-special functions, and outlines a concrete path toward a complete n-point function framework via quantum-group representations.

Abstract

We compute the exact, all energy scale, 4-point function of the large $N$ double-scaled SYK model, by using only combinatorial tools and relating the correlation functions to sums over chord diagrams. We apply the result to obtain corrections to the maximal Lyapunov exponent at low temperatures. We present the rules for the non-perturbative diagrammatic description of correlation functions of the entire model. The latter indicate that the model can be solved by a reduction of a quantum deformation of SL$(2)$, that generalizes the Schwarzian to the complete range of energies.

Figures (14)

• Figure 1: An example of a chord diagram for $k=8$. With the numbering in the figure, this diagram means that $I_1=I_2$, $I_3=I_6$, $I_4=I_8$ and $I_5=I_7$ (and otherwise all of these are distinct).
• Figure 2: A chord diagram contributing to $\langle \mathop{\mathrm{tr}}\nolimits M H^2 M H^4 \rangle_J$. A chord connecting 2 $M$-operators is distinguished from the Hamiltonian chords, and is represented here by a dashed line.
• Figure 3: Cutting open a chord diagram.
• Figure 4: An example for partial open chords, until stage 2.
• Figure 5: At any given node, either a chord opens (as in the left), or a chord is closed (as in the right), crossing all the chords below it.