The double scaled limit of Super--Symmetric SYK models

TL;DR

The paper solves the double-scaled limit of the ${\cal N}=2$ SUSY-SYK model by encoding moments of the Hamiltonian through oriented chord diagrams and a local transfer-matrix formalism built from $q$-deformed oscillators. It provides exact results for the asymptotic density of states, ground-state counting in fixed charge sectors, and two-point functions, revealing substantial ground-state contributions that modify the conformal (super-Schwarzian) picture and enabling a Liouville/Super-Liouville interpretation in the IR. It also uncovers a quantum-group structure, showing the transfer-matrix algebra is a contraction of $sl_q(2|1)$, which hints at a deeper algebraic-gravitational dual description and constrains higher-point correlators in this highly symmetric double-scaled regime.

Abstract

We compute the exact density of states and 2-point function of the $\mathcal{N} =2$ super-symmetric SYK model in the large $N$ double-scaled limit, by using combinatorial tools that relate the moments of the distribution to sums over oriented chord diagrams. In particular we show how SUSY is realized on the (highly degenerate) Hilbert space of chords. We further calculate analytically the number of ground states of the model in each charge sector at finite $N$, and compare it to the results from the double-scaled limit. Our results reduce to the super-Schwarzian action in the low energy short interaction length limit. They imply that the conformal ansatz of the 2-point function is inconsistent due to the large number of ground states, and we show how to add this contribution. We also discuss the relation of the model to $SL_q(2|1)$. For completeness we present an overview of the $\mathcal{N}=1$ super-symmetric SYK model in the large $N$ double-scaled limit.

Figures (12)

• Figure 1: An example of a chord diagram (left: on a circle, right: on a line).
• Figure 3: Disentanglement of a specific chord diagram, according to the algorithm presented above, going from top to bottom. The minimal chords - ones that are separated only by enemy chords, are colored in orange. In the next step these chords taken to the left edge, and we have new minimal chords. For example, see that in the first step chord 2 is not minimal, as it is friends with chord 4, nested in it. In the second step chords 1 and 2 are both minimal, as they are enemies. Primed notation means that a chord has the original indices, excluding the ones it shared when passing through friends. For example, the indices of $3'$ are the indices of $3$, excluding the ones shared with $1$.
• Figure 5: An example of a vector in ${\cal H}_{\text{aux}}$, and its representation in terms of chords.
• Figure 6: An example for a product of 3 flavors of chords for (\ref{['eq:inn_oneq']}), denoted by $X,O,\Delta$. The left diagram has a single intersection, and the right one has four, which means that $\left<XO\Delta X|OX\Delta X\right>=q+q^4$.
• Figure 7: An example for two diagrams contributing to the state $Q^{\dagger} \left |\cdots OO\cdots\right >$. As can be seen in section \ref{['sec:TMrules']}, the two diagrams have the same contribution, up to a minus sign coming from the intersection in the right diagram. This means that their sum will vanish. We see that we cannot bring diagrams of this type to an empty chord diagram. This means that any diagram with two consecutive $X$'s or $O$'s will not contribute to the element $\left< \emptyset |T^k_s|\emptyset \right>$.