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 SLq(2|1). For completeness we present an overview of the mathcal{N} = 1 super-symmetric SYK model in the large N double-scaled limit.
Highlights
The Sachdev-Ye-Kitaev (SYK) model consists of N Majorana fermions with random allto-all interactions [1,2,3]
We compute the exact density of states and 2-point function of the 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
The main result of this paper is the derivation of an analytical expression for the asymptotic spectrum of the N = 2 SYK model in the double scaled limit, both in fixed charge sectors and in the presence of a chemical potential
Summary
The Sachdev-Ye-Kitaev (SYK) model consists of N Majorana fermions with random allto-all interactions [1,2,3]. In this paper we primarily focus on the N = 2 supersymmetric SYK model in the double scaled limit, extending the chord diagram and transfer matrix methods in [40, 41] to treat this model as well. We present an analysis of the number of ground states at finite N We use this formalism to calculate the 2-point functions in the double scaled limit. Recent progress has been made in analyzing the level spacing in the SYK model, JT gravity, and their relation to random matrix theory ensembles [17, 33,34,35, 37, 47] Such analysis requires considering double trace quantities, which is beyond the scope of this paper
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