Three Dimensional View of Arbitrary $q$ SYK models

TL;DR

This paper extends the 3D geometric realization of SYK models to arbitrary q-fermion interactions, showing that their spectra and propagators can be exactly reproduced in a conformally related space with specific boundary conditions.

Contribution

It generalizes the 3D holographic description of SYK models to arbitrary q, including a detailed analysis of the spectrum, propagators, and the q→∞ limit.

Findings

Exact spectrum matches between SYK and 3D model

Bilocal propagator reproduces SYK results

Wave functions vanish as 1/q and 1/q^2 in the large q limit

Abstract

In \url{arXiv:1704.07208} it was shown that the spectrum and bilocal propagator of SYK model with four fermion interactions can be realized as a three dimensional model in $AdS_2 \times S^1/Z_2$ with nontrivial boundary conditions in the additional dimension. In this paper we show that a similar picture holds for generalizations of the SYK model with $q$-fermion interactions. The 3D realization is now given on a space whose metric is conformal to $AdS_2 \times S^1/Z_2$ and is subject to a non-trivial potential in addition to a delta function at the center of the interval. It is shown that a Horava-Witten compactification reproduces the exact SYK spectrum and a non-standard propagator between points which lie at the center of the interval exactly agrees with the bilocal propagator. As $q \rightarrow \infty$, the wave function of one of the modes at the center of the interval vanish as $1/q$, while the others vanish as $1/q^2$, in a way consistent with the fact that in the SYK model only one of the modes contributes to the bilocal propagator in this limit.