Three Dimensional View of Arbitrary $q$ SYK models

TL;DR

The work shows that for arbitrary even $q$, a three dimensional bulk model with a metric conformal to $AdS_2 \times S^1/Z_2$ and a central delta potential reproduces the exact SYK spectrum via a spectral equation $k_c(h,q)=1$. A Horava-Witten compactification yields the correct spectrum, and a nonstandard $AdS_2$-like propagator between central points matches the SYK bilocal propagator up to a $q$-dependent factor. In the large $q$ limit, only the $p_m=3/2$ mode contributes to the bilocal propagator, consistent with SYK, while the others decouple as $1/q^2$. Overall, the results provide a KK-like three dimensional interpretation of the SYK spectrum and its bilocal structure, suggesting a essential role for a 3D bulk in the holographic dual of SYK models.

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.

Figures (1)

• Figure 1: Plot of $\log f(q)$ for $p_m=3/2$. For large $q$ the data fits well with the function $f(q) \sim \frac{2}{q}$