Three dimensional view of arbitrary q SYK models

In [15] 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) x 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) x 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 compacti fi cation 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 -> infinity, 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.