The bulk dual of SYK: cubic couplings

Abstract

The SYK model, a quantum mechanical model of N 1 Majorana fermions χ i , with a q-body, random interaction, is a novel realization of holography. It is known that the AdS 2 dual contains a tower of massive particles, yet there is at present no proposal for the bulk theory. As SYK is solvable in the 1/N expansion, one can systematically derive the bulk. We initiate such a program, by analyzing the fermion two, four and six-point functions, from which we extract the tower of singlet, large N dominant, operators, their dimensions, and their three-point correlation functions. These determine the masses of the bulk fields and their cubic couplings. We present these couplings, analyze their structure and discuss the simplifications that arise for large q.

Correlators in the N = 2 $$ \mathcal{N}=2 $$ supersymmetric SYK model

“…Finally in section 5 we briefly comment on the operator product expansion (OPE) between operators that are bilinear in the fundamental fields. These OPE coefficients may be extracted from our 4-point function results in a manner almost identical to the analysis of [32].…”

“…This would be a supersymmetric generalization of the work [32], and we will follow many of the notations employed there. Given all the 4-point functions that we have worked out, we can take their OPE limits and read out the structure constants c With these coefficients extracted out, one can easily check that there are again two different types of contributing diagrams to the leading order in the large-N limit of the supersymmetric SYK model.…”

“…The model develops an emergent (approximate) reparametrization symmetry at low energy [6,[17][18][19] that is also present in dilaton gravity theories on AdS 2 [6,[20][21][22][23]. It has intimate relations with wellstudied random matrix models [6,18,[24][25][26][27][28][29][30][31][32], it further boosts the study of a different type of the large-N limit [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48], and it is closely related to vector models [49]. The SYK model can be generalized to include extra symmetries [50] or to live in higher dimensions [51][52][53][54][55][56].…”

Space-time in the SYK model

“…Detailed investigations of the Sachdev-Ye-Kitaev (SYK) model [1][2][3][4][5][6][7][8][9][10][11][12] have given an interesting, highly nontrivial example of the AdS/CFT duality and a potential framework for quantum black holes. The model, which can be studied at Large N , features emergent reparametrization invariance at the IR critical point, with related out-of-time-order correlators exhibiting quantum chaos, with a Lyapunov exponent characteristic of black holes [2], providing the butterfly effect [13][14][15][16][17][18][19][20].…”

“…The kinetic terms are now standard: the nontrivial residues at the poles in the SYK propagator now appear as JHEP07 (2018)184 nontrivial wave functions. However, one should not expect that higher point functions [10] can be reproduced by local interactions in this 3D picture [11]. The usefulness of an additional dimension also appears in the description of Higher Spin theories [80].…”