Multi-trace Correlators in the SYK Model and Non-geometric Wormholes

TL;DR

This paper investigates multi-trace correlators in the SYK model, focusing on global (spectral) fluctuations of the density of states that survive at large N and dominate early-time dynamics over standard RMT correlations. The authors develop a diagrammatic framework where leading connected contributions are captured by 1PI cactus diagrams and show these can be encoded by a zero-dimensional vector model for the random couplings, extending naturally to non-Gaussian distributions. They identify fluctuation parameters, starting with h2 and h3, that correspond to light bulk fields in a holographic dual and reinterpret connected multi-trace correlators as effects of these fluctuations rather than geometric wormholes. The work also extends to multi-trace operator correlations, replica path integrals, and a G–Σ formulation, laying groundwork for a gravitational description that includes non-geometric bulk degrees of freedom and potentially a dual realization for a single chaotic realization. Overall, the study reveals a structured hierarchy of ensemble fluctuations that inform both the gravity dual and the interpretation of SYK as a quantum-gravitational system beyond JT-like wormholes, with implications for the holographic dictionary and the role of ultra-light bulk modes.

Abstract

We consider multi-energy level distributions in the SYK model, and in particular, the role of global fluctuations in the density of states of the SYK model. The connected contributions to the moments of the density of states go to zero as $N \to \infty$, however, they are much larger than the standard RMT correlations. We provide a diagrammatic description of the leading behavior of these connected moments, showing that the dominant diagrams are given by 1PI cactus graphs, and derive a vector model of the couplings which reproduces these results. We generalize these results to the first subleading corrections, and to fluctuations of correlation functions. In either case, the new set of correlations between traces (i.e. between boundaries) are not associated with, and are much larger than, the ones given by topological wormholes. The connected contributions that we discuss are the beginning of an infinite series of terms, associated with more and more information about the ensemble of couplings, which hints towards the dual of a single realization. In particular, we suggest that incorporating them in the gravity description requires the introduction of new, lighter and lighter, fields in the bulk with fluctuating boundary couplings.

Figures (23)

• Figure 1: Two different chord diagrams that contribute to the moment $M(4,4,4)$. (a) and (b) are the full chord diagrams, while (c) and (d) are their contracted diagrams respectively. Diagram (b) (or it's contraction (d)) is connected while diagram (a) (or its contraction (c)) is not.
• Figure 2: An example of a diagram contributing to the double trace with $k_1=4$ and $k_2=8$.
• Figure 3: The leading diagram contributing to the double trace, where the edges represent the chords connecting the different traces.
• Figure 4: A cyclic diagram for three traces.
• Figure 5: Examples of diagrams going as $\epsilon ^3$ and sub-leading ones.