Towards a String Dual of SYK

TL;DR

This work embeds the SYK model into string theory by placing a large number $Q$ of FZZT branes in $(p,1)$ minimal string theory. Through the two-matrix model description and a color-flavor transformation, the open-string sector between FZZT and ZZ branes yields a discretized SYK action with $G_{ab}=\frac{1}{N}\sum_i \psi_{ia}^\dagger \psi_{ib}$ and a disorder-averaged $J$-dependent interaction $G^{*p}$; the continuum limit at large $Q$ reproduces the standard SYK quantum mechanics with emergent time. A rich phase diagram emerges, linking the minimal-string $(p,q)$ spectral curve to SYK: minimal strings reside in the ZZ regime ($N>pQ$), while continuum SYK sits in the FZZT regime ($N<pQ$), connected by a critical line at $N=pQ$; in the doubled-scaled limit, a Liouville-like 2D theory suggests a non-critical-string dual to SYK. The construction also informs debates on ensemble averaging in holography and points to an emergent-time, UV-regularized perspective on low-dimensional quantum gravity systems.

Abstract

We propose a paradigm for realizing the SYK model within string theory. Using the large $N$ matrix description of $c<1$ string theory, we show that the effective theory on a large number $Q$ of FZZT D-branes in $(p,1)$ minimal string theory takes the form of the disorder averaged SYK model with $J ψ^{p}$ interaction. The SYK fermions represent open strings between the FZZT branes and the ZZ branes that underly the matrix model. The continuum SYK dynamics arises upon taking the large $Q$ limit. We observe several qualitative and quantitative links between the SYK model and $(p,q)$ minimal string theory and propose that the two describe different phases of a single system. We comment on the dual string interpretation of double scaled SYK and on the relevance of our results to the recent discussion of the role of ensemble averaging in holography.

Figures (2)

• Figure 1: The phase diagram of $(p,1)$ minimal string theory with $Q$ FZZT branes. The $(p,q)$ minimal string emerges in the regime where the number of ZZ-branes $N$ is much larger than $p \, Q$. The continuum SYK model with ${\cal J} \psi^p$ interaction resides in the regime when $p\, Q\gg N$.
• Figure 2: