[Paper Review] Towards a String Dual of SYK

Akash Goel, Herman Verlinde|arXiv (Cornell University)|Mar 4, 2021

Black Holes and Theoretical Physics40 references4 citations

TL;DR

This paper proposes a string-theoretic realization of the Sachdev-Ye-Kitaev (SYK) model by embedding Q FZZT D-branes in (p,1) minimal string theory. Using the large N matrix model of c < 1 string theory, it derives the effective worldvolume theory on these branes as a disorder-averaged SYK model with Jψp interactions. The key result is that the continuum SYK dynamics emerges in the large Q limit, suggesting SYK and (p,q) minimal string theory describe different phases of a unified system governed by three integers: (p,1), N (ZZ branes), and Q (FZZT branes).

ABSTRACT

We propose a paradigm for realizing the SYK model within string theory. Using the large $N$ matrix description of $c&lt;1$ 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.

Motivation & Objective

• To establish a concrete string theory realization of the SYK model, which has so far lacked a UV completion within string theory.
• To explore the connection between the SYK model and (p,q) minimal string theory through the framework of open-closed string duality.
• To demonstrate that the effective theory on Q FZZT branes in (p,1) minimal string theory takes the form of the disorder-averaged SYK model with Jψp interaction.
• To clarify the role of ensemble averaging in holography by showing that string theory naturally produces the averaged SYK Hamiltonian rather than a specific instance.

Proposed method

• Using the large N matrix model description of (p,1) minimal string theory, the authors model the system via two matrices A and B with potentials tuned to a p/q multi-critical point.
• The FZZT branes are coupled to ZZ branes via open strings, whose dynamics are encoded in the matrix model partition function.
• Applying a color-flavor transformation to the matrix model, the partition function is mapped to a matrix model form of the SYK Hamiltonian with Jψp interaction.
• Taking the large Q limit (number of FZZT branes) leads to the continuum SYK quantum mechanics, with time emerging from the large N limit.
• The spectral curve of the (p,q) minimal string is used to relate the FZZT brane correlation functions to SYK correlation functions.
• The phase diagram of the system is constructed by varying N (ZZ branes), Q (FZZT branes), and the (p,1) minimal model, identifying distinct phases: ZZ-dominated (minimal string) and FZZT-dominated (SYK).

Experimental results

Research questions

• RQ1 Can the SYK model be realized as a worldvolume theory of D-branes in string theory, providing a UV completion?
• RQ2 How does the disorder-averaged SYK Hamiltonian arise naturally from a string-theoretic construction?
• RQ3 What is the relationship between (p,q) minimal string theory and the SYK model in terms of phase structure and duality?
• RQ4 How does time emerge in the SYK model from a non-perturbative string theory framework?
• RQ5 What is the role of ensemble averaging in holography, and how is it realized in this string-theoretic construction?

Key findings

• The effective theory on Q FZZT branes in (p,1) minimal string theory is shown to be a disorder-averaged SYK model with Jψp interaction, derived from the matrix model partition function.
• The continuum SYK dynamics arises in the large Q limit, with time emerging as a collective phenomenon from the matrix model structure.
• The system exhibits a phase diagram with two distinct regimes: the ZZ-regime (N ≫ pQ) corresponding to minimal string theory, and the FZZT-regime (pQ ≫ N) corresponding to the SYK model.
• The FZZT brane correlation functions in the matrix model match the correlation functions of the SYK model, establishing a quantitative link between the two theories.
• The double-scaled SYK model is interpreted as the worldsheet theory of a non-critical string, suggesting a deeper duality between SYK and string theory.
• The construction naturally produces ensemble averaging: the string theory computes the average over random couplings, not individual realizations, resolving a long-standing puzzle in holographic duality.