The classical bulk dual of the SYK model beyond cubic order: quartic and higher couplings, locality, the dilaton sector, and the organizing principle

The Sachdev-Ye-Kitaev (SYK) model — $N$ Majorana fermions with a random $q$ -body interaction, solvable at large $N$ — is a solvable realization of AdS $_2$ /CFT $_1$ holography, but the bulk theory it is dual to has never been constructed. Gross and Rosenhaus (JHEP 05 (2017) 092) initiated the systematic derivation of this classical (large- $N$ ) bulk dual: from the fermion six-point function they extracted the tower of massive scalars $\phi_n$ dual to the bilinear operators $O_n$ , their masses, and all their cubic couplings, and they posed the completion of the program as an open problem. One sub-item posed there — the closed-form evaluation of the large- $q$ planar triple sum $s^{(2)}_{nmk}$ — was resolved by the authors' own follow-up work (the sum is a terminating ${}_4F_3(1)$ hypergeometric), while the planar contribution was later identified as a conformal 6j symbol by Liu, Perlmutter, Rosenhaus, and Simmons-Duffin. The open problem is the completion of the bulk dual itself: determine the quartic and higher couplings of the $\phi_n$ tower (including the derivative interactions that field redefinitions cannot remove) by matching to the now-available fermion eight-point and higher correlation functions; decide whether the resulting bulk Lagrangian is local or effectively nonlocal through the decay rate of the higher-derivative coefficients; compute the coupling of the dimension-two (Schwarzian/dilaton) sector to the massive scalars; and identify the organizing principle behind the two classes (contact versus planar) of contributions, which the authors conjectured to be string-like.

Background

The SYK model is a zero-dimensional quantum mechanical model of $N\gg 1$ Majorana fermions $\chi_i$ with a random all-to-all interaction of strength set by $q$ fermions per vertex (the $q$ -body coupling $J_{i_1\ldots i_q}$ ). At large $N$ it is solvable in a $1/N$ expansion: the fermion two-point function is dominated by melon diagrams (the iterated one-loop self-energy insertions that are the only diagrams surviving at leading order in $1/N$ ), and in the infrared the model flows to a nearly conformal, nearly diffeomorphism-invariant theory whose Lyapunov exponent — the exponential growth rate of out-of-time-order correlators, the standard diagnostic of quantum chaos — equals that of a black hole. These properties suggest a gravitational dual in $\mathrm{AdS}_2$ (two-dimensional anti-de Sitter space), specifically a dilaton gravity theory — Jackiw-Teitelboim-type gravity, in which a scalar dilaton controls the deviation from exact $\mathrm{AdS}_2$ — coupled to a tower of massive particles. In a conformal field theory in one dimension (CFT $_1$ ), the natural operators are the $O(N)$ -singlet fermion bilinears $O_n=\sum_i\chi_i\partial_\tau^{2n+1}\chi_i$ , with infrared scaling dimensions $h_n$ ; by the AdS/CFT dictionary each $O_n$ is dual to a bulk scalar field $\phi_n$ of mass $m_n^2=h_n(h_n-1)$ in units of the AdS radius. What was not known, prior to the source paper, was any proposal for the bulk interaction Lagrangian of this tower.

Gross and Rosenhaus, in the source paper, initiated the systematic construction of this classical (large- $N$ ) bulk dual. They analyzed the fermion two-, four-, and six-point functions, extracted the dimensions $h_n$ and the three-point coefficients $c_{nmk}$ of the bilinears, and, by evaluating the bulk Witten diagrams (the AdS/CFT analogues of Feynman diagrams for boundary correlation functions, built from bulk-to-boundary and bulk-to-bulk propagators) for a cubic interaction $\lambda_{nmk}\phi_n\phi_m\phi_k$ , obtained the cubic couplings via $\lambda_{nmk}=c_{nmk}N_nN_mN_k/a_{nmk}$ , where $a_{nmk}$ is the standard AdS $_2$ Witten-diagram coefficient built from gamma functions and $N_n$ the bilinear normalization. The six-point function receives two classes of contributions: contact diagrams, in which three four-point ladders meet at a hard vertex, contributing $c^{(1)}_{nmk}$ ; and planar diagrams, in which the ladders join smoothly, contributing $c^{(2)}_{nmk}$ . The two classes behave very differently: at large indices $n,m,k$ the contact piece decays as a power law while the planar piece grows exponentially, tracking the cubic couplings of a generalized free field theory (a theory whose correlators follow from Wick contractions with a nonstandard propagator). At large $q$ the planar piece reduces to a finite triple sum of products of binomial coefficients, $s^{(2)}_{nmk}$ , which the authors could not evaluate in closed form in general; they also noted the near-coincidence $\lambda^{(2)}_{nmk}\approx\lambda^{\mathrm{free}}_{nmk}$ at large indices.

In their discussion the authors posed the completion of the program: (i) the quartic and higher bulk couplings, which require the fermion eight-point and higher functions (at quartic order, derivative interactions can no longer be removed by field redefinitions, so there are generically infinitely many independent terms); (ii) whether the dual is a local quantum field theory — a Lagrangian all of whose terms are local but with arbitrarily many derivatives can still be effectively nonlocal, the criterion being how rapidly the coefficients of the higher-derivative terms decay; (iii) a closed form for the triple sum $s^{(2)}_{nmk}$ ; (iv) the coupling of the dimension-two bilinear — the mode whose effective action is the Schwarzian action (the low-energy action of the reparametrization soft mode of nearly- $\mathrm{AdS}_2$ gravity, dual to the dilaton) — to the massive scalars $\phi_n$ ; and (v) an interpretation of the contact/planar decomposition and an organizing principle for the bulk dual, by analogy with how the string worldsheet organizes the infinite tower of string modes beyond any field-theory Lagrangian.

Substantial subsequent work has transformed the boundary-side data but not completed the bulk-side construction. Gross and Rosenhaus computed all large- $N$ correlation functions of the bilinears: the planar three-point coefficient at any $q$ is a sum of four generalized hypergeometric functions ${}_4F_3$ at unit argument, and in the large- $q$ limit the triple sum $s^{(2)}_{nmk}$ takes the closed form of a terminating ${}_4F_3(1)$ ; the bilinear four-point function and all higher-point functions follow from the analytic structure of $c_{123}$ and were written in Mellin space (a representation of conformal correlators in which Witten diagrams become scattering-like amplitudes). They also explained the planar/free-field near-coincidence via a line of fixed points (cSYK) interpolating between generalized free fields and SYK. Liu, Perlmutter, Rosenhaus, and Simmons-Duffin then showed that the planar three-point function is exactly a conformal 6j symbol — the recoupling coefficient of conformal-group representations — giving the planar class a group-theoretic identity. In parallel, Kitaev and Suh derived the soft-mode (Schwarzian) effective action including its nonlocal correction from integrating out the massive modes, and the near-AdS $_2$ literature established the general form of a generic matter coupling to the gravitational (Schwarzian) sector (refs 14, 15) — a general form that does not fix the explicit coupling of the dimension-two mode to the tower $\phi_n$ . However, the bulk Lagrangian beyond cubic order has never been extracted: the authors of the program explicitly left the quartic-level analysis as future work, conjecturing the bulk dual to be a theory of extended objects rather than local fields, and the quartic couplings, the locality question, the explicit coupling of the dimension-two mode to the tower $\phi_n$ , and the organizing principle for the fixed- $q$ model remain open (the active chord-diagram bulk programs — chord diagrams being the pairings of the $q$ fermion indices at each interaction vertex that encode the disorder-averaged correlators in that limit — concern the different double-scaled limit, in which $q$ grows with $N$ at fixed $q^2/N$ ).

Problem Statement

Construct the classical (large- $N$ ) bulk dual of the $q$ -body SYK model in its infrared conformal limit, beyond the cubic order already determined, in the sense posed by Gross and Rosenhaus, "The bulk dual of SYK: cubic couplings" (JHEP 05 (2017) 092), and using the now-available boundary data. Specifically, determine: (1) the quartic and higher couplings of the tower of massive scalars $\phi_n$ dual to the bilinear operators $O_n$ — including the derivative interactions that, unlike at cubic order, cannot be removed by field redefinitions — by matching the sum of exchange and contact Witten diagrams built from the bulk Lagrangian to the large- $N$ bilinear four-point function derived from the fermion eight-point function (equivalently, matching all double-trace conformal-block coefficients — the coefficients, in the conformal-block decomposition of the four-point function, of the composite operators built from two bilinear primaries — or the Mellin amplitude); (2) whether the resulting bulk Lagrangian defines a local quantum field theory or an effectively nonlocal one, i.e., how rapidly the coefficients of the arbitrarily-many-derivative interaction terms decay at large index; (3) the coupling of the dimension-two (Schwarzian/dilaton) sector — the non-conformal mode regulating $\mathrm{AdS}_2$ — to the massive scalars $\phi_n$ , going beyond the purely conformal part of SYK; and (4) the organizing principle underlying the bulk dual, and in particular the interpretation of the two distinct classes of contributions (contact versus planar) to the correlation functions, for which a string-like structure has been conjectured. An organizing principle adequate to (4) must be precise enough to generate the bulk interactions — in particular to entail the quartic and higher couplings of (1) — and must be consistent with the bilinear spectrum, the cubic couplings with their contact/planar split, and the planar 6j-symbol identity. The spectrum ( $m_n^2=h_n(h_n-1)$ ), all cubic couplings $\lambda_{nmk}$ , the closed-form planar coefficients (including the formerly open large- $q$ triple sum $s^{(2)}_{nmk}$ , now a terminating ${}_4F_3(1)$ ), and all large- $N$ boundary correlation functions are given as inputs. The problem is posed at general fixed $q$ (the large- $q$ limit may be used as a controlled regime, but a large- $q$ -only answer does not resolve the general- $q$ question).

References

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