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The Bulk Dual of SYK: Cubic Couplings

David J. Gross, Vladimir Rosenhaus

arXiv:1702.08016v2hep-thcond-mat.str-el

TL;DR

The SYK model has an AdS2 dual with a tower of massive particles but no proposed bulk theory. The paper systematically analyzes large-N fermion correlators to derive the bulk spectrum and cubic interactions, finding a diagrammatic six-point structure and simplified couplings at large q. The resulting program initiates a systematic construction of the large-N bulk dual, while leaving open the question of whether it is local.

  • Problem

    SYK has a tower of massive AdS2-dual particles but no proposed bulk theory, motivating a systematic construction of its dual.

  • Method

    The paper analyzes fermion two-, four-, and six-point functions at large N to extract singlet dimensions, three-point functions, bulk masses, and cubic couplings.

  • Results

    The fermion six-point function is organized by contact and planar diagrams, yielding bilinear three-point functions and the cubic couplings of the dual bulk theory.

  • Takeaways & Limitations

    The results initiate a systematic program for constructing the large-N bulk dual of SYK from its correlators.

  • Takeaways & Limitations

    The analysis focuses on the purely conformal part of the theory, and the locality of the SYK dual remains an open question.

Abstract

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The SYK model, a quantum mechanical model of $N \gg 1$ 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 $q$.

1. Introduction

SYK is a solvable holographic model whose AdS2 dual lacks a proposed bulk theory and must contain infinitely many massive fields. This paper uses large-N fermion correlators to extract bulk masses and cubic couplings, finding marked simplifications at large q.

  • Motivation: SYK has no known bulk construction, while its dual must include a tower of massive particles associated with bilinear singlet operators.The model’s solvability at large N provides a route to constructing this dual theory.
  • Motivation: Unlike canonical AdS/CFT examples, SYK has no limit in which only finitely many finite-dimension operators survive, requiring infinitely many local bulk fields.Adding flavor or supersymmetry does not alter this conclusion.
  • Bulk spectrum: The fermion four-point function decomposes into conformal two-point functions of bilinear operators, whose infrared dimensions determine the masses of the bulk fields.These bilinears are primary O(N) singlets in the SYK sector.
  • Cubic couplings: At order 1/N^2, the fermion six-point function receives contact and planar diagram contributions, reducible to three four-point functions glued together.Its conformal limit yields three-point functions of the bilinears.
  • Cubic couplings: The extracted bilinear three-point coefficients determine bulk cubic couplings through the tree-level Witten diagram and the AdS/CFT dictionary.A cubic bulk interaction produces the corresponding boundary CFT three-point function.
  • Large-q structure: The cubic-coupling expressions are complicated at finite q but simplify dramatically as q →∞, where an explicit analytic expression is obtained.Contact contributions can be written for any q and simplify at large q, while planar contributions dominate at large n, m, k and relate simply to generalized-free-field bulk couplings.

2. A Tower of Particles

The large-N SYK correlators reveal a tower of O(N)-singlet bilinear operators whose dimensions and conformal blocks determine the bulk particle spectrum. In the large-q limit, both their dimensions and OPE data simplify, while the four-point function is organized by ladder diagrams and a distinguished dimension-two contribution.

  • 2.2. Fermion four-point function: The operator dimensions approach h_n = 2n + 1 + 2ε_n at large q, with ε_n scaling as 1/q.The large-q kernel equation yields the simplified dimensions for n ≥ 1.
  • 2.1. Fermion two-point function: The infrared fermion two-point function is obtained by summing melon diagrams at leading order in 1/N.At large q, combinatorial dominance allows only a particular subset of melon diagrams to be retained for a simple analytic treatment.
  • 2.2. Fermion four-point function: The infrared four-point function is a sum of conformal blocks for bilinear operators O_n with dimensions h_n, after resumming ladder diagrams through the kernel.The ladder sum is F = (1 − K)^−1F_0, with h_n determined by kernel eigenvalues equal to one.
  • 2.2. Fermion four-point function: The four-point function contains a dimension-two block that breaks conformal invariance and is separated from the conformally invariant sector.The remaining conformal-block sum contains operators with h_n > 2, while the h = 2 eigenvalue is present for every q.
  • 2.2. Fermion four-point function: In the large-q limit, the OPE coefficients c_n vanish, so the operators O_n decouple from the fermions even though their correlators and dual fields remain studyable.The four-point analysis nevertheless identifies the bilinear operator tower and its conformal data.

The dimension-two block

The dimension-two block is the nonconformal sector associated with the Schwarzian dynamics of nearly-AdS2 SYK. Focusing instead on the conformal sector exposes the bilinear operators and maps their dimensions to bulk particle masses.

  • The dimension-two block: The dimension-two contribution breaks conformal invariance and can be viewed as arising from the Schwarzian action, reflecting that infrared SYK is nearly rather than exactly a CFT1.The corresponding bulk geometry is nearly AdS2 and requires regulation because finite-energy excitations have large back-reaction.
  • The dimension-two block: The analysis excludes the h = 2 block to focus on higher-dimension operators and interactions of their dual bulk fields.The authors state that the purely conformal sector is their object of interest.
  • 2.3. The operator product expansion of the fermions: In the double short-time limit, the fermion four-point function becomes a sum of two-point functions of primary, O(N)-invariant bilinear operators with dimensions h_n.The operator coefficients are constructed so that the bilinears are primary, and resumming the corresponding series reproduces the hypergeometric functions.
  • 2.3. The operator product expansion of the fermions: At weak coupling the bilinear dimensions are 2n + 1, while strong-coupling shifts are order one and decay as 1/q at large q.This identifies the h_n as the dimensions of the singlet bilinear tower extracted from the fermion OPE.
  • 2.4. Particles in the bulk: The AdS/CFT dictionary maps the O(N)-invariant bilinears to bulk fields whose masses are related to h_n, with free fields at leading order in 1/N.At the next order, their cubic coupling is expected to be fixed by the fermion six-point function.

3. Three-Point Function of Bilinears

This section computes bilinear three-point functions from the fermion six-point function by separating contact and planar contributions and evaluating the resulting conformal integrals. It analyzes their UV-divergent regions to obtain finite large- results, along with exact finite- and symmetric triple-sum representations for the relevant integrals.

  • 3.1. Fermion six-point function: The fermion six-point function’s leading nontrivial contribution consists of contact and planar diagram classes, whose sums are denoted S1 and S2.Contact diagrams glue three four-point functions through interaction vertices, while planar diagrams glue partially amputated four-point functions to avoid double counting.
  • 3. The short time limit: The bilinear three-point coefficient cnmk is fixed in conformal form and decomposes into contact and planar terms, c(1)nmk and c(2)nmk, represented by three- and four-loop integrals for general q.The coefficient is extracted from the six-point function by taking three fermion pairs to coincident times and applying the bilinear operator product expansion.
  • 3.2. The short time limit: The triple short-time limit τ1→τ2, τ3→τ4, and τ5→τ6 converts the fermion six-point function into bilinear three-point functions, with both contributions transforming conformally.The contact and planar terms are evaluated separately using the corresponding short-time limits of the four-point function and partially amputated four-point function.
  • 3.3. Evaluating the integrals: For general q-body SYK, bilinear dimensions have no explicit form and are determined implicitly by kc(h)=1; at large q, simplification occurs because the IR fermion dimension Δ=1/q is small and bilinear dimensions are close to odd integers.
  • 3.3. Evaluating the integrals: At large q, the divergent τ_a and τ_b integrations cancel explicit q factors, leaving a finite result for c^(1).For I^(1), the relevant expansion confirms the conformal three-point-function form, while the remaining coincident-time regions contribute no 1/(τ_1aτ_1b) term.
  • 3.3. Evaluating the integrals: The large-q coefficient s^(1)_nmk has a relatively simple Gamma-function form, and I^(1) can also be evaluated exactly at finite q through a generalized Selberg integral.A change of variables puts the finite-q integral into generalized Selberg form, producing a result involving the function ρ and permutations of (h_n,h_m,h_k).
  • 3.3. Evaluating the integrals: For I^(2) at large q, summing all eight UV-divergent regions yields the conformal three-point-function structure with coefficient s^(2)_nmk.When two integration variables approach the same time, retaining numerator epsilon factors is essential; dropping them would give a result smaller by a factor of two.
  • 3.3. Evaluating the integrals: The coefficient s^(2)_nmk is represented by a triple sum that is symmetric under permutations of n, m, and k and independent of z = τ_12/τ_31.These properties are not manifest in the sum but can be verified.

4. The Bulk Cubic Couplings

The paper determines bulk cubic couplings by matching AdS/CFT three-point functions to SYK bilinear three-point functions, using coefficients extracted from contact and planar diagrams. At large q, contact and planar contributions exhibit distinct structures, while finite-q contact couplings are also obtained.

  • 4. The Bulk Cubic Couplings: At leading order in 1/N, derivative cubic terms are equivalent to non-derivative terms up to a field redefinition.The analysis therefore uses a bulk Lagrangian with non-derivative cubic interactions at this order.
  • 4. The Bulk Cubic Couplings: The bulk cubic couplings λ_nmk are fixed by matching the Witten-diagram three-point function to the SYK three-point function of bilinears O_n.The couplings are split into contributions from contact and planar diagrams.
  • 4.1.1. Large q: At large q, contact diagrams give λ_nmk = −(−1)^(n+m+k)16√π/q(ε_n + ε_m + ε_k)α_nα_mα_k, and the coupling decays for large n, m, k.The coupling is also expressed in terms of the masses of the dual fields for selected cases.
  • 4.1.2. Finite q: At finite q, the contact contribution λ^(1)_nmk is written using A(h_n) and R(h_n, h_m, h_k), with c(h_n) defined from the derivative of k_c(h).The finite-q expression follows from evaluating the contact-diagram contribution to the three-point function.
  • 4.2. Planar diagrams, large q: For large n, m, k, the planar contribution approaches the cubic coupling of the generalized free field theory, while λ^(2)_kkk grows exponentially with k.The generalized-free-field relation is obtained from the singlet-sector bilinear three-point function.

5. Discussion

The paper initiates a systematic construction of the large-N bulk dual of SYK by extracting bilinear correlators and translating them into bulk cubic couplings. It separates contact and planar contributions, identifies large-q simplifications, and outlines higher-point extensions and remaining questions about locality.

  • 5. Discussion: The large-N bulk dual is constructed systematically from fermion correlation functions, with bilinear three-point coefficients determining bulk cubic couplings.The six-point function yields bilinear three-point functions, which are converted into cubic bulk interactions using the AdS/CFT dictionary.
  • 5. Discussion: At order 1/N^2, contact and planar diagrams contribute separately, but their coefficients behave very differently at large operator indices.The planar contribution becomes much larger than the contact contribution for large n, m, k, and the paper leaves the interpretation of these pieces open.
  • 5. Discussion: Contact diagrams provide a novel contribution that generalizes to higher 2p-point functions when q > p, suggesting bulk interactions polynomial in multiple fields.In the q → ∞ limit, the authors conjecture an infinite polynomial in bulk fields, with terms through p = q − 1 calculable by analogous methods.
  • 5. Discussion: In the large-q limit, planar couplings reduce to finite triple sums, while bilinear dimensions approach odd integers and their bulk interactions remain finite.The bilinears decouple from the fermions because their operator-product coefficients vanish, yet their correlation functions and bulk interactions remain finite.
  • 5. Discussion: The planar contribution is related to three-point coefficients in a generalized non-local quadratic fermion theory and becomes equal to them at large n, m, k.The paper computes explicit generalized-free-field coefficients, although the triple sums for the planar SYK coefficients are not generally evaluated.
  • 5. Discussion: The bilinear dimensions approach 2∆ + 2n + 1 at large n, with ∆ = 1/q, and their three-point coefficients approach free-field values for large q and large indices.This behavior is consistent with the constituent-fermion and derivative structure of the bilinears.
  • 5. Discussion: The next step is the fermion eight-point function, whose contact diagrams contribute to quartic bulk couplings and whose exchange diagrams may generate derivative interactions.At quartic order, matching bulk and boundary terms will determine whether derivative couplings are required.
  • 5. Discussion: A central open question is whether the SYK bulk is local, which depends on how rapidly coefficients of arbitrarily high-derivative terms decay.The first terms of the bulk Lagrangian may provide clues to the organizing principle of the dual theory.

A. Generalized Free Fields

The appendix studies singlet bilinears in a generalized free fermion theory, computing their two- and three-point functions by Wick contraction. Its three-point coefficient reproduces a factor in the planar large-q SYK result.

  • A. Generalized Free Fields: The appendix computes singlet-sector bilinear two-point functions and constructs their primary operators using derivative combinations and normalization.The calculation proceeds through derivatives of the fermion two-point function, Wick contractions, and explicit sums.
  • A. Generalized Free Fields: The generalized free theory is quadratic, so all correlation functions follow from Wick contractions, while finite ∆ makes it non-local in time.The free-Majorana limit is recovered only after taking ∆ to zero at the end of the calculation.
  • A. Generalized Free Fields: The resulting generalized-free-field bilinear three-point function has a coefficient matching a factor in the planar large-q SYK three-point function.The comparison is made after evaluating the relevant sums and relating the generalized free coefficient to the SYK planar contribution.

Summary

The summary identifies the generalized free-field three-point function as similar to the planar SYK contribution and presents their coefficient ratio. It thereby isolates a simple comparison between the two theories.

  • Summary: The generalized free-field bilinear three-point coefficient is similar to the planar contribution in SYK.The comparison concerns the coefficient obtained after summing planar diagrams.
  • Summary: The ratio of the generalized free-field and planar SYK coefficients is computed explicitly.The supplied passage identifies the ratio as the final comparison in this summary section.

B. Integrals

The appendix develops standard two- and three-point integrals used in the calculation. Two-point integrals can be evaluated by Fourier transformation, while conformal covariance fixes the functional form of a three-point integral.

  • B. Integrals: Two-point integrals are evaluated using Fourier transformation of both sides.
  • B. Integrals: For a three-point integral with a1 + a2 + a3 = 1, conformal covariance fixes its functional form and a limiting procedure determines the constant.The constant is fixed by taking τ3 to infinity and using the two-point integral.

B.1. Selberg Integral

This appendix develops generalized Selberg integrals over mixed integration regions and relates them recursively to the ordinary Selberg integral.

  • B.1. Selberg Integral: The integral evaluates to a Gamma-function expression involving α, β, γ, n, and j.
  • B.1. Selberg Integral: The generalized integral S_n,p uses the Selberg integrand while integrating n−p variables over 1 < τ_i < ∞, with S_n,n recovering the ordinary Selberg integral.An inversion τ_i → 1/τ_i relates S_n,p to S_n,n−p with shifted parameters.
  • B.1. Selberg Integral: A recursion relation expresses S_n,p through S_n,p−1, so successive applications reduce all such integrals to S_n,n.
  • B.1. Selberg Integral: For the two-variable plane integral, partitioning the domain and changing variables expresses it as a sum of S_2,2 and S_2,1 terms with shifted parameters.

C. Triple Sum

This section analyzes a triple sum arising in SYK and related planar or free-field calculations, using recursion relations and generating functions to characterize its complexity.

  • C. Triple Sum: For arbitrary n and m with small fixed k, the sum has an explicit expression, while the n = m = k specialization can be examined at low n.
  • C. Triple Sum: The triple sum does not take a simple form, and its complexity increases as k increases.
  • C. Triple Sum: Viewing n, m, and k as half-integers yields two recursion relations for the auxiliary function F(n,m,k).
  • C. Triple Sum: These recursion relations alone are insufficient to determine s^(2)_nmk completely, although they still constrain the sum.
  • C. Triple Sum: The authors do not pursue a substantially simpler form for a related sum because it is unlikely to simplify significantly.
  • C. Triple Sum: A generating function is known for the sum, which appears in planar diagrams at large q and in the free-field theory.

D. Field Redefinition

The appendix shows that derivative cubic couplings in the bulk can be transformed by field redefinitions into equivalent couplings without derivatives at this order in 1/N.

  • D. Field Redefinition: The paper derives the bulk SYK cubic couplings by computing bilinear three-point functions and applying the AdS/CFT dictionary.
  • D. Field Redefinition: CFT three-point functions alone cannot determine whether a bulk cubic coupling contains derivatives under the assumed action framework.
  • D. Field Redefinition: At this order in 1/N, derivative and non-derivative cubic couplings are equivalent.
  • D. Field Redefinition: A field redefinition removes derivative terms from the cubic interaction while modifying the mass term correspondingly.
  • D. Field Redefinition: The redefinition can be continued until all derivatives are removed from the cubic coupling.
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