PaperlayerView exact revision on arXiv

Source-linked AI summary

Bi-Local Holography in the SYK Model

Antal Jevicki, Kenta Suzuki, Junggi Yoon

arXiv:1603.06246v7hep-th

TL;DR

The paper asks how to formulate SYK holography as explicit 1/N bi-local Feynman rules, in a model used to study holography and black-hole physics. It constructs those rules and interprets them as Witten-type AdS diagrams, defining the theory at and away from the IR fixed point.

  • Problem

    The paper addresses how SYK holography can be expressed through explicit 1/N bi-local Feynman rules in a model used to study holography and black-hole physics.

    Source: §1 · p. 3

  • Method

    It formulates bi-local propagators and vertices associated with O(N) symmetry after disorder averaging, and adds a dynamical time coordinate with reparametrization gauge symmetry.

    Source: §1 · p. 3, §4 · p. 17

  • Results

    The bi-local rules admit a Witten-type AdS interpretation, and the dynamical time coordinate lets the theory be defined at the IR fixed point and away from it.

    Source: §1 · p. 3, §4 · p. 17

  • Takeaways & Limitations

    The representation has reparametrization symmetry even away from the IR, a feature the authors identify as relevant to establishing the dual correspondence.

    Source: §4 · p. 17

    Corrections in ε=(q − 2)/2 and to the J-dependent background are expected only to change the overall Schwarzian coefficient to α(ε).

    Source: §3.1 · p. 11

Abstract

from arXiv · show

We discuss large $N$ rules of the Sachdev-Ye-Kitaev model and the bi-local representation of holography of this theory. This is done by establishing $1/N$ Feynman rules in terms of bi-local propagators and vertices, which can be evaluated following the recent procedure of Polchinski and Rosenhaus. These rules can be interpreted as Witten type diagrams of the dual AdS theory, which we are able to define at IR fixed point and off.

1 Introduction

The paper develops explicit 1/N rules for describing SYK through bi-local fields, extending its large-N holographic description. These rules are presented as Witten-type diagrams for a dual AdS theory, including at and away from the IR fixed point.

1 Introduction

  • The work builds on SYK studies of large-N strong coupling, including the two-particle spectrum obtained by solving Schwinger-Dyson equations for the four-point function [8].The introduction also situates SYK as a one-dimensional disordered-fermion model with a potentially interesting AdS2 dual.Source: §1 · p. 2
  • The paper formulates SYK 1/N Feynman rules using bi-local propagators and vertices associated with O(N) symmetry after quenched disorder averaging.The collective representation incorporates arbitrary n-point bi-local correlators and generates Schwinger-Dyson equations.Source: §1 · p. 3
  • The rules are interpreted as Witten-type diagrams for the dual AdS theory, which the paper defines at the IR fixed point and away from it.A dynamical time coordinate is introduced to eliminate the zero-mode problem and define the theory in the IR limit.Source: §1 · p. 3

2 Replica Collective Field Theory

The replica formulation yields a bi-local collective theory with concrete 1/N propagators and vertices. In the replica limit, only diagonal propagators contribute; the bi-local two- and three-point functions connect to fermion four- and six-point functions.

2 Replica Collective Field Theory

  • The collective-field formulation replaces the original degrees of freedom with bi-local fields and generates concrete 1/N rules, whose vertices take a star-product form.The bi-local fields form a closed set under the large-N Schwinger-Dyson equations.Source: §2 · pp. 4–5

2.1 1/N Diagrams

  • In the n → 0 replica limit, only the replica-diagonal bi-local propagator is relevant, with each diagram loop requiring a replica-index twist to contribute at order n.The one-loop vacuum diagrams are shown as n-dependent in Figure 1.Source: §2.1 · pp. 7–8, Figure 1 · p. 8

2.2 Two-point Function

  • The two-point bi-local correlation function plays a special role as a propagator and, in the fermion description, represents a four-point function.The paper notes a correspondence with the 1PI four-point function considered in [8].Source: §2.2 · pp. 8–9
  • The three-point function of diagonal bi-local fluctuations can be compared with the SYK fermion six-point function.Figure 3 depicts the three-point function; its differently colored double lines intersect only at four-point vertices in non-planar diagrams.Source: §2.2 · p. 10, Figure 3 · p. 10

3 AdS Bulk

The AdS bulk description treats time reparameterization as a dynamical degree of freedom and identifies bi-local propagator poles with AdS-type contributions. Its interactions include both bi-local and bulk-local terms.

3 AdS Bulk

  • At the IR critical theory, emergent t → f(t) reparameterization symmetry is responsible for zero modes, which the authors address by quantizing symmetry modes.Source: §3 · pp. 10–11

3.1 Time as a Dynamical Variable

  • The reparameterization f(t) becomes dynamical through a Faddeev–Popov construction, whose delta-function condition projects out a chosen wavefunction to eliminate the IR zero mode.Source: §3.1 · p. 11
  • The action for f(t) takes Schwarzian form in an ε expansion starting from q = 2; corrections are expected to change its overall coefficient to α(ε).Source: §3.1 · p. 11

3.2 The Propagator

  • In the strong-coupling limit J|t| ≫ 1, the ν = 3/2 zero-mode divergence associated with conformal reparameterization symmetry is eliminated by the delta-function constraint; the first-line contribution exactly cancels the discrete sum, leaving p_m (m > 0) poles.The propagator defines the 1/N Feynman rules, and each pole has one AdS-type contribution.Source: §3.2 · pp. 14–15
  • A nonlocal field redefinition relates collective-field propagation to an effective scalar action, whose pole at p_m yields the on-shell propagator.Source: §3.2 · p. 16
  • The cubic interaction combines a bi-local term with a bulk-local term, while interactions with n = 5, 6, 7, … contain only the bi-local trace (star-product) term.Source: §3.2 · p. 16

4 Conclusion

The paper formulates the SYK O(N) singlet sector using a bi-local field and a dynamical time coordinate with reparametrization symmetry. This representation is presented as a basis for establishing the dual correspondence, while locality of higher-point vertices and finite-temperature extension remain future questions.

4 Conclusion

  • The bi-local field Ψ and dynamical time coordinate f(t) define the O(N) singlet sector, with reparametrization symmetry also present away from the IR.In the IR, the symmetry is used to define the theory and eliminate the associated zero-mode problem.Source: §4 · p. 17
  • Whether the three-point and higher-point vertices admit AdS locality remains an open question, with only some indications that it may be possible.The issue is closely related to external-leg redefinitions of the scalar field.Source: §4 · p. 17
  • A finite-temperature extension using Thermo-Field Dynamics is identified as a relevant direction for further study.The authors note that this formalism can be implemented alongside recent TFD O(N) vector-model studies.Source: §4 · p. 17

A Higher Loop Diagrams

The appendix lists replica-collective-theory diagrams and their n-dependence through two loops. Twisting diagonalizes replica indices, leaving only the replica-diagonal propagator relevant after the replica trick.

A Higher Loop Diagrams

  • Twisting the bi-local propagator diagonalizes replica space, so O(n) diagrams contain one replica index and only the replica-diagonal propagator is relevant after the replica trick.The appendix lists the diagrams and their n-dependence through two-loop order.Source: A Higher Loop Diagrams · p. 17
  • Figure 4 presents the first kind of two-loop vacuum diagram and its n-dependence.The passage does not state a specific n-dependence value.Source: Figure 4 · p. 18
  • Figure 5 presents the second kind of two-loop vacuum diagram and its n-dependence.The passage does not state a specific n-dependence value.Source: Figure 5 · p. 18
  • Figure 6 presents the third kind of two-loop vacuum diagram and its n-dependence.The passage does not state a specific n-dependence value.Source: Figure 6 · p. 19

B Two- and Three-point Function

This appendix develops two- and three-point functions in the bi-local fluctuation framework. It distinguishes disconnected and connected contributions, then relates connected functions to amputated vertices and diagrammatic expressions.

B Two- and Three-point Function

  • The first three terms in the cubic-action identity are disconnected propagators from the bi-local-fluctuation viewpoint, while its last term is connected.The connected three-point function of diagonal fluctuations is then considered after assigning replica labels to the external pairs.Source: B Two- and Three-point Function · p. 20
  • After amputation of external legs, the calculation defines an amputated cubic vertex for the diagonal-fluctuation three-point function.The passage also separates three-point contributions into connected and disconnected parts from the fermion viewpoint.Source: B Two- and Three-point Function · pp. 20–21
  • The connected six-point function from the fermion viewpoint is written using bi-local propagator and vertex contributions.The displayed expression includes cyclic terms in the three external time-pair labels.Source: B Two- and Three-point Function · pp. 19, 21

C Evaluation of the Time Coordinate Action

The appendix evaluates the action for the collective time coordinate f(t) using an expansion around q = 2. It removes a divergent term in obtaining the action and expects epsilon and J corrections to alter only the Schwarzian coefficient.

C Evaluation of the Time Coordinate Action

  • For q = 2 + 2ε, the action is evaluated through an ε expansion whose leading approximation is the q = 2 case.The generalized q-fermion coupling changes the power in the bi-local action.Source: C Evaluation of the Time Coordinate… · p. 22
  • The first term in the short-distance expansion is eliminated because it gives a divergence before taking the derivative and limit to obtain the action.The evaluation expands the critical solution as t2 approaches t1.Source: C Evaluation of the Time Coordinate… · p. 22
  • Higher-order ε corrections are expected to retain Schwarzian form and change only its overall coefficient; evaluations including J-corrections are left for future work.The passage presents this correction form as an expectation, not a completed evaluation.Source: C Evaluation of the Time Coordinate… · p. 22

D Identity

The appendix derives an identity using Fourier transformation of the background field and integral representations of Bessel functions. It also uses a change from bi-local coordinates to (τ, z).

D Identity

  • The derivation Fourier-transforms the background field into momentum space and writes its inverse.The passages refer to the inverse when the bi-local field is treated as a matrix in bi-local space.Source: D Identity · p. 23
  • The derivation transforms bi-local coordinates (t1, t2) to (τ, z) and uses variables ui ≡ wzi for i = 1, 2.It also states parameter conditions involving n = 0, 1, 2, … and ν = ir with r > 0.Source: D Identity · pp. 23–24
  • The appendix derives an identity after evaluating an integral over p using integral representations of Bessel functions.The supplied passages do not state the identity itself.Source: D Identity · pp. 23–24
Loading 1603.06246v7…