# A Cordial Introduction to Double Scaled SYK

Micha Berkooz  Department of Particle Physics and Astrophysics, Weizmann
Institute of Science, Rehovot 7610001, Israel Ohad Mamroud  SISSA, via Bonomea
265, 34136 Trieste, Italy  INFN, Sezione di Trieste, Via Valerio 2, 34127
Trieste, Italy

###### Abstract

We review recent progress regarding the double scaled Sachdev-Ye-Kitaev model
and other $p$-local quantum mechanical random Hamiltonians. These models
exhibit an expansion using chord diagrams, which can be solved by
combinatorial methods. We describe exact results in these models, including
their spectrum, correlation functions, and Lyapunov exponent. In a certain
limit, these techniques manifest the relation to the Schwarzian quantum
mechanics, a theory of quantum gravity in $AdS_{2}$. More generally, the
theory is controlled by a rigid algebraic structure of a quantum group,
suggesting a theory of quantum gravity on non-commutative $q$-deformed
$AdS_{2}$. We conclude with discussion of related universality classes, and
survey some of the current research directions.

###### Contents

  1. 1 Introduction
    1. 1.0.0.0.1 For $p$-local systems:
    2. 1.0.0.0.2 For Holography:
    3. 1.0.0.0.3 For RMT:
  2. 2 Chord Diagrams and the partition function
    1. 2.0.0.0.1 Chord diagram:
    2. 2.1 Chord diagrams
    3. 2.2 The transfer matrix
      1. 2.2.0.0.1 The thermofield double
      2. 2.2.0.0.2 Diagonalizing the transfer matrix
      3. 2.2.0.0.3 The spectrum
    4. 2.3 Adding matter and two point functions
    5. 2.4 The $p$-spin model
  3. 3 Semiclassics (bulk I)
    1. 3.1 The triple scaling limit and the Schwarzian theory
    2. 3.2 Finite and fake temperature
      1. 3.2.0.0.1 The partition function
      2. 3.2.0.0.2 The two point function
      3. 3.2.0.0.3 The chaos exponent
  4. 4 The four point function and $SU_{\hat{q}}(1,1)$
    1. 4.1 The diagrammatic rules of the full model
  5. 5 Non-commutative AdS (bulk 2)
    1. 5.1 Non-commutative $AdS_{2}$
      1. 5.1.0.0.1 Some ad-hoc realizations
      2. 5.1.0.0.2 Non-commuting coordinates
      3. 5.1.0.0.3 Particle on a group manifold
    2. 5.2 The chord algebra
  6. 6 Generalizations and variations
    1. 6.1 The Parisi hypercube model
    2. 6.2 Models with additional symmetries
    3. 6.3 Multi chord problems
  7. 7 Some additional frontiers
    1. 7.1 The bulk
      1. 7.1.0.0.1 A gravitational path integral
      2. 7.1.0.0.2 Yang-Baxter moves and geometry
    2. 7.2 level statistics, multi trace correlators and wormholes
    3. 7.3 Relation to (operator) complexity
    4. 7.4 de Sitter space
      1. 7.4.0.0.1 Acknowledgments
  8. A Some $q$-functions
    1. A.0.0.0.1 The $q$-Pochhammer symbol
    2. A.0.0.0.2 The Jacobi theta function
    3. A.0.0.0.3 $q$-numbers
    4. A.0.0.0.4 The $q$-factorial and $q$-Gamma function
    5. A.0.0.0.5 The $q$-exponential
    6. A.0.0.0.6 The continuous $q$-Hermite polynomials
    7. A.0.0.0.7 The $R$-matrix

##  1 Introduction

The Sachdev-Ye-Kitaev (SYK) model [1, 2, 3, 4] is a quantum mechanical model
of $N$ Majorana fermions with all-to-all $p$-local interactions. The model is
quantum chaotic, as indicated by level repulsion [10, 11, 12] and its maximal
chaos exponent [1, 2, 13]. Variants of the SYK model appeared early in nuclear
physics [14, 15] but it became particularly useful in condensed matter physics
as a model for strange metals [3, 4, 5]. Its maximal chaos exponent made it
important as a controllable model of holography [1, 6, 12, 16, 17, 18, 19, 20,
4] (and many others).

This review concerns the double scaled SYK model (DS-SYK), where instead of a
standard large $N$ limit one takes a _double scaling limit_ , in which the
length of the interaction also scales with $N$. The model then describes a
generalization of the Schwarzian theory which holds at all energy scales. We
review how this structure naturally emerges from DS-SYK via the diagrammatics
of _chord diagrams_ , point at its underlying algebraic structure and the role
of non-commutative geometry, and comment on possible holographic realizations.

Actually, much of what we will say also applies to $p$-local systems in
general, encompassing a rich _universality class_. But let us for now focus on
the Majorana SYK model:

$\ H=i^{p/2}\sum_{1\leq i_{1}<i_{2}<\cdots<i_{p}\leq N}J_{i_{1}\cdots
i_{p}}\psi_{i_{1}}\cdots\psi_{i_{p}}\,,\qquad\\{\psi_{i},\psi_{j}\\}=2\delta_{ij}\,\qquad
i,j=1,\dots,N\,.$ (1.1)

$H$ is a $p$-local Hamiltonian111For finite $p$ the SYK model is special in
the sense that it has a non-Fermi-liquid phase at low energies rather than a
spin glass phase., as it is a sum of terms, each acting on $p$ out of the $N$
available degrees of freedom. At large $N$ the theory is in fact nearly
conformal in the IR, which matches the symmetry of JT gravity on
near-$AdS_{2}$ and results in the Schwarzian theory.

We will be interested in a different limit of the model – the _double scaling_
limit

$p,\,N\rightarrow\infty,\quad\lambda\propto p^{2}/N\ fixed\,.$ (1.2)

Throughout the review we will take the strict limit without any $1/N$
corrections. This limit has advantages for the study of the thermodynamics and
chaos properties of the SYK model, and $p$-local systems in general; for
holography; and for random matrix theory (RMT).

####  1.0.0.0.1 For $p$-local systems:

The model can be solved efficiently using combinatorial tools. We can write
closed expressions for the free energy and correlators at all energy
scales222At times that remain finite as we take $N\rightarrow\infty$., rather
than just at low energies or having to solve additional differential equations
(sections 2 and 4). Furthermore, the rules for this limit are the same for a
large class of $p$-local systems, the only difference being the precise
relation between $\lambda$ and the underlying microscopic model, exhibiting
the universal nature of the construction (sections 2 and 6).

In addition, there is a natural set of local observables which are fairly
universal across many $p$-local models (Section 2). Together with the
Hamiltonian, they form multi-matrix models where non-trivial correlation
appear just because of the $p$-local structure, rather then due to direct
couplings in a potential. In particular one can compute their out-of-time-
order correlators exactly for cases with both maximal and submaximal chaos
(Section 4).

It is possible to generalize the theory to sums of $p$-local Hamiltonians with
different spectral properties, and solve it in some limits. This allows the
study of RG flows, chaos-integrability transitions, and quantum critical
points (Section 6.3).

####  1.0.0.0.2 For Holography:

An interesting feature of the DS-SYK model is that it makes the SYK/2D gravity
duality manifest. The combinatorial tools that we will present start from the
microscopic data and, after ensemble averaging, produce the bulk gravitational
Hilbert space and Hamiltonian. For finite $\lambda$ it could be interpreted as
having a minimal length in the bulk, so we can think about it as a full model
of quantum gravity.

When $\lambda\to 0$ the low energy spectrum reproduces the Schwarzian theory.
Thus in the IR the theory is holographically dual to JT gravity in
near-$AdS_{2}$. But in DS-SYK we can also access the rest of the spectrum. It
keeps exhibiting certain holographic features, the only difference being that
the temperature is replaced by a “fake” temperature (Section 3). The chords
provide an underlying symmetry that enforces these properties (Section 5.2).

At finite $\lambda$ the theory is “algebraically rigid”, as correlation
functions, and the Lyapunov exponent, are given by group theoretic objects
related to the quantum group $SU_{\hat{q}}(1,1)$ which is a $q$-deformation of
$SU(1,1)$ (Section 4). This algebraic structure seems to be reflected in the
bulk by changing the familiar $AdS_{2}$ into a non-commutative space (Section
5).

####  1.0.0.0.3 For RMT:

Following ’t Hooft we are used to 2D surfaces arising from planar diagrams and
large $N$ limits. The way it is done is similar to free probability theory and
exploits the features of the standard large $N$ Wigner-Dyson
$\beta$-ensembles, where the Hamiltonians $H$ are drawn from a distribution
function of the form $\exp(-\dim({\cal H})V(H))$ where $V$ is invariant under
$SU(\dim{\cal H})$ or other such groups. The $p$-local models that we discuss
here do not fall into this class, yet they still give rise to 2D surfaces.
They are solvable using manageable combinatorial tools, a fact that was
already appreciated in the mathematics community [21, 22, 23], with the
structure of chords (that we describe below) allowing for an elegant way of
going beyond free probability theory.

Finally, Section 7 mentions some open problems: a gravitational path integral
formulation, the relation to de-Sitter space (and type II1 von-Neumann
algebras), multi-trace correlators and Euclidean wormholes, how local geometry
arises in the bulk and the relation to operator complexity.

All of of these topics have been developed in quite a few papers. Above we
only indicated where the different topics appear in the review, and more
detailed references appear in each section. It is unlikely that we have
managed to do justice to the different topics, or to the publications and
authors who worked them out and we apologize for this.

##  2 Chord Diagrams and the partition function

The main technical tool used to solve the model are _chord diagrams_ :

####  2.0.0.0.1 Chord diagram:

Consider $k$ (even) dots on a circle. A chord diagram is a pairing of these
points. We denote it by connecting each pair of points by a chord, see Figure
1.

These appear quite naturally when evaluating the moments
$\operatorname{Tr}(H_{rand}^{k})$ of random Hamiltonians of the form

$H_{rand}=\sum_{\alpha}J_{\alpha}{\cal O}_{\alpha}\,,$ (2.1)

where $J_{\alpha}$ are Gaussian random variables, and ${\cal O}_{\alpha}$ is
any set of operators acting on the Hilbert space. When evaluating the ensemble
average over the couplings
$\langle\operatorname{Tr}(H^{k}_{rand})\rangle_{J}$, the chord diagrams just
encode the Gaussian averages of the $J_{\alpha}$ via Wick’s theorem. The
choice of ${\cal O}_{\alpha}$ then determines the weight of the chord diagram,
and changes from model to model. Some choices of the weights give rise to
universality classes – the ordinary Wigner-Dyson (Gaussian) $\beta$-ensembles
are just those with only non-intersecting chords, while our universality class
is that in which there is a fixed weight for each chord intersection [21, 22,
23]. Let us see how this works for the DS-SYK model.

###  2.1 Chord diagrams

The SYK model is a microscopic model of $N$ Majorana fermions $\psi_{i}$,
$\\{\psi_{i},\psi_{j}\\}=2\delta_{ij}$, with

$H=i^{p/2}\sum_{i_{1}<i_{2}<\cdots<i_{p}=1}^{N}J_{i_{1}\cdots
i_{p}}\psi_{i_{1}}\cdots\psi_{i_{p}}\equiv i^{p/2}\sum_{I}J_{I}\Psi_{I}\,,$
(2.2)

where the $J$’s are random couplings drawn from a Gaussian ensemble. $I$’s
denote ordered sets of $p$ indices, $\\{i_{1},\cdots,i_{p}\\}$, and $\Psi_{I}$
denotes the product of $p$ fermions with indices in $I$. At large $N$ the
model is self-averaging, allowing us to study annealed averages over the
couplings, denoted by $\langle\cdot\rangle_{J}$. The statistics of the
couplings $J$ are

$\left\langle J_{I}\right\rangle_{J}=0\,,\qquad\left\langle
J_{I}J_{K}\right\rangle_{J}=\binom{N}{p}^{-1}\mathcal{J}^{2}\delta_{IK}=\frac{N}{2p^{2}}\binom{N}{p}^{-1}\mathbb{J}^{2}\delta_{IK}\,,$
(2.3)

where both the $\mathcal{J}$ [9, 8] and the $\mathbb{J}$ convention [2, 24,
25] exist in the literature, and are related by
$\mathbb{J}^{2}=\frac{2p^{2}}{N}\mathcal{J}^{2}$. With the normalized333One
could also use a physical convention, where
$\operatorname{Tr}(\mathbbm{1})=2^{N/2}$. This extra factor simply multiplies
quantities such as the partition function. trace,
$\operatorname{Tr}(\mathbbm{1})=1$, these choices imply
$\left\langle\operatorname{Tr}\left(H^{2}\right)\right\rangle_{J}=\mathcal{J}^{2}$.
From now on we set $\mathcal{J}=1$, and restore it by dimensional analysis
when needed. We will be interested in the double scaling limit

$N,p\to\infty\,,\qquad\lambda\equiv 2p^{2}/N\quad fixed.$ (2.4)

We’ll start by analyzing the partition function of the model by expanding it
into moments,

$\displaystyle Z\left(\beta\right)$
$\displaystyle=\left\langle\operatorname{Tr}\left(e^{-\beta
H}\right)\right\rangle_{J}=\sum_{k=0}^{\infty}\frac{\left(-\beta\right)^{k}}{k!}m_{k}\,,$
(2.5) $\displaystyle m_{k}$
$\displaystyle=\left\langle\operatorname{Tr}\left(H^{k}\right)\right\rangle_{J}=i^{kp/2}\sum_{I_{1},\dots,I_{k}}\left\langle
J_{I_{1}}\cdots
J_{I_{k}}\right\rangle_{J}\operatorname{Tr}\left(\Psi_{I_{1}}\cdots\Psi_{I_{k}}\right)\,.$
(2.6)

Since the $J$’s are Gaussian variables we use Wick’s theorem for the ensemble
average. (2.3) implies that the indices of pairs of $\Psi$’s are identified,
which we also represent as contractions,

$m_{k}=i^{kp/2}\sum_{I_{1},\dots,I_{k}}\binom{N}{p}^{-k/2}\sum_{I_{1},\cdots,I_{k/2}}\sum_{\text{Wick
cont.}}\operatorname{Tr}\left(\leavevmode\hbox to7.78pt{\vbox
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{I_{2}}\cdots\leavevmode\hbox
to7.78pt{\vbox to6.83pt{\pgfpicture\makeatletter\hbox{\hskip
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to19.00002pt{}{}\cdots\right)\,.$ (2.7)

As the trace is cyclic, we can represent these contractions by a _chord
diagram_ as in Figure 1. The sum over all Wick contractions becomes a sum over
all chord diagrams.

Figure 1: The chord diagram for (2.7) (left) and for
${\operatorname{Tr}\left(\leavevmode\hbox to7pt{\vbox
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to9.00002pt{}{}}$ (right). Here we also highlighed the nodes at the ends of
each chord, but we will suppress them later.

Next we evaluate the trace with the prescribed ordering of the $\Psi_{I}$’s.
We permute the $\Psi_{I}$’s until two with the same index set $I$ are
adjacent, and then annihilate them as $\Psi_{I}^{2}\propto\mathbbm{1}$.
Commuting $\Psi_{I}$ and $\Psi_{K}$ introduces a sign that depends on the
number of indices in common to $I$ and $K$, $(-1)^{\left|I\cap K\right|}$. In
the double scaling limit $\left|I\cap K\right|$ is Poisson distributed with
mean $\frac{p^{2}}{N}$ since $I$ and $K$ are two independent lists of numbers
of length $p$, drawn from $\\{1,\cdots,N\\}$ [7, 12], while intersections of
three or more index sets are empty with probability 1. The exchange of each
$\Psi_{I}$ and $\Psi_{K}$ – each intersection of chords in the diagram –
therefore gives a factor of $\langle(-1)^{|I\cap K|}\rangle_{Poisson}$ [12, 9]

$\sum_{m=0}^{\infty}\frac{\left(p^{2}/N\right)^{m}}{m!}e^{-p^{2}/N}\left(-1\right)^{m}=e^{-\lambda}\equiv
q\,.$ (2.8)

Summing over all chord diagrams, we find that each moment gives444In the
$\mathbb{J}$ normalization of (2.3) there is an additional factor of
$\left(\frac{\mathbb{J}}{\sqrt{\lambda}}\right)^{k}$ appearing in the moment
$m_{k}$.

$m_{k}=\sum_{\begin{subarray}{c}\text{chord diagrams}\\\ \text{with $k/2$
chords}\end{subarray}}q^{\,\text{no. of intersections}}\,.$ (2.9)

This expression is valid when $k$ is held finite555Perhaps it can be extended
to $k\sim N^{\alpha}$, $\alpha$ positive, small enough. as
$N\rightarrow\infty$.

Formally666The action is not bounded from below, and so the correct
integration contour for $g$ is unclear. Similar action can be found when $q\to
1$ by coarse graining chord diagrams [26], where the integral is better
defined., one can write a generating function for chord diagrams777In SYK, the
expression can be directly derived from the partition function of the
underlying fermionic model by the $G\Sigma$ formalism [12]. In the double
scaling limit
$G(\tau_{1},\tau_{2})=\operatorname{sign}(\tau_{1}-\tau_{2})e^{g(\tau_{1},\tau_{2})/p}$,
$\Sigma(\tau_{1},\tau_{2})=\sigma(\tau_{1},\tau_{2})/p$, and integrating out
the latter gives (2.10). Yet, the generating function holds for any model
described by chord diagrams, even without direct $G\Sigma$ like derivation.
[12, 25, 27],

$Z(\beta)=\int\mathcal{D}g\exp\left[-\frac{1}{\lambda}\int_{0}^{\beta}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\left[\frac{1}{4}\partial_{1}g(\tau_{1},\tau_{2})\partial_{2}g(\tau_{1},\tau_{2})-\mathbb{J}^{2}e^{g(\tau_{1},\tau_{2})}\right]\right]\,,$
(2.10)

The $k$-th order in the perturbative expansion in $\mathbb{J}$ reproduces the
contribution of the $k$-th moment to the partition function of temperature
$\beta$, see Appendix H of [25] for details.

###  2.2 The transfer matrix

The number of chord diagrams increases rapidly with $k$, so we need an
efficient way of evaluating the sum over them. We do so by building all the
chord diagrams iteratively with a _transfer matrix_ [8] in the following way.
First, choose some point on the circle of Figure 1 between the $k$ nodes, and
declare that at this point there are no open chords. Next move clockwise
around the circle and at each node either open a new chord or close an already
open one. After $k$ steps we must return to the same point, and therefore must
again have no open chords. All chord diagrams are uniquely generated by this
process. An illustration of the process for a single diagram is given in
Figure 2.

Figure 2: Starting from the blue line, we rewrite the chord diagram (left) by
opening and closing chords at each node (right). We noted the state in the
chord Hilbert space after each step for this particular diagram. The total
weight of the diagram is $q^{2}$.

Suppose we have $n$ open chords as we go along the diagram. If a chord opens,
then we put it at the bottom of the stack of open chords. We have no new
intersections and have $n+1$ chords for the next step888There are other
conventions for when intersections happen, resulting in conjugate transfer
matrices.. If a chord closes, it could be either the bottom one, adding no
intersections, or the one above it, which would contribute a factor of $q$, or
all the way to the top one, adding a factor of $q^{n-1}$. So summing over all
possible diagrams, if a chord closes, we multiply by
$1+q+\cdots+q^{n-1}=\frac{1-q^{n}}{1-q}$, and reduce the number of chords by
1.

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{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{7.08745pt}{-34.5pt}\pgfsys@lineto{31.72478pt}{-34.5pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}\pgfsys@moveto{0.0pt}{-53.25pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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{}{{}}{}{{}}{}\pgfsys@moveto{16.40237pt}{-24.2625pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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This process is an evolution in the _chord Hilbert space_

$\mathcal{H}=\left\\{\ket{n}\big{|}n\in\mathbb{Z}_{+}\right\\}\,,$ (2.12)

which describes how many open chords there are at any given step. We start
from ”no chords”, $\ket{0}$, and end with $\bra{0}$. Defining

$\displaystyle\alpha\ket{n}$ $\displaystyle=\ket{n-1}\,,$
$\displaystyle\alpha^{+}\ket{n}$ $\displaystyle=\ket{n+1}\,,$
$\displaystyle\hat{n}\ket{n}$ $\displaystyle=n\ket{n}\,,$ (2.13)
$\displaystyle a$ $\displaystyle=\alpha\frac{1-q^{\hat{n}}}{1-q}\,,$
$\displaystyle a^{\dagger}$ $\displaystyle=\alpha^{+}\,,$ $\displaystyle T$
$\displaystyle=a^{\dagger}+a\,.$

the moments and partition function are

$m_{k}=\left\langle\operatorname{Tr}(H^{k})\right\rangle_{J}=\langle
0|T^{k}|0\rangle\,,\qquad Z=\langle 0|e^{-\beta T}|0\rangle\,.$ (2.14)

$T$ is called the _transfer matrix_ , and we can think about it as the
Hamiltonian acting on the chord Hilbert space. Its spectrum and eigenstates
are known, allowing us to compute this expression, as we will do shortly.

More conceptually, by the same arguments any quantum mechanical time evolution
$e^{-iHt}$ inside an ensemble averaged correlation function can be replaced by
$e^{-iTt}$ acting on the chord Hilbert space. So we have derived a duality
explicitly – computations in the microscopic Hilbert space are equal to those
in the chord Hilbert space. This is just the AdS/CFT duality in this case, as
we will see shortly that $T$ is the gravitational Hamiltonian.

This also is the first appearance of non-commutative geometry as the operators
$a$, $a^{\dagger}$ are the Arik-Coon $q$-oscillators which satisfy

$[a,a^{\dagger}]_{q}\equiv
aa^{\dagger}-qa^{\dagger}a=1\,,\qquad[{\hat{n}},a]=-a,\qquad[{\hat{n}},a^{\dagger}]=a^{\dagger}\,.$
(2.15)

We also choose an inner product on $\mathcal{H}$ such that $a$ and
$a^{\dagger}$ are Hermitian conjugates,

$\langle m|n\rangle=\delta_{mn}[n]_{q}!\,.$ (2.16)

####  2.2.0.0.1 The thermofield double

Even though we have a Hilbert space, we are not computing traces over it but
rather transition amplitudes. A more careful interpretation, due to Lin [24],
is that the chord Hilbert space describes the dynamics on the doubled Hilbert
space of the system999See also [28] in the context of DS-SYK. – a tensor
product of two copies of its Hilbert space, named the left- and the right-
copy. We briefly introduce the construction, and return to it later in Section
7.3.

Consider the state $\ket{0}=\sum_{k=1}^{\infty}\ket{k}_{L}\ket{k}_{R}$ on the
doubled space, where $\\{\ket{k}\\}_{L,R}$ are two copies of a basis of the
single space. Given two copies of the one-sided Hamiltonians $H_{L,R}$, there
is another useful state, the thermofield double state
$\ket{\text{TFD}}=e^{-\frac{1}{4}(H_{L}+H_{R})}\ket{0}$. These can be used to
express the partition function of the theory as an amplitude, and in DS-SYK
once we ensemble average over the couplings it is

$Z=\big{\langle}\braket{0}{e^{-\frac{\beta}{2}\left(H_{L}+H_{R}\right)}}{0}\big{\rangle}_{J}=\big{\langle}\braket{\text{TFD}}{\text{TFD}}\big{\rangle}_{J}\,,$
(2.17)

where the couplings of both Hamiltonians are drawn from the same distribution.
This again results in a sum over chord diagrams, and has the pictorial
representation of cutting them open in two places, 3(a).

(a) The cuts to the diagram.

(b) The contribution of a single diagram to the bra- and ket-states (blue) and
their inner-product (red).

Figure 3: Chord diagrams as transition amplitude in the two-sided Hilbert
space.

As before, all chord diagrams are generated by a transfer matrix [24]. At each
step, the state of the system is given by the number of open chords,
$\ket{n}$. Then, a chord can either open from the left boundary or close into
it, and similarly from the right. As a chord closes, it crosses all the chords
between itself and the appropriate boundary. Our transfer matrix now has two
parts, $T_{L}$ and $T_{R}$, in terms of which the partition function and TFD
are

$Z=\braket{0}{e^{-\frac{\beta}{2}(T_{L}+T_{R})}}{0}\,,\qquad\ket{\rm
TFD}=e^{-\frac{\beta}{4}(T_{L}+T_{R})}\ket{0}\,.$ (2.18)

The TFD state can be thought of as a (weigthed) sum over all “bottom halves”
of a chord diagram, without accounting for the intersections between the
chords that leave them. In the amplitude $\braket{\text{TFD}}{\text{TFD}}$ the
inner product (2.16) accounts for these intersections given two such
halves101010Say there are $n$ chords coming out of each half, and look at them
from left to right. When the first chord on the bottom connects to the first
chord on the top, it does not intersect any of the other chords. When it
connects to the second top chord, it intersects exactly one chord and gives a
factor of $q$. Summing over the ways to connect it gives a factor of
$[n]_{q}$. The second bottom chord then gives an additional factor of
$[n-1]_{q}$, and so it goes until we get an overall factor of $[n]_{q}!$.,
3(b), thus summing over all diagrams.

####  2.2.0.0.2 Diagonalizing the transfer matrix

The transfer matrix $T$ is Hermitian under the inner product (2.16), and
admits a complete set of eigenstates with real eigenvalues. The spectrum is
continuous and bounded, such that the eigenvalues are in
$[-\frac{2}{\sqrt{1-q}},\frac{2}{\sqrt{1-q}}]$, and it can be parameterized by
an angle $\theta\in[0,\pi]$. We denote by $\ket{\theta}$ the eigenvector and
$\psi_{n}(\theta)$ is its $n^{\text{th}}$ component in the chord basis, such
that

$T\ket{\theta}=\frac{2\cos(\theta)}{\sqrt{1-q}}\ket{\theta}\,,\qquad\psi_{n}(\theta)=\sqrt{\rho(\theta)}\frac{(1-q)^{n/2}}{(q;q)_{n}}H_{n}\left(\cos(\theta)|q\right)\,,$
(2.19)

where $(q;q)_{n}$ denotes the $q$-Pochhammer symbol, $H_{n}(x|q)$ the
continuous $q$-Hermite polynomials, and $\rho(\theta)=\frac{(q,e^{\pm
2i\theta};q)_{\infty}}{2\pi}$, which we will soon interpret as the density of
states, is a function that ensures the completeness relation

$\int_{0}^{\pi}d\theta\,\langle
n|\theta\rangle\langle\theta|m\rangle=[n]_{q}!\delta_{nm}\quad\implies\quad\int_{0}^{\pi}d\theta|\theta\rangle\langle\theta|=\mathbbm{1}\,.$
(2.20)

The list of relevant mathematical definitions is in Appendix A.

####  2.2.0.0.3 The spectrum

The partition function can now be phrased as a one dimensional integral by
inserting complete sets of eigenstates of $T$ into (2.14),

$Z=\int_{0}^{\pi}d\theta\int_{0}^{\pi}d\theta^{\prime}\,\langle
0|\theta\rangle\langle\theta|e^{-\beta
T}|\theta^{\prime}\rangle\langle\theta^{\prime}|0\rangle=\int_{0}^{\pi}\frac{d\theta}{2\pi}(q,e^{\pm
2i\theta};q)_{\infty}e^{-\frac{2\beta\mathbb{J}\cos(\theta)}{\sqrt{\lambda(1-q)}}}\,,$
(2.21)

where we reintroduced the dimensionful coupling
$\mathcal{J}=\frac{\mathbb{J}}{\sqrt{\lambda}}$ that was set to $1$ earlier.
The spectrum is bounded and parameterized by $\theta$, and the density of
states is [29, 12, 9]

$\begin{gathered}\rho(\theta)d\theta=\frac{1}{2\pi}(q,e^{\pm
2i\theta};q)_{\infty}d\theta\,,\qquad
E(\theta)=\frac{2\mathbb{J}\cos(\theta)}{\sqrt{\lambda(1-q)}}\,,\qquad\theta\in[0,\pi]\,.\\\
\text{or}\quad\rho(E)dE=\frac{\sqrt{\lambda(1-q)}}{2\pi\mathbb{J}}\frac{1}{q^{1/8}}\theta_{1}\left(\frac{\theta(E)}{\pi}\bigg{|}\frac{\lambda
i}{2\pi}\right)dE\,,\end{gathered}$ (2.22)

where we used (A.3). Figure 4 shows the shape of $\rho(E)$ for different
values of $q$. After using the modular transform (A.4) followed by the
expansion (A.3) we find the form [30]

$\rho(\theta)=\frac{\sin\left(\theta\right)}{\pi
q^{1/8}}\sqrt{\frac{2\pi}{\lambda}}\sum_{k=-\infty}^{\infty}\left(-1\right)^{k}e^{-\frac{2}{\lambda}\left(\theta-\pi\left(k+\frac{1}{2}\right)\right)^{2}}\,,$
(2.23)

which is valid for any $q$, and almost suggests a full semi-classical
interpretation.

(a) $q=0.01$

(b) $q=0.8$

(c) $q=0.99$

Figure 4: The density of states (top) in units of $2^{N/2}$ and the entropy
(bottom) for different values of $q$. The energies are in units of
$\frac{2\mathbb{J}}{\sqrt{\lambda(1-q)}}$.

###  2.3 Adding matter and two point functions

To probe the quantum state of the system further we would like to come up with
a suitable set of probes [8]. We have already considered one operator – the
Hamiltonian (2.2) – which is drawn from a specific statistical ensemble of
operators. Next we would like to define the notion of an appropriate probe
operator. The correct, or at least convenient, choice is to draw probe
operators from similar ensembles modified to encode the fact that they may
have different dimensions (and other quantum numbers, like charges, if present
in the model).

In the context of the $AdS/CFT$ duality, and in particular if we think of a
dual 2D black holes as arising from a near horizon S-wave compactification of
a higher dimensional black hole, then these probes should be the analogues of
the ”single trace” operators. In this context, a simple “single trace”
operator in the UV theory, such as the Hamiltonian, flows to a very
complicated operator (2.2) in terms of the effective description of the IR
theory, which we take to be the fermions of the SYK model. Similarly, other
simple “single trace” operators in the UV will look complicated in the IR, but
their overall renormalized appearance need not be that different from that of
the Hamiltonian.

The easiest way to achieve this is to take operators similar to (2.2), except
with a different length $\tilde{p}$ instead of $p$ and with a new set of
independent Gaussian couplings,

$M_{\Delta}=i^{\tilde{p}/2}\sum_{i_{1}<i_{2}<\cdots<i_{\tilde{p}}=1}^{N}{\tilde{J}}_{i_{1}\cdots
i_{\tilde{p}}}\psi_{i_{1}}\cdots\psi_{i_{\tilde{p}}}\equiv
i^{\tilde{p}/2}\sum_{I^{\prime}}\tilde{J}_{I^{\prime}}\Psi_{I^{\prime}}\,.$
(2.24)

We denoted this operator by $\Delta=\tilde{p}/p$ since in the JT gravity limit
of Section 3.1 it corresponds to an operator with conformal dimension
$\Delta$. However, we can evaluate its exact (unnormalized) thermal two point
function before taking such a limit. We want to compute

$G_{\Delta}(\beta_{1},\beta_{2})=\left\langle\operatorname{Tr}\left[e^{-\beta_{1}H}Me^{-\beta_{2}H}M\right]\right\rangle_{J,\tilde{J}}=\big{\langle}\operatorname{Tr}\big{[}e^{-(\beta_{1}+\beta_{2})H}M(\beta_{2})M(0)\big{]}\big{\rangle}_{J,\tilde{J}}\,,$
(2.25)

or, after expanding into moments,
$\left\langle\operatorname{Tr}(H^{k_{1}}MH^{k_{2}}M)\right\rangle_{J,\tilde{J}}$,
where the ensemble average is over both $J$ and $\tilde{J}$. The chord diagram
associated with this observable is one in which we pair the $H$’s as before,
but there is one distinct chord which connects the two $M$’s – we will call
these $H$-chords and $M$-chord. Again as before, the intersection of two
$H$-chords comes with a weight $q$ but the intersection of an $H$-chord with
the $M$-chord comes with a new weight $\tilde{q}$

$q=e^{-{2p^{2}\over N}},\qquad{\tilde{q}}=e^{-{2p\tilde{p}\over
N}}=q^{\Delta}\,.$ (2.26)

A variant of the transfer matrix method works here just as well. We cut the
circular diagram such that one $M$ appears just to the right of the cut, and
the other $M$ somewhere in the interior, as in Figure 5. The transfer matrix
takes care of the “bubbling” of $H$-chords and it is the same as before. We
start with the leftmost insertion of $H$ and propagate the system through the
first $k_{2}$ steps, i.e. the $k_{2}$ $H$’s between the two $M$ operators,
which gives the same contribution as before. However, when we hop over the
second insertion of $M$, the open $H$-chords cross the $M$-chord picking up an
additional factor of ${\tilde{q}}^{\,\text{no.\ of\ lines}}$. Finally, we
propagate the chords for the remaining $k_{1}$ steps. The expression for 2-pt
function is therefore

$\left\langle\operatorname{Tr}\bigl{(}e^{-\beta_{1}H}MH^{-\beta_{2}H}M\bigr{)}\right\rangle_{J,\tilde{J}}=\langle
0|e^{-\beta_{1}T}\tilde{q}^{\,\hat{n}}e^{-\beta_{2}T}|0\rangle\,.$ (2.27)

Figure 5: Marked chord diagram for a two point function.

After inserting complete sets of energy states (2.19) and using (A.12) one
finds [8]

$G_{\Delta}(\beta_{1},\beta_{2})=\int_{0}^{\pi}d\theta_{1}\int_{0}^{\pi}d\theta_{2}\,\rho(\theta_{1})\rho(\theta_{2})\,e^{-\beta_{1}E(\theta_{1})-\beta_{2}E(\theta_{2})}\frac{({\tilde{q}}^{2};q)_{\infty}}{(\tilde{q}e^{i(\pm\theta_{1}\pm\theta_{2})};q)_{\infty}}\,.$
(2.28)

###  2.4 The $p$-spin model

Whenever the dynamics of a system is described via chord diagrams, then the
transfer matrix and the subsequent machinery can be applied to it. This is
expected to be the case for many double scaled $p$-local systems described in
Section 6 (and even beyond, see Section 6.1), so it should be viewed more as a
broad universality class, rather than a trick for a specific model. In fact,
equation (2.9) was actually written first for the $p$-spin model [7], and only
then adapted to the SYK model [12]. The entire machinery of the transfer
matrix was applied to the $p$-spin model first in [8].

The $p$-spin model is the following. Consider $N$ qubits, and Pauli matrices
$\sigma^{(a_{i})}_{i}$ with $a_{i}=1,2,3$ acting on the $i$-th qubit,
$i=1,\dots,N$. Denote by $e=(i_{1},\dots,i_{p})$ some subset of the qubits,
and by $A=(a_{1},\dots,a_{p})$ the choice of each Pauli matrix acts on each of
them. Denoting the pair $(e,A)$ by $I$, and a tensor product of the matrices
by

$\sigma_{I}=\sigma_{(e,A)}=\sigma_{i_{1}}^{(a_{1})}\sigma_{i_{2}}^{(a_{2})}\cdots\sigma_{i_{p}}^{(a_{p})}\,,$
(2.29)

we define the random Hamiltonian with Gaussian random couplings $J_{I}$ to be

$H_{p\text{-spin}}=3^{-p/2}{n\choose p}^{-1/2}\sum_{I}J_{I}\sigma_{I}\,.$
(2.30)

The relevant parameter controlling the double scaling limit
$N\rightarrow\infty$, $\lambda$ fixed is [7]

$q=e^{-\lambda}\quad\text{with}\quad\lambda=\frac{4}{3}\ \frac{p^{2}}{N}\ ,$
(2.31)

The computation of the moments,
$\sum_{I_{1},\cdots,I_{k}}\bigl{\langle}J_{I_{1}}\cdots
J_{I_{k}}\bigr{\rangle}_{J}\operatorname{Tr}\bigl{(}\sigma_{I_{1}}\cdots\sigma_{I_{k}}\bigr{)}$,
proceeds very similarly to the SYK model. First we carry out the ensemble
average by identifying pairs of $I$’s, which brings us to the chord picture.
Next, however, we do not “pass one interaction term across another”, but
recall that in the double scaling limit each qubit appears in at most two
chords. The number of times the same qubit is shared by two index sets $I_{1}$
and $I_{2}$ is Poisson distributed with expectation value
${\frac{3\lambda}{4}}$.

Now we carry out the trace on each qubit Hilbert space, with four cases to
consider. The first is when a qubit does not appear in any operator. It then
contributes 1 to the normalized trace. Second, a qubit that only appears in
one chord gives a weight111111The factor of $1/2$ is because we normalize
$\operatorname{tr}_{i}(\mathbbm{1})=1$. of $3^{-1}\sum_{a}{1\over
2}\operatorname{tr}_{i}\left(\sigma^{(a)}\sigma^{(a)}\right)=1$. Next, a qubit
that appears in two non intersecting chord gives
$3^{-2}\sum_{a,b=1}^{3}{1\over
2}\operatorname{tr}_{i}\bigl{(}\sigma^{(a)}\sigma^{(a)}\sigma^{(b)}\sigma^{(b)}\bigr{)}~{}=~{}1$.
Finally, a qubit that appears in two intersecting chords gives
$3^{-2}\sum_{a,b=1}^{3}{1\over
2}\operatorname{tr}_{i}\bigl{(}\sigma^{(a)}\sigma^{(b)}\sigma^{(a)}\sigma^{(b)}\bigr{)}=-{1\over
3}\,.$ Recalling the Poisson nature of $|I_{1}\cap I_{2}|$, each pairwise
intersection gives a factor of $\langle(-{1\over 3})^{|I_{1}\cap
I_{2}|}\rangle_{\text{Poiss},{3\lambda\over 4}}=q$, and we are back in
equation (2.9) for this model as well.

##  3 Semiclassics (bulk I)

###  3.1 The triple scaling limit and the Schwarzian theory

At low energies the SYK model with finite $p$ is described by the Schwarzian
theory (and JT gravity) [1, 2]. This limit also arises in our case by taking
the _triple scaling limit_ [12], which is when we take $\lambda\ll 1$ while
concentrating on energies near the edge of the spectrum.

More precisely, suppose we write $\theta=\pi-\lambda\sqrt{\varepsilon}$, such
that $E-E_{\text{min}}=\lambda\mathbb{J}\varepsilon$ when $\lambda\to 0$.
After the modular transformation (A.4) and the series expansion (A.3) the
density of states, (2.22), reproduces that of the Schwarzian theory up to an
overall factor,

$\rho(\varepsilon)=\sqrt{\frac{2\lambda^{3}}{\pi}}e^{-\frac{\pi^{2}}{2\lambda}}\sinh\left(2\pi\sqrt{\varepsilon}\right)\,.$
(3.1)

The two-point function for an operator of length $\tilde{p}$ can also be
computed in this limit, and it is proportional to
$Z^{-1}G_{\Delta}(\beta-\tau,\tau)=\sin^{-2\Delta}(2\pi\tau/\beta)$, i.e. the
two point function of a conformal operator of dimension $\Delta=\tilde{p}/p$.

We can do more than just reproduce the density of states, and match with a
(form of) the Schwarzian directly [8]. First conjugate the transfer matrix to
the form

$T={1\over\sqrt{1-q}}\begin{pmatrix}0&1&0&0&\cdots\\\ 1&0&1-q&0&\ddots\\\
0&1&0&1-q^{2}&\ddots\\\ \vdots&\ddots&\ddots&\ddots&\ddots\end{pmatrix}\,.$
(3.2)

When we go to large values of $n$ (way down the diagonal) and replace it by a
continuous variable, the 1’s above and below the diagonal go over the 2nd
derivative, and the $q^{n}$ becomes a potential. The transfer matrix
essentially describes dynamics on a discrete one sided lattice.

Using the notation of [24, 31] we can define the new operators, $\ell$, and
its (formal) conjugate $\pi_{\ell}$, and write the transfer matrix
as121212Formally, this is related to (2.13) by
$\alpha=-\sqrt{1-q}\,e^{-i\lambda\pi_{\ell}}$,
$\alpha^{+}=-\frac{1}{\sqrt{1-q}}\,e^{i\lambda\pi_{\ell}}$, but $\pi_{\ell}$
is not entirely well defined at any finite $\lambda$, since $\ell$ is
discrete.

$\ell=\lambda\hat{n}\,,\qquad[\pi_{\ell},\ell]=i\,,\qquad
T=-\frac{\mathbb{J}}{\sqrt{\lambda(1-q)}}\left(e^{i\lambda\pi_{\ell}}+e^{-i\lambda\pi_{\ell}}\left(1-e^{-\ell}\right)\right)\,.$
(3.3)

Going to low energies/long wavelength/$\ell\to\infty$ on the lattice, we
define $\tilde{\ell}=\ell+2\log\lambda$ and keep it fixed when
$\lambda\rightarrow 0$. The transfer matrix then becomes

$T=E_{\text{min}}+\frac{\lambda\mathbb{J}}{2}\left(\pi_{\tilde{\ell}}^{2}+e^{-\tilde{\ell}}\right)\,,\qquad
E_{\text{min}}=-\frac{2\mathbb{J}}{\lambda}\,.$ (3.4)

This is exactly the Liouville Hamiltonian which is a gauge fixed version of
the Schwarzian quantum mechanics [32, 33], and so the transfer matrix that
captures the dynamics on this new emergent chord degrees of freedom becomes
the gravitational Hamiltonian. In the thermofield double interpretation, the
Hilbert space is that of JT-gravity on a strip with two boundaries [34], where
$\ell$ and $\tilde{\ell}$ play the role of the bare and renormalized lengths
between the two boundaries, respectively.

Note that for finite $\lambda$ this length is quantized by $\lambda$. A theory
which has a minimal length scale manifest throughout the computation is
entitled to be considered a full theory of _Quantum Gravity_. This short
distance is usually expected to the Planck scale131313The role of a string
scale here is unclear since there is no obvious stringy description of this
background.. A little more precisely, the semiclassical two point function of
an operator of dimension $\Delta$ is

${\tilde{q}}^{\,\hat{n}}=e^{-\lambda\Delta\cdot\hat{n}}\sim
e^{-mR_{AdS}\cdot\ell}$ (3.5)

so changing $n$ by 1 means that we change the distance by $\lambda R_{AdS}$,
i.e. $\ell_{Planck}=\lambda R_{AdS}$.

###  3.2 Finite and fake temperature

We will now summarize some features of the model when $\lambda\to 0$ but the
temperature $\beta\mathbb{J}$ is finite, such that we are sensitive to the
entire spectrum. The two-point function and the Lyapunov exponent will behave
almost holographically, the only difference being that the temperature
dependence will be replaced by a “fake” temperature. The leading order of
these quantities reproduces141414At high temperatures
$\beta\mathbb{J}\ll\lambda$, the $\beta\mathbb{J}$ expansion is just the
moment expansion, allowing to compare the subleading orders in $\lambda$ for
the double scaling limit and in $1/p$ for the large $p$ limit, and to verify
that they differ [35, 36, 9].  the large $p$ limit of SYK [2]. At low
temperatures, the fake temperature converges to the real one, and the triple
scaling limit is reproduced.

####  3.2.0.0.1 The partition function

Consider the density of states in the form (2.23), and focus on the
dominant151515For other $k$’s the story is similar when defining
$\theta-\frac{\pi\left(2k+1\right)}{2}=\left(-1\right)^{k}\frac{\pi v_{k}}{2}$
with $\theta\in[0,\pi]$. term, $k=0$. At the saddle point, the average energy
can be parameterized by the variable $v$, and the equations become

$\frac{\pi v}{\cos\left(\frac{\pi
v}{2}\right)}=\beta\mathbb{J}\,,\qquad\theta=\frac{\pi}{2}+\frac{\pi
v}{2}\,,\qquad v\in[-1,1]\,,$ (3.6)

resulting in [37, 38, 39]

$Z(\beta)=\frac{\cos\left(\frac{\pi v}{2}\right)}{\sqrt{1+\frac{\pi
v}{2}\tan\left(\frac{\pi
v}{2}\right)}}\exp\left[-\frac{1}{\lambda}\left(\frac{\pi^{2}v^{2}}{2}-2\pi
v\tan\left(\frac{\pi v}{2}\right)\right)+\frac{1}{2}\pi v\tan\left(\frac{\pi
v}{2}\right)\right]\,.$ (3.7)

At low temperatures $\beta\mathbb{J}\rightarrow\infty$ it agrees with the
Schwarzian result [2] with coefficient $C=2\lambda\mathbb{J}$,

$Z\left(\beta\right)=\frac{\pi\sqrt{2}}{\left(\beta\mathbb{J}\right)^{3/2}}e^{\left(\frac{2}{\lambda}+\frac{1}{2}\right)\beta\mathbb{J}-\frac{\pi^{2}}{2\lambda}+\frac{\pi^{2}}{\beta\mathbb{J}\lambda}}\,.$
(3.8)

####  3.2.0.0.2 The two point function

Again by saddle point methods, from (2.28) follows the normalized two-point
function of a random operator of finite dimension
$\Delta=\frac{\tilde{\lambda}}{\lambda}$ [37, 38, 39],

$\frac{G_{\Delta}(\beta-\tau,\tau)}{Z}=\left[\frac{\cos\left(\frac{\pi
v}{2}\right)}{\cos\left[\pi
v\left(\frac{1}{2}-\frac{\tau}{\beta}\right)\right]}\right]^{2\Delta}\,.$
(3.9)

For the first correction in $\lambda$ see [39]. At low temperatures $v\to 1$
the numerator vanishes, forcing us to renormalize our operators, which is
related to the renormalization of the bulk length as we will see below. Up to
this factor, (3.9) matches the Schwarzian theory [39].

As observed in [40, 25], this looks like the conformal two point function,
$\sin^{-2\Delta}(2\pi s_{12}/\beta_{\text{fake}})$ when written not in terms
of the coordinates on the thermal circle, but rather on a “fake” circle of
circumference $\beta_{\text{fake}}=\beta/v$, Figure 6, via the mapping

$s_{1}=\tau_{1}+\frac{\beta-\beta_{\text{fake}}}{4}\,,\qquad
s_{2}=\tau_{2}-\frac{\beta-\beta_{\text{fake}}}{4}\,,\qquad
s_{12}=s_{1}-s_{2}\,.$ (3.10)

where $\tau_{1}=\frac{\beta-\tau}{2}$, $\tau_{2}=\frac{\beta+\tau}{2}$ are the
Euclidean times in which the operators are inserted. The two point function
(3.9) does not diverge when $\tau_{1}=\tau_{2}$, as the probes annihilate. In
the fake disk, the divergence is evaded as the operators are mapped to
different points.

$\displaystyle\tau_{2}$$\displaystyle\tau_{1}$$\displaystyle
s_{2}$$\displaystyle
s_{1}$$\displaystyle\beta_{\text{fake}}=\beta/v$$\displaystyle\beta$ Figure 6:
The fake circle.

Another manifestation of the fake temperature appears in a Lorenzian two-sided
correlator, where $\tau_{1}=it$, $\tau_{2}=\beta/2-it$. Usually in holographic
systems its decay rate is proportional to the temperature, dictated by the
quasinormal modes of the dual black hole. However here
$G_{\Delta}\sim\cosh^{-2\Delta}(2\pi t/\beta_{\text{fake}})$, and the rate is
proportional to the fake temperature [25, 31, 41]. At high temperatures the
decay rate approaches a constant, $\beta_{\text{fake}}\to 1/\mathbb{J}$ (see
[42]).

This form of the two point function is protected by a symmetry that originates
at finite $\lambda$ [25], and in the limit $\lambda\to 0$ becomes an
$\mathfrak{sl}_{2}$ symmetry that acts naturally on the fake circle. We will
come back to this in Section 5.2.

The fake temperature also appears in the expectation value of $\hat{n}$ in
some TFD state161616See [43] for the finite $\lambda$ distribution., i.e. in
its thermal expectation value. Since the gravitational interpretation of the
state is a two-sided geometry, this measures the length $\ell=\lambda\hat{n}$
of the wormhole connecting the two sides. It is computed by taking derivatives
of the two point function (2.27)

$\left\langle\text{TFD}|\hat{n}|\text{TFD}\right\rangle=-\frac{1}{Z}\frac{1}{\lambda}\frac{\partial}{\partial\Delta}G_{\Delta}\left(\tfrac{\beta}{2},\tfrac{\beta}{2}\right)\Big{|}_{\Delta=0}=-\frac{2}{\lambda}\log\left(\beta_{\text{fake}}\cdot\mathbb{J}/\pi\right)\,.$
(3.11)

In the triple scaling limit, when real and fake temperatures are the same and
$\beta\mathbb{J}\propto 1/\lambda$, the renormalized length
$\tilde{\ell}=\ell+2\log\lambda$ is finite, as we previously claimed. On the
other hand, at high temperatures the typical number of chords shrinks to zero
[44].

####  3.2.0.0.3 The chaos exponent

The four point function, reviewed in Section 4, is more difficult to tackle in
the semi-classical limit. We are especially interested in the out-of-time-
order correlator, whose connected part is

$\big{\langle}\operatorname{Tr}\big{[}e^{-\beta
H}M_{1}(0)M_{2}(it)M_{1}(0)M_{2}(it)\big{]}\big{\rangle}_{J,\tilde{J}_{1},\tilde{J}_{2}}^{\text{connected}}\propto\lambda\cdot
e^{\lambda_{L}t}\,,\quad\lambda_{L}=\frac{2\pi
v}{\beta}=\frac{2\pi}{\beta_{\text{fake}}}\,,$ (3.12)

The Lyapunov exponent, $\lambda_{L}$, was computed in the large $p$ limit of
SYK [2] to be smaller171717Actually, with respect to the bound derived in
[45], the Lyapunov exponent here is still maximal. than the maximal value [13]
$\lambda_{max}=\frac{2\pi}{\beta}$. The leading deviation from $\lambda_{max}$
at low temperatures was computed directly from the exact double scaled four
point function (4.5) [9]. The expression for generic $v$ was obtained by [25]
based on the $\mathfrak{sl}_{2}$ algebra reviewed in Section 5.2. The algebra
also implies the four point function has a simple closed form for any $v$,
reproducing the large $p$ result of [40, 46].

##  4 The four point function and $SU_{\hat{q}}(1,1)$

Earlier we obtained JT gravity in the $\lambda\rightarrow 0$ and low energy
limit. However, the main test is to compute the 4-pt function and chaos
exponent. Then we can ask what that means for gravity at finite $\lambda$,
which we suggested to be the quantum gravity regime.

Our goal is to compute the four point function [9]. We concentrate on the
crossed ordering

$G(\tau_{1},\tau_{3};\tau_{2},\tau_{4})=\big{\langle}\operatorname{Tr}\big{(}e^{-\beta
H}\leavevmode\hbox to10.79pt{\vbox
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{2}(\tau_{2})\leavevmode\hbox
to10.79pt{\vbox to6.83pt{\pgfpicture\makeatletter\hbox{\hskip
5.39583pt\lower-3.41666pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle M$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\leavevmode\hbox
to0.4pt{\vbox to9.57pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
6.21666pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}{}{}}{}{{}{}}{}{{}}{}{
{}{}{}{}{}}{}{{}{}}{{}{}} {}{}{ {}{}{}{}{}}{}{{}{}}{{}{}} {}{}{
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{}{}{}\pgfsys@moveto{0.0pt}{6.41666pt}\pgfsys@lineto{0.0pt}{15.58336pt}\pgfsys@lineto{0.0pt}{15.58336pt}\pgfsys@lineto{0.0pt}{6.41666pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{1}(\tau_{3})\leavevmode\hbox
to10.79pt{\vbox to6.83pt{\pgfpicture\makeatletter\hbox{\hskip
5.39583pt\lower-3.41666pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\leavevmode\hbox
to0.4pt{\vbox to6.57pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{2}(t_{4})\vbox
to19.00002pt{}{}\big{)}\big{\rangle}_{J,\tilde{J}_{1},\tilde{J}_{2}}\,.$ (4.1)

The technical novelty is that even if we “cut open” the circular diagram at an
insertion point of one of the operators, say $M_{1}$, as we did for the
computation of the two point function, we still need to deal with the double
insertion of $M_{2}$ at two arbitrary point as in figure 7(a). This is the
analogue of the bi-local operators in the Schwarzian theory [6].

(a) A region between contracted $M$ nodes.

(b) The last case is incorrectly assigned a value of $\tilde{q}^{2}$ when
naively inserting $\tilde{q}^{\,\hat{n}}$ at $M$ nodes.

Figure 7: Computation of the weight of bi-local operators.

For starters, we expect that at each of the insertions of $M_{i}$ we will need
to count the number of chords, so we expect that at each $M_{i}$ insertion we
will need to insert $\tilde{q}_{i}^{\,\hat{n}}$, where $\hat{n}$ counts the
number of open Hamiltonian chords as before (and ${\tilde{q}}_{i}$ are the
weights associated with $M_{i}$). However, if we just insert this matrix at
the two point we will be giving a wrong factor ${\tilde{q}_{i}}^{\,2}$ to
chords that enter and leave the region without intersecting the $M_{i}$-chord,
see 7(b). We can correct the error by adding $(1-\tilde{q}_{i}^{2})$
multiplied by the weight of $n-1$ chords entering the region. But we still
pick up an error when 2 chords remain above the $M_{i}$-chord, which we will
need to correct again by the weight of $n-2$ chords entering the region, and
so on. Since $n$ is finite this is a finite set of corrections.

Going from $l$ chords to $l-1$ chords and vice versa, and adding factors of
$\tilde{q}_{i}$, are conveniently implemented using the matrices

$U=\begin{pmatrix}0&1&0&\cdots\\\ 0&0&1&\cdots\\\ 0&0&0&\ddots\\\
\vdots&\vdots&\ddots&\ddots\end{pmatrix},\quad
D=\begin{pmatrix}0&0&0&\cdots\\\ 1&0&0&\cdots\\\ 0&1&0&\ddots\\\
\vdots&\vdots&\ddots&\ddots\end{pmatrix},\quad
S_{i}=\begin{pmatrix}1&0&0&\cdots\\\ 0&\tilde{q}_{i}&0&\cdots\\\
0&0&\tilde{q}_{i}^{2}&\ddots\\\
\vdots&\vdots&\ddots&\ddots\end{pmatrix}=\tilde{q}_{i}^{\,\hat{n}}$ (4.2)

For $l$ incoming chords into the $M_{i}$-region consisting of $k$ Hamiltonian
nodes, the transition matrix through the entire $M_{i}$-region can be written
as

$S_{i}T^{k}S_{i}+P^{(l)}_{1}DS_{i}T^{k}S_{i}U+\cdots+P^{(l)}_{l}D^{l}S_{i}T^{k}S_{i}U^{l}\,,\quad
P_{j,i}^{(l)}=\frac{(q;q)_{l}(\tilde{q}_{i}^{\,2};q)_{j}}{(q;q)_{l-j}(q;q)_{j}},\quad
P_{0,i}^{(l)}=1\,.$ (4.3)

where $P^{(l)}_{j}$ guarantee there we are not overcounting or undercounting
of the diagrams [9]. This bi-local operator can be inserted in any correlation
function, but specifically for the crossed 4-pt function it gives

$\langle\operatorname{Tr}\big{(}H^{k_{4}}\contraction[2ex]{}{M_{2}}{H^{k_{3}}M_{1}H^{k_{2}}}{M_{2}}\contraction{M_{2}H^{k_{3}}}{M_{1}}{H^{k_{2}}M_{2}H^{k_{1}}}{M_{1}}M_{2}H^{k_{3}}M_{1}H^{k_{2}}M_{2}H^{k_{1}}M_{1}\big{)}\big{\rangle}_{J,\tilde{J}_{1},\tilde{J}_{2}}=\\\
=\sum_{l,m,n=0}^{\infty}\sum_{j=0}^{l}P^{(l)}_{j,2}\tilde{q}_{1}^{\,j}\langle
0|T^{k_{4}}|m\rangle\langle m|D^{j}S_{2}T^{k_{3}}S_{1}|n\rangle\langle
n|T^{k_{2}}S_{2}U^{j}|l\rangle\langle l|T^{k_{1}}|0\rangle\,.$ (4.4)

After some algebraic manipulation, the crossed four point function becomes [9]

$G(\tau_{1},\tau_{3};\tau_{2},\tau_{4})=\int_{0}^{\pi}\prod_{j=1}^{4}\big{[}d\theta_{j}\rho(\theta_{j})\big{]}e^{-\beta_{1}E(\theta_{1})-\beta_{2}E(\theta_{2})-\beta_{3}E(\theta_{3})-\beta_{4}E(\theta_{4})}\\\
\times\langle\theta_{1}|M_{1}|\theta_{4}\rangle\langle\theta_{3}|M_{2}|\theta_{2}\rangle\langle\theta_{2}|M_{1}|\theta_{1}\rangle\langle\theta_{4}|M_{2}|\theta_{3}\rangle\times
R^{(q)}_{\theta_{4}\theta_{2}}\begin{bmatrix}\theta_{3}&l_{2}\\\
\theta_{1}&l_{1}\end{bmatrix}\,,$ (4.5)

where we denoted $\beta_{1}=\beta+\tau_{41}$, $\beta_{2}=\tau_{12}$,
$\beta_{3}=\tau_{23}$, $\beta_{4}=\tau_{34}$, $\tau_{ij}=\tau_{i}-\tau_{j}$,
and $\tilde{q}_{i}=q^{\,l_{i}}$. The matrix elements already appeared
implicitly in the computation of the 2-pt function in Section 2 and are given
explicitly in equation (4.7). The $R$-matrix is defined in (A.13), and is the
$6j$-symbol of $SU_{q}(1,1)$, as we will discuss more below.

###  4.1 The diagrammatic rules of the full model

While the expressions above look complicated, they are naturally organized in
terms of “skeleton digrams” that keep only the information about the probe
operators, and encode the sum over the insertions of the Hamiltonians [9].
Their diagrammatic rules allow for the calculation of general correlation
functions, analogously to the Schwarzian theory [47, 48, 49, 50]:

  * •

Each segment along the circle represents an evolution using $T$ during a
Euclidean time $\tau$, and we therefore associate with it a factor of
$e^{-\tau E(\theta)}$. Summation over energy eigenstates is done with the
measure $d\theta\rho(\theta)$.

$\begin{gathered}\raisebox{0.0pt}[27.98602pt][0.0pt]{\ignorespaces\fmffile{prop}
\fmfgraph*(50,0) \fmfleft{i1} \fmfright{o1} \fmfdot{i1,o1}
\fmflabel{$\tau_{1}$}{i1} \fmflabel{$\tau_{2}$}{o1}
\fmf{plain,left,label=$\theta$}{i1,o1}
}\end{gathered}\qquad=e^{-(\tau_{2}-\tau_{1})\cdot E(\theta)}.$ (4.6)

  * •

Each operator insertion between two segments $\theta_{1}$, $\theta_{2}$ is
naturally thought of as a matrix element
$\langle\theta_{1}|M_{A}|\theta_{2}\rangle$. In order to reproduce the two
point function (2.28), the corresponding vertex is

$\begin{gathered}\fmffile{vertex}\fmfgraph*(50,50)\fmfleft{i1}\fmfright{o1,o2,o3}\fmfdot{o2}\fmf{dashes,label=$l_A$}{o2,i1}\fmf{plain,right=0.1,label=$\theta_2$}{o1,o2}\fmf{plain,right=0.1,label=$\theta_1$}{o2,o3}\end{gathered}\qquad=\langle\theta_{1}|M_{A}|\theta_{2}\rangle=\sqrt{\frac{(\tilde{q}_{A}^{2};q)_{\infty}}{\left(\tilde{q}_{A}e^{i(\pm\theta_{1}\pm\theta_{2})};q\right)_{\infty}}}\,,$
(4.7)

  * •

A contracted pair of operators conserves the energy before and after the
insertion [9]. Thus, the same $\theta$ variable is used before and after such
a contracted pair if there is no additional operator insertions in between, as
in 8(a),

$\begin{gathered}\fmffile{en_{c}ons}\fmfgraph*(50,50)\fmfleft{i1,i2,i3}\fmfright{o1,o2,o3}\fmfdot{i2,o2}\fmf{dashes,label=$l_A$,label.side=left}{o2,i2}\fmf{plain,right=0.2,label=$\theta_2$}{o1,o2}\fmf{plain,right=0.7,label=$\theta_1$}{o2,i2}\fmf{plain,right=0.2,label=$\theta_2$}{i2,i1}\end{gathered}\qquad.$
(4.8)

  * •

Every crossing of two internal lines comes with a factor of the $R$-matrix,
(A.13),

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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\vbox{\halign{#\hfil\cr\cr\vskip
23.0pt\cr\hbox{{$\displaystyle\theta_{1}$}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{116.25pt}{-47.77478pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{117.2pt}{-148.47365pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\vbox{\halign{#\hfil\cr\cr\vskip
91.0pt\cr\hbox{{$\displaystyle\theta_{2}$}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{70.5pt}{-86.09962pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{71.45pt}{-259.7985pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\vbox{\halign{#\hfil\cr\cr\vskip
164.0pt\cr\hbox{{$\displaystyle\theta_{3}$}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{22.5pt}{-47.77478pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.45pt}{-148.47365pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\vbox{\halign{#\hfil\cr\cr\vskip
91.0pt\cr\hbox{{$\displaystyle\theta_{2}$}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\,,$ (4.9)

(a) Uncrossed.

(b) Crossed.

Figure 8: Skeleton diagrams of the 4-point functions $\langle
M_{1}M_{1}M_{2}M_{2}\rangle$ and $\langle M_{1}M_{2}M_{1}M_{2}\rangle$.

Applying these rules to the crossed diagram Figure 8(b), reproduces the
explicit result (4.5).

In the triple scaling limit of Section 3.1, these diagrammatic rules become
those of the Schwarzian theory, described in [47, 48]. Yet, the rules
described here are exact at all energy scales and for any $q=e^{-\lambda}$,
not just semiclassically.

In the finite $p$ SYK the connected 4-pt function from which the chaos
exponent is extracted is suppressed by $1/N$, reflecting the fact that the
interaction is gravitational. Here the expression is finite, and goes to zero
when $\lambda\rightarrow 0$ supporting the idea that the theory has a finite
Planck scale, controlled by $\lambda$.

Finally, the core of the 4-pt function is the $R$-matrix. In the Schwarzian
case, it is the $6j$ symbol of $SU(1,1)$ [47, 48, 51]. In our case, the
$R$-matrix is the $6j$ symbol of the quantum group $SU_{\sqrt{q}}(1,1)$ [9,
52]. In addition, the chord wavefunctions (2.19) are representation matrix
elements181818A different $q$ deformation of $\mathfrak{sl}_{2}$,
$SL_{q}(2,\mathbb{R})$, is exhibited by Liouville gravity [53, 54, 55]. of
$SU_{\sqrt{q}}(1,1)$, and the vertex (4.7) originates from their Clebsch-
Gordan coefficients [56]. This is the key to the non-commutative geometry
structure of the model, which we turn to now.

##  5 Non-commutative AdS (bulk 2)

Just like groups describe the symmetries of usual geometric spaces, quantum
groups capture the symmetries of non-commutative geometries. So far, we have
seen two appearances of objects from the realm of non-commutative geometry.
First, the $q$-oscillator algebra (2.15). Second, the four point function is
related to the $6j$ symbol of the quantum group $SU_{\hat{q}}(1,1)$, which
parallels a similar role as the Lie group $SU(1,1)$ in the Schwarzian theory.
But the latter also controls the full geometry, so one then wonders whether
$SU_{\hat{q}}(1,1)$ can be related to the full geometry also for DS-SYK, and
in particular whether non-commutative geometry replaces ordinary geometry in
the bulk. The answer seems to be yes, with some guesswork.

The main points of Section 5.1 is that one can construct a set of non-
commutative $AdS_{2}$ spaces as homogeneous spaces under the action of
$SU_{\hat{q}}(1,1)$. The transfer matrix is then a reduction of a Casimir for
this action, and appears as the Hamiltonian of a particle on this non-
commutative space [57, 56]. This generalizes the construction of Schwarzian as
the action of the boundary particle moving on $AdS_{2}$.

For proper disclosure – the picture is not complete. One immediate problem is
that there is more than one $q$-deformed $AdS_{2}$, so in any approach one
needs to add some more assumptions in order the get “the correct” one.
However, the main problem in our opinion is that it is not clear what is the
microscopic origin of any of the non-commutative geometry objects that we will
discuss191919of course still after ensemble average and not realization-by-
realization..

In Section 5.2 we review the generalization of the $q$-oscillator algebra in
the presence of matter chords [42], and highlight a specific subalgebra that
generalizes the global $\mathfrak{su}(1,1)$ symmetry of the Schwarzian theory
when coupled to matter. The semi-classical limit of this algebra is the origin
of the fake temperature of Section 3 [25].

But first, we would like to provide a (brief and non-systematic) definition of
the quantum algebra $\mathcal{U}_{\hat{q}}(\mathfrak{s}\mathfrak{u}(1,1))$ and
its associated quantum group $SU_{\hat{q}}(1,1)$. In order to conform with the
standard notation on quantum groups, we define

${\hat{q}}=q^{1\over 2}=e^{-\frac{1}{2}\lambda}\,.$ (5.1)

The algebra $\mathcal{U}_{\hat{q}}(\mathfrak{s}\mathfrak{u}(1,1))$ is
generated by elements $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$,
$\mathcal{D}$, modulo the following relations:

$\mathcal{A}\mathcal{D}=\mathcal{D}\mathcal{A}=1,\qquad\mathcal{A}\mathcal{B}={\hat{q}}\mathcal{B}\mathcal{A},\qquad\mathcal{A}\mathcal{C}={\hat{q}}^{-1}\mathcal{C}\mathcal{A},\qquad[\mathcal{B},\mathcal{C}]={\mathcal{A}^{2}-\mathcal{D}^{2}\over{\hat{q}}-{\hat{q}}^{-1}}\,.$
(5.2)

The Casimir element, generating the center of this algebra, is given by

$\Omega=\left({{\hat{q}}^{-1}\mathcal{A}^{2}+{\hat{q}}\mathcal{D}^{2}-2\over({\hat{q}}^{-1}-{\hat{q}})^{2}}+\mathcal{B}\mathcal{C}\right)\,.$
(5.3)

In the limit $\hat{q}\to 1$, formally defining
$\mathcal{A}=\hat{q}^{\mathcal{E}}$, the algebra reduces to
$\mathfrak{su}(1,1)$ as $[\mathcal{E},\mathcal{B}]=\mathcal{B}$,
$[\mathcal{E},\mathcal{C}]=-\mathcal{C}$,
$[\mathcal{B},\mathcal{C}]=-2\mathcal{E}$.

One way [56], which will be used below, to write elements of the quantum group
$SU_{\hat{q}}(1,1)$ in terms of these generators is by a Gauss decomposition
using non-commutative coordinates,

$e_{{\hat{q}}^{-2}}^{\gamma\mathcal{C}}\mathcal{A}^{2\phi}e_{{\hat{q}}^{2}}^{\beta\mathcal{B}}\in
SU_{\hat{q}}(1,1)\,,\qquad\left[\gamma,\phi\right]=\frac{\lambda}{2}\gamma\,,\qquad\left[\beta,\phi\right]=\frac{\lambda}{2}\beta\,,\qquad\left[\beta,\gamma\right]=0\,.$
(5.4)

where $e_{q}^{x}$ is the $q$-exponential (A.7) – a $q$-deformation of the
exponential function, motivated by the action of the $q$-derivative
$D_{q}e_{q}^{z}\equiv{e_{q}(qz)-e_{q}(z)\over qz-z}=e_{q}^{z}$ and by
$e_{q}(z/(1-q))\rightarrow e^{z}$.

###  5.1 Non-commutative $AdS_{2}$

In the triple scaling limit the transfer matrix reduces to the Liouville form
of the Schwarzian theory [33]. There are many different formulations of the
latter – from JT gravity on the disk topology, to a BF theory with an
$\mathfrak{sl}_{2}(\mathbb{R})$ gauge algebra on that space. All formulations
reduce to a boundary action, which is that of a particle moving on $AdS_{2}$
(with an effective large imaginary magnetic field).

Since we lack a rigorous analogue for the gravitational or BF path integral
for DS-SYK (see however recent advances in Section 7.1.0.0.1), we aim for a
generalization of the boundary particle action. The key point is that
$AdS_{2}$ is an homogeneous space of $SL(2,\mathbb{R})$, and the Hamiltonian
which controls the motion on it is the Casimir of the group on an appropriate
set of representations. In the context of DS-SYK, the transfer matrix that we
constructed above will turn out to be a Casimir of $SU_{\hat{q}}(1,1)$ on such
a set.

We will briefly go through the construction for the case of ordinary $AdS_{2}$
mainly because we will see the same components in the non-commutative version,
in an almost one to one fashion. The reduction from the motion of the boundary
particles on $AdS_{2}$ to the Schwarzian action is encoded in the formula
$Z(\beta)=\langle X|e^{-\beta H_{AdS_{2}}}|X\rangle$ where $Z(\beta)$ is the
partition function of the SYK model at low energies, or equivalently the
Schwarzian action [58] (see also [47]). On the RHS we have the propagator of
the boundary particle from point X back to X, and $H_{AdS_{2}}$ is its
Hamiltonian (which is a particle moving on $AdS_{2}$ with a large imaginary
magnetic field). $X$ is any point on $AdS_{2}$, and all choices of $X$’s are
gauge equivalent.

In Euclidean Poincare coordinates a particle moving on $AdS_{2}$ with
coordinates $ds^{2}=d\phi^{2}+e^{2\phi}df^{2}$ in the presence of a large
magentic field is given by [47]

$S=\int dt\left({1\over 4}{\dot{\phi}}^{2}+{1\over
4}e^{2\phi}{\dot{f}}^{2}+B{\dot{f}}e^{\phi}\right)\,.$ (5.5)

Introducing the canonical momenta $\pi_{f},\pi_{\phi}$, the Hamiltonian for
the system is

$H_{B}=\pi_{\phi}^{2}+(\pi_{f}e^{-\phi}-B)^{2}\longrightarrow\pi_{\phi}^{2}+e^{-\phi}\,,$
(5.6)

On the right side we have fixed $\pi_{f}$ (which is an invariant of the
motion), shifted $\phi$ and taken the $B\to\infty$ limit, resulting in the
Liouville form of the Schwarzian. The $SL(2,\mathbb{R})$ action on this
$AdS_{2}$ is a composition of spatial translation and a gauge symmetry

$l_{-1}=\pi_{f}\,,\quad l_{0}=f\pi_{f}-\pi_{\phi}\,\quad
l_{1}=f^{2}\pi_{f}-2f\pi_{\phi}-\pi_{f}e^{-2\phi}+2Be^{-\phi}\,,$ (5.7)

and $H$ is just the Casimir of this geometric $SL(2)$. We will next see the
non-commutative versions of these expressions.

####  5.1.0.0.1 Some ad-hoc realizations

There could be many different202020 It is worth mentioning a different non-
commutative space which is the analogue of the Euclidean disk with
rotationally invariant coordinates, $\frac{dz\otimes dz^{*}}{(1-zz^{*})^{2}}$,
which is natural in holography. The non-commutative analog was analyzed in
[59], focusing on wave functions of ordinary particles on this space, but not
on the boundary particle. This would be necessary next step to connect with
the transfer matrix. realizations of non-commutative $AdS_{2}$. We are going
to begin with some heuristics which explains the basic structure and then
switch to the more canonical non-commutative geometry point of view.

Non-commutative spaces are usually defined by deforming the algebra of
functions on the space. So to define a non-commutative $AdS_{2}$ as a
homogeneous space of (5.2), we are interested in deforming functions of two
variables on which 1) we realize this algebra and 2) have an $AdS_{2}$ limit
when $\hat{q}\rightarrow 1$. Sometimes, however, one can get away with simpler
constructions, such as functions on a 2D lattice. The lattice appears here
naturally because ${\cal A}$ is a group like element and not an infinitesimal
generator (it’s roughly $\hat{q}^{\,l_{0}}$), and we expect that ${\cal A}$
will act discretely on a lattice of points. This actually turns all the
generators to ones that act on a lattice of points. If we define the discrete
translation operators $\mathcal{T}$, $\mathcal{R}$,

$\mathcal{T}\ F(\tilde{H},\tilde{z})=F(\hat{q}^{1\over
2}\tilde{H},\tilde{z})\,,\qquad\mathcal{R}\
F(\tilde{H},\tilde{z})=F(\tilde{H},\hat{q}\tilde{z})\,,$ (5.8)

where $F$ is a function on lattice of points
$({\tilde{H}},{\tilde{z}})\in\left\\{(\hat{q}^{m_{1}/2},\pm\hat{q}^{m_{2}})\,|\,m_{1},m_{2}\in\mathbb{Z}\right\\}\equiv\mathbb{R}^{2}_{\hat{q}}\subset\mathbb{R}^{2}\,,$
then the following generators satisfy (5.2) for any integers $a$,
${\tilde{a}}$

$\displaystyle\mathcal{A}$
$\displaystyle=\mathcal{T}\mathcal{R}^{-1}\,,\quad\mathcal{B}=\frac{1}{\tilde{z}}{\mathcal{R}^{a+2}-\mathcal{R}^{a}\over\hat{q}^{a+2}-\hat{q}^{a}}\,,\quad\mathcal{C}={\tilde{z}\over\hat{q}^{-1}-\hat{q}}\left(\mathcal{T}^{-2}\mathcal{R}^{-a}-\mathcal{T}^{2}\mathcal{R}^{-a-2}\right)+i\mu\tilde{H}^{-2}\mathcal{T}^{\tilde{a}}\,,$
(5.9) $\displaystyle\Omega$
$\displaystyle\equiv\left({\hat{q}^{-1}\mathcal{A}^{2}+\hat{q}\mathcal{D}^{2}-2\over(\hat{q}^{-1}-\hat{q})^{2}}+\mathcal{B}\mathcal{C}\right)={\hat{q}^{-1}\mathcal{T}^{-2}+\hat{q}\mathcal{T}^{2}-2\over(\hat{q}^{-1}-\hat{q})^{2}}+{i\mu\tilde{H}^{-2}}\mathcal{T}^{\tilde{a}}\mathcal{B}\,,$

where we have also written the Casimir $\Omega$. In the limit $\hat{q}\to 1$,
the $AdS_{2}$ coordinates are identified as $\tilde{H}=e^{\phi}$ and
$\tilde{z}=f$, and the translation operators formally become
$\mathcal{T}=e^{\lambda\partial_{\phi}}$ and $\mathcal{R}=e^{-\lambda
f\partial_{f}}$, such that $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$
become212121In the limit where $B=i\mu$ is large. The canonical momenta
$\pi_{f}$, $\pi_{\phi}$ act as derivatives $\partial_{f}$, $\partial_{\phi}$.
$\hat{q}^{l_{0}}$, $l_{-1}$, $l_{1}$ of (5.6).

To go from $\Omega$ in the (5.9) to the transfer matrix we mimic the steps
around (5.6). There we fixed $\pi_{f}$, the eigenvalue of $l_{-1}$, which in
(5.9) corresponds to the operator $\mathcal{B}$. Its eigenvectors are some
$q$-deformations of the exponential function222222However, orthogonality and
completeness relations are not known for most values of $a$. that depend on
$a$. For example for $a=0$ these are $e_{{\hat{q}}^{2}}^{c\tilde{z}}$. Once we
digonalize $\mathcal{B}$ we are left with a Casimir which only has
$\mathcal{T}$ and $\tilde{H}$ – i.e., we reduced $\Omega$ to a finite
difference operator on the discrete lattice of ${\tilde{H}}$. By properly
choosing the eigenvalue of $\mathcal{B}$, as well as $\mu$ (which plays the
role of the magnetic field), and also setting ${\tilde{a}}=2$, it is easy to
see that we obtain the DS-SYK transfer matrix (2.13) (after a simple
conjugation) acting on the discrete lattice of ${\tilde{H}}$.

So we can construct realizations of the quantum group which reproduce the
correct transfer matrix and go over to $AdS_{2}$ when $q\rightarrow 1$. In
this approach, however, we had to do some tinkering and
guesswork232323Although it could be that different realizations are actually
the same under complicated conjugation.. Below we will see other types of
constructions but they will also, by and large, suffer from similar (and
partially overlapping) arbitrariness.

####  5.1.0.0.2 Non-commuting coordinates

One way of obtaining (5.5) is to formally start with a Minkowskian $AdS_{3}$
and take $B$ to be the momenta along one of the null coordinates, leaving us
with the other null coordinate, which is $f$, and with the radial coordinate
$\phi$ [47]. So a possible starting point for DS-SYK is to $q$-deform this
$AdS_{3}$, then reduce to a 2D space, and then to a transfer matrix acting on
a 1D lattice. We will denote the commuting $AdS_{3}$ by

$\displaystyle ds^{2}_{\mathbb{H}^{3}}={dH^{2}\over H^{2}}+H^{2}dzdz^{*}\,.$
(5.10)

After the reduction we identify $z\rightarrow f$ and $H\rightarrow e^{\phi}$
to match with (5.5).

$H$, $z$, $z^{*}$ are now promoted to non-commuting variables, and some known
non-commuting spaces that generalize $AdS_{3}$ include [60, 61, 57]

$\displaystyle{\mathbb{H}}_{\hat{q}}^{3}:$ $\displaystyle H=H^{*},\quad
Hz=\hat{q}^{2}zH,\quad z^{*}H=\hat{q}^{2}Hz^{*},\quad
zz^{*}=\hat{q}^{2}z^{*}z+(\hat{q}^{2}-1)(1-H^{-2})\,,$ (5.11)
$\displaystyle{\mathbb{H}}_{\hat{q},\kappa}^{3}:$ $\displaystyle
zH=\kappa^{-1}Hz,\quad zz^{*}=az^{*}z-bH^{-2},\quad
a={\kappa^{2}\over\hat{q}^{2}}\,,\quad
b={\kappa\over\hat{q}^{2}}(1-{\hat{q}^{2}})\,.$

The next step is to define functions on these spaces. The construction starts
with polynomials of the non-commuting variables, and then extend it to a
richer family of functions. After the deformation the main subtlety is to deal
with the ordering of the non-commuting coordinates, but there will be a
natural action of $SU_{\hat{q}}(1,1)$ on these functions. We then carry out
the reduction to an $AdS_{2}$ by diagonalizing the appropriate generator
giving us a non-commutative $AdS_{2,\hat{q}}$. So far all the non-commutative
objects are now in correspondence to those of ordinary $AdS_{2}$. We then
further reduce to a 1D system by diagonlizing the non-commutative analogue of
$l_{-1}$ in (5.7), as before.

In slightly more detail: a convenient set of normal ordered monomials is

$v^{m,r,n}\equiv(\hat{q}^{1\over 2}z^{*})^{m}H^{r}(\hat{q}^{1\over
2}z)^{n}\,.$ (5.12)

The reduction to $AdS_{2,\hat{q}}$, which means for us functions of $H$ and
$z$, is done by the ansatz

$F_{\mu}(z^{*},H,z)\colonequals e_{\hat{q}^{2}}(i\mu\hat{q}^{1\over
2}z^{*})\cdot F_{\mu}(H,z),\quad
F_{\mu}(H,z)=\sum_{r,n}a_{r+1,n}H^{r}(\hat{q}^{1\over 2}z)^{n}\,,$ (5.13)

where we kept the ordering of the symbols in $F_{\mu}$, and the $a_{r+1,n}$
are some constants.

The action of the quantum group on the $q$-deformed $AdS_{3}$ now descends to
an action on $F_{\mu}$, so these $(H,z)$ have full claim to be considered a
$AdS_{2,\hat{q}}$ $q$-deformed homogeneous space. In fact, the generators that
one obtains in this approach are very similar to (5.9), except that now the
symbols are non-commuting variables and one needs to be careful about their
ordering (for complete details see [57]). The subsequent steps are also very
similar. We replace the general $z$ dependence by a $q$-exponential, which is
the same as diagonalizing $\mathcal{B}$, and obtain an action on a discrete
lattice of $H$, again with some free parameters that we have to choose
correctly. If we now go to the space of eigenvalues of $H$ we get the transfer
matrix acting on a discrete lattice.

####  5.1.0.0.3 Particle on a group manifold

Another way of getting a particle on non-commutative $AdS_{3}$ is to start
with a particle on the quantum group $SU_{\hat{q}}(1,1)$ [56]. This is more
compact than the constructions above, and an easier reduction from $AdS_{3}$
to $AdS_{2}$, but it only gives one of the options, and still some guesswork
is needed. The particle moves in a 6D phase-space with coordinates $\gamma$,
$\phi$, $\beta$, $\pi_{\gamma}$, $\pi_{\phi}$, $\pi_{\beta}$ and equipped with
the symplectic two-form

$\omega=d\gamma\wedge d\pi_{\gamma}+d\beta\wedge
d\pi_{\beta}+d\left(\phi+i\lambda\left(\gamma\pi_{\gamma}+\beta\pi_{\beta}\right)\right)\wedge
d\pi_{\phi}\,,$ (5.14)

i.e. the Darboux coordinates are
$\varphi=\phi+i\lambda\left(\gamma\pi_{\gamma}+\beta\pi_{\beta}\right)$,
$\pi_{\varphi}=\pi_{\phi}$. Upon quantization, these satisfy the commutation
relations (5.4) for the non-commutative coordinates on
$SU_{\hat{q}}\left(1,1\right)$.

In242424Their $q$ is our $\hat{q}$, and therfore we rescaled their $\phi$ by a
factor of $2$ and their $\pi_{\phi}$ by a factor of $1/2$. [56] the authors
constructed left- and right-regular representations of the algebra
$\mathcal{U}_{\hat{q}}(\mathfrak{su}(1,1))$ that act on this space, and a
Hamiltonian that commutes with them

$H=-\frac{1}{\lambda^{2}}\left[e^{-i\lambda\pi_{\phi}}+e^{i\lambda\pi_{\phi}}+e^{-\phi}e^{i\lambda\pi_{\phi}}\frac{\left(e^{-i\lambda\beta\pi_{\beta}}-1\right)\left(e^{-i\lambda\gamma\pi_{\gamma}}-1\right)}{\beta\gamma}\right]\,.$
(5.15)

which is (5.9) for the specific values of $a=0$ and $\tilde{a}=2$. In the
$\lambda\to 0$ limit we reproduce (5.6) after treating $\pi_{\gamma}$ as the
magnetic field $B\to\infty$, $\pi_{\beta}$ as $\pi_{f}$, and shifting $\phi$.

In the quantization of JT gravity a similar picture arises, where additional
constraints coming from the boundary conditions in the gravitational theory
are imposed. [56] suggested their analog would be

$\frac{1}{\gamma}\left(e^{i\lambda\gamma\pi_{\gamma}}-1\right)e^{\frac{i}{2}\lambda\gamma\pi_{\gamma}}e^{-\frac{i}{2}\lambda\pi_{\varphi}}=i\,,\quad\frac{1}{\beta}\left(e^{-\frac{i}{2}\lambda\beta\pi_{\beta}}-e^{\frac{i}{2}\lambda\beta\pi_{\beta}}\right)e^{\frac{i}{2}\lambda\pi_{\varphi}}=-i\,.$
(5.16)

After a shift of $\phi$, the Hamiltonian turns into

$H=-\frac{1}{\lambda^{2}}\left[\left(1-e^{-\phi}\right)e^{i\lambda\pi_{\phi}}+e^{-i\lambda\pi_{\phi}}\right]\,,$
(5.17)

exactly252525Up to an overall factor, which could be absorbed into the
definition of the dimensionful parameter. as the chord Hamiltonian (3.3). A
generalization of the path integral form of the Schwarzian, named the
$q$-Schwarzian [56], can be found as a phase space path integral by similar
constraints.

###  5.2 The chord algebra

Another avatar of the algebraic structure of the theory is the $q$-oscillator
algebra of the chords, (2.15). This algebraic structure only becomes richer
once we consider not only two sided states that contain Hamiltonian chords, as
we did in Section 2, but also matter chords [24, 25]. The advantage of this is
that the $q$-deformed SL(2) manifests itself as a global symmetry. In the
semi-classical limit, it naturally acts on the “fake disk” geometries, and
implies a sub-maximal chaos exponent.

Let’s start with the (two-sided) Hilbert space containing a single particle,
which corresponds to an operator of dimension $\Delta$, such that an
intersection of a Hamiltonian chord with the matter chord results in a factor
of $q^{\Delta}$. We can build all chord diagrams that contribute to the
thermal 2-pt function step-by-step, as in the TFD analysis of Section 2, by
evolving time on the left and on the right. We start by cutting the diagram
around the matter chord, Figure 9. At each step we open (or close) a
Hamiltonian chord from one of the boundaries,

Figure 9: Cutting the chord diagrams for the 1-particle Transfer matrix.

and state of the system is given by the number of chords that are still open
to the left ($n$) and to the right ($m$) of the matter chord, $\ket{n,m}$. As
before, we choose a convention where chords intersect when they close. We can
summarize the evolution by two transfer matrices, describing chords that open
and close from the left (right) boundary262626Under a convention where chords
intersect when they open, related to ours by conjugation, the 1-particle
eigenstates are the bi-variate $q$-Hermite polynomials, [62, 63], up to this
conjugation issue. [24],

$\displaystyle T_{L}$ $\displaystyle=a^{\dagger}_{L}+a_{L}\,,$ $\displaystyle
T_{R}$ $\displaystyle=a^{\dagger}_{R}+a_{R}\,,$ (5.18) $\displaystyle
a^{\dagger}_{L}\ket{n,m}$ $\displaystyle=\ket{n+1,m}\,,$ $\displaystyle
a_{L}\ket{n,m}$
$\displaystyle=q^{\Delta+n}[m]_{q}\ket{n,m-1}+[n]_{q}\ket{n-1,m}\,,$ (5.19)
$\displaystyle a^{\dagger}_{R}\ket{n,m}$ $\displaystyle=\ket{n,m+1}\,,$
$\displaystyle a_{R}\ket{n,m}$
$\displaystyle=q^{\Delta+m}[n]_{q}\ket{n-1,m}+[m]_{q}\ket{n,m-1}\,.$ (5.20)

As with the TFD state, we can view
$e^{\frac{1}{2}(\beta_{L}T_{L}+\beta_{R}T_{R})}|0,0\rangle$ as generating all
“bottom halves” of diagrams with a single matter chord. In order to reproduce
the two point function from an inner product of two such states, we will need
it to account for the intersections of the chords that leave these two halves,
see Figure 9,

$\braket{n_{2},m_{2}}{n_{1},m_{1}}=\delta_{n_{2}+m_{2},n_{1}+m_{1}}\sum_{\text{pairings}}q^{\text{no.
of $H$-$H$ intersections}}q^{\Delta\times\text{ no. of $H$-$M$
intersections}}\,.$ (5.21)

The inner product272727A closed formula can be found in [25]. allows
$H$-chords to cross from one side of the matter chord to the other, and the
basis $\ket{n,m}$ is not orthogonal. Yet, the creation and annihilation
operators $a_{L/R}$ and $a^{\dagger}_{L/R}$ are Hermitian conjugates, and the
inner product is positive definite [21, 64, 65].

States with different $n+m$ are orthogonal, though, and so we define an
operator that measures this number, the _size operator_ , $\hat{n}$. Together
with the chord creation and annihilation operators it generates the _chord
algebra_ [25],

$\displaystyle\hat{n}\ket{n,m}$ $\displaystyle=n+\Delta+m\,,$
$\displaystyle[\hat{n},a_{L,R}]$ $\displaystyle=-a_{L,R}\,,$
$\displaystyle[\hat{n},a^{\dagger}_{L,R}]$
$\displaystyle=a^{\dagger}_{L,R}\,,$ (5.22)
$\displaystyle[a_{L},a_{R}^{\dagger}]$
$\displaystyle=[a_{R},a_{L}^{\dagger}]=q^{\hat{n}}\,,$
$\displaystyle[a_{L},a_{R}]$
$\displaystyle=[a_{L}^{\dagger},a_{R}^{\dagger}]=0\,,$
$\displaystyle[a_{L},a_{L}^{\dagger}]_{q}$
$\displaystyle=[a_{R},a_{R}^{\dagger}]_{q}=1\,.$ (5.23)

which generalizes the $q$-oscillator algebra (2.15). In fact, the latter could
be thought of as a short representation of the chord algebra in which
$a_{L}=a_{R}$ and $a_{L}^{\dagger}=a_{R}^{\dagger}$, and $\Delta=0$. The only
[25] other highest weight irrep for the algebra is the one with a single
matter chord discussed above. There are, however, reducible representations in
which there are several matter chords, but only $H$-chords open and close
dynamically.

The commutation relations imply $[T_{L},T_{R}]=0$ and the operatorial equation
[25]

$\left[T_{L},\left[T_{R},\hat{n}\right]\right]=-2q^{\hat{n}}\quad\implies\quad\partial_{\tau_{1}}\partial_{\tau_{2}}\hat{n}=-2q^{\,\hat{n}}\,,$
(5.24)

when the evolution along the two boundaries is generated by the respective
transfer matrix. This generalizes the evolution using the Liouville
Hamiltonian (3.4) to finite $\lambda$, when $\ell=\lambda\hat{n}$.

A curious feature is a sub-algebra [25], $U(J)$, that commutes with $\hat{n}$
and is generated by

$J_{ij}=a^{\dagger}_{i}a_{j}-[\hat{n}]_{q}\,,\qquad i,j\in\\{L,R\\}\,,$ (5.25)

In Appendix C of [25] it was shown that it generates a proper subalgebra of
$\mathcal{U}_{\hat{q}}(\mathfrak{sl}_{2})$. The $U(J)$ algebra, together with
$c=q^{\hat{n}/2}$, can be used to define the generators

$B=\frac{1}{2c}\left(J_{LL}-J_{RR}\right)\,,\quad
E=-\frac{1}{2c}\left(J_{LL}+J_{RR}\right)\,,\quad
P=\frac{i}{2}\left(J_{LR}-J_{RL}\right)=iq[B,E]\,.$ (5.26)

When $\lambda\to 0$, (3.11) implies that to leading order $c=\frac{\pi
v}{\beta\mathbb{J}}=\frac{\pi}{\beta_{\text{fake}}\cdot\mathbb{J}}$ is a