# Understanding Higher-Spin Gravity through Bilocal Holography for Free
Fermions

Tomer Solberg

###### Abstract

We consider a simple theory of $N$ free fermions in $d$ dimensions with
$O\left(N\right)$ or $U\left(N\right)$ symmetry. The singlet sector of this
theory is expected from holography to be dual to the notoriously complicated
Vasiliev gravity. By defining bilocal fields we find an explicit holographic
mapping between the two theories, similar to what was done previously for
scalars, which allows us to construct a quantum version of Vasiliev gravity,
including loop ($=1/N$) corrections. Further more, such a mapping establishes
a non-supersymmetric instance of the AdS/CFT duality.

Department of Particle Physics and Astrophysics, Weizmann Institute of
Science, Rehovot 76100, Israel

  * E-mail<EMAIL_ADDRESS>
##  1 Introduction

It has long been known that some $d$-dimensional quantum field theories are
dual to theories of gravity in $\left(d+1\right)$-dimensions [1]. This kind of
duality is known as holography, and its most known and studied example is the
AdS/CFT correspondence, in which the gravitational theory (for example a
string theory) is living in anti-de Sitter (AdS) spacetime, and it is dual to
a conformal field theory on a space that is conformally equivalent to the
boundary of AdS [3]. This is especially useful in the case where the field
theory has some gauge group with a fundamental representation of size $N$,
where $N$ is taken to be large. In the large $N$ limit, the dual gravitational
theory is then weakly coupled, with a coupling proportional to $1/N$ [2].
Generally speaking, the AdS/CFT correspondence is a weak/strong duality. If we
focus on the large $N$ limit then the gravitational theories are weakly
coupled. However, these gravitational theories can look approximately local,
in the sense that higher derivative corrections to their interactions are
small at the scale of the curvature of space-time, only when the field
theories are strongly coupled (in the sense of having a large ‘t Hooft
coupling). This is the regime where most of the research on holography is
focused, since it can be used to relate standard, almost classical,
gravitational theories, to strongly coupled field theories (some of which are
interesting for applications like the study of QCD).

In this work we study the opposite regime – the extreme limit where we have a
free field theory on the field theory side of the equivalence. Somewhat
surprisingly, such theories are still equivalent (in an appropriate sense) to
gravitational theories, which can be weakly coupled if the free field theory
has a large number of degrees of freedom. The first such example of the
AdS/CFT duality was suggested by Klebanov and Polyakov [4], who looked at the
simple theory of $N$ real free fields
$\mathcal{L}=\frac{1}{2}\left(\partial\phi_{i}\right)^{2}$ which has an
$O\left(N\right)$ symmetry, and infinitely many conserved currents which are
$O\left(N\right)$ singlets

$J_{\left(\mu_{1}\cdots\mu_{s}\right)}=\phi_{i}\left(\overleftrightarrow{\partial}_{\mu_{1}}\cdots\overleftrightarrow{\partial}_{\mu_{s}}\right)\phi_{i},$
(1.1)

and observed that these are in one-to-one correspondence with with the
spectrum of massless higher-spin fields in the minimal bosonic theory in AdS,
which is known as Vasiliev gravity [6]. The latter is a generalization of
Einstein’s gravity in AdS space, which includes a tower of massless degrees of
freedom of spins $s=1,2,3,4,\dots$ (for $O\left(N\right)$ models only even
spins appear, while all integer spins appear for $U\left(N\right)$ models),
and is considered to be immensely complicated. In this paper we discuss
similarly the fermionic case, first studied by Sezgin and Sundell [7]. We
continue the work by Aharony, Chester and Urbach [8], which have explicitly
found this mapping for scalar theories, and generalize it from free scalars to
free fermions. This case is more complicated because of the extra spinor
indices of the fermions.

###  1.1 The story with scalars so far

We begin by reviewing the explicit mapping for the scalar case [8]. Our goal
is to try and translate explicitly from the CFT side to the bulk in order to
build a quantum model of Vasiliev gravity. To achieve this, we follow a recent
proposal made by Jevicki et al. [5] for a way to analyze this duality using
bilocal fields. The bilocal fields are

$G\left(x_{1},x_{2}\right)=\frac{1}{N}\sum_{i}\phi_{i}\left(x_{1}\right)\phi_{i}\left(x_{2}\right)$
(1.2)

and they contain all information on $O\left(N\right)$-invariant operators,
including the local operators
$\mathcal{O}_{s}=\phi_{i}\underset{{}_{s}}{\underbrace{\partial\cdots\partial}}\phi_{i}$
which appear in the expansion of $G$ as $x_{2}\rightarrow x_{1}$. We can
rewrite the partition function in terms of the bilocal fields

$\intop\mathcal{D}\phi_{i}e^{-\intop
d^{d}x\frac{1}{2}\left(\partial\phi_{i}\right)^{2}}\rightarrow\intop\mathcal{D}Ge^{-S\left(G\right)}.$
(1.3)

To achieve this, we discretize space-time to $V=M^{d}$ sites, such that
$\phi_{i}\left(x\right)$ is a length $V$ vector and
$G\left(x_{1},x_{2}\right)$ is a $V\times V$ matrix. We work in the large $N$
limit in which we also assume $N\geq V$. We introduce the bilocal field matrix
multiplication which is defined by

$\left(GH\right)\left(x_{1},x_{2}\right)=\intop
d^{d}x_{3}G\left(x_{1},x_{3}\right)H\left(x_{3},x_{2}\right).$ (1.4)

We then manipulate the path integral in the following way

$\displaystyle\int\mathcal{D}\phi_{i}e^{-S_{0}\left(G\left(\phi\right)\right)}$
$\displaystyle=\int\mathcal{D}G\mathcal{D}\phi_{i}e^{-S_{0}\left(G\right)}\delta\left(G-\frac{1}{N}\sum_{i}\phi_{i}\phi_{i}\right)$
$\displaystyle=\int\mathcal{D}Ge^{-S_{0}\left(G\right)}\int\mathcal{D}\Sigma\mathcal{D}\phi_{i}e^{\mbox{Tr}\left(i\frac{N}{2}\Sigma
G-\frac{i}{2}\phi_{i}\Sigma\phi_{i}\right)}$
$\displaystyle=\int\mathcal{D}Ge^{-S_{0}\left(G\right)}\int\mathcal{D}\Sigma\left(\det\Sigma\right)^{-N}e^{\mbox{Tr}\left(i\frac{N}{2}\Sigma
G\right)}$ (1.5)

where we have introduced another bilocal field $\Sigma$. The integral over
$\Sigma$ can be carried out by a change of variables $\tilde{\Sigma}=\Sigma G$
so that $\mathcal{D}\Sigma=\left(\det G\right)^{-V}\mathcal{D}\tilde{\Sigma}$,
and so up to a constant function of $N,V$

$\int\mathcal{D}\phi_{i}e^{-S_{0}\left(G\left(\phi\right)\right)}=C_{N}\int\mathcal{D}Ge^{-S_{0}\left(G\right)}\left(\det
G\right)^{N-V}.$ (1.6)

Plugging in $S_{0}\left(G\right)$ which is just the original action of
$\phi_{i}$ expressed in terms of $G$, we get

$S\left(G\right)=\frac{1}{2}\intop
d^{d}x\left[-N\left(\partial_{x}^{2}G\left(x,x^{\prime}\right)\right)_{x^{\prime}=x}\right]-\left(N-V\right)\mbox{Tr}\log
G.$ (1.7)

One should note that $V$ is an arbitrary regulator. For other regulators $V$
will be replaced by an appropriate cut-off dependent constant, which can be
viewed as a one-loop renormalization of an action that is proportional to $N$.
Thus, the loop expansion of (1.7) is an expansion in powers of $1/N$. This
action can be expanded around a simple saddle point

$\displaystyle G_{0}\left(x_{1},x_{2}\right)$
$\displaystyle=\frac{1}{N}\sum_{i}\left\langle\phi_{i}\left(x_{1}\right)\phi_{i}\left(x_{2}\right)\right\rangle,$
(1.8) $\displaystyle G$ $\displaystyle=G_{0}+\frac{1}{\sqrt{N}}\eta,$ (1.9)

so that we can write the action for $\eta$. This action will have a Gaussian
part as well as perturbation terms of the form $\eta^{n}$ for any $n>2$ due to
the $\mbox{Tr}\log G$ term in $S\left[G\right]$. This will give us an
intricate perturbation theory in the parameter $1/N$ which we can work out.
Our advantage is that we know what the final results must be, as this is
equivalent to the free theory which we can work out to any order. For example
in $d=3$ we know that $G_{0}$ and the two point function in $\eta$ are exactly
(denoting $x_{ij}=x_{i}-x_{j}$)

$\displaystyle G_{0}\left(x_{1},x_{2}\right)$
$\displaystyle\sim\frac{1}{\left|x_{12}\right|}$ (1.10)
$\displaystyle\left\langle\eta\left(x_{1},x_{2}\right)\eta\left(x_{3},x_{4}\right)\right\rangle$
$\displaystyle\sim\frac{1}{\left|x_{13}\right|\left|x_{24}\right|}+\frac{1}{\left|x_{14}\right|\left|x_{23}\right|}$
(1.11)

with no further dependence on $N$.

This means that all the loop diagrams in the $\eta$ theory have to cancel out,
which has indeed been shown. The $\eta$ field can then be expanded in the
basis of the conformal 3-pt. functions

$\displaystyle\left\langle\phi\left(x_{1}\right)\phi\left(x_{2}\right)\mathcal{O}_{\Delta}^{\mu_{1}\dots\mu_{J}}\left(x_{3}\right)\right\rangle$
$\displaystyle=\frac{Z^{\mu_{1}}\cdots
Z^{\mu_{J}}-\text{traces}}{\left|x_{12}\right|^{2\Delta_{0}-\Delta}\left|x_{13}\right|^{\Delta}\left|x_{23}\right|^{\Delta}}$
(1.12) $\displaystyle Z^{\mu}$
$\displaystyle=\frac{\left|x_{13}\right|\left|x_{23}\right|}{\left|x_{12}\right|}\left(\frac{x_{13}^{\mu}}{x_{13}^{2}}-\frac{x_{23}^{\mu}}{x_{23}^{2}}\right)$
(1.13)

where $\Delta_{0}=\frac{d-2}{2}$ is the conformal weight of a free field
$\phi$. The expansion is [9]

$\eta\left(x_{1},x_{2}\right)=\sum_{J=0}^{\infty}\intop_{C_{J}}\frac{d\Delta}{2\pi
i}\intop\frac{d^{d}y}{J!\left(\frac{d}{2}-1\right)_{J}}c_{\Delta,J}^{\mu_{1}\dots\mu_{J}}\left(y\right)\left\langle\phi\left(x_{1}\right)\phi\left(x_{2}\right)\mathcal{O}_{\mu_{1}\dots\mu_{J}}^{\Delta,J}\left(y\right)\right\rangle$
(1.14)

where the $c_{\Delta,J}^{\mu_{1}\dots\mu_{J}}\left(y\right)$ are the
coefficients of the decomposition, and the contour $C_{J}$ in the $\Delta$
integral runs over the principal series $\text{Re}(\Delta)=d/2$, as well as
around possible poles in the discrete series when $d<4$. Using this expansion,
we can write the quadratic action using the action of the bilocal Laplacian
(which appears in the quadratic action for $\eta$),

$\nabla_{1}^{2}\nabla_{2}^{2}\left\langle\phi\left(x_{1}\right)\phi\left(x_{2}\right)\mathcal{O}_{\Delta,J}\left(y\right)\right\rangle=\lambda_{\Delta,J}\left\langle\tilde{\phi}\left(x_{1}\right)\tilde{\phi}\left(x_{2}\right)\mathcal{O}_{\Delta,J}\left(y\right)\right\rangle,$
(1.15)

where $\tilde{\phi}$ denotes the shadow transform of $\phi$ with conformal
weight $\tilde{\Delta}_{0}\equiv d-\Delta_{0}$, and

$\displaystyle\lambda_{\Delta,J}$
$\displaystyle=\left(M_{\Delta,J}^{2}-M_{d+J,J}^{2}\right)\left(M_{\Delta,J}^{2}-M_{d+J-2,J}^{2}\right),$
(1.16) $\displaystyle M_{\Delta,J}^{2}$
$\displaystyle=\Delta\left(\Delta-d\right)-J.$ (1.17)

One can write a similar expansion of transverse traceless bulk fields living
on a fixed AdS background $\Phi_{J}\left(X,S\right)$, where $X_{a}$ is a bulk
coordinate and $S^{a}$ is an auxiliary null vector variable keeping track of
spin indices (this notation is explained in detail in appendix A), using the
bulk-to-boundary propagators of massive fields on $AdS$ with spin $J$

$G_{\Delta,J}\left(X_{1},X_{2};S_{1},S_{2}\right)=\frac{\left(X_{12}\left(S_{1}\cdot
S_{2}\right)+2\left(S_{1}\cdot X_{2}\right)\left(S_{2}\cdot
X_{1}\right)\right)^{J}}{X_{12}^{\Delta+J}}.$ (1.18)

Here $\left(X_{2},S_{2}\right)$ is a coordinate on the boundary, given in bulk
notation, and $X_{12}=-2X_{1}\cdot X_{2}$. These are unique solutions to the
$AdS$ Laplace equation

$\nabla_{X_{1}}^{2}G_{\Delta,J}\left(X_{1},X_{2};S_{1},S_{2}\right)=M_{\Delta,J}^{2}G_{\Delta,J}\left(X_{1},X_{2};S_{1},S_{2}\right),$
(1.19)

which satisfy the transversality condition

$\nabla_{X_{1}}\cdot
K_{S_{1}}G_{\Delta,J}\left(X_{1},X_{2};S_{1},S_{2}\right)=0,$ (1.20)

where

$\displaystyle K_{S,a}$
$\displaystyle=\left(\frac{d-1}{2}+S\cdot\frac{\partial}{\partial
S}\right)\frac{\partial}{\partial S^{a}},$ (1.21)

and the boundary condition (as $\left(X_{1},S_{1}\right)$ approaches the
boundary)

$G_{\Delta,J}\left(\frac{1}{z}X_{1},X_{2};\frac{1}{z}S_{1},S_{2}\right)=z^{\Delta-J}\left\langle\mathcal{O}\left(X_{2},S_{2}\right)\mathcal{O}\left(X_{1},S_{1}\right)\right\rangle+z^{d-\Delta-J}S_{B}^{\Delta,J}\delta\left(X_{1},X_{2}\right)\left(S_{1}\cdot
S_{2}\right)^{J}+\dots$ (1.22)

Here we used an AdS metric $ds^{2}=\left(dx_{\mu}^{2}+dz^{2}\right)/z^{2}$ and
denoted the bulk shadow coefficient

$S_{B}^{\Delta,J}=\frac{\pi^{d/2}\Gamma\left(\Delta-d/2\right)}{\left(J+\Delta-1\right)\Gamma\left(\Delta-1\right)}.$
(1.23)

The expansion is given by

$\Phi_{J}\left(X,S\right)=\intop_{P.S.}\frac{d\Delta}{2\pi
i}\intop\frac{dY}{J!\left(\frac{d}{2}-1\right)_{J}}c_{\Delta,J}^{bulk}\left(Y,D_{Z}\right)G_{\Delta,J}\left(X,Y;S,Z\right).$
(1.24)

By writing the CFT coefficients
$c_{\Delta,J}^{\mu_{1}\dots\mu_{J}}\left(x\right)$ in embedding space
coordinates $c_{\Delta,J}\left(X,S\right)$, we can find an explicit mapping
between the boundary and bulk theories through

$c_{\Delta,J}^{bulk}\left(X,S\right)=f_{\Delta,J}c_{\Delta,J}\left(X,S\right)$
(1.25)

for some arbitrary (up to certain consistency constraints) normalization
$f_{\Delta,J}$. Plugging this in gives us a linear mapping between the bilocal
and bulk fields

$\displaystyle\Phi_{J}\left(X,W\right)$ $\displaystyle=\intop
d^{d}x_{1}d^{d}x_{2}\mathcal{M}_{J}\left(X,W;x_{1},x_{2}\right)\eta\left(x_{1},x_{2}\right)$
(1.26) $\displaystyle\eta\left(x_{1},x_{2}\right)$
$\displaystyle=\sum_{J}\intop
dX\mathcal{M}_{J}^{-1}\left(x_{1},x_{2};X,K_{W}\right)\Phi_{J}\left(X,W\right),$
(1.27)

The bulk quadratic action is then (for a specific choice of $f_{\Delta,J}$)

$S^{\left(2\right)}\left[\Phi_{J}\right]=\sum_{J}\frac{\pi^{d/2}\Gamma\left(J+1\right)}{2^{J}\Gamma\left(J+\frac{d}{2}\right)}\intop\frac{dX}{\left(\frac{d-1}{2}\right)_{J}J!}\Phi_{J}\left(X,K_{W}\right)\left(\nabla_{X}^{2}-M_{d+J-2,J}^{2}\right)\left(\nabla_{X}^{2}-M_{d+J,J}^{2}\right)\Phi_{J}\left(X,W\right),$
(1.28)

and indeed the full bilocal action can be translated into a full (non-local)
bulk action which is an explicit quantum generalization of Vasiliev gravity.

##  2 Bi-local holography with fermions

###  2.1 Bilocal fermion action

We start in $d\geq 2$ dimensions, with $N$ fermion fields $\psi_{\alpha}^{i}$
with $i=1,\dots,N$ and $\alpha$ is a spinor index (assumed to be Dirac for
now). We have a global $U\left(N\right)$ symmetry, and all singlets are given
by the bilocal fields

$\sigma_{\alpha\beta}\left(x_{1},x_{2}\right)=\frac{1}{N}\overline{\psi}_{\alpha}^{i}\left(x_{1}\right)\psi_{\beta}^{i}\left(x_{2}\right).$
(2.1)

Note that for Majorana fermions we would have $O\left(N\right)$ symmetry. The
free fermion Lagrangian can be expressed in terms of $\sigma$ as

$\overline{\psi}_{\alpha}^{i}\gamma_{\alpha\beta}^{\mu}\partial_{\mu}\psi_{\beta}^{i}\left(x\right)=\left.N\gamma_{\alpha\beta}^{\mu}\partial_{\mu}^{\prime}\sigma_{\alpha\beta}\left(x,x^{\prime}\right)\right|_{x^{\prime}\rightarrow
x}=\left.N\text{tr}\left(\cancel{\partial}\sigma\right)\right|_{x^{\prime}\rightarrow
x}$ (2.2)

where the trace is taken over spinor indices. The path integral is

$\displaystyle Z$
$\displaystyle=\intop\mathcal{D}\overline{\psi}\mathcal{D}\psi
e^{-S\left(\overline{\psi},\psi\right)}$ (2.3)
$\displaystyle=\intop\mathcal{D}\sigma\mathcal{D}\overline{\psi}\mathcal{D}\psi
e^{-S_{0}\left(\sigma\right)}\prod_{x_{1},x_{2},\alpha,\beta}\delta\left(\sigma_{\alpha\beta}\left(x_{1},x_{2}\right)-\frac{1}{N}\overline{\psi}_{\alpha}^{i}\left(x_{1}\right)\psi_{\beta}^{i}\left(x_{2}\right)\right)$
(2.4) $\displaystyle=\intop\mathcal{D}\sigma
e^{-S_{0}\left(\sigma\right)}\intop\mathcal{D}\Sigma\mathcal{D}\overline{\psi}\mathcal{D}\psi
e^{i\mbox{Tr}\left(\Sigma\sigma-\frac{1}{N}\overline{\psi}^{i}\Sigma\psi^{i}\right)}$
(2.5) $\displaystyle=\intop\mathcal{D}\sigma
e^{-S_{0}\left(\sigma\right)}\intop\mathcal{D}\Sigma\left(\det\Sigma\right)^{N}e^{i\mbox{Tr}\left(\Sigma\sigma\right)},$
(2.6)

where $\Sigma$ is now a matrix both in position space and in spinor space, and
the trace is taken over both types of indices. Now define
$\tilde{\Sigma}=\Sigma\sigma$ so that
$\mathcal{D}\Sigma=\mathcal{D}\tilde{\Sigma}\left|\text{det}\sigma\right|^{-K}$
with $K$ the number of (spacetime+spinor) DOF of $\sigma$. Integrating over
$\tilde{\Sigma}$ we get

$Z=\\#\intop\mathcal{D}\sigma
e^{-S_{0}\left(\sigma\right)}\left|\det\sigma\right|^{-\left(N+K\right)}=\\#\intop\mathcal{D}\sigma
e^{-S_{0}\left(\sigma\right)-\left(N+K\right)\mbox{Tr}\log\sigma}$ (2.7)

where the Trace is taken over both spacetime and spinor indices. We will
ignore the $K$ factor as it can be removed with proper counter terms as we did
in the bosonic case. We can now expand $\sigma=\sigma_{0}+\rho/\sqrt{N}$ where
$\sigma_{0}=\left\langle\overline{\psi}^{i}\psi^{i}\right\rangle/N$ so that we
get $S\left(\rho\right)=S_{quad}+S_{int}$ where the quadratic action is

$S_{quad}=\frac{1}{2}\mbox{Tr}\left(\sigma_{0}^{-1}\rho\right)^{2}=\frac{1}{2}\mbox{Tr}\left(\cancel{\partial}\rho\right)^{2}=-\frac{1}{2}\intop
d^{d}x_{1}d^{d}x_{2}\rho_{\gamma\delta}\left(x_{1},x_{2}\right)\cancel{\partial}_{1\beta}^{\gamma}\cancel{\partial}_{2\alpha}^{\delta}\rho^{\alpha\beta}\left(x_{1},x_{2}\right).$
(2.8)

The 2-pt. correlation function of the free fermions is given by

$\displaystyle\left\langle\psi\left(x\right)\overline{\psi}\left(0\right)\right\rangle_{\alpha\beta}$
$\displaystyle=\intop\frac{d^{d}p}{\left(2\pi\right)^{d}}\frac{i\cancel{p}_{\alpha\beta}}{p^{2}}e^{-ip\cdot
x}=-\cancel{\partial}\intop\frac{d^{d}p}{\left(2\pi\right)^{d}}\frac{1}{p^{2}}e^{-ip\cdot
x}$ (2.9)
$\displaystyle=\cancel{\partial}i\frac{\Gamma\left(\frac{d}{2}-1\right)}{4\pi^{d/2}}\left(x^{2}\right)^{-\frac{d-2}{2}}=-i\gamma_{\alpha\beta}^{\mu}\frac{\Gamma\left(\frac{d}{2}\right)}{2\pi^{d/2}}x_{\mu}\left(x^{2}\right)^{-d/2}.$
(2.10)

In order to make the conformal symmetry explicit at the level of the path
integral, we would like to use a basis that diagonalizes the Cartan subalgebra
of the conformal algebra. In a fermion mean field theory (MFT),
$\rho\left(x_{1},x_{2}\right)$ is in the tensor product of two representations
with a fermion primary of dimension $\Delta_{\psi}$ (for free fermions
$\Delta_{\psi}=\frac{d-1}{2}$). Because we want to decompose it into
irreducible representations of the conformal group, the basis is by definition
the Clebsch-Gordan coefficients of the conformal group. By known harmonic
analysis [9], the coefficients are exactly three-point functions (in some
arbitrary normalization), namely, for a representation with a primary of spin
$J$ (for $d\geq 4$, we may have to use to quantum numbers
$\left(\ell,\overline{\ell}\right)$ for the spin representations, but we soon
restrict our attention to $d=3$ in which a single quantum number is
sufficient) and dimension $\Delta$ they are

$\left\langle\psi\left(x_{1}\right)\overline{\psi}\left(x_{2}\right)O^{\Delta,J}\left(x_{3}\right)\right\rangle.$
(2.11)

These have a few complexities that were absent in the scalar case:

  1. 1.

In the scalar case we only had 3-pt. functions in which the scalars coupled to
symmetric traceless tensors (in $d=4$ these are $\left(\ell,\ell\right)$).
Fermions in $d\geq 4$ can couple also to other operators, for example in $d=4$
they can couple to $\left(\ell,\ell+2\right),\left(\ell+2,\ell\right)$.

  2. 2.

In the scalar case we had two options for the scalar representations \- real
or complex. Fermions can come in a few different representations, such as
Dirac, Majorana, (real or complex) Weyl. The different representations (in
addition to Dirac fermions) that are allowed also depend on the spacetime
dimension

  3. 3.

Unlike in the scalar case, we now have four different tensor structures (this
is true for any dimension) that can appear in the 3-pt. function. The 3-pt.
function is given by

$\left\langle\psi\left(x_{1}\right)\overline{\psi}\left(x_{2}\right)O^{\Delta,J}\left(x_{3}\right)\right\rangle=\lambda_{\psi\overline{\psi}O}^{m}\left\langle\psi\left(x_{1}\right)\overline{\psi}\left(x_{2}\right)O^{\Delta,J}\left(x_{3}\right)\right\rangle^{m}$
(2.12)

where $m=1,2,3,4$ runs over the different structures, which in turn depend on
the fermion representation.

To deal with these complications, we will specify the dimension $d$ and the
fermion representation. The simplest case would be the $d=3$ Majorana theory
$\left(\overline{\psi}=\psi\right)$. These are in fact expected to be
continuously connected to the $d=3$ $O\left(N\right)$ critical scalar theory,
and to have the same $O\left(N\right)$-invariant operator spectrum (for free
fermions at large $N$, after imposing their equation of motion). The tensor
structures in this case are [11] (Here and from now on we make use of the
embedding space formalism, and use for short
$\left\langle\psi_{1}^{\Delta_{\psi}}\psi_{2}^{\Delta_{\psi}}O^{\Delta,J}\right\rangle=\left\langle\psi\left(X_{1},S_{1}\right)\psi\left(X_{2},S_{2}\right)O\left(X_{3},S_{3}\right)\right\rangle$.
Details can be found in appendix A)

$\displaystyle\left\langle\psi_{1}^{\Delta_{\psi}}\psi_{2}^{\Delta_{\psi}}O^{\Delta,J}\right\rangle^{1}$
$\displaystyle=\frac{\left\langle S_{1}S_{2}\right\rangle\left\langle
S_{3}X_{1}X_{2}S_{3}\right\rangle^{J}}{X_{12}^{\left(2\Delta_{\psi}-\Delta+J+1\right)/2}X_{23}^{\left(\Delta+J\right)/2}X_{31}^{\left(\Delta+J\right)/2}},$
(2.13)
$\displaystyle\left\langle\psi_{1}^{\Delta_{\psi}}\psi_{2}^{\Delta_{\psi}}O^{\Delta,J}\right\rangle^{2}$
$\displaystyle=\frac{\left\langle S_{2}S_{3}\right\rangle\left\langle
S_{1}S_{3}\right\rangle\left\langle
S_{3}X_{1}X_{2}S_{3}\right\rangle^{J-1}}{X_{12}^{\left(2\Delta_{\psi}-\Delta+J-1\right)/2}X_{23}^{\left(\Delta+J\right)/2}X_{31}^{\left(\Delta+J\right)/2}},$
(2.14)
$\displaystyle\left\langle\psi_{1}^{\Delta_{\psi}}\psi_{2}^{\Delta_{\psi}}O^{\Delta,J}\right\rangle^{3}$
$\displaystyle=\frac{\left\langle
S_{3}X_{1}X_{2}S_{3}\right\rangle^{J-1}\left\langle
S_{1}S_{3}\right\rangle\left\langle
S_{2}X_{1}S_{3}\right\rangle}{X_{12}^{\left(2\Delta_{\psi}-\Delta+J\right)/2}X_{23}^{\left(\Delta+J-1\right)/2}X_{31}^{\left(\Delta+J+1\right)/2}}+\frac{\left\langle
S_{3}X_{1}X_{2}S_{3}\right\rangle^{J-1}\left\langle
S_{2}S_{3}\right\rangle\left\langle
S_{1}X_{2}S_{3}\right\rangle}{X_{12}^{\left(2\Delta_{\psi}-\Delta+J\right)/2}X_{23}^{\left(\Delta+J+1\right)/2}X_{31}^{\left(\Delta+J-1\right)/2}},$
(2.15)
$\displaystyle\left\langle\psi_{1}^{\Delta_{\psi}}\psi_{2}^{\Delta_{\psi}}O^{\Delta,J}\right\rangle^{4}$
$\displaystyle=\frac{\left\langle
S_{3}X_{1}X_{2}S_{3}\right\rangle^{J-1}\left\langle
S_{1}S_{3}\right\rangle\left\langle
S_{2}X_{1}S_{3}\right\rangle}{X_{12}^{\left(2\Delta_{\psi}-\Delta+J\right)/2}X_{23}^{\left(\Delta+J-1\right)/2}X_{31}^{\left(\Delta+J+1\right)/2}}-\frac{\left\langle
S_{3}X_{1}X_{2}S_{3}\right\rangle^{J-1}\left\langle
S_{2}S_{3}\right\rangle\left\langle
S_{1}X_{2}S_{3}\right\rangle}{X_{12}^{\left(2\Delta_{\psi}-\Delta+J\right)/2}X_{23}^{\left(\Delta+J+1\right)/2}X_{31}^{\left(\Delta+J-1\right)/2}}.$
(2.16)

Where the first two structures are parity-even and the latter two are parity-
odd. Note that not all of these may appear independently in correlation
functions. First, for $J=0$ only the first and third exist (the existence of
the third may be seen by use of the Fierz identities). Moreover, for operators
with even $J$ we have $\lambda_{\psi\psi O^{-}}^{4}=0$ and for operators with
odd $J$ we have $\lambda_{\psi\psi O^{+}}^{1}=\lambda_{\psi\psi
O^{+}}^{2}=\lambda_{\psi\psi O^{-}}^{3}=0$ (here the $\pm$ in $O^{\pm}$
denotes parity). Also, since
$\left\langle\psi_{1}^{\Delta_{\psi}}\psi_{2}^{\Delta_{\psi}}O^{\Delta,J}\right\rangle^{*}=-\left\langle\psi_{1}^{\Delta_{\psi}}\psi_{2}^{\Delta_{\psi}}O^{\Delta,J}\right\rangle$,
all $\lambda_{\psi\psi O}^{m}$ must be pure imaginary. For a free fermion
($\Delta_{\psi}=\frac{d-1}{2}=1$), the three point functions must satisfy the
EOM which will relate some of the coefficients

$0=\cancel{\partial}_{1}\left\langle\psi^{\Delta_{\psi}=1}\left(x_{1}\right)\psi^{\Delta_{\psi}=1}\left(x_{2}\right)O^{\Delta,J}\left(x_{3}\right)\right\rangle=\lambda_{\psi\psi
O}^{m}\cancel{\partial}_{1}\left\langle\psi^{\Delta_{\psi}=1}\left(x_{1}\right)\psi^{\Delta_{\psi}=1}\left(x_{2}\right)O^{\Delta,J}\left(x_{3}\right)\right\rangle^{m},$
(2.17)

and this relation is useful for showing that on the gravity side one gets on-
shell a single tower of higher-spin degrees of freedom as expected for
Vasiliev gravity. We can use the embedding space notations we introduced above
to calculate the physical space expressions for the tensor structures. For
example

$\displaystyle\left\langle S_{1}S_{2}\right\rangle$
$\displaystyle=S_{1I}\Omega^{IJ}S_{2J}=s_{2\alpha}x_{1}^{\mu}\gamma_{\mu\beta}^{\alpha}s_{1}^{\beta}-s_{1\alpha}x_{2}^{\mu}\gamma_{\mu\beta}^{\alpha}s_{2}^{\beta},$
(2.18) $\displaystyle\left\langle S_{1}X_{2}S_{3}\right\rangle$
$\displaystyle=4x_{12}^{\mu}x_{32}^{\nu}s_{1}\left(\frac{1}{4}\left[\gamma_{\mu},\gamma_{\nu}\right]\right)s_{3},$
(2.19) $\displaystyle\left\langle S_{3}X_{1}X_{2}S_{3}\right\rangle$
$\displaystyle=x_{13}^{2}x_{23}^{2}\left(\frac{x_{23}^{\mu}}{x_{23}^{2}}-\frac{x_{13}^{\mu}}{x_{13}^{2}}\right)s_{3}\gamma_{\mu}s_{3},$
(2.20)
$\displaystyle\left\langle\psi^{\alpha}\left(x_{1}\right)\psi^{\beta}\left(x_{2}\right)O_{J}^{\mu_{1}\dots\mu_{J}}\left(x_{3}\right)\right\rangle^{1}$
$\displaystyle=\frac{\left(-1\right)^{J}}{2^{J}\left(2J\right)!}\gamma_{\alpha_{1}\alpha_{2}}^{\mu_{1}}\cdots\gamma_{\alpha_{2J-1,\alpha_{2J}}}^{\mu_{J}}\frac{\partial^{2J}}{\partial
s_{\alpha_{1}}\cdots\partial s_{\alpha_{2J}}}\times$
$\displaystyle\qquad\times\left(\frac{x_{13}^{2}x_{23}^{2}}{x_{12}^{2}}\left(\frac{x_{23}^{\mu}}{x_{23}^{2}}-\frac{x_{13}^{\mu}}{x_{13}^{2}}\right)s\gamma_{\mu}s\right)^{J}\frac{x_{12}^{\mu}\gamma_{\mu}^{\alpha\beta}}{x_{12}^{2\Delta_{\psi}-\Delta-J+1}x_{23}^{\Delta+J}x_{31}^{\Delta+J}},$
(2.21)

etc. We define the pairing of $n$-pt. functions as

$\left(\mathcal{O}\left(x_{1},\dots,x_{n}\right),\mathcal{O}^{\prime}\left(x_{1},\dots,x_{n}\right)\right)=\intop\frac{d^{d}x_{1}\cdots
d^{d}x_{n}}{\text{vol}\left(SO\left(d+1,1\right)\right)}\mathcal{O}\left(x_{1},\dots,x_{n}\right)\mathcal{O}^{\prime}\left(x_{1},\dots,x_{n}\right).$
(2.22)

The pairing
$\left(\left\langle\psi_{1}^{\Delta_{1}}\psi_{2}^{\Delta_{2}}O^{\Delta,J}\right\rangle^{m},\left\langle\psi_{1}^{\tilde{\Delta}_{1}}\psi_{2}^{\tilde{\Delta}_{2}}O^{\tilde{\Delta},J}\right\rangle^{n}\right)$
of 3-pt. functions is calculated in appendix B.1. We can use it to define the
bubble coefficients

$\mathcal{B}_{\Delta,J}^{mn}=\frac{\left(\left\langle\psi_{1}\psi_{2}O^{\Delta,J}\right\rangle^{m},\left\langle\tilde{\psi}_{1}\tilde{\psi}_{2}O^{\tilde{\Delta},J}\right\rangle^{n}\right)}{\mu\left(\Delta,J\right)}$
(2.23)

where $\mu\left(\Delta,J\right)$ is the Plancherel measure for the
representation $\Delta,J$ of the Euclidean conformal group
$SO\left(4,1\right)$. The Plancherel measure for a group $G$ is a measure on
the space of unitary irreducible representations of $G.$ For compact groups
and for a unitary irreducible rep. $\pi$, it is given by

$\mu\left(\pi\right)=\frac{\text{dim}\pi}{\text{Vol}G}$ (2.24)

where $\text{Vol}G$ is defined using the Haar measure on $G$. For non-compact
groups, both $\text{dim}\pi$ and $\text{Vol}G$ can be infinite, and one can
think of the Plancherel measure as a regularized quotient. In our case one can
calculate and find

$\mu\left(\Delta,J\right)=\frac{\left(2\Delta-3\right)\left(J+\Delta-1\right)\left(J+\tilde{\Delta}-1\right)\Gamma\left(2J+2\right)\cot\left(\pi\left(\Delta+J\right)\right)}{128\pi^{5}}.$
(2.25)

We can write an expression for the orthogonality of the 3-pt. functions

$\displaystyle\intop d^{d}x_{1}d^{d}x_{2}\frac{\partial^{2}}{\partial
s_{1\alpha}\partial s_{1}^{\alpha}}\frac{\partial^{2}}{\partial
s_{2\beta}\partial
s_{2}^{\beta}}\left\langle\psi\left(x_{1},s_{1}\right)\psi\left(x_{2},s_{2}\right)O^{\Delta,J}\left(y,s\right)\right\rangle^{m}\left\langle\tilde{\psi}\left(x_{1},s_{1}\right)\tilde{\psi}\left(x_{2},s_{2}\right)O^{\tilde{\Delta^{\prime}},J^{\prime}}\left(y^{\prime},s^{\prime}\right)\right\rangle^{n}$
$\displaystyle\qquad=2\pi
i\mathcal{B}_{\Delta,J}^{mk}\delta_{JJ^{\prime}}\left(\delta_{k}^{n}\delta\left(\Delta-\Delta^{\prime}\right)\delta^{\left(d\right)}\left(y-y^{\prime}\right)\left(s\cdot
s^{\prime}\right)^{J}+\left(S_{k}^{n}\left(\psi\psi\left[\mathcal{O}_{\Delta,J}\right]\right)\right)^{-1}\delta\left(\Delta-\tilde{\Delta^{\prime}}\right)\left\langle
O^{\Delta,J}\left(y,s\right)O^{\Delta,J}\left(y^{\prime},s^{\prime}\right)\right\rangle\right)$
(2.26)

where $S_{k}^{n}\left(\psi\psi\left[\mathcal{O}_{\Delta,J}\right]\right)$ are
the shadow coefficients whose explicit expressions are given in appendix B.2.
Our bilocal fields can then be expressed as the expansion

$\rho_{\alpha\beta}\left(x_{1},x_{2}\right)=\sum_{m=1}^{4}\sum_{J=0}^{\infty}\intop_{C_{J}}\frac{d\Delta}{2\pi
i}\intop
d^{d}yc_{\Delta,J}^{m}\left(y,\partial_{s}\right)\left\langle\psi_{\alpha}\psi_{\beta}O^{\Delta,J}\left(y,s\right)\right\rangle^{m}$
(2.27)

which is invertible by

$c_{\Delta,J}^{m}\left(y,s\right)=\left(\mathcal{B}_{\Delta,J}^{mn}\right)^{-1}\intop
d^{d}x_{1}d^{d}x_{2}\rho^{\alpha\beta}\left(x_{1},x_{2}\right)\left\langle\tilde{\psi}_{\alpha}\tilde{\psi}_{\beta}O^{\tilde{\Delta},J}\left(y,s\right)\right\rangle^{n}.$
(2.28)

Note that the coefficients $c_{\Delta,J}^{m}$ have a spinor index which is
encoded above using the auxiliary variable $s$. The $\partial_{s}$ argument
contracts this index with the spinor indices of the operator $O^{\Delta,J}$.
We will suppress these indices in the following analysis to avoid cluttering
of the equations. Completeness is given by

$\delta\left(x_{13}\right)\delta\left(x_{24}\right)\Omega_{\alpha\gamma}\Omega_{\beta\delta}=\sum_{m,n1}^{4}\sum_{J=0}^{\infty}\intop_{C_{J}}\frac{d\Delta}{2\pi
i}\intop
d^{d}y\left(\mathcal{B}_{\Delta,J}^{mn}\right)^{-1}\left\langle\psi_{\alpha}\psi_{\beta}O^{\Delta,J}\right\rangle^{m}\left\langle\tilde{\psi}_{\gamma}\tilde{\psi}_{\delta}O^{\tilde{\Delta},J}\right\rangle^{n}.$
(2.29)

The second term in the orthogonality theorem tells us that there is a relation
between the basis elements for $\Delta$ and $\tilde{\Delta}$. They are not
independent of each other, but instead are related by the shadow transform

$\left\langle\psi_{\alpha}\left(x_{1}\right)\psi_{\beta}\left(x_{2}\right)O^{\Delta,J}\left(y\right)\right\rangle^{m}=\left(S_{n}^{m}\left(\psi\psi\left[\mathcal{O}_{\Delta,J}\right]\right)\right)^{-1}\intop
d^{d}y^{\prime}\left\langle
O^{\Delta,J}\left(y\right)O^{\Delta,J}\left(y^{\prime}\right)\right\rangle\left\langle\psi_{\alpha}\left(x_{1}\right)\psi_{\beta}\left(x_{2}\right)O^{\tilde{\Delta},J}\left(y^{\prime}\right)\right\rangle^{n}.$
(2.30)

Which gives us a relation between the coefficients $c_{\Delta,J}^{m}$ and
their shadow counterparts

$c_{\tilde{\Delta},J}^{m}\left(y\right)=\left(S_{n}^{m}\left(\psi\psi\left[\mathcal{O}_{\Delta,J}\right]\right)\right)^{-1}\intop
d^{d}y^{\prime}\left\langle
O^{\Delta,J}\left(y\right)O^{\Delta,J}\left(y^{\prime}\right)\right\rangle
c_{\Delta,J}^{n}\left(y^{\prime}\right).$ (2.31)

A subtlety of this construction is that the contour $C_{J}$ we used in the
expansion of $\rho_{\alpha\beta}$ (2.27) and the completeness relation, was
shown in the literature to hold on the principal series $\Delta=d/2+is$ with
$s\in\mathbb{R}$. However, we would like to use it for real values of
$\Delta_{\psi}$ as those are the conformal weights of our physical operators.
We can do this by analytically continuing our expressions to real values of
$\Delta$, but we need to carefully check for which values of
$\Delta,\Delta_{\psi}$ the integral in (2.28) still converges. The convergence
conditions are:

  1. 1.

In the limit of large $x_{1}$ (or equivalently large $x_{2}$) we use
$\rho_{\alpha\beta}\sim\left|x_{1}\right|^{-2\Delta_{\psi}}$ and
$\left\langle\tilde{\psi}_{\alpha}\tilde{\psi}_{\beta}O^{\tilde{\Delta},J}\right\rangle\sim\left|x_{1}\right|^{-2\tilde{\Delta}_{\psi}}$
and get powers of $\left|x_{1}\right|$ that are equal to
$d-2\text{Re}\left(\Delta_{\psi}\right)-2\text{Re}\left(\tilde{\Delta}_{\psi}\right)=-d$,
which is convergent for any value of $\Delta_{\psi}$.

  2. 2.

In the limit $x_{1}\sim x_{2}$ (assuming a finite limit of
$\rho_{\alpha\beta}\left(x_{1},x_{2}\right)$) we get powers of
$\left|x_{1}-x_{2}\right|$ equal to
$d+\text{Re}\left(\tilde{\Delta}\right)-2\text{Re}\left(\tilde{\Delta}_{\psi}\right)=2\text{Re}\left(\Delta_{\psi}\right)-\text{Re}\left(\Delta\right)$
which for $\Delta=d/2+is$ gives the convergence condition $\Delta_{\psi}>d/4$.

It is clear that for the free fermions where $\Delta_{\psi}=\frac{d-1}{2}$
there is no need to worry about convergence in $d>2$. In the scalar case, one
had to define a double shadow transform of the bilocal field $\eta$ to ensure
convergence. In the fermion case there is no need to worry about convergence
as we have seen, but we will keep these definitions to get a nicer expression
for our quadratic action, as we will see in the following subsection. We
define a new function $\tilde{\rho}_{\alpha\beta}$ which is the double shadow
transform of $\rho_{\alpha\beta}$ by

$\rho_{\alpha\beta}\left(x_{1},x_{2}\right)=\intop
d^{d}xd^{d}x^{\prime}\left\langle\psi_{\alpha}^{\Delta_{\psi}}\left(x_{1}\right)\psi_{\gamma}^{\Delta_{\psi}}\left(x\right)\right\rangle\left\langle\psi_{\beta}^{\Delta_{\psi}}\left(x_{2}\right)\psi_{\delta}^{\Delta_{\psi}}\left(x^{\prime}\right)\right\rangle\tilde{\rho}^{\gamma\delta}\left(x,x^{\prime}\right).$
(2.32)

Now, $\tilde{\rho}_{\alpha\beta}$ transforms under the tensor product of two
$\tilde{\Delta}_{\psi}$ representations, and obeys the same boundary
conditions as $\rho_{\alpha\beta}$, except replacing $\Delta_{\psi}$ with
$\tilde{\Delta}_{\psi}$. We can now expand

$\tilde{\rho}_{\alpha\beta}\left(x_{1},x_{2}\right)=\sum_{m=1}^{4}\sum_{J=0}^{\infty}\intop_{C_{J}}\frac{d\Delta}{2\pi
i}\intop
d^{d}y\tilde{c}_{\Delta,J}^{m}\left(y\right)\left\langle\tilde{\psi}_{\alpha}\tilde{\psi}_{\beta}O^{\Delta,J}\left(y\right)\right\rangle^{m}$
(2.33)

with the coefficients

$\tilde{c}_{\Delta,J}^{m}\left(y\right)=\left(\mathcal{B}_{\Delta,J}^{mn}\right)^{-1}\intop
d^{d}x_{1}d^{d}x_{2}\tilde{\rho}^{\alpha\beta}\left(x_{1},x_{2}\right)\left\langle\psi_{\alpha}\psi_{\beta}O^{\tilde{\Delta},J}\left(y\right)\right\rangle^{n}.$
(2.34)

Where the integral is now convergent. We can now plug in (2.33) into (2.32) to
get

$\displaystyle\rho_{\alpha\beta}\left(x_{1},x_{2}\right)$
$\displaystyle=\sum_{m=1}^{4}\sum_{J=0}^{\infty}\intop_{C_{J}}\frac{d\Delta}{2\pi
i}\intop d^{d}y\intop
d^{d}xd^{d}x^{\prime}\left\langle\psi_{\alpha}^{\Delta_{\psi}}\left(x_{1}\right)\psi_{\gamma}^{\Delta_{\psi}}\left(x\right)\right\rangle\left\langle\psi_{\beta}^{\Delta_{\psi}}\left(x_{2}\right)\psi_{\delta}^{\Delta_{\psi}}\left(x^{\prime}\right)\right\rangle\times$
$\displaystyle\quad\times\tilde{c}_{\Delta,J}^{m}\left(y\right)\left\langle\tilde{\psi}^{\gamma}\left(x\right)\tilde{\psi}^{\delta}\left(x^{\prime}\right)O^{\Delta,J}\left(y\right)\right\rangle^{m}$
(2.35)
$\displaystyle=\sum_{m=1}^{4}\sum_{J=0}^{\infty}\intop_{C_{J}}\frac{d\Delta}{2\pi
i}\intop
d^{d}y\tilde{c}_{\Delta,J}^{m}\left(y\right)S_{n}^{m}\left(\left[\tilde{\psi}\right]\tilde{\psi}\mathcal{O}^{\Delta,J}\right)S_{k}^{n}\left(\psi\left[\tilde{\psi}\right]\mathcal{O}^{\Delta,J}\right)\left\langle\psi_{\alpha}\left(x_{1}\right)\psi_{\beta}\left(x_{2}\right)O^{\Delta,J}\left(y\right)\right\rangle^{k},$
(2.36)

where we used twice the shadow transform (with the short hand notation
$\psi=\psi^{\Delta_{\psi}},\ \tilde{\psi}=\psi^{\tilde{\Delta}_{\psi}}$)

$\intop
d^{d}x\left\langle\psi_{\alpha}^{\Delta_{1}}\left(x_{1}\right)\psi_{\gamma}^{\Delta_{1}}\left(x\right)\right\rangle\left\langle\psi^{\tilde{\Delta}_{1}\gamma}\left(x\right)\psi^{\Delta_{2}\delta}\left(x^{\prime}\right)O^{\Delta,J}\left(y\right)\right\rangle^{m}=S_{n}^{m}\left(\left[\psi^{\tilde{\Delta}_{1}}\right]\psi^{\Delta_{2}}\mathcal{O}^{\Delta,J}\right)\left\langle\psi^{\Delta_{1}\gamma}\left(x\right)\psi^{\Delta_{2}\delta}\left(x^{\prime}\right)O^{\Delta,J}\left(y\right)\right\rangle^{n}.$
(2.37)

Note that the shadow coefficients we need here are different than the ones we
defined earlier. However, they can be obtained from the literature [10] and
appear explicitly in appendix B.3. Note that we get a block anti-diagonal form
for the shadow coefficients due to parity. We then get that the matrix

$A_{k;\Delta,J}^{m}=S_{n}^{m}\left(\left[\tilde{\psi}\right]\tilde{\psi}\mathcal{O}^{\Delta,J}\right)S_{k}^{n}\left(\psi\left[\tilde{\psi}\right]\mathcal{O}^{\Delta,J}\right)$
(2.38)

is block diagonal, with the blocks

$A_{k;\Delta,J}^{m}=\begin{cases}\frac{\pi^{3}\Gamma^{2}\left(2-\Delta_{\psi}\right)\Gamma\left(\Delta_{\psi}-\frac{\Delta-J+1}{2}\right)\Gamma\left(\Delta_{\psi}-\frac{\tilde{\Delta}-J+1}{2}\right)}{\Gamma^{2}\left(\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(\tilde{\Delta}_{\psi}-\frac{\Delta-J-1}{2}\right)\Gamma\left(\tilde{\Delta}_{\psi}-\frac{\tilde{\Delta}-J-1}{2}\right)}\times\\\
\quad\times\left(\begin{array}[]{cc}-\left(\tilde{\Delta}_{\psi}-\frac{\Delta-J+2}{2}\right)\left(\Delta_{\psi}-\frac{\Delta-J+1}{2}\right)&J(\Delta-1)\\\
\left(\tilde{\Delta}_{\psi}-\frac{\Delta-J+1}{2}\right)\left(\Delta_{\psi}-\frac{\Delta-J+1}{2}\right)&\left(\tilde{\Delta}_{\psi}-\frac{\Delta-J+1}{2}\right)\left(\Delta_{\psi}-\frac{\Delta-J+2}{2}\right)\end{array}\right)&m,k=1,2\\\
\frac{\pi^{3}\Gamma^{2}\left(2-\Delta_{\psi}\right)\Gamma\left(\Delta_{\psi}-\frac{\Delta-J}{2}\right)\Gamma\left(\Delta_{\psi}-\frac{\tilde{\Delta}-J}{2}\right)}{\Gamma^{2}\left(\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(\tilde{\Delta}_{\psi}-\frac{\Delta-J}{2}\right)\Gamma\left(\tilde{\Delta}_{\psi}-\frac{\tilde{\Delta}-J}{2}\right)}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}&m,k=3,4\end{cases}.$ (2.39)

We can now analytically continue our expression for $\rho_{\alpha\beta}$, with
the only problem coming from the poles of
$\left(A_{k;\Delta,J}^{m}\right)^{-1}$. For odd parity operators
$O^{\Delta,J}$ this function has the pole structure we expect from the
dimensions of the physical primary operators of the generalized free field
theory

$\Delta^{\left(n,J\right)}=2\Delta_{\psi}+2n+J$ (2.40)

and their appropriate $\tilde{\Delta}^{\left(n,J\right)}$. For even parity
operators the pole structure is a bit more complicated. The idea is that we
can now deform the contour $C_{J}$ to a new one $\gamma_{J}$ which includes
the poles $\Delta^{\left(n,J\right)}$ that are to the right of the principal
series ($\text{Re}\left(\Delta\right)=d/2$) and excludes the poles
$\tilde{\Delta}^{\left(n,J\right)}$ to its left (which appear at high enough
$n,J$). We then get a convergent definition for

$\rho_{\alpha\beta}\left(x_{1},x_{2}\right)=\sum_{m=1}^{4}\sum_{J=0}^{\infty}\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\intop
d^{d}yc_{\Delta,J}^{m}\left(y\right)\left\langle\psi_{\alpha}\psi_{\beta}O^{\Delta,J}\left(y\right)\right\rangle^{m}$
(2.41)

where we define

$c_{\Delta,J}^{m}=A_{k;\Delta,J}^{m}\tilde{c}_{\Delta,J}^{k}.$ (2.42)

We now turn our attention to the free fermion case, for which
$\Delta_{\psi}=\frac{d-1}{2}$, and in our case $d=3$ and
$\Delta_{\psi}=1,\tilde{\Delta}_{\psi}=2$. Here we have

$A_{k;\Delta,J}^{m}=\begin{cases}\frac{\left(4\pi\right)^{2}}{\left(1-\Delta+J\right)\left(1-\tilde{\Delta}+J\right)\left(\Delta+J\right)\left(\tilde{\Delta}+J\right)}\left(\begin{array}[]{cc}-\left(2-\Delta+J\right)\left(1-\Delta+J\right)&4J(\Delta-1)\\\
\left(\tilde{\Delta}+J\right)\left(1-\Delta+J\right)&\left(\tilde{\Delta}+J\right)\left(-\Delta+J\right)\end{array}\right)&m,k=1,2\\\
\frac{\left(4\pi\right)^{2}}{\left(2-\Delta+J\right)\left(2-\tilde{\Delta}+J\right)}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}&m,k=3,4\end{cases}.$ (2.43)

And taking the inverse we get

$\left(A_{\Delta,J}^{-1}\right)_{\,k}^{m}=\frac{1}{\left(4\pi\right)^{2}}\begin{cases}\left(\begin{array}[]{cc}-\left(3-\Delta+J\right)\left(-\Delta+J\right)&4J\left(\Delta-1\right)\\\
\left(3-\Delta+J\right)\left(1-\Delta+J\right)&\
\left(2-\Delta+J\right)\left(1-\Delta+J\right)\end{array}\right)&m,k=1,2\\\
\left(\begin{array}[]{cc}\left(2-\Delta+J\right)\left(-1+\Delta+J\right)&0\\\
0&-\left(2-\Delta+J\right)\left(-1+\Delta+J\right)\end{array}\right)&m,k=3,4\end{cases}.$
(2.44)

###  2.2 The quadratic bilocal action

We can now write our quadratic bilocal action (2.8) in this formalism

$\displaystyle S_{quad}$ $\displaystyle=-\frac{1}{2}\intop
d^{d}x_{1}d^{d}x_{2}\rho_{\gamma\delta}\left(x_{1},x_{2}\right)\cancel{\partial}_{1\beta}^{\gamma}\cancel{\partial}_{2\alpha}^{\delta}\rho^{\alpha\beta}\left(x_{1},x_{2}\right)$
$\displaystyle=-\frac{1}{2}\intop
d^{d}x_{1}d^{d}x_{2}\tilde{\rho}_{\gamma\delta}\left(x_{1},x_{2}\right)x_{12}^{2}\cancel{\partial}_{1\beta}^{\gamma}\cancel{\partial}_{2\alpha}^{\delta}\rho^{\alpha\beta}\left(x_{1},x_{2}\right)$
$\displaystyle=-\frac{1}{2}\sum_{m,m^{\prime}=1}^{4}\sum_{J,J^{\prime}=0}^{\infty}\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\intop_{\gamma_{J^{\prime}}}\frac{d\Delta^{\prime}}{2\pi i}\intop
d^{d}yd^{d}y^{\prime}\tilde{c}_{\Delta^{\prime},J^{\prime}}^{m^{\prime}}c_{\Delta,J}^{m}\times$
$\displaystyle\ \ \times\intop
d^{d}x_{1}d^{d}x_{2}\left\langle\tilde{\psi}^{\gamma}\tilde{\psi}^{\delta}O^{\Delta^{\prime},J^{\prime}}\right\rangle^{m^{\prime}}x_{12}^{2}\cancel{\partial}_{1\beta}^{\gamma}\cancel{\partial}_{2\alpha}^{\delta}\left\langle\psi^{\alpha}\psi^{\beta}O^{\Delta,J}\right\rangle^{m}.$
(2.45)

One can note here the quadratic action operator

$S_{qa}=x_{12}^{2}\cancel{\partial}_{1\beta}^{\gamma}\cancel{\partial}_{2\alpha}^{\delta}.$
(2.46)

This operator is manifestly conformally invariant, as can be checked
explicitly by commuting it with the bilocal conformal generators. Acting with
$S_{qa}$ on the 3-pt. function basis with $\Delta_{\psi}=1$ was done using
Wolfram Mathematica. The code is given in appendix C. it gives the result

$\displaystyle\left(S_{qa}\right)_{\gamma\delta}^{\alpha\beta}\left\langle\psi_{\alpha}\psi_{\beta}O^{\Delta,J}\right\rangle^{m}$
$\displaystyle\equiv
B_{k;\Delta,J}^{m}\left\langle\psi_{\gamma}\psi_{\delta}O^{\Delta,J}\right\rangle^{k}$
$\displaystyle=-\left(4\pi\right)^{2}\left(A_{\Delta,J}^{-1}\right)_{\,k}^{m}\left\langle\psi_{\gamma}\psi_{\delta}O^{\Delta,J}\right\rangle^{k}.$
(2.47)

We see that the action matrix in the 3-pt. basis is exactly the same as the
matrix which relates between $c_{\Delta,J}^{m},\tilde{c}_{\Delta,J}^{k}$. We
can use this fact to derive the relation

$\displaystyle\tilde{\rho}_{\alpha\beta}\left(x_{1},x_{2}\right)$
$\displaystyle=\sum_{m=1}^{4}\sum_{J=0}^{\infty}\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\intop
d^{d}y\tilde{c}_{\Delta,J}^{m}\left\langle\tilde{\psi}_{\alpha}\tilde{\psi}_{\beta}O^{\Delta,J}\right\rangle^{m}$
$\displaystyle=\sum_{m=1}^{4}\sum_{J=0}^{\infty}\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\intop
d^{d}yc_{\Delta,J}^{m}\left(A_{\Delta,J}^{-1}\right)^{mk}\left\langle\tilde{\psi}_{\alpha}\tilde{\psi}_{\beta}O^{\Delta,J}\right\rangle^{k}$
$\displaystyle=\frac{1}{x_{12}^{2}}\sum_{m=1}^{4}\sum_{J=0}^{\infty}\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\intop
d^{d}yc_{\Delta,J}^{m}\left(A_{\Delta,J}^{-1}\right)^{mk}\left\langle\psi_{\alpha}\psi_{\beta}O^{\Delta,J}\right\rangle^{k}$
$\displaystyle=\frac{-1}{\left(4\pi\right)^{2}}\cancel{\partial}_{1\beta}^{\gamma}\cancel{\partial}_{2\alpha}^{\delta}\rho_{\gamma\delta}\left(x_{1},x_{2}\right).$
(2.48)

Note that this also follows directly from (2.32). We can write our quadratic
action in terms of the coefficients

$\displaystyle S_{quad}\left[c_{\Delta,J}^{m}\right]$
$\displaystyle=\frac{1}{2\left(4\pi\right)^{2}}\intop
d^{d}x_{1}d^{d}x_{2}\rho_{\alpha\beta}\left(x_{1},x_{2}\right)\tilde{\rho}^{\alpha\beta}\left(x_{1},x_{2}\right)$
$\displaystyle=\frac{1}{2\left(4\pi\right)^{2}}\sum_{J,J^{\prime}=0}^{\infty}\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\intop_{\gamma_{J^{\prime}}}\frac{d\Delta^{\prime}}{2\pi i}\intop
d^{d}yd^{d}y^{\prime}c_{\Delta,J}^{m}\left(A_{\tilde{\Delta}^{\prime},J}^{-1}\right)_{m^{\prime}k}c_{\tilde{\Delta}^{\prime},J^{\prime}}^{k}\times$
$\displaystyle\ \times\intop
d^{d}x_{1}d^{d}x_{2}\left\langle\psi_{\alpha}\psi_{\beta}O^{\Delta,J}\right\rangle^{m}\left\langle\tilde{\psi}^{\alpha}\tilde{\psi}^{\beta}O^{\tilde{\Delta}^{\prime},J^{\prime}}\right\rangle^{m^{\prime}}$
(2.49)
$\displaystyle=\frac{1}{\left(4\pi\right)^{2}}\sum_{J=0}^{\infty}\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\intop
d^{d}yc_{\Delta,J}^{m}\mathcal{B}_{\Delta,J}^{mm^{\prime}}\left(A_{\Delta,J}^{-1}\right)_{m^{\prime}k}c_{\tilde{\Delta},J}^{k}.$
(2.50)

Where in the second equality we used the relation
$c_{\Delta,J}^{m}=A_{k;\Delta,J}^{m}\tilde{c}_{\Delta,J}^{k}$, and in the last
equality we used the orthogonality relation (2.26).

###  2.3 Bilocal theory 2-point function

We can now calculate the bilocal theory 2-pt. function, and see that it indeed
reproduces the 4-pt. function of the free theory, as must be the case since

$\displaystyle\left\langle\rho_{\alpha\beta}\left(x_{1},x_{2}\right)\rho_{\gamma\delta}\left(x_{3},x_{4}\right)\right\rangle$
$\displaystyle=\frac{1}{N}\left\langle\left(\psi_{\alpha}^{i}\left(x_{1}\right)\psi_{\beta}^{i}\left(x_{2}\right)-\left\langle\psi_{\alpha}^{i}\left(x_{1}\right)\psi_{\beta}^{i}\left(x_{2}\right)\right\rangle\right)\left(\psi_{\gamma}^{j}\left(x_{3}\right)\psi_{\delta}^{j}\left(x_{4}\right)-\left\langle\psi_{\gamma}^{j}\left(x_{3}\right)\psi_{\delta}^{j}\left(x_{4}\right)\right\rangle\right)\right\rangle$
$\displaystyle=\frac{1}{N}\left(\left\langle\psi_{\alpha}^{i}\left(x_{1}\right)\psi_{\beta}^{i}\left(x_{2}\right)\psi_{\gamma}^{j}\left(x_{3}\right)\psi_{\delta}^{j}\left(x_{4}\right)\right\rangle-\left\langle\psi_{\alpha}^{i}\left(x_{1}\right)\psi_{\beta}^{i}\left(x_{2}\right)\right\rangle\left\langle\psi_{\gamma}^{j}\left(x_{3}\right)\psi_{\delta}^{j}\left(x_{4}\right)\right\rangle\right).$
(2.51)

Using only the quadratic term in the action, we have

$\displaystyle\left\langle\rho_{\alpha\beta}\left(x_{1},x_{2}\right)\rho_{\gamma\delta}\left(x_{3},x_{4}\right)\right\rangle_{\text{tree}}$
$\displaystyle=\sum_{J,J^{\prime}=0}^{\infty}\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\intop_{\gamma_{J^{\prime}}}\frac{d\Delta^{\prime}}{2\pi i}\intop
d^{d}yd^{d}y^{\prime}\left\langle\psi_{\alpha}\psi_{\beta}O^{\Delta,J}\right\rangle^{m}\left\langle\psi^{\alpha}\psi^{\beta}O^{\tilde{\Delta}^{\prime},J^{\prime}}\right\rangle^{m^{\prime}}\times$
$\displaystyle\
\times\intop\left(dc_{\Delta,J}^{m}\right)c_{\Delta,J}^{m}c_{\tilde{\Delta}^{\prime},J^{\prime}}^{m^{\prime}}e^{-S_{quad}\left[c_{\Delta,J}^{m}\right]}.$
(2.52)

This is a Gaussian path integral. It is useful to define the conformal partial
wave [12]

$\Psi_{\Delta,J,\alpha\beta\gamma\delta}^{\Delta_{\psi}\left(mk\right)}\left(x_{1},\dots,x_{4}\right)=\intop
d^{d}y\left\langle\psi_{\alpha}\psi_{\beta}O^{\Delta,J}\left(y\right)\right\rangle^{m}\left\langle\psi_{\gamma}\psi_{\delta}O^{\tilde{\Delta},J}\left(y\right)\right\rangle^{k}.$
(2.53)

So we get

$\displaystyle\left\langle\rho_{\alpha\beta}\left(x_{1},x_{2}\right)\rho_{\gamma\delta}\left(x_{3},x_{4}\right)\right\rangle_{\text{tree}}$
$\displaystyle=\sum_{J=0}^{\infty}\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\mathcal{B}_{\Delta,J}^{mm^{\prime}}\left(A_{\Delta,J}^{-1}\right)_{m^{\prime}k}\Psi_{\Delta,J,\alpha\beta\gamma\delta}^{\Delta_{\psi}\left(mk\right)}\left(x_{1},\dots,x_{4}\right).$
(2.54)

Recalling our derivation (2.39), one can note that the coefficients
$\mathcal{B}_{\Delta,J}^{mm^{\prime}}\left(A_{\Delta,J}^{-1}\right)_{m^{\prime}k}$
are exactly what one would get from bootstrapping the local fermionic MFT,
without going through the bilocal theory. This calculation was done in detail
by [11, 12] and it indeed gives the fermionic free field 4-pt. function as
expected

$\left\langle\rho_{\alpha\beta}\left(x_{1},x_{2}\right)\rho_{\gamma\delta}\left(x_{3},x_{4}\right)\right\rangle=\left(\frac{\Gamma\left(\frac{d}{2}\right)}{2\pi^{d/2}}\right)^{2}\left(\frac{\gamma_{\alpha\gamma}^{\mu}x_{13\mu}}{\left|x_{13}\right|^{d}}\frac{\gamma_{\beta\delta}^{\nu}x_{24\nu}}{\left|x_{24}\right|^{d}}-\frac{\gamma_{\alpha\delta}^{\mu}x_{14\mu}}{\left|x_{14}\right|^{d}}\frac{\gamma_{\beta\gamma}^{\nu}x_{23\nu}}{\left|x_{23}\right|^{d}}\right).$
(2.55)

###  2.4 Mapping to AdS

We now attempt to map our bilocal degrees of freedom into AdS spacetime. In
the bosonic field theory this was achieved by mapping the eigenvalues of the
action $\lambda_{\Delta,J}$ to the masses of AdS fields $M_{\Delta,J}$. In the
bosonic free theory

$\displaystyle\lambda_{\Delta,J}$
$\displaystyle=\left(\Delta+J\right)\left(d-\Delta+J\right)\left(d-2-\Delta+J\right)\left(\Delta+J-2\right)$
$\displaystyle=\left(M_{\Delta,J}^{2}-M_{d+J,J}^{2}\right)\left(M_{\Delta,J}^{2}-M_{d+J-2,J}^{2}\right),$
(2.56)

where

$M_{\Delta,J}^{2}=\Delta\left(\Delta-d\right)-J.$ (2.57)

The analogue of $\lambda_{\Delta,J}$ in the fermionic free theory is
$\left(\lambda_{\Delta,J}\right)_{\,k}^{m}=\left(4\pi\right)^{2}\left(A_{\Delta,J}^{-1}\right)_{\,k}^{m}$.
Diagonalizing it we get

$\left(\lambda_{\Delta,J}\right)_{\,k}^{m}=\begin{cases}\left(\begin{array}[]{cc}-\left(3-\Delta+J\right)\left(-\Delta+J\right)&4J\left(\Delta-1\right)\\\
\left(3-\Delta+J\right)\left(1-\Delta+J\right)&\
\left(2-\Delta+J\right)\left(1-\Delta+J\right)\end{array}\right)&m,k=1,2\\\
\left(\begin{array}[]{cc}\left(2-\Delta+J\right)\left(-1+\Delta+J\right)&0\\\
0&-\left(2-\Delta+J\right)\left(-1+\Delta+J\right)\end{array}\right)&m,k=3,4\end{cases}$
(2.58)

then

$\displaystyle 0$
$\displaystyle=\det\left(\begin{array}[]{cc}\lambda+\left(3-\Delta+J\right)\left(-\Delta+J\right)&-4J\left(\Delta-1\right)\\\
-\left(3-\Delta+J\right)\left(1-\Delta+J\right)&\
\lambda-\left(2-\Delta+J\right)\left(1-\Delta+J\right)\end{array}\right)$
(2.61)
$\displaystyle=\left(\lambda+\left(3-\Delta+J\right)\left(-\Delta+J\right)\right)\left(\lambda-\left(2-\Delta+J\right)\left(1-\Delta+J\right)\right)-4J\left(\Delta-1\right)\left(3-\Delta+J\right)\left(1-\Delta+J\right)$
$\displaystyle=\lambda^{2}-\left[\left(2-\Delta+J\right)\left(1-\Delta+J\right)-\left(3-\Delta+J\right)\left(-\Delta+J\right)\right]\lambda+$
$\displaystyle\
-\left(3-\Delta+J\right)\left(1-\Delta+J\right)\left[\left(-\Delta+J\right)\left(2-\Delta+J\right)+4J\left(\Delta-1\right)\right]$
$\displaystyle=\lambda^{2}-2\lambda-\left(3-\Delta+J\right)\left(1-\Delta+J\right)\left(\Delta+J\right)\left(J+\Delta-2\right),$
(2.62) $\displaystyle\lambda_{\pm}$
$\displaystyle=1\pm\sqrt{1+\left(3-\Delta+J\right)\left(1-\Delta+J\right)\left(\Delta+J\right)\left(-2+\Delta+J\right)}.$
(2.63)

Our action then has 4 eigenvalues

$\lambda_{\Delta,J}=\text{diag}\begin{pmatrix}1+\sqrt{1+\left(3-\Delta+J\right)\left(1-\Delta+J\right)\left(\Delta+J\right)\left(-2+\Delta+J\right)}\\\
1-\sqrt{1+\left(3-\Delta+J\right)\left(1-\Delta+J\right)\left(\Delta+J\right)\left(-2+\Delta+J\right)}\\\
\left(2-\Delta+J\right)\left(-1+\Delta+J\right)\\\
-\left(2-\Delta+J\right)\left(-1+\Delta+J\right)\end{pmatrix}.$ (2.64)

Let’s try to express these in terms of the AdS masses. We can see that

$\displaystyle-\left(2-\Delta+J\right)\left(-1+\Delta+J\right)$
$\displaystyle=M_{\Delta,J}^{2}-M_{J+2,J}^{2},$ (2.65)
$\displaystyle\left(2-\Delta+J\right)\left(-1+\Delta+J\right)$
$\displaystyle=M_{J+2,J}^{2}-M_{\Delta,J}^{2},$ (2.66)
$\displaystyle\left(3-\Delta+J\right)\left(1-\Delta+J\right)\left(\Delta+J\right)\left(-2+\Delta+J\right)$
$\displaystyle=\left(M_{\Delta,J}^{2}-M_{3+J,J}^{2}\right)\left(M_{\Delta,J}^{2}-M_{1+J,J}^{2}\right).$
(2.67)

We can see here the physical degrees of freedom which are the zeros of the
quadratic action. For $J=0$ we have the parity odd operator $\psi\psi$ with
$\Delta=J+2=2$. Recall that for $J=0$ the fourth structure does not exist, so
this is matched with the third structure. Recall also that for odd $J$’s we
only have the fourth structure, while for even $J$’s we have the other three.
For $J\geq 1$ we have the parity even operators
$\psi\gamma^{\mu}\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{J-1}}\psi$
which have $\Delta=J+1$ for which the second eigenvalue is zero. Another zero
mode exists for $\Delta=J+3$, which together with $\Delta=J+1$ both also exist
in the scalar theory. We know from the scalar theory that while $\Delta=J+1$
is a physical state with a positive norm, the $\Delta=J+3$ is a negative norm
state and is not physical.

Our CFT has four bilocal fields for every spin $J$, and we expect that these
will be related to bulk fields of spin $J$. To do so, we follow the derivation
of the mapping of scalar fields, given in section 1. Given
$G_{\Delta,J}\left(X,Y;S,Z\right)$, the basis of bulk-to-boundary propagators
of massive fields on $AdS$ with spin $J$ (1.18), one can write the expansion

$\Phi_{J}^{m}\left(X,S\right)=\intop_{P.S.}\frac{d\Delta}{2\pi
i}\intop\frac{dY}{J!\left(\frac{d}{2}-1\right)_{J}}C_{\Delta,J}^{bulk,m}\left(Y,D_{Z}\right)G_{\Delta,J}\left(X,Y;S,Z\right).$
(2.68)

One can invert this relation by writing a completeness relation for our basis.
Such a relation is given in [14] by defining the AdS harmonic function

$\Omega_{\Delta,J}\left(X_{1},X_{2},S_{1},S_{2}\right)=\intop\frac{dY}{J!\left(\frac{d}{2}-1\right)_{J}}\frac{G_{\Delta,J}\left(X_{1},Y,S_{1},D_{Z}\right)G_{\tilde{\Delta},J}\left(X_{2},Y,S_{3},Z\right)}{N_{\Delta,J}}$
(2.69)

where

$\displaystyle N_{\Delta,J}$
$\displaystyle=\frac{2\pi\Gamma\left(\Delta-\frac{d}{2}\right)\Gamma\left(\tilde{\Delta}-\frac{d}{2}\right)}{\Gamma\left(\Delta-1\right)\left(\Delta+J-1\right)\Gamma\left(\tilde{\Delta}-1\right)\left(\tilde{\Delta}+J-1\right)}.$
(2.70)

And the completeness relation is given by

$\displaystyle\delta\left(X_{1},X_{2}\right)\left\langle
S_{12}\right\rangle^{J}$ $\displaystyle=\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\Omega_{\Delta,J}\left(X_{1},X_{2},S_{1},S_{2}\right)+$ $\displaystyle\
+\sum_{\ell=1}^{J}\left(S_{1}\cdot\partial_{X_{1}}\right)^{\ell}\left(S_{2}\cdot\partial_{X_{2}}\right)^{\ell}\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}A_{\Delta,j,\ell}\Omega_{\Delta,J-\ell}\left(X_{1},X_{2},S_{1},S_{2}\right),$
(2.71) $\displaystyle A_{\Delta,j,\ell}$
$\displaystyle=\frac{2^{\ell}\left(J-\ell+1\right)_{\ell}\left(\frac{d}{2}+J-\ell-\frac{1}{2}\right)_{\ell}}{\ell!\left(d+2J-2\ell-1\right)_{\ell}\left(d+J-\ell-\Delta\right)_{\ell}\left(\Delta+J-\ell\right)_{\ell}}.$
(2.72)

When acting on traceless transverse $\left(TT\right)$ functions only, as is
our case, the second term can be discarded and we are left with just

$\delta^{\left(TT\right)}\left(X_{1},X_{2}\right)\left\langle
S_{12}\right\rangle^{J}=\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\Omega_{\Delta,J}\left(X_{1},X_{2},S_{1},S_{2}\right).$ (2.73)

We can now write

$C_{\Delta,J}^{bulk,m}\left(Y,Z\right)=\frac{1}{N_{\Delta,J}}\intop\frac{dX}{J!\left(\frac{d}{2}-1\right)_{J}}\Phi_{J}^{m}\left(X,K_{S}\right)G_{\tilde{\Delta},J}\left(X,Y;S,Z\right)$
(2.74)

where $K_{S}$ was defined in (1.21). we can now find an explicit mapping
between the boundary and bulk theories through

$C_{\Delta,J}^{bulk,m}\left(X,S\right)=f_{\Delta,J}^{m}c_{\Delta,J}^{m}\left(X,S\right)$
(2.75)

for some normalization functions $f_{\Delta,J}^{m}$. This normalization must
obey certain consistency conditions. For example, we can get such a condition
by applying the shadow relations to $c_{\Delta,J}^{m},C_{\Delta,J}^{bulk,m}$.
The shadow relation of $c_{\Delta,J}^{m}$ is given by (2.31), and we can write
a similar relation for the bulk

$G_{\Delta,J}\left(X,Y;S,Z\right)=\frac{1}{S_{B}^{\tilde{\Delta},J}}\intop\frac{dY^{\prime}}{J!\left(\frac{d}{2}-1\right)_{J}}G_{\Delta,J}\left(X,Y^{\prime};S,D_{Z^{\prime}}\right)\left\langle
O_{\Delta,J}\left(Y^{\prime},Z^{\prime}\right)O_{\Delta,J}\left(Y,Z\right)\right\rangle$
(2.76)

where the bulk shadow coefficients are given by (1.23). We then get

$\frac{f_{\Delta,J}^{m}}{f_{\tilde{\Delta},J}^{k}}=\left(\left(S_{\Delta_{\psi},\Delta_{\psi}}^{\tilde{\Delta},J}\right)^{-1}\right)_{\,k}^{m}S_{B}^{\tilde{\Delta},J}.$
(2.77)

Another condition is given by hermiticity of the CFT which implies

$f_{\Delta,J}^{m*}=\left(-1\right)^{J}f_{\Delta^{*},J}^{m},$ (2.78)

which can be satisfied by requiring $f_{\Delta,J}^{m}$ to be real analytic
functions times a factor of $\left(i\right)^{J}$. We can then write an
explicit mapping from $\rho\left(X_{1},X_{2},S_{1},S_{2}\right)$ to
$\Phi_{J}\left(X,W\right)$ by plugging (2.28) into (2.68) to get

$\displaystyle\Phi_{J}^{m}\left(X,W\right)$
$\displaystyle=\frac{1}{2}\sum_{k=1}^{4}\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}f_{\Delta,J}^{m}\left(\mathcal{B}_{\Delta,J}^{-1}\right)_{\,k}^{m}\intop\frac{dY}{J!\left(\frac{d}{2}-1\right)_{J}}G_{\Delta,J}\left(X,Y;W,D_{Z}\right)\times$
$\displaystyle\ \times\left\langle
O_{\tilde{\Delta}_{\psi}}\left(X_{1},D_{S_{1}}\right)O_{\tilde{\Delta}_{\psi}}\left(X_{2},D_{S_{2}}\right)O_{\tilde{\Delta},J}\left(Y,Z\right)\right\rangle^{k}\rho\left(X_{1},X_{2},S_{1},S_{2}\right).$
(2.79)

From here on we will write the summation over structures explicitly to avoid
confusion. This is an explicit mapping between the CFT degrees of freedom to
the bulk degrees of freedom. Our last step would be to write the bulk
quadratic action. First, we write the bilocal quadratic action (2.50) Using
embedding space coordinates, restoring spinor indices

$S^{\left(2\right)}=\left(4\pi\right)^{2}\sum_{m,n,k=1}^{4}\sum_{J=0}^{\infty}\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\intop\frac{dY}{J!\left(\frac{d}{2}-1\right)_{J}}c_{\Delta,J}^{m}\left(Y,D_{S}\right)\mathcal{B}_{\Delta,J,n}^{m}A_{\Delta,J,k}^{n}c_{\tilde{\Delta},J}^{k}\left(Y,S\right).$
(2.80)

We can now plug in the value of $c_{\Delta,J}^{m}$ in terms of the bulk
fields, to get

$S^{\left(2\right)}\left[\Phi_{J}\right]=\left(4\pi\right)^{2}\sum_{m,n,k=1}^{4}\sum_{J=0}^{\infty}\intop\frac{dX_{1}dX_{2}}{\left(J!\left(\frac{d}{2}-1\right)_{J}\right)^{2}}\Phi_{J}^{m}\left(X_{1},K_{S_{1}}\right)\Phi_{J}^{k}\left(X_{2},K_{S_{2}}\right)\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\frac{\mathcal{B}_{\Delta,J,n}^{m}A_{\Delta,J,k}^{n}}{N_{\Delta,J}f_{\Delta,J}^{m},f_{\tilde{\Delta},J}^{k}}\Omega_{\Delta,J}\left(X_{1},X_{2};S_{1},S_{2}\right).$
(2.81)

Now, recall from (2.65,2.66) that for the odd parity structures ($n,k=3,4$) we
can write

$\left(4\pi\right)^{2}A_{\Delta,J,k}^{n}=\left(-1\right)^{n}\left(M_{\Delta,J}^{2}-M_{J+2,J}^{2}\right)\delta_{k}^{n},$
(2.82)

recall that

$\nabla_{X_{1}}^{2}G_{\Delta,J}\left(X_{1},X_{2};S_{1},S_{2}\right)=M_{\Delta,J}^{2}G_{\Delta,J}\left(X_{1},X_{2};S_{1},S_{2}\right),$
(2.83)

so we can write for $n=3,4$

$\left(4\pi\right)^{2}\sum_{k=3,4}A_{\Delta,J,k}^{n}\Phi_{J}^{k}\left(X,S\right)=\left(-1\right)^{n}\left(\nabla_{X}^{2}-M_{J+2,J}^{2}\right)\Phi_{J}^{n}\left(X,S\right).$
(2.84)

It would be nice to have such a differential operator for the even parity
structures as well. That is, a differential operator in embedding space that
would give us the correct eigenvalues when acting on the even parity
structures. Assuming such an operator can be found, we could write

$\left(4\pi\right)^{2}\sum_{k=1}^{4}A_{\Delta,J,k}^{n}\Phi_{J}^{k}\Phi_{J}\left(X,S\right)=\mathcal{D}_{k}^{n}\Phi_{J}^{k}\left(X,S\right)$
(2.85)

where for $n=3,4$ we have
$\mathcal{D}_{k}^{n}=\left(\nabla_{X}^{2}-M_{J+2,J}^{2}\right)\delta_{k}^{n}$
and and for $n=1,2$ some yet unknown $\mathcal{D}_{k}^{n}$. We will then get

$S^{\left(2\right)}\left[\Phi_{J}\right]=\sum_{m,n,k=1}^{4}\sum_{J=0}^{\infty}\intop\frac{dX_{1}dX_{2}}{\left(J!\left(\frac{d}{2}-1\right)_{J}\right)^{2}}\Phi_{J}^{m}\left(X_{1},K_{S_{1}}\right)\mathcal{D}_{k}^{n}\Phi_{J}^{k}\left(X_{2},K_{S_{2}}\right)\intop_{\gamma_{J}}\frac{d\Delta}{2\pi
i}\frac{\mathcal{B}_{\Delta,J,n}^{m}}{N_{\Delta,J}f_{\Delta,J}^{m},f_{\tilde{\Delta},J}^{n}}\Omega_{\Delta,J}\left(X_{1},X_{2};S_{1},S_{2}\right).$
(2.86)

For generic $f_{\Delta,J}^{m}$, this quadratic action is non-local. However,
we can choose $f_{\Delta,J}^{m,\text{local}}$ such that

$\frac{\mathcal{B}_{\Delta,J,n}^{m}}{N_{\Delta,J}f_{\Delta,J}^{m},f_{\tilde{\Delta},J}^{n}}=\delta_{n}^{m},$
(2.87)

Which, from bulk completeness, would give us the local quadratic action

$S_{\text{local}}^{\left(2\right)}\left[\Phi_{J}\right]=\sum_{m=1}^{4}\sum_{J=0}^{\infty}\intop\frac{dX}{J!\left(\frac{d}{2}-1\right)_{J}}\Phi_{J}^{m}\left(X,K_{S}\right)\mathcal{D}_{k}^{n}\Phi_{J}^{k}\left(X,S\right).$
(2.88)

###  2.5 Finding a suitable $\mathcal{D}_{k}^{1,2}$ operator

The translation of the first two rows of (2.58) to bulk operators is not
straightforward, as they are not simply functions of $M_{\Delta,J}^{2}$ which
we can map to the bulk Laplacian. However their eigenvalues are such
functions, so we try to first change the basis so that it takes a simpler
form. We wish to write a $2\times 2$ matrix $\lambda$ such that

$\displaystyle\det\lambda$
$\displaystyle=-\left(M_{\Delta,J}^{2}-M_{3+J,J}^{2}\right)\left(M_{\Delta,J}^{2}-M_{1+J,J}^{2}\right),$
(2.89) $\displaystyle\mbox{Tr}\lambda$ $\displaystyle=2.$ (2.90)

Take the ansatz

$\lambda=\begin{pmatrix}1+\frac{1}{\sqrt{2}}\left(M_{\Delta,J}^{2}-N_{1}\right)&1+\frac{i}{\sqrt{2}}\left(M_{\Delta,J}^{2}-N_{2}\right)\\\
1-\frac{i}{\sqrt{2}}\left(M_{\Delta,J}^{2}-N_{2}\right)&1-\frac{1}{\sqrt{2}}\left(M_{\Delta,J}^{2}-N_{1}\right)\end{pmatrix}.$
(2.91)

It is clear that the trace condition is satisfied. The determinant is

$\displaystyle\det\lambda$
$\displaystyle=-\frac{1}{2}\left(M_{\Delta,J}^{2}-N_{1}\right)^{2}-\frac{1}{2}\left(M_{\Delta,J}^{2}-N_{2}\right)^{2}$
$\displaystyle=-M_{\Delta,J}^{4}+M_{\Delta,J}^{2}\left(N_{1}+N_{2}\right)-\frac{1}{2}\left(N_{1}^{2}+N_{2}^{2}\right)$
$\displaystyle=-M_{\Delta,J}^{4}+M_{\Delta,J}^{2}\left(M_{1+J,J}^{2}+M_{3+J,J}^{2}\right)-M_{3+J,J}^{2}M_{1+J,J}^{2},$
(2.92)

so we have

$\displaystyle
N_{1}^{2}-N_{1}\left(M_{1+J,J}^{2}+M_{3+J,J}^{2}\right)+\frac{1}{2}\left(M_{1+J,J}^{2}-M_{3+J,J}^{2}\right)^{2}$
$\displaystyle=0,$ (2.93)

which gives

$\displaystyle N_{1}$
$\displaystyle=\frac{1}{2}\left(M_{1+J,J}^{2}+M_{3+J,J}^{2}\right)+\frac{i}{2}\left(M_{1+J,J}^{2}-M_{3+J,J}^{2}\right),$
(2.94) $\displaystyle N_{2}$
$\displaystyle=\frac{1}{2}\left(M_{1+J,J}^{2}+M_{3+J,J}^{2}\right)-\frac{i}{2}\left(M_{1+J,J}^{2}-M_{3+J,J}^{2}\right)$
(2.95)

or equivalently the other way around. This gives us an explicit local operator
in the bulk, just by replacing $M_{\Delta,J}^{2}$ with $\nabla_{X_{1}}^{2}$.
The price we pay is that the operator is complex, and the meaning of that is
unclear, however we know it will not cause problems because it is equivalent
to the original CFT action which makes sense.

We make another attempt of finding such an operator, by turning our attention
back to the CFT. We wish to find an embedding space operator, which when
acting on even parity structures will give us the coefficients matrix which
appears in the first case of (2.58). As our coordinate space action is given
by
$S_{qa}=x_{12}^{2}\cancel{\partial}_{1\beta}^{\gamma}\cancel{\partial}_{2\alpha}^{\delta}$,
a natural candidate would be

$S_{emb}=-S_{1K}S_{2L}\Gamma_{AJ}^{K}\Gamma_{BI}^{L}\frac{\partial}{\partial
S_{1I}}\frac{\partial}{\partial
S_{2J}}X_{12}\partial_{1}^{A}\partial_{2}^{B}.$ (2.96)

Let’s see how this acts on our four 3-pt. structures. On the first structure
we get

$S_{emb}\left\langle\Psi_{1}\Psi_{2}O^{\Delta,J}\right\rangle^{1}=\left(d-\Delta+J\right)\left(J-\Delta\right)\left\langle\Psi_{1}\Psi_{2}O^{\Delta,J}\right\rangle^{1}-4J\left(\Delta-1\right)\left\langle\Psi_{1}\Psi_{2}O^{\Delta,J}\right\rangle^{2}.$
(2.97)

Recall that we had

$s_{qa}\left\langle\psi\psi
O^{\Delta,J}\right\rangle^{1}=\left(d-\Delta+J\right)\left(J-\Delta\right)\left\langle\psi\psi
O^{\Delta,J}\right\rangle^{1}-4J\left(\Delta-1\right)\left\langle\psi\psi
O^{\Delta,J}\right\rangle^{2},$ (2.98)

so this indeed seems to match what we expect from $S_{emb}$. For the second
structure

$\displaystyle
S_{emb}\left\langle\Psi_{1}\Psi_{2}O^{\Delta,J}\right\rangle^{2}$
$\displaystyle=\left(1-\Delta+J\right)\left(2-\Delta+J\right)\left\langle\Psi_{1}\Psi_{2}O^{\Delta,J}\right\rangle^{2}+\left(1-\Delta+J\right)\left(3-\Delta+J\right)\left\langle\Psi_{1}\Psi_{2}O^{\Delta,J}\right\rangle^{1}$
(2.99)

also, as expected from $S_{qa}$. For the third and fourth structures we get

$S_{emb}\left\langle\Psi_{1}\Psi_{2}O^{\Delta,J}\right\rangle^{3,4}=0.$
(2.100)

The fact that this operator maps the space spanned by the 3-pt. function into
a subspace of it, tells us that it is well-defined in embedding space, i.e.
that it preserves the ideal generated by the nullness and traverseness
conditions of the embedding space variables. We conclude that there is indeed
an embedding space differential operator which can act on the 3-pt. structures
and reproduce the correct coefficients $\lambda_{\Delta,J}$ on the even parity
structures, and in addition contributes nothing when acts on the odd parity
structures. We can hope that this operator can take a local form in AdS, by
acting with it on the bulk to boundary propagator basis and restricting the
embedding space coordinates to the AdS hyperboloid. This will be done in
future research.

##  3 Discussion

To summarize, this paper achieves an explicit mapping between the theory of
free Majorana fermions in 3 dimensions, and a high-spin gravity theory in
$AdS_{4}$, which we believe should be equivalent to type II Vasiliev gravity,
since the latter was also shown to lead to the same correlation functions as
the free fermion theory (at least in some cases that were tested, see [13]).
This mapping gives a quantization of a notoriously difficult theory of
gravity. There are however many questions that remain open and allow for much
future research:

  1. 1.

What is the explicit form of the bulk quadratic action for parity even
operators?

  2. 2.

Can this be generalized to other spacetime dimensions and fermion
representations? It would appear that this is the case as we have mainly used
3D Majorana fermions for simplicity and not for any specific properties they
have.

  3. 3.

How do the interactions of the bilocal theory map to the bulk and what are the
Feynman rules?

  4. 4.

What happens when we add deformations (mass, double trace, etc.) to the
theory?

  5. 5.

Can we show that the theory we get is equivalent to an off-shell version of
Vasiliev’s theory (which is so far known only as on-shell equations of
motion)?

  6. 6.

Can these results be generalized to finite $N$, volume and temperature

The last point was achieved recently by Aharony et al. [15] for the bosonic
case, and it stands to reason that it can similarly be done for the fermionic
case.

### Acknowledgments

I thank Ofer Aharony for advising me during this work. This work was supported
in part by an Israel Science Foundation (ISF) center for excellence grant
(grant number 2289/18), by ISF grant no. 2159/22, by Simons Foundation grant
994296 (Simons Collaboration on Confinement and QCD Strings), by grant no.
2018068 from the United States-Israel Binational Science Foundation (BSF), by
the Minerva foundation with funding from the Federal German Ministry for
Education and Research, by the German Research Foundation through a German-
Israeli Project Cooperation (DIP) grant “Holography and the Swampland”, and by
a research grant from Martin Eisenstein.

##  Appendix A Embedding space formalism

In this work we make use of the embedding space formalism, following the
notations of [11]. The $d$ dimensional Lorentzian conformal group is
isomorphic to $SO\left(d,2\right)$, which is the rotation group of $d+2$
dimensional flat spacetime with metric
$\eta_{AB}=\eta^{AB}=\text{diag}\left(-1,1,\cdots,1,-1\right)$,
$A,B=0,\cdots,d+1$. The $SO\left(d,2\right)$ generators $J^{AB}$ can be
related to the $d$ dimensional conformal group generators through
($\mu,\nu=0,\dots,d-1$)

$\displaystyle D$ $\displaystyle=-J^{d,d+1},\ P^{\mu}=J^{d,\mu}+J^{d+1,\mu},$
(A.1) $\displaystyle K^{\mu}$ $\displaystyle=-J^{d,\mu}+J^{d+1,\mu},\
M^{\mu\nu}=J^{\mu\nu}.$ (A.2)

We denote the embedding space coordinates $X^{A}$, and note that our $d$
dimensional spacetime is embedded as the projective null cone, which are the
points that satisfy $X\cdot X=0$ and are identified up to the rescaling
$X^{A}\sim\lambda X^{A}$. In light-cone coordinates
$X=\left(X^{\mu},X^{+},X^{-}\right)$ where $X^{\pm}=X^{d+1}\pm X^{d}$ we can
parameterize our light cone as

$x^{\mu}=\frac{X^{\mu}}{X^{+}},\ X=X^{+}\left(x^{\mu},1,x^{2}\right)$ (A.3)

where $x^{2}=\eta_{\mu\nu}x^{\mu}x^{\nu}$. Without loss of generality we can
take $X^{+}=1$. We define

$X_{ij}\equiv-2X_{i}\cdot X_{j}=-2x_{i}\cdot
x_{j}+x_{i}^{2}+x_{j}^{2}=\left(x_{i}-x_{j}\right)^{2}\equiv x_{ij}^{2}.$
(A.4)

We will be interested in spinor fields that are primaries of the conformal
group. We embed primary spinor fields by noting how the conformal group acts
on them

$\displaystyle i\left[M^{\mu\nu},\psi^{\alpha}\left(x\right)\right]$
$\displaystyle=\left(x^{\nu}\partial^{\mu}-x^{\mu}\partial^{\nu}\right)\psi^{\alpha}\left(x\right)-i\left(\mathcal{M}^{\mu\nu}\right)_{\,\beta}^{\alpha}\psi^{\beta}\left(x\right),$
$\displaystyle i\left[P^{\mu},\psi^{\alpha}\left(x\right)\right]$
$\displaystyle=-\partial^{\mu}\psi^{\alpha}\left(x\right),$ $\displaystyle
i\left[K^{\mu},\psi^{\alpha}\left(x\right)\right]$
$\displaystyle=\left(2x^{\mu}x^{\nu}\partial_{\nu}-x^{2}\partial^{\mu}+2\Delta_{\psi}x^{\mu}\right)\psi^{\alpha}\left(x\right)+2ix_{\nu}\left(\mathcal{M}^{\nu\mu}\right)_{\,\beta}^{\alpha}\psi^{\beta}\left(x\right),$
$\displaystyle i\left[D,\psi^{\alpha}\left(x\right)\right]$
$\displaystyle=\left(x^{\mu}\partial_{\mu}+\Delta_{\psi}\right)\psi^{\alpha}\left(x\right).$
(A.5)

Where
$\mathcal{M}^{\mu\nu}=-\frac{i}{4}\left[\gamma^{\mu},\gamma^{\nu}\right]$, and
$\alpha$,$\beta$ are spinor indices, which can be raised and lowered with the
symplectic form $\Omega^{\alpha\beta}=\Omega_{\alpha\beta}$ which preserves
$\Omega M^{\mu\nu}+\left(M^{\mu\nu}\right)^{T}\Omega=0$. To keep track of
spinor indices, we will introduce a set of auxiliary commuting variables
$s_{\alpha},\overline{s}^{\dot{\alpha}}$ such that

$\displaystyle\psi\left(x,s\right)$ $\displaystyle\equiv
s_{\alpha}\psi^{\alpha}\left(x\right),$ (A.6)
$\displaystyle\overline{\psi}\left(x,\overline{s}\right)$
$\displaystyle\equiv\overline{s}^{\dot{\alpha}}\overline{\psi}_{\dot{\alpha}}\left(x\right).$
(A.7)

In general we can write operators $\mathcal{O}$ in the
$\left(\ell,\overline{\ell}\right)$ spin representations (note that these
representations only exist in certain spacetime dimensions, but we will use
them for general dimensions and set $\ell=\overline{\ell},\
\alpha=\dot{\alpha}$ in spacetime dimensions where there is only a single spin
quantum number)

$\displaystyle\mathcal{O}\left(x,s,\overline{s}\right)$
$\displaystyle=s_{\alpha_{1}}\cdots
s_{\alpha_{\ell}}\overline{s}^{\dot{\alpha}_{1}}\cdots\overline{s}^{\dot{\alpha}_{\overline{\ell}}}\mathcal{O}_{\dot{\alpha}_{1}\cdots\dot{\alpha}_{\overline{\ell}}}^{\alpha_{1}\dots\alpha_{\ell}}\left(x\right),$
(A.8)
$\displaystyle\mathcal{O}_{\dot{\alpha}_{1}\cdots\dot{\alpha}_{\overline{\ell}}}^{\alpha_{1}\dots\alpha_{\ell}}\left(x\right)$
$\displaystyle=\frac{1}{\ell!\overline{\ell}!}\frac{\partial^{\ell}}{\partial
s_{\alpha_{1}}\cdots\partial
s_{\alpha_{\ell}}}\frac{\partial^{\overline{\ell}}}{\partial
s_{\dot{\alpha}_{1}}\cdots\partial
s_{\dot{\alpha}_{\overline{\ell}}}}\mathcal{O}\left(x,s,\overline{s}\right).$
(A.9)

Traceless symmetric tensors $\left(\ell,\ell\right)$ will have integer spin
and can be written as

$\mathcal{O}^{\mu_{1}\cdots\mu_{\ell}}\left(x\right)=\frac{\left(-1\right)^{\ell}}{2^{\ell}}\gamma_{\alpha_{1}}^{\mu_{1}\dot{\alpha}_{1}}\cdots\gamma_{\alpha_{\ell}}^{\mu_{\ell}\dot{\alpha}_{\ell}}\mathcal{O}_{\dot{\alpha}_{1}\cdots\dot{\alpha}_{\ell}}^{\alpha_{1}\dots\alpha_{\ell}}\left(x\right).$
(A.10)

Note that for real representations (e.g. Majorana fermions) we will have
$s_{\alpha}=\overline{s}^{\dot{\alpha}}$. In this work we use only these
representations, so we assume this from now on. Operators in embedding space
are identified with their physical space counterparts (up to a power of
$X^{+}$ which we will ignore)

$\mathcal{O}\left(X,S\right)=\mathcal{O}\left(x,s\right)$ (A.11)

with

$S_{I}=\begin{pmatrix}s_{\alpha}\\\ -x^{\alpha\beta}s_{\beta}\end{pmatrix},\
S^{I}=\begin{pmatrix}s_{\beta}x^{\beta\alpha}\\\ s_{\alpha}\end{pmatrix}$
(A.12)

where $x^{\alpha\beta}=x^{\mu}\gamma_{\mu}^{\alpha\beta}$. $S_{I}$ satisfies
the transversality condition $S_{I}X_{\,J}^{I}=0$ where
$X_{\,J}^{I}=X^{A}\Gamma_{AJ}^{I}$ and $\Gamma_{A}$ are the embedding space
Dirac matrices which satisfy the Clifford algebra
$\left\\{\Gamma_{A},\Gamma_{B}\right\\}=2\eta_{AB}$. We will define for 2 spin
variables $S_{i},S_{j}$ and a set of embedding space coordinates
$X_{1},\dots,X_{k}$

$\left\langle S_{i}X_{1}\cdots X_{k}S_{j}\right\rangle\equiv
S_{iI}X_{1J}^{I}\cdots X_{kL}^{K}\Omega^{LM}S_{jM}$ (A.13)

where $\Omega^{LM}$ is the embedding space symplectic form. Note that this
satisfies

$\displaystyle\left\langle S_{i}S_{j}\right\rangle$
$\displaystyle=-\left\langle S_{j}S_{i}\right\rangle,$ (A.14)
$\displaystyle\left\langle S_{i}S_{i}\right\rangle$ $\displaystyle=0,$ (A.15)
$\displaystyle\left\langle S_{k}X_{i}S_{k}\right\rangle$ $\displaystyle=0,$
(A.16) $\displaystyle\left\langle S_{k}X_{i}X_{j}S_{k}\right\rangle$
$\displaystyle=-\left\langle S_{k}X_{j}X_{i}S_{k}\right\rangle,$ (A.17)

as well as the Fierz identities (here we assume transversality
$S_{iI}X_{iJ}^{I}=0$)

$\displaystyle\left\langle S_{1}X_{2}S_{3}\right\rangle\left\langle
S_{2}X_{1}S_{3}\right\rangle$ $\displaystyle=-\left\langle
S_{1}S_{2}\right\rangle\left\langle
S_{3}X_{1}X_{2}S_{3}\right\rangle+2\left\langle
S_{2}S_{3}\right\rangle\left\langle S_{1}S_{3}\right\rangle X_{1}\cdot X_{2},$
(A.18) $\displaystyle\left\langle S_{1}X_{3}S_{2}\right\rangle\left\langle
S_{3}X_{1}X_{2}S_{3}\right\rangle$ $\displaystyle=-2\left\langle
S_{1}S_{3}\right\rangle\left\langle S_{2}X_{1}S_{3}\right\rangle X_{2}\cdot
X_{3}-2\left\langle S_{2}S_{3}\right\rangle\left\langle
S_{1}X_{2}S_{3}\right\rangle X_{1}\cdot X_{3}.$ (A.19)

Included here are some more useful identities, mostly used in section 2.5.

$\displaystyle\left\langle
S_{1}\Gamma_{A}\Gamma_{B}S_{2}\right\rangle\left\langle
S_{3}X_{1}\Gamma^{B}S_{3}\right\rangle\left\langle
S_{3}\Gamma^{A}X_{2}S_{3}\right\rangle$ $\displaystyle=4\left\langle
S_{3}X_{1}X_{2}S_{3}\right\rangle\left\langle
S_{1}S_{3}\right\rangle\left\langle S_{2}S_{3}\right\rangle,$ (A.20)
$\displaystyle\left\langle
S_{1}\Gamma_{A}\Gamma_{B}S_{2}\right\rangle\left\langle
S_{3}\Gamma^{A}\Gamma^{B}S_{3}\right\rangle$ $\displaystyle=8\left\langle
S_{1}S_{3}\right\rangle\left\langle S_{2}S_{3}\right\rangle,$ (A.21)
$\displaystyle\left\langle
S_{1}\Gamma_{A}\Gamma_{B}S_{2}\right\rangle\left(\left\langle
S_{3}\Gamma^{A}X_{2}S_{3}\right\rangle X_{1}^{B}+X_{2}^{A}\left\langle
S_{3}X_{1}\Gamma^{B}S_{3}\right\rangle\right)$ $\displaystyle=4\left\langle
S_{3}X_{1}X_{2}S_{3}\right\rangle\left\langle S_{1}S_{2}\right\rangle,$ (A.22)
$\displaystyle\left\langle
S_{1}\Gamma_{A}\Gamma_{B}S_{2}\right\rangle\left\langle
S_{3}\Gamma^{A}X_{2}S_{3}\right\rangle X_{3}^{B}$
$\displaystyle=-2\left\langle S_{1}S_{3}\right\rangle\left\langle
S_{2}S_{3}\right\rangle X_{23},$ (A.23) $\displaystyle\left\langle
S_{1}\Gamma_{A}\Gamma_{B}S_{2}\right\rangle\left\langle
S_{3}X_{1}\Gamma^{B}S_{3}\right\rangle X_{3}^{A}$
$\displaystyle=-2\left\langle S_{1}S_{3}\right\rangle\left\langle
S_{2}S_{3}\right\rangle X_{31},$ (A.24) $\displaystyle\left\langle
S_{1}\Gamma_{A}\Gamma_{B}S_{2}\right\rangle\eta^{AB}$
$\displaystyle=D\left\langle S_{1}S_{2}\right\rangle,$ (A.25)
$\displaystyle\left\langle S_{1}\Gamma_{A}\Gamma_{B}S_{2}\right\rangle
X_{3}^{A}X_{1}^{B}$ $\displaystyle=-\left\langle S_{1}S_{2}\right\rangle
X_{31},$ (A.26) $\displaystyle\left\langle
S_{1}\Gamma_{A}\Gamma_{B}S_{2}\right\rangle X_{3}^{A}X_{3}^{B}$
$\displaystyle=0,$ (A.27) $\displaystyle\left\langle
S_{1}\Gamma_{A}\Gamma_{B}S_{2}\right\rangle X_{2}^{A}X_{1}^{B}$
$\displaystyle=-\left\langle S_{1}S_{2}\right\rangle X_{12},$ (A.28)
$\displaystyle\left\langle
S_{1}\Gamma_{A}\Gamma_{B}S_{2}\right\rangle\left(X_{2}^{A}X_{3}^{B}+X_{2}^{B}X_{3}^{A}\right)$
$\displaystyle=-\left\langle S_{1}S_{2}\right\rangle X_{23},$ (A.29)
$\displaystyle\left\langle S_{1}\Gamma_{A}S_{3}\right\rangle\left\langle
S_{2}\Gamma_{B}S_{3}\right\rangle\eta^{AB}$ $\displaystyle=\left\langle
S_{1}S_{3}\right\rangle\left\langle S_{2}S_{3}\right\rangle,$ (A.30)
$\displaystyle\left\langle S_{1}\Gamma_{A}S_{3}\right\rangle\left\langle
S_{2}\Gamma_{B}S_{3}\right\rangle X_{2}^{A}X_{1}^{B}$
$\displaystyle=\left\langle S_{1}X_{2}S_{3}\right\rangle\left\langle
S_{2}X_{1}S_{3}\right\rangle.$ (A.31)

Another useful identity for $D=5$ Dirac matrices is

$\Gamma_{\
\,J}^{AI}\Gamma_{AL}^{K}=2\delta_{\,J}^{K}\delta_{\,L}^{I}-\delta_{\,J}^{I}\delta_{\,L}^{K}+2\Omega^{IK}\Omega_{JL}.$
(A.32)

##  Appendix B Useful properties of the fermion 3-pt. functions

###  B.1 Calculation of 3-pt. function pairings

This appendix contains properties of the fermion 3-pt. functions used
throughout the paper. We work with these functions by noting that they can be
expanded using the scalar 3-pt. functions when operated on by weight shifting
operators. The expansion is given by [10]

$\left\langle\psi_{1}^{\Delta_{1}}\psi_{2}^{\Delta_{2}}O^{\Delta,J}\right\rangle^{m}=\sum_{p,q=\pm
1}\kappa_{3,pq}^{m}\left(\psi_{1}^{\Delta_{1}}\psi_{2}^{\Delta_{2}}O^{\Delta,J}\right)\mathcal{D}_{1}^{-p,+}\mathcal{D}_{2}^{-q,+}\left\langle\phi_{1}^{\Delta_{1}+p/2}\phi_{2}^{\Delta_{2}+q/2}O^{\Delta,J}\right\rangle$
(B.1)

where
$\kappa_{3,pq}^{m}\left(\psi_{1}^{\Delta_{1}}\psi_{2}^{\Delta_{2}}O^{\Delta,J}\right)$
are the expansion coefficients and can be expressed as an invertible $4\times
4$ matrix, and $\mathcal{D}_{i}^{pq}$ are the weight shifting operators acting
on the $i$th field at $\left(X_{i},S_{i}\right)$ and are given in embedding
space by

$\displaystyle\mathcal{D}_{I}^{-+}$ $\displaystyle=S_{I},$ (B.2)
$\displaystyle\mathcal{D}_{I}^{++}$
$\displaystyle=-2\left(\Delta-1\right)S_{J}\left(\partial_{X}\right)_{\,I}^{J}-S_{I}S_{J}\left(\partial_{X}\right)_{\,K}^{J}\frac{\partial}{\partial
S_{K}},$ (B.3) $\displaystyle\mathcal{D}_{I}^{--}$
$\displaystyle=X_{IJ}\frac{\partial}{\partial S_{J}},$ (B.4)
$\displaystyle\mathcal{D}_{I}^{+-}$
$\displaystyle=4\left(1+\ell-\Delta\right)\left(\Delta-1\right)\frac{\partial}{\partial
S^{I}}+2\left(1+\ell-\Delta\right)X_{IJ}\left(\partial_{X}\right)_{\,K}^{J}\frac{\partial}{\partial
S_{K}}-S_{I}\frac{\partial}{\partial
S_{J}}X_{JK}\left(\partial_{X}\right)_{\,L}^{K}\frac{\partial}{\partial
S_{L}},$ (B.5)

where $A,B,C,D$ are embedding space spinor indices, and
$\left(\partial_{X}\right)_{\,J}^{I}=\Gamma_{AJ}^{I}\frac{\partial}{\partial
X_{A}}$. We can calculate the coefficients $\kappa_{3}$ using

$\displaystyle\mathcal{D}_{1}^{-+}\mathcal{D}_{2}^{-+}\left\langle\phi_{1}^{\Delta_{\psi}+1/2}\phi_{2}^{\Delta_{\psi}+1/2}O^{\Delta,J}\right\rangle$
$\displaystyle=\left\langle\psi_{1}\psi_{2}O^{\Delta,J}\right\rangle^{1},$
(B.6)
$\displaystyle\mathcal{D}_{1}^{++}\mathcal{D}_{2}^{-+}\left\langle\phi_{1}^{\Delta_{\psi}-1/2}\phi_{2}^{\Delta_{\psi}+1/2}O^{\Delta,J}\right\rangle$
$\displaystyle=2\left(\Delta_{\psi}-3/2\right)\left(\left(\Delta+2J-1\right)\left\langle\psi_{1}\psi_{2}O^{\Delta,J}\right\rangle^{3}+J\left\langle\psi_{1}\psi_{2}O^{\Delta,J}\right\rangle^{4}\right),$
(B.7)
$\displaystyle\mathcal{D}_{1}^{-+}\mathcal{D}_{2}^{++}\left\langle\phi_{1}^{\Delta_{\psi}+1/2}\phi_{2}^{\Delta_{\psi}-1/2}O^{\Delta,J}\right\rangle$
$\displaystyle=2\left(\Delta_{\psi}-3/2\right)\left(\left(\Delta+2J-1\right)\left\langle\psi_{1}\psi_{2}O^{\Delta,J}\right\rangle^{3}-J\left\langle\psi_{1}\psi_{2}O^{\Delta,J}\right\rangle^{4}\right),$
(B.8)
$\displaystyle\mathcal{D}_{1}^{++}\mathcal{D}_{2}^{++}\left\langle\phi_{1}^{\Delta_{\psi}-1/2}\phi_{2}^{\Delta_{\psi}-1/2}O^{\Delta,J}\right\rangle$
$\displaystyle=-4\left(\Delta_{\psi}-3/2\right)^{2}\left(4J\left(J-3\right)\left\langle\psi_{1}\psi_{2}O^{\Delta,J}\right\rangle^{2}+\right.$
$\displaystyle\quad\left.+\left(2\Delta_{\psi}-\Delta+J+1\right)\left(2\Delta_{\psi}-\Delta+J-2\right)\left\langle\psi_{1}\psi_{2}O^{\Delta,J}\right\rangle^{1}\right).$
(B.9)

This expansion can be used to calculate pairings of 3-pt. functions, which are
defined by These obey the property of integration by parts of the weight
shifting operators

$\displaystyle\left(\mathcal{D}_{\Delta,J}^{pq}\mathcal{O},\tilde{\mathcal{O}}^{\prime}\right)$
$\displaystyle=\left(\mathcal{O},\left(\mathcal{D}_{\Delta,J}^{pq}\right)^{*}\tilde{\mathcal{O}}^{\prime}\right)$
$\displaystyle=\left(\mathcal{O},\zeta_{J}^{pq}\mathcal{D}_{\tilde{\Delta}-\frac{p}{2},J+\frac{q}{2}}^{p,-q}\tilde{\mathcal{O}}^{\prime}\right)$
(B.10)

where

$\displaystyle\zeta_{J}^{--}$ $\displaystyle=-\zeta_{J}^{+-}=-2J,$ (B.11)
$\displaystyle\zeta_{J}^{-+}$ $\displaystyle=-\zeta_{J}^{++}=\frac{1}{2J+1}.$
(B.12)

And we get

$\left(\left\langle\psi_{1}^{\Delta_{1}}\psi_{2}^{\Delta_{2}}O^{\Delta,J}\right\rangle^{m},\left\langle\psi_{1}^{\tilde{\Delta}_{1}}\psi_{2}^{\tilde{\Delta}_{2}}O^{\tilde{\Delta},J}\right\rangle^{n}\right)=\begin{cases}\frac{1}{8\pi}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}&J=0\\\
\frac{\left(-1\right)^{J}\Gamma\left(J+1\right)}{8\sqrt{\pi}\Gamma\left(J+\frac{1}{2}\right)}\begin{pmatrix}-1&1/2&0&0\\\
1/2&-\frac{2J+1}{4J}&0&0\\\ 0&0&1&0\\\
0&0&0&-\frac{J+1}{J}\end{pmatrix}&J>0\end{cases}.$ (B.13)

###  B.2 Shadow coefficients for the orthogonality of the 3-pt. functions

The shadow coefficients that can also calculated using the expansion method
above. This gives (Note that these generally depend on the difference between
the conformal weights of the 2 fermions. We assume here that the fermions are
identical so there is no dependence on $\Delta_{\psi}$)

$\displaystyle
S_{1}^{1}\left(\psi\psi\left[\mathcal{O}_{\Delta,J}\right]\right)$
$\displaystyle=\frac{\pi^{3/2}\left(-1\right)^{J}\Gamma\left(J+\Delta-1\right)\Gamma\left(\frac{J+\tilde{\Delta}}{2}\right)^{2}}{\Gamma\left(\Delta-1\right)\Gamma\left(J+\tilde{\Delta}\right)\Gamma\left(\frac{J+\Delta}{2}\right)^{2}},$
(B.14) $\displaystyle
S_{1}^{2}\left(\psi\psi\left[\mathcal{O}_{\Delta,J}\right]\right)$
$\displaystyle=-\frac{\Delta-3/2}{\Delta-1}S_{1}^{1}\left(\psi\psi\left[\mathcal{O}_{\Delta,J}\right]\right),$
(B.15) $\displaystyle
S_{2}^{2}\left(\psi\psi\left[\mathcal{O}_{\Delta,J}\right]\right)$
$\displaystyle=-\frac{\Delta-2}{\Delta-1}S_{1}^{1}\left(\psi\psi\left[\mathcal{O}_{\Delta,J}\right]\right),$
(B.16) $\displaystyle
S_{3}^{3}\left(\psi\psi\left[\mathcal{O}_{\Delta,J}\right]\right)$
$\displaystyle=\frac{\left(\Delta-1\right)\left(\Delta-2\right)^{2}-\left(\Delta-2\right)J\left(J+1\right)}{2\left(2\Delta-3\right)\left(-\Delta+J+2\right)\left(\Delta+J-1\right)}S_{1}^{1}\left(\psi\psi\left[\mathcal{O}_{\Delta+1,J}\right]\right),$
(B.17) $\displaystyle
S_{4}^{4}\left(\psi\psi\left[\mathcal{O}_{\Delta,J}\right]\right)$
$\displaystyle=\frac{-\left(\Delta-2\right)\left(\Delta-1\right)^{2}+\left(\Delta-1\right)J\left(J+1\right)}{2\left(2\Delta-3\right)\left(-\Delta+J+2\right)\left(\Delta+J-1\right)}S_{1}^{1}\left(\psi\psi\left[\mathcal{O}_{\Delta+1,J}\right]\right),$
(B.18)

and the rest vanish.

###  B.3 Shadow coefficients for the shadow transform of the 3-pt. functions

The shadow coefficients for equation (2.37) are

$\displaystyle
S_{3}^{1}\left(\left[\psi^{\Delta_{1}}\right]\psi^{\Delta_{2}}\mathcal{O}^{\Delta,J}\right)$
$\displaystyle=\frac{i\pi^{3/2}\left(-\Delta+\Delta_{1}+\Delta_{2}-2\right)\Gamma\left(\Delta_{1}-1\right)\Gamma\left(\frac{J+\Delta-\Delta_{1}-\Delta_{2}+2}{2}\right)\Gamma\left(\frac{J-\Delta-\Delta_{1}+\Delta_{2}+3}{2}\right)}{2\Gamma\left(\frac{7}{2}-\Delta_{1}\right)\Gamma\left(\frac{J+\Delta+\Delta_{1}-\Delta_{2}}{2}\right)\Gamma\left(\frac{J-\Delta+\Delta_{1}+\Delta_{2}+1}{2}\right)},$
(B.19) $\displaystyle
S_{3}^{1}\left(\psi^{\Delta_{1}}\left[\psi^{\Delta_{2}}\right]\mathcal{O}^{\Delta,J}\right)$
$\displaystyle=S_{3}^{1}\left(\left[\psi^{\Delta_{2}}\right]\psi^{\Delta_{1}}\mathcal{O}^{\Delta,J}\right),$
(B.20)
$\displaystyle\frac{S_{4}^{1}\left(\left[\psi^{\Delta_{1}}\right]\psi^{\Delta_{2}}\mathcal{O}^{\Delta,J}\right)}{\frac{J}{\Delta-\Delta_{1}-\Delta_{2}+2}}$
$\displaystyle=\frac{S_{3}^{2}\left(\left[\psi^{\Delta_{1}}\right]\psi^{\Delta_{2}}\mathcal{O}^{\Delta,J}\right)}{\frac{-\Delta+\Delta_{1}+\Delta_{2}+J-1}{2\left(\Delta-\Delta_{1}-\Delta_{2}+2\right)}}=\frac{S_{4}^{2}\left(\left[\psi^{\Delta_{1}}\right]\psi^{\Delta_{2}}\mathcal{O}^{\Delta,J}\right)}{\frac{\Delta-\Delta_{1}-\Delta_{2}-J+1}{2\left(\Delta-\Delta_{1}-\Delta_{2}+2\right)}}=S_{3}^{1}\left(\left[\psi^{\Delta_{1}}\right]\psi^{\Delta_{2}}\mathcal{O}^{\Delta,J}\right),$
(B.21)
$\displaystyle\frac{S_{1}^{3}\left(\left[\psi^{\Delta_{1}}\right]\psi^{\Delta_{2}}\mathcal{O}^{\Delta,J}\right)}{\frac{-\Delta-\Delta_{1}+\Delta_{2}+J+2}{\Delta+\Delta_{1}-\Delta_{2}-2}}$
$\displaystyle=\frac{S_{2}^{3}\left(\left[\psi^{\Delta_{1}}\right]\psi^{\Delta_{2}}\mathcal{O}^{\Delta,J}\right)}{\frac{2J}{\Delta+\Delta_{1}-\Delta_{2}-2}}=\frac{S_{2}^{4}\left(\left[\psi^{\Delta_{1}}\right]\psi^{\Delta_{2}}\mathcal{O}^{\Delta,J}\right)}{\frac{-2\Delta-2\Delta_{1}+2\Delta_{2}+2}{\Delta+\Delta_{1}-\Delta_{2}-2}}=S_{3}^{1}\left(\left[\psi^{\Delta_{1}}\right]\psi^{-\Delta_{2}}\mathcal{O}^{-\Delta,J}\right),$
(B.22) $\displaystyle
S_{1}^{4}\left(\left[\psi^{\Delta_{1}}\right]\psi^{\Delta_{2}}\mathcal{O}^{\Delta,J}\right)$
$\displaystyle=S_{1}^{3}\left(\left[\psi^{\Delta_{1}}\right]\psi^{\Delta_{2}}\mathcal{O}^{\Delta,J}\right),$
(B.23)
$\displaystyle\frac{S_{4}^{1}\left(\psi^{\Delta_{1}}\left[\psi^{\Delta_{2}}\right]\mathcal{O}^{\Delta,J}\right)}{\frac{-J}{\Delta-\Delta_{1}-\Delta_{2}+2}}$
$\displaystyle=\frac{S_{3}^{2}\left(\psi^{\Delta_{1}}\left[\psi^{\Delta_{2}}\right]\mathcal{O}^{\Delta,J}\right)}{\frac{-\Delta+\Delta_{1}+\Delta_{2}+J-1}{2\left(\Delta-\Delta_{1}-\Delta_{2}+2\right)}}=\frac{S_{4}^{2}\left(\psi^{\Delta_{1}}\left[\psi^{\Delta_{2}}\right]\mathcal{O}^{\Delta,J}\right)}{\frac{-\Delta+\Delta_{1}+\Delta_{2}+J-1}{2\left(\Delta-\Delta_{1}-\Delta_{2}+2\right)}}=S_{3}^{1}\left(\psi^{\Delta_{1}}\left[\psi^{\Delta_{2}}\right]\mathcal{O}^{\Delta,J}\right),$
(B.24)
$\displaystyle\frac{S_{1}^{3}\left(\psi^{\Delta_{1}}\left[\psi^{\Delta_{2}}\right]\mathcal{O}^{\Delta,J}\right)}{\frac{-\Delta+\Delta_{1}-\Delta_{2}+J+2}{\Delta-\Delta_{1}+\Delta_{2}-2}}$
$\displaystyle=\frac{S_{2}^{3}\left(\psi^{\Delta_{1}}\left[\psi^{\Delta_{2}}\right]\mathcal{O}^{\Delta,J}\right)}{\frac{2J}{\Delta-\Delta_{1}+\Delta_{2}-2}}=\frac{S_{2}^{4}\left(\psi^{\Delta_{1}}\left[\psi^{\Delta_{2}}\right]\mathcal{O}^{\Delta,J}\right)}{\frac{2\Delta-2\Delta_{1}+2\Delta_{2}-2}{\Delta-\Delta_{1}+\Delta_{2}-2}}=S_{3}^{1}\left(\psi^{-\Delta_{1}}\left[\psi^{\Delta_{2}}\right]\mathcal{O}^{-\Delta,J}\right),$
(B.25) $\displaystyle
S_{1}^{4}\left(\psi^{\Delta_{1}}\left[\psi^{\Delta_{2}}\right]\mathcal{O}^{\Delta,J}\right)$
$\displaystyle=-S_{1}^{3}\left(\psi^{\Delta_{1}}\left[\psi^{\Delta_{2}}\right]\mathcal{O}^{\Delta,J}\right).$
(B.26)

The other coefficients vanish.

##  Appendix C Mathematica code

The following is a Mathematica code which defines the 3-point structures and
acts on them with the operator $S_{qa}$ to give (2.47).

Embedding space $d=5$

  * d = 5;

Metric $eta=\eta_{AB}$ and inverse metric $etaa$

  * eta = {{-1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0},

{0, 0, 0, 0, -1/2}, {0, 0, 0, -1/2, 0}};

etaa = Inverse[eta];

Defining the spacetime coordinates $X_{1}^{A},X_{2}^{A},X_{3}^{A}$

  * x1 = Table[X1[i], {i, 0, d - 1, 1}];

x2 = Table[X2[i], {i, 0, d - 1, 1}];

x3 = Table[X3[i], {i, 0, d - 1, 1}];

Defining the spin coordinates $S_{1I},S_{2I},S_{3I}$

  * s1 = Table[S1[i], {i, 1, 4, 1}];

s2 = Table[S2[i], {i, 1, 4, 1}];

s3 = Table[S3[i], {i, 1, 4, 1}];

Defining the Dirac matrices

  * G[0] = {{0, 1, 0, 0}, {-1, 0, 0, 0}, {0, 0, 0, -1}, {0, 0, 1, 0}};

G[1] = {{0, 1, 0, 0}, {1, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 1, 0}};

G[2] = {{1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, -1}};

G[3] = {{0, 0, 0, 1}, {0, 0, -1, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}};

G[4] = {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 1, 0, 0}, {-1, 0, 0, 0}};

g = {G[0], G[1], G[2], G[3], G[4]};(*These are $\Gamma_{A}$*)

gg = {-G[0], G[1], G[2], -2 G[4], -2 G[3]};(*These are $\Gamma^{A}$*)

Defining the spacetime coordinates in spin space $X_{\,J}^{I}$

  * X1IJ = Sum[X1[i] G[i], {i, 0, 4}];

X2IJ = Sum[X2[i] G[i], {i, 0, 4}];

X3IJ = Sum[X3[i] G[i], {i, 0, 4}];

Defining the spin space symplectic form $\Omega^{IJ}=\Omega_{IJ}$

  * Om = {{0, 0, 1, 0}, {0, 0, 0, 1}, {-1, 0, 0, 0}, {0, -1, 0, 0}};

Defining spin bilinears

  * S1S2 = s1.Om.s2;(*$\left\langle S_{1}S_{2}\right\rangle$*)

S3X1X2S3 = s3.X1IJ.X2IJ.Om.s3;(*$\left\langle
S_{3}X_{1}X_{2}S_{3}\right\rangle$*)

S1X2S3 = s1.X2IJ.Om.s3;(*$\left\langle S_{1}X_{2}S_{3}\right\rangle$*)

S2X1S3 = s2.X1IJ.Om.s3;(*$\left\langle S_{2}X_{1}S_{3}\right\rangle$*)

S1S3 = s1.Om.s3;(*$\left\langle S_{1}S_{3}\right\rangle$*)

S2S3 = s2.Om.s3;(*$\left\langle S_{2}S_{3}\right\rangle$*)

S1X3S2 = s1.X3IJ.Om.s2;(*$\left\langle S_{1}X_{3}S_{2}\right\rangle$*)

Defining Light-cone ($X^{2}=0$) and transversality ($S_{I}X_{\,J}^{I}=0$)
constraints

  * r1lc = {X1[3] -> 1, X1[4] -> -X1[0]^2 + X1[1]^2 + X1[2]^2};

r1tr = Solve[(s1.X1IJ /. r1lc) == {0, 0, 0, 0}, {S1[1], S1[2]}];

r2lc = {X2[3] -> 1, X2[4] -> -X2[0]^2 + X2[1]^2 + X2[2]^2};

r2tr = Solve[(s2.X2IJ /. r2lc) == {0, 0, 0, 0}, {S2[1], S2[2]}];

r3lc = {X3[3] -> 1, X3[4] -> -X3[0]^2 + X3[1]^2 + X3[2]^2};

r3tr = Solve[(s3.X3IJ /. r3lc) == {0, 0, 0, 0}, {S3[1], S3[2]}];

r = Join[r1lc, r1tr[[1]], r2lc, r2tr[[1]], r3lc, r3tr[[1]]];

Defining 3-pt correlation functions

  * X12 = -2 x1.eta.x2;

X23 = -2 x2.eta.x3;

X31 = -2 x3.eta.x1;

$\phi$1$\phi$2O = Simplify[(S3X1X2S3)^J/

(X12^((Δ1+Δ2-Δ+J)/2) X23^((Δ2+Δ-Δ1+J)/2) X31^((Δ1+Δ-Δ2+J)/2))];

ψ1ψ2O1 = Simplify[S1S2 (S3X1X2S3)^J/

(X12^((2Δψ-Δ+J+1)/2) X23^((Δ+J)/2) X31^((Δ+J)/2))];

ψ1ψ2O2 = Simplify[S2S3 S1S3 (S3X1X2S3)^(J-1)/

(X12^((2Δψ-Δ+J-1)/2) X23^((Δ+J)/2) X31^((Δ+J)/2))];

ψ1ψ2O3 = Simplify[S2X1S3 S1S3 (S3X1X2S3)^(J-1)/

(X12^((2Δψ-Δ+J)/2) X23^((Δ+J-1)/2) X31^((Δ+J+1)/2)) +

S1X2S3 S2S3 (S3X1X2S3)^(J-1)/

(X12^((2Δψ-Δ+J)/2) X23^((Δ+J+1)/2) X31^((Δ+J-1)/2))];

ψ1ψ2O4 = Simplify[S2X1S3 S1S3 (S3X1X2S3)^(J-1)/

(X12^((2Δψ-Δ+J)/2) X23^((Δ+J-1)/2) X31^((Δ+J+1)/2)) -

S1X2S3 S2S3 (S3X1X2S3)^(J-1)/

(X12^((2Δψ-Δ+J)/2) X23^((Δ+J+1)/2) X31^((Δ+J-1)/2))];

Checking the Fierz identities

  * F1 = Simplify[S1X2S3 S2X1S3 + S1S2 S3X1X2S3 + S2S3 S1S3 X12 /.r]

>>> 0

F2 = Simplify[S1X3S2 S3X1X2S3 - S1S3 S2X1S3 X23 - S2S3 S1X2S3 X13 /.r]

>>> 0

Test points

  * p1 = {X1[0] -> 1, X1[1] -> 0, X1[2] -> 1, X1[3] -> 1, X1[4] -> 0,

X2[0] -> 0,X2[1] -> 1, X2[2] -> 0, X2[3] -> 1,X2[4] -> 1,

X3[0] -> 0, X3[1] -> 0,X3[2] -> 1, X3[3] -> 1, X3[4] -> 1};

pp = {X1[0] -> 0, X1[1] -> 0, X1[2] -> 0, X1[3] -> 1, X1[4] -> 0,

X2[0] -> 0,X2[1] -> 1, X2[2] -> 0, X2[3] -> 1, X2[4] -> 1};

ppp = {X1[0] -> 0, X1[1] -> 0, X1[2] -> 0, X1[3] -> 1, X1[4] -> 0};

p = {X1[0] -> 0, X1[1] -> 0, X1[2] -> 0, X1[3] -> 1, X1[4] -> 0,

X2[0] -> 0,X2[1] -> 1, X2[2] -> 0, X2[3] -> 1,X2[4] -> 1,

X3[0] -> 0, X3[1] -> 1/2,X3[2] -> Sqrt[3]/2, X3[3] -> 1,X3[4] -> 1};

(*Three equidistant points in the t=0 plane in 3D spacetime*)

Defining 3D Gamma matrices, spinors and bispinors

  * G3[0] = {{0, -1}, {1, 0}};

G3[1] = {{0, 1}, {1, 0}};

G3[2] = {{1, 0}, {0, -1}};

g3 = {G3[0], G3[1], G3[2]}; (*These are $\gamma_{\alpha}$*)

gg3 = {-G3[0], G3[1], G3[2]}; (*These are $\gamma^{\alpha}$*)

s13 = {S1[3], S1[4]};

s23 = {S2[3], S2[4]};

eab = {{0, -1}, {1, 0}};

s2gs1 = {s23.G3[0].eab.s13, s23.G3[1].eab.s13, s23.G3[2].eab.s13};

s1gs2 = {s13.G3[0].eab.s23, s13.G3[1].eab.s23, s13.G3[2].eab.s23};

3D bilocal action: $Action=x_{12}^{2}\left\langle
s_{1}\partial_{1}\partial_{s_{2}}\right\rangle\left\langle
s_{2}\partial_{2}\partial_{s_{1}}\right\rangle$

  * Action[x_] := X12 Sum[(s13.(gg3[[m]]))[[b]] (s23.(gg3[[n]]))[[a]]

D[x /.r, s13[[b]], s23[[a]], x1[[m]], x2[[n]]],

{a, 1, 2}, {b, 1, 2}, {m, 1, 3}, {n, 1, 3}]

A1 = Simplify[Action[ψ1ψ2O1 /.r] -

((3+J-Δ)(J-Δ)ψ1ψ2O1 - 4J(Δ-1)ψ1ψ2O2) /.r /.pp /.Δψ -> 1]

>>> 0 (*This shows $S_{qa}\left\langle\psi\psi
O\right\rangle^{1}=\left(3+J-\Delta\right)\left(J-\Delta\right)\left\langle\psi\psi
O\right\rangle^{1}-4J\left(\Delta-1\right)\left\langle\psi\psi
O\right\rangle^{2}$*)

A2 = Simplify[Action[ψ1ψ2O2 /.r] +

((3+J-Δ)(1+J-Δ)ψ1ψ2O1 + (2+J-Δ)(1+J-Δ)ψ1ψ2O2) /.r /.pp /.Δψ -> 1]

>>> 0 (*This shows $S_{qa}\left\langle\psi\psi
O\right\rangle^{2}=-\left(3+J-\Delta\right)\left(1+J-\Delta\right)\left\langle\psi\psi
O\right\rangle^{1}-(2+J-\text{\leavevmode{D}})(1+J-\text{\leavevmode{D}})\left\langle\psi\psi
O\right\rangle^{2}$*)

A3 = Factor[Simplify[Action[ψ1ψ2O3 /. r]/ψ1ψ2O3 /.r /.pp /.Δψ -> 1]]

>>> -(2 + J - Δ) (-1 + J + Δ)

(*This shows $S_{qa}\left\langle\psi\psi
O\right\rangle^{3}=-\left(2+J-\Delta\right)\left(-1+J+\Delta\right)\left\langle\psi\psi
O\right\rangle^{3}$*)

A4 = Factor[Simplify[Action[ψ1ψ2O4 /. r]/ψ1ψ2O4 /.r /.pp /.Δψ -> 1]]

>>> (2 + J - Δ) (-1 + J + Δ)

(*This shows $S_{qa}\left\langle\psi\psi
O\right\rangle^{4}=\left(2+J-\Delta\right)\left(-1+J+\Delta\right)\left\langle\psi\psi
O\right\rangle^{4}$*)

Checking the action of $S_{emb}$ on the 3-pt. structures

  * ActionEmb[x_] := Sum[-X12 (s1.(gg[[a]]))[[j]] (s2.(gg[[b]]))[[i]]

D[x, s1[[i]], s2[[j]], x1[[a]], x2[[b]]],

{a, 1, d}, {b, 1, d}, {i, 1, 4}, {j, 1, 4}]

AE1 = Simplify[ActionEmb[ψ1ψ2O1] -

((3+J-Δ)(J-Δ)ψ1ψ2O1 - 4J(Δ-1)ψ1ψ2O2) /.r /.p /.Δψ -> 1]

>>> 0 (*This shows $S_{emb}\left\langle\psi\psi
O\right\rangle^{1}=\left(3+J-\Delta\right)\left(J-\Delta\right)\left\langle\psi\psi
O\right\rangle^{1}-4J\left(\Delta-1\right)\left\langle\psi\psi
O\right\rangle^{2}$*)

AE2 = Simplify[ActionEmb[ψ1ψ2O2] -

((3+J-Δ)(1+J-Δ)ψ1ψ2O1 + (2+J-Δ)(1+J-Δ)ψ1ψ2O2) /.r /.p /.Δψ -> 1]

>>> 0 (*This shows $S_{emb}\left\langle\psi\psi
O\right\rangle^{2}=\left(3+J-\Delta\right)\left(1+J-\Delta\right)\left\langle\psi\psi
O\right\rangle^{1}+(2+J-\text{\leavevmode{D}})(1+J-\text{\leavevmode{D}})\left\langle\psi\psi
O\right\rangle^{2}$*)

AE3 = Factor[Simplify[ActionEmb[ψ1ψ2O3]/ψ1ψ2O3 /.r /.p /.Δψ -> 1]]

>>> 0 (*This shows $S_{qa}\left\langle\psi\psi O\right\rangle^{3}=0$*)

AE4 = Factor[Simplify[ActionEmb[ψ1ψ2O4]/ψ1ψ2O4 /.r /.p /.Δψ -> 1]]

>>> 0 (*This shows $S_{qa}\left\langle\psi\psi O\right\rangle^{4}=0$*)

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