1,21,2institutetext: Indian Institute of Science Education and Research,
Dr Homi Bhabha Road, Pashan, Pune, India33institutetext: Service de Physique
de l’Univers, Champs et Gravitation,
Université de Mons, 20 place du Parc, 7000 Mons, Belgium

# Hidden sectors of Chern–Simons Matter theories and Exact Holography

Sachin Jain 2 Dhruva K.S 3 Evgeny Skvortsov

###### Abstract

Chiral higher-spin gravity is a higher-spin extension of both self-dual Yang-
Mills and self-dual gravity and is a unique local higher-spin gravity in four
dimensions. Its existence implies that there are two closed subsectors in
Chern-Simons matter theories. We make first steps in identifying these
(anti-)chiral subsectors directly on the CFT side, which should result in a
holographically dual pair where both sides are nontrivial, complete, yet
exactly soluble. We also discuss closely related theories: self-dual Yang-
Mills (SDYM) and self-dual gravity (SDGR) in the holographic context.

##  1 Introduction

AdS/CFT duality Maldacena:1997re ; Witten:1998qj ; Gubser:1998bc was born
inside string theory and extended later with many examples that are not
necessarily easy to embed into string theory. Most generally, AdS/CFT is the
relation between theories of quantum gravity with negative cosmological
constant and CFTs. Even more generally bulk theories can be dual to ensembles
of CFTs.

AdS/CFT is often applied as a “knowledge transfer” tool to deduce some
nontrivial properties on one side of the duality from another one. One can
also combine pieces of information coming from both sides of duality.
Nevertheless, for many reasons it would be important to have an example of a
simple AdS/CFT duality where both sides are easy to define independently of
each other and are simple enough to compute any relevant observable as a
matter of principle. Ideally, this should lead to a direct proof of the
holographic duality in such cases, which should shed more light on the
mechanism of AdS/CFT and on the quantum gravity problem.

It has been surprisingly hard to establish such simple AdS/CFT pairs, if we
consider the canonical stringy dualities to be beyond technical capabilities
to explore them in the whole space of couplings. For example, the supergravity
limit where string theory is tractable corresponds to the regime of classical
tensionful strings. Even putting the string loops aside, the opposite regime
of tensionless strings is surprisingly hard to understand. In the canonical
example of Type-IIB strings on $AdS_{5}\times S^{5}$ the tensionless limit
corresponds to the weakly coupled $\mathcal{N}=4$ SYM HaggiMani:2000ru ;
Sundborg:2000wp ; Sezgin:2001yf ; Sezgin:2002rt ; Beisert:2003te ;
Beisert:2004di , i.e. a simple regime on the CFT side, but it has so far been
difficult to take the tensionless limit directly on the string side.111It has
been possible to directly land on the tensionless limit worldsheet theory in
the case of string theory on $AdS_{3}$, see e.g. Eberhardt:2018ouy . The
tensionful limit, on the other hand, corresponds to the strongly coupled SYM
and can only be understood thanks to the integrability, see e.g.
Beisert:2010jr , which still gives access to a limited set of observables at
present and is also hard to extend beyond the large-$N$ limit.

A natural idea to find tractable AdS/CFT pairs is to look for CFTs
simpler/smaller than SYM and for bulk theories that are simpler than string
theory. Among the latter, it is hard to find potentially UV-complete theories
since it is related to the quantum gravity problem.222It is possible to
consider examples without a dynamical graviton, e.g. just the $\phi^{4}$
theory in $AdS_{4}$, thereby avoiding the quantum gravity problem. On the CFT
side, one has to look for CFTs without the stress-tensor, which is not
impossible, e.g. the long-range Ising model.  Some examples that have
extensively been studied over the years include $3d$-gravity as Chern-Simons
theory, SYK, and JT models. Here, higher-spin gravities could be of some help.
On the CFT side, the smallest possible conformal field theories are $3d$
vector models, in particular the $O(N)$ vector model, which describes second
order phase transitions of many physical systems, e.g. the Ising critical
point.

The space of $3d$ vector models can be substantially enriched by gauging some
global symmetries therein and adding the Chern-Simons term, which results in
Chern-Simons matter theories. One can also add supersymmetry and pass to
models with bi-fundamental matter that extends to ABJ(M) theories
Aharony:2008ug ; Aharony:2008gk . Remarkably, Chern-Simons matter theories are
related to each other by a web of dualities, see e.g. Giombi:2011kc ;
Aharony:2011jz ; Maldacena:2012sf ; Aharony:2012nh ; Aharony:2015mjs ;
Karch:2016sxi ; Seiberg:2016gmd , of which, perhaps, the most remarkable is
the $3d$ bosonization duality Giombi:2011kc ; Aharony:2011jz ; Aharony:2012nh
since it remains to be nontrivial in the large-$N$ limit.

Since (Chern-Simons) vector models are the smallest $3d$ CFTs it is natural to
ask how the AdS/CFT dual thereof can look like Klebanov:2002ja ; Sezgin:2003pt
; Leigh:2003gk . In the large-$N$ limit, it is easy to see the signs of an
infinite-dimensional extension of conformal symmetry, called higher-spin
symmetry. It is immediately visible in free vector models. One manifestation
of the higher-spin symmetry is the presence of infinitely many conserved
higher-spin tensors, $J_{a_{1}...a_{s}}$, of which the stress tensor is a
particular member. By the standard AdS/CFT dictionary a spin-$s$ conserved
tensor is dual to a gauge/massless spin-$s$ field on the AdS side. Therefore,
the dual theory must feature an infinite multiplet of massless fields with
spins from $0$ to $\infty$. Theories with massless higher-spin fields had been
studied long before AdS/CFT correspondence and are usually called higher-spin
gravities, see e.g. Bekaert:2022poo .

Many properties of higher-spin gravities can be deduced before even
constructing a theory. A pro example of AdS/CFT correspondence is the Flato-
Fronsdal theorem Flato:1978qz that, in the modern language, states that the
single trace operators in the free vector model are dual to the spectrum built
from massless fields of all spins. However, constructing interactions faced a
number of difficulties, the main being that higher-spin symmetry mixes spins
and derivatives, thereby, rendering any generic higher-spin gravity too non-
local, see e.g. Bekaert:2010hp ; Bekaert:2015tva ; Maldacena:2015iua ;
Sleight:2017pcz ; Ponomarev:2017nrr ; Ponomarev:2017qab ; Neiman:2023orj .
There are very few exceptions: topological $3d$ higher-spin gravities
Blencowe:1988gj ; Bergshoeff:1989ns ; Campoleoni:2010zq ; Henneaux:2010xg ;
Pope:1989vj ; Fradkin:1989xt ; Grigoriev:2019xmp and conformal higher-spin
gravity Segal:2002gd ; Tseytlin:2002gz ; Bekaert:2010ky . The former do not
have propagating degrees of freedom and the latter extends conformal gravity.
The AdS/CFT itself implies that the holographic duals of vector models do not
obey the usual locality assumptions in field theory Bekaert:2015tva ;
Maldacena:2015iua ; Sleight:2017pcz ; Ponomarev:2017qab and, hence, cannot be
constructed by following, say, the Noether procedure. Nevertheless, certain
structures that are stable under nonlocal field-redefinitions can be
formulated as an $L_{\infty}$-algebra or as formally consistent classical
equations of motion Vasiliev:1990en ; Prokushkin:1998bq ; Vasiliev:2003ev ;
Sagnotti:2005ns ; Bonezzi:2016ttk ; Bekaert:2017bpy ; Arias:2017bvi ;
Grigoriev:2018wrx ; Sharapov:2019vyd and the dual of free/critical vector
models can, in some sense, be reconstructed deMelloKoch:2018ivk ;
Aharony:2020omh . The former leaves infinitely many coefficients unfixed and
does not allow for systematic holographic calculations, while the latter does
not give an independent definition of the bulk theory and cannot at present be
extended to Chern-Simons matter theories.

Therefore, it turns out that simple CFTs, like free vector models, do not have
simple holographic duals in the sense that the bulk theories have to be too
nonlocal for the standard field theory rules to apply and new techniques are
yet to be developed to construct and deal with them.

A new impulse into this puzzle came from the flat space, an unusual place for
AdS/CFT duality, when it was shown that there exists a local theory with the
required spectrum Metsaev:1991nb ; Metsaev:1991mt ; Ponomarev:2016lrm — chiral
higher-spin gravity. It was conjectured already in Ponomarev:2016lrm that
there should exist a smooth deformation to $(A)dS_{4}$, which was constructed
later in Metsaev:2018xip ; Skvortsov:2018uru ; Sharapov:2022wpz ;
Sharapov:2022awp . The theory can be interpreted as a higher-spin extension of
both self-dual Yang-Mills and self-dual gravity. It was shown to be at least
one-loop finite Skvortsov:2018jea ; Skvortsov:2020gpn . It is also integrable
at least in flat space Ponomarev:2017nrr . Therefore, there exists a tame
theory in $AdS_{4}$ with the right spectrum to be dual to Chern-Simons matter
theories. However, it cannot be dual to the full Chern-Simons matter theory.
For example, the chirality of interactions is such that the sum of helicities
must be positive (or negative for the anti-chiral theory) at each cubic
vertex.

Given there is a well-defined (anti-)chiral higher-spin gravity in $AdS_{4}$
we can apply the $AdS/CFT$ correspondence to conclude that there should exist
two closed hidden subsectors in Chern-Simons matter theories that are
consistent on their own.

Figure 1: Chern-Simons vector models vs. higher spin gravity: the structure of
AdS/CFT duality.

Another simple consequence of there being an $AdS$ theory is the $3d$
bosonization at the level of $3$-point correlators Skvortsov:2018uru . Indeed,
chiral and anti-chiral interactions form a complete basis of cubic
interactions and all the couplings are completely fixed in each of the
sectors. Therefore, in order to bootstrap the $3$-point correlators in a
theory with the same spectrum of higher-spin fields, one needs to see how to
glue (anti-)chiral pieces together. The gluing depends on one phase-like
parameter that gives the structure observed by Maldacena and Zhiboedov in
Maldacena:2012sf . Beyond $3$-point (anti-)chiral interactions do not cover
everything possible, but see Skvortsov:2022wzo . The structure of the duality
is illustrated on the figure.

On the CFT side there has been a lot of progress in understanding the Chern-
Simons matter theories in the large-$N$ limit, a significant part of which was
based on the idea of the slightly-broken higher-spin symmetry Giombi:2011kc ;
Aharony:2011jz ; Maldacena:2011jn ; Maldacena:2012sf . Several remarkable
achievements in these theories include exact computations of the partition
function Giombi:2011kc ; Aharony:2011jz ; Jain:2012qi ; Aharony:2012ns ;
Jain:2013py ; Jain:2013gza ; Minwalla:2023esg , understanding the Hilbert
space structure Minwalla:2022sef and the higher-spin spectrum GurAri:2012is ;
Takimi:2013zca ; Giombi:2016zwa ; Giombi:2017rhm ; Charan:2017jyc ;
Jain:2019fja ; Minwalla:2020ysu , determination of the exact S matrix
Jain:2014nza ; Inbasekar:2015tsa ; Mehta:2022lgq , showing the existence of
BCFW recursion relations and dual superconformal invariance Inbasekar:2017ieo
; Inbasekar:2017sqp .

An important set of observables in these theories are the correlation
functions of gauge invariant single trace operators, aka higher-spin currents.
It was found in Giombi:2011rz that three-point functions in general three-
dimensional CFTs involving higher-spin currents contain three pieces: two of
which are realized by the free bosonic and free fermionic theories and a
third, which is parity violating and not realized by free fields. In
Maldacena:2011jn ; Maldacena:2012sf it was found that if one imposes in
addition, the Ward identities due to the slightly-broken higher-spin symmetry,
the relative coefficients of these structures get determined in terms of a
single parameter. This parameter was then related to the Chern-Simons matter
theory ’t Hooft coupling in Aharony:2012nh . Since then, there have been many
attempts to compute three- and four-point correlators in position space and
Mellin space Chowdhury:2017vel ; Sezgin:2017jgm ; Skvortsov:2018uru ;
Yacoby:2018yvy ; Chowdhury:2018uyv ; Aharony:2018pjn ; Gerasimenko:2021sxj ;
Scalea:2023dpw ; Bedhotiya:2015uga ; Turiaci:2018nua ; Li:2019twz ;
Silva:2021ece ; Kalloor:2019xjb ; Trivko:2024thesis ; Trivko:2024 and also in
momentum space Jain:2020rmw ; Jain:2020puw . It was subsequently realized by
working in momentum space, that the parity odd correlators can be obtained
from the difference of the free fermionic and free bosonic correlators by the
so-called epsilon transform Jain:2021gwa ; Jain:2021whr ; Caron-Huot:2021kjy .
Further, by employing the technology of three-dimensional spinor-helicity
variables, this relation becomes extremely simple Jain:2021vrv ; Jain:2021wyn
; Jain:2021qcl . Spinor-helicity variables also revealed the anyonic nature of
the correlators Gandhi:2021gwn . Eventually, these analyses culminated in the
computation of all four- and higher-point functions in the Chern-Simons matter
theories Jain:2022ajd using the slightly-broken higher-spin symmetry. Based on
these results, the implications on the bulk dual of Chern-Simons matter theory
were analyzed in Jain:2023juk where it was found that there exists the
possibility of a sub-sector of the theory that has a local bulk dual.

Since chiral higher-spin gravity is a perturbatively local field theory, one
can directly compute holographic correlators to define the hidden subsectors
of Chern-Simons matter theories. However, it would be important to identify
the hidden sectors directly on the CFT side. This is what the present paper
attempts to do. Crucial ingredients for our analysis are the epsilon transform
and the usage of three-dimensional spinor-helicity variables which together
manifest the anyonic nature of correlation functions. We shall show that
taking appropriate limits of the ’t Hooft coupling of the theory projects us
into a chiral or anti-chiral sector at the level of three-point correlators.
We also derive some simple constraints for the structure of higher-order
correlators and loop corrections from the bulk. The loop corrections are shown
to truncate at a finite order in $1/N$-expansion, which makes the latter
convergent. All of this paves the way to constructing exact models of AdS/CFT
correspondence.

##### Outline:

In section 2, we discuss and derive some general facts about three-point
correlators in three-dimensional CFTs. Using these results, we define a
(anti-)chiral limit for the theory of fundamental fermions minimally coupled
to the Chern-Simons gauge field in section 3. In section 4, we see the
implications of the (anti-)chiral limit on the anomalous dimensions of the
currents in our theory and higher-order correlation functions. In section 5 we
present some simple predictions from Chiral higher-spin gravity for
holographic correlation functions. Conclusions and discussion can be found in
section 6. The paper is supplemented with a number of technical appendices. In
appendix A we explore the possibility of a chiral truncation from the bulk
point of view. Appendix B is a guide to some of the notation used throughout
the paper. In appendix C, we consider the (anti-)chiral limit for the theory
of bosons coupled to Chern-Simons. Appendix D deals with the generalization of
some of the results of section 2 to correlators involving fermionic operators.
In appendix E, we provide an argument for some of the statements made in
section 2 using the CFT/S-Matrix correspondence. In appendix F, we provide
evidence for some of the statements made in section 2 for correlators outside
the triangle. In appendix G, we derive the correct prescription to obtain the
parity odd contributions to three-point correlators that are outside the
triangle. In appendix H, we bootstrap some four-point correlators using
conformal partial waves using our (anti-)chiral three-point functions as
seeds. Finally, in appendix I, we discuss more about taking the (anti-)chiral
limit of four-point functions.

Note added : During the work on this paper, we became aware of Aharony:2024 ,
which contains some overlapping results.

##  2 Some generalities about correlators in $\text{CFT}_{3}$

In this section, we shall discuss and derive some general facts about two and
three-point correlators in three-dimensional CFT involving scalars and
conserved currents. We work throughout in momentum space and in three-
dimensional spinor helicity variables Bzowski:2013sza ; Coriano:2013jba ;
Bzowski:2015pba ; Bzowski:2017poo ; Jain:2020rmw ; Jain:2020puw ; Jain:2021wyn
; Jain:2021vrv ; Jain:2021gwa ; Jain:2021whr ; Maldacena:2011nz . We provide a
guide to our notation and conventions in appendix B.

###  2.1 Two-point functions

The two-point function of a primary scalar operator $O_{\Delta}$ is fixed by
the conformal ward identities up to an overall constant. In momentum space, it
is given by the following expression Bzowski:2013sza ,333In this equation and
the ones to follow, we suppress the overall momentum conserving delta
functions.

$\displaystyle\langle
O_{\Delta}(p)O_{\Delta}(-p)\rangle=c_{\Delta}~{}p^{2\Delta-3}.$ (1)

Two-point functions of conserved currents are similarly constrained and
determined up to two constants, one of which multiplies the usual parity even
expression while the other, multiplies a parity odd contact term Jain:2021gwa
. This is an artefact of three dimensions, due to the presence of the three
indexed Levi-Civita symbol. For example, the two-point function of a spin-one
conserved current is given by,444Given a spin $s$ current
$J_{s}^{\mu_{1}\cdots\mu_{s}}(p_{i})$, it is often convenient to contract it
with the polarization vectors $z_{i\mu_{1}}\cdots z_{i\mu_{s}}$. These
polarization vectors are null ($z_{i}^{2}=0$) and transverse ($z_{i}\cdot
p_{i}=0$).

$\displaystyle z_{1\mu}z_{2\nu}\langle
J^{\mu}(p)J^{\nu}(-p)\rangle=c_{1,\text{even}}~{}p~{}(z_{1}\cdot
z_{2})+c_{1,\text{odd}}~{}\epsilon^{z_{1}z_{2}p}.$ (2)

This expression is easily generalized to the spin $s$ case:

$\displaystyle\begin{aligned} z_{1\mu_{1}}\cdots
z_{1\mu_{s}}z_{2\nu_{1}}\cdots z_{2\nu_{s}}\langle
J_{s}^{\mu_{1}\cdots\mu_{s}}(p)J_{s}^{\nu_{1}\cdots\nu_{s}}(-p)\rangle&=c_{s,\text{even}}~{}p^{2s-1}~{}(z_{1}\cdot
z_{2})^{s}+\\\
&+c_{s,\text{odd}}~{}p^{2s-2}~{}\epsilon^{z_{1}z_{2}p}(z_{1}\cdot
z_{2})^{s-1}\,,\end{aligned}$ (3)

where we used the notation
$\epsilon^{\mu\nu\rho}a_{\mu}b_{\nu}c_{\rho}:=\epsilon^{abc}$. It is
interesting to note that the parity odd expression can be obtained from the
parity even one via the epsilon transform as follows Jain:2021gwa :

$\displaystyle\langle
J_{s}^{\mu_{1}\cdots\mu_{s}}(p)J_{s}^{\nu_{1}\cdots\nu_{s}}(-p)\rangle_{\text{odd}}=\frac{1}{p}{\epsilon^{p\alpha(\mu_{1}}}\langle
J_{s}^{\mu_{1}\cdots\mu_{s})}(p)J_{s}^{\nu_{1}\cdots\nu_{s}}(-p)\rangle_{\text{even}}.$
(4)

One may think that the $c_{s,\text{odd}}$ contribution can be removed via the
addition of suitable counter-terms as it multiplies a contact term
contribution.555$p^{2(s-1)}\epsilon^{z_{1}z_{2}p}(z_{1}\cdot z_{2})^{s-1}$ is
the Fourier transform of the following contact term:
$(z_{1}\cdot
z_{2})^{s-1}\epsilon^{z_{1}z_{2}\mu}\partial_{1\mu}\Box_{1}^{2(s-1)}\delta^{3}(x_{1}-x_{2})$.
This, however, is not completely true as there could exist a scheme-
independent part of the parity odd two-point functions and hence no counter-
terms can remove them. This is explicitly realized in Chern-Simons theory with
matter Aharony:2012nh . In fact, these contact terms will play a crucial role
in section 3 while defining the chiral limit of Chern-Simons matter theory.

It turns out that (3) takes an interesting form in the language of three-
dimensional spinor-helicity variables. For a review of the same with our
conventions please refer to appendix B. In the two nonzero helicity
configurations, i.e., $(-~{}-)$ and $(+~{}+)$, we have,

$\displaystyle\langle
J_{s}^{-}(p)J_{s}^{-}(-p)\rangle\sim(c_{s,\text{E}}-ic_{s,\text{O}})\frac{\langle
12\rangle^{2s}}{p}~{},~{}\langle
J_{s}^{+}(p)J_{s}^{+}(-p)\rangle\sim(c_{s,\text{E}}+ic_{s,\text{O}})\frac{\langle\bar{1}\bar{2}\rangle^{2s}}{p}.$
(5)

We see that in spinor-helicity variables, the even and odd terms are identical
up to a helicity (but not momentum) dependent factor. This factor of $\mp i$
can be understood by looking at the epsilon transform (4). Effectively, the
epsilon transform with respect to a conserved current $J_{s}$ becomes a simple
multiplicative transformation in spinor-helicity variables.666Consider the
$s=1$ case for instance with a negative helicity polarization.  $\displaystyle
z_{\mu}^{-}(\epsilon\cdot J_{s})^{\mu}:=z_{\mu}^{-}\frac{\epsilon^{\mu
p\alpha}}{p}J_{\alpha}.$ (6) By using the fact that
$\epsilon^{\mu\nu\rho}=\frac{1}{2i}\Tr{\sigma^{\mu}\sigma^{\nu}\sigma^{\rho}}$
and
$z_{\mu}^{-}=\frac{(\sigma_{\mu})^{a}_{b}\lambda_{a}\lambda^{b}}{2p},p^{\mu}=\frac{(\sigma^{\mu})^{a}_{b}\lambda_{a}\bar{\lambda}^{b}}{2}$,
we find that,  $\displaystyle z_{\mu}^{-}(\epsilon\cdot
J_{s})^{\mu}=-iz^{\alpha-}J_{\alpha}=-iJ^{-}.$ (7) For the case when the
polarization has a positive helicity, we obtain a $+i$ on the RHS instead of
the $-i$. Since the polarization vectors for higher-spins are formed out of
products of the spin one polarization, the same results hold there. In short,
if the polarization has helicity $h$ we obtain a $\text{sign}(h)i$ factor
while performing the epsilon transform.

In section 3, we will see that it is the structure of (5) that will enable us
to define the chiral limit of Chern-Simons matter theory by appropriately
relating the odd coefficient that appears there to the even one.

###  2.2 Three-point functions

Three-point correlators of conserved currents can be grouped into two broad
categories. Given a three-point function $\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle$ such that $s_{i}+s_{j}\geq s_{k}~{}\forall
i,j,k\in\\{1,2,3\\}$, we say that the correlator is inside the (spin)
triangle. If this inequality is violated, we say that it is outisde the (spin)
triangle. In this subsection, we first review some general facts about
correlators both inside and outside the triangle. We then proceed to derive
some new results which will play an important role in section 3.

####  2.2.1 Three-point functions inside the triangle

The most general three-point function that is inside the triangle takes the
following form Giombi:2011rz :777There could also be parity odd nonhomogeneous
contributions but they are all contact terms. Since the FF and FB Ward-
Takahashi identities are equal inside the triangle, we can form one parity odd
nonhomogeneous structure by replacing the two-point function that appears
there by its parity odd counterpart Jain:2021vrv .

$\displaystyle\langle J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle=n_{f}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FF}+n_{odd}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{odd}+n_{b}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FB}\,,$ (8)

where we have used the abbreviations $FF=$free fermion and $FB=$free boson
that we shall employ throughout this paper. It was found in Jain:2021gwa that
the odd part can be written in terms of the FF and FB results via an epsilon
transformation.

$\displaystyle\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{odd}=\langle\epsilon\cdot
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FF-FB}=\langle\epsilon\cdot
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FF}-\langle\epsilon\cdot
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FB}.$ (9)

For example, (9) in more detail is as follows:

$\displaystyle\begin{aligned} \langle
T_{\mu\nu}(p_{1})T_{\alpha\beta}(p_{2})T_{\gamma\theta}(p_{3})\rangle_{\text{odd}}&=\langle\epsilon\cdot
T_{\mu\nu}(p_{1})T_{\alpha\beta}(p_{2})T_{\gamma\theta}(p_{3})\rangle_{FF-
FB}=\\\ &=\frac{\epsilon_{\mu ab}p_{1a}}{p_{1}}\langle
T_{b\nu}(p_{1})T_{\alpha\beta}(p_{2})T_{\gamma\theta}(p_{3})\rangle_{FF-
FB}.\end{aligned}$ (10)

It is important to note that for correlators inside the triangle, one can
obtain the odd correlator via the epsilon transform with respect to any of the
currents in the $FF-FB$ correlator.888This is due to the fact that the $FF-FB$
correlators are non-zero only when all their helicities have the same sign as
we shall see in (2.2.1). Thus, the epsilon transform, which gives a factor of
$i~{}\text{sign}(h)$ can be performed with respect to any of the operators. We
will see that the same cannot be said for correlators that violate the
triangle inequality in the next subsection.

Further, we define the homogeneous and nonhomogeneous correlators Jain:2021gwa
:999The homogeneous and nonhomogeneous correlators have nice bulk
interpretations. The former comes due to higher derivative interactions
whereas the latter comes due to the term that also gives rise to the
propagators of the involved bulk fields.

$\displaystyle\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{h}=\frac{1}{2}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FF-FB}=\frac{1}{2}(\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FF}-\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FB}),$ $\displaystyle\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{nh}=\frac{1}{2}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FF-FB}=\frac{1}{2}(\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FF}+\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FB}),$ (11)

that satisfy,

$\displaystyle p_{1\mu}\langle
J_{s_{1}}^{\mu\cdots}J_{s_{2}}J_{s_{3}}\rangle_{h}=0,$ $\displaystyle
p_{1\mu}\langle
J_{s_{1}}^{\mu\cdots}J_{s_{2}}J_{s_{3}}\rangle_{nh}=\text{Ward-Takahashi
identity terms}.$ (12)

Since the Ward-Takahashi identity is made out of two-point functions, the
coefficient of the nonhomogeneous correlator is that of the two-point
function.

Now, using (9) and (2.2.1), (8) can be written as,

$\displaystyle\langle J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle=(n_{f}+n_{b})\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{nh}+(n_{f}-n_{b})\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{h}+2n_{\text{odd}}~{}\langle\epsilon\cdot
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{h}.$ (13)

As mentioned just above this equation, the coefficient of the nonhomogeneous
correlator viz $n_{f}+n_{b}$ is proportional to the two-point function
coefficient.101010For instance, consider the $\langle TT\rangle$ two-point
function: $\langle TT\rangle\propto C_{T}$. We then have $\langle
TTT\rangle_{nh}\propto C_{T}$ since $p_{1}\cdot\langle TTT\rangle\sim\langle
TT\rangle$. This also implies in (13) that $n_{f}+n_{b}=C_{T}$. $n_{f}-n_{b}$
and $n_{\text{odd}}$ on the other hand are truly three-point data. Thus we see
that in contrast to the three structures appearing in (8), only two appear in
(13) thanks to the epsilon transformation that related the odd part to the
$FF-FB$ (homogeneous) correlator. Further, these homogeneous and
nonhomogeneous correlators obey many interesting properties. However, for the
best way to see these properties, we must turn to the language of three-
dimensional spinor-helicity variables as we did even for the two-point
functions in (5). For instance, (13) in spinor-helicity variables takes the
following form (where $h_{i}$ is the helicity of $J_{s_{i}}$) in spinor-
helicity variables:

$\displaystyle\langle
J_{s_{1}}^{h_{1}}J_{s_{2}}^{h_{2}}J_{s_{3}}^{h_{3}}\rangle=(n_{f}+n_{b})\langle
J_{s_{1}}^{h_{1}}J_{s_{2}}^{h_{2}}J_{s_{3}}^{h_{3}}\rangle_{nh}+(n_{f}-n_{b}\pm
2in_{\text{odd}})\langle
J_{s_{1}}^{h_{1}}J_{s_{2}}^{h_{2}}J_{s_{3}}^{h_{3}}\rangle_{h},$ (14)

where the $\pm i$ multiplying $n_{\text{odd}}$ depends on the helicity
configuration in question.

We shall now state some facts about the (non)homogeneous correlators and
subsequently prove them.

Statement 1: For correlators inside the triangle, the homogeneous parts,
$\langle J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{h}$ are only nonzero in the
$(---)$ and $(+++)$ helicity configurations Jain:2021vrv . It is important to
note that the odd correlators in spinor-helicity variables are equal to the
even homogeneous correlators up to factors of $\pm 2i$ that depend on the
helicity111111The fact that homogeneous correlators are dual to higher
derivative interactions in the bulk supports this statement. Such interactions
are nonzero only when the helicities of all three particles coincide. For
example, consider $\langle TTT\rangle_{h}$. It can be obtained from the
graviton $W^{3}$ (Weyl tensor cubed) interaction in the bulk which vanishes
except when the gravitons all have the same helicity.

$\displaystyle\langle J_{s_{1}}^{-}J_{s_{2}}^{-}J_{s_{3}}^{-}\rangle_{h}\neq
0\,,\qquad\langle J_{s_{1}}^{+}J_{s_{2}}^{+}J_{s_{3}}^{+}\rangle_{h}\neq 0\,,$
$\displaystyle\langle
J_{s_{1}}^{h_{1}}J_{s_{2}}^{h_{2}}J_{s_{3}}^{h_{3}}\rangle_{h}=0~{}\text{for
all other helicities}.$ (15)

Statement 2: The nonhomogeneous correlators inside the triangle, $\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{nh}$ are only nonzero in the mixed
helicity configurations, i.e. they are zero in the $(---)$ and $(+++)$
helicity configurations. We show that the nonhomogeneous correlators that
appear in Jain:2021vrv can be modified via the addition of contact terms to
obey this statement121212Nonhomogeneous correlators arise from the $s$
derivative interactions in the bulk which have support only if one of the
particles has a different helicity than the other two. For instance consider
$\langle JJJ\rangle_{nh}$ (non-Abelian currents). This correlator is dual to
the usual Yang-Mills interaction in the bulk which is nonzero only in the
$(--+)$ configuration and those obtained via exchanges or complex conjugation.

$\displaystyle\langle
J_{s_{1}}^{-}J_{s_{2}}^{-}J_{s_{3}}^{-}\rangle_{nh}=\langle
J_{s_{1}}^{+}J_{s_{2}}^{+}J_{s_{3}}^{+}\rangle_{nh}=0\,,~{}~{}~{}$
$\displaystyle\langle
J_{s_{1}}^{h_{1}}J_{s_{2}}^{h_{2}}J_{s_{3}}^{h_{3}}\rangle_{nh}\neq
0~{}\text{for all other (net nonzero) helicity configurations}.$ (16)

Statement 3: The zero-helicity sectors (i.e. the sector where the net helicity
of the correlator is zero) of the nonhomogeneous correlators is identically
zero. Similar to what we discussed in the above statement, this statement too
will be true modulo contact term contributions. Put another way, we will show
that we can find contact terms to remove such contributions to the
nonhomogeneous correlators131313As the CFT correlators give rise to S-matrices
in one higher dimension when the total energy goes to zero and zero-helicity
three-point S-matrices are identically zero, the CFT correlator should be free
of total energy poles. Thus, it is plausible (as we shall also show) that the
zero-helicity CFT correlators can be removed via suitable contact terms. For a
slightly more detailed discussion regarding this as well as an argument for
statement-2 from the CFT-S Matrix correspondence, please refer to appendix E.

$\displaystyle\langle
J_{s_{1}}^{h_{1}}J_{s_{2}}^{h_{2}}J_{s_{3}}^{h_{3}}\rangle=0~{}~{}\text{if
$s_{1}h_{1}+s_{2}h_{2}+s_{3}h_{3}=0$}.$ (17)

Let us now present an argument for statement 1, (2.2.1): consistency of the
momentum space correlator with the operator product expansion forces it to
only have total energy poles Maldacena:2011nz . The other type of poles which
are of the form $p_{i}+p_{j}-p_{k}$, are inconsistent with the OPE. By solving
the conformal Ward identity in all the helicity configurations, we see that
only in the $(---)$ and $(+++)$ helicities the homogeneous solution is
consistent with the OPE Jain:2021whr . All other helicity configurations lead
to solutions with poles of the form $p_{i}+p_{j}-p_{k}$ which we must discard
for OPE consistency to be maintained.

We now present a few examples to illustrate the validity of statements 2 and
3.

### $\textbf{Example~{}1}:\langle JJJ\rangle$

This correlator in the two independent helicity configurations takes the form
Jain:2021vrv :141414In this example and the ones to follow, we focus on
nonhomogeneous correlators that are parity even. Parity odd nonhomogeneous
correlators are contact terms Jain:2021vrv and hence we do not keep them in
the analysis to follow.

$\displaystyle\langle
J^{-A}J^{-B}J^{-C}\rangle=f^{ABC}\big{(}\frac{c_{E}+ic_{O}}{E^{3}}+\frac{c_{J}}{p_{1}p_{2}p_{3}}\big{)}\langle
12\rangle\langle 23\rangle\langle 31\rangle\,,$ $\displaystyle\langle
J^{-A}J^{-B}J^{+C}\rangle=f^{ABC}\frac{c_{J}}{p_{1}p_{2}p_{3}}\big{(}1-\frac{2p_{3}}{E}\big{)}\langle
12\rangle\langle 2\bar{3}\rangle\langle\bar{3}1\rangle\,,$
$\displaystyle\text{where}~{}E=p_{1}+p_{2}+p_{3}\,,$ (18)

where $c_{E}$ ($ic_{O}$) multiplies the parity even (parity odd) homogeneous
contribution and $c_{J}$ is the coefficient of the parity even nonhomogeneous
term. The $(+++)$ and $(++-)$ configurations can be obtained from the above
expression by complex conjugation (which replaces $\langle
ij\rangle\to\langle\bar{i}\bar{j}\rangle$). Finally, the remaining helicities
are obtained via exchanges of labels.

Looking at (2), we see that the homogeneous solution is present only in the
$(---)$ and $(+++)$ helicities which is in accordance to statement $1$.
However, it appears as though there is a nonhomogeneous contribution in the
$(---)$ and $(+++)$ helicities which should not be there according to
statement $2$ (2.2.1). We shall show however, that via the addition of a
suitable contact term, we can cancel these contributions. Indeed, consider the
following contact term contribution:

$\displaystyle\begin{aligned} \langle
J^{A}J^{B}J^{C}\rangle_{\text{contact}}=\frac{2c_{J}f^{ABC}}{3p_{1}p_{2}p_{3}}\bigg{(}&(z_{1}\cdot
z_{2})(z_{3}\cdot p_{1})(p_{1}+p_{2})p_{3}+\\\ -&(z_{1}\cdot z_{3})(z_{2}\cdot
p_{1})p_{2}(p_{1}+p_{3})+(z_{1}\cdot p_{2})(z_{2}\cdot
z_{3})p_{1}(p_{2}+p_{3})\bigg{)}.\end{aligned}$ (19)

We then redefine the correlator via the addition of (19) as follows:

$\displaystyle\langle J^{A}J^{B}J^{C}\rangle_{new}=\langle
J^{A}J^{B}J^{C}\rangle-\langle J^{A}J^{B}J^{C}\rangle_{\text{contact}}.$ (20)

This results in the following expression for $\langle JJJ\rangle_{new}$ in the
two independent helicity configurations:

$\displaystyle\langle
J^{-A}J^{-B}J^{-C}\rangle_{new}=f^{ABC}\big{(}\frac{c_{1}+ic_{1}^{\prime}}{E^{3}}\big{)}\langle
12\rangle\langle 23\rangle\langle 31\rangle,$ $\displaystyle\langle
J^{-A}J^{-B}J^{+C}\rangle_{new}=f^{ABC}\frac{2c_{J}}{3p_{1}p_{2}p_{3}}\big{(}1-\frac{3p_{3}}{E}\big{)}\langle
12\rangle\langle 2\bar{3}\rangle\langle\bar{3}1\rangle,$ (21)

proving that the nonhomogeneous contribution to $\langle JJJ\rangle$ can be
tuned to zero in the $(---)$ and $(+++)$ helicities by the addition of the
contact term (19).151515The same is true in the $(+++)$ helicity as it is the
complex conjugate of the $(---)$ helicity.

### $\textbf{Example~{}2}:\langle TTT\rangle$

Let us now consider the three-point function of the stress tensor. This
correlator takes the following form in the two independent helicities
Jain:2021vrv :

$\displaystyle\langle T^{-}T^{-}T^{-}\rangle$
$\displaystyle=\bigg{(}\frac{(c_{E}+ic_{O})p_{1}p_{2}p_{3}}{E^{6}}+$
$\displaystyle+c_{T}$
$\displaystyle\frac{1}{(p_{1}p_{2}p_{3})^{2}}\big{(}E^{3}-E(p_{1}p_{2}+p_{2}p_{3}+p_{1}p_{3})-p_{1}p_{2}p_{3}\big{)}\bigg{)}\langle
12\rangle^{2}\langle 23\rangle^{2}\langle 31\rangle^{2},$ (22)
$\displaystyle\langle T^{-}T^{-}T^{+}\rangle$
$\displaystyle=c_{T}\frac{(E-2p_{3})^{2}(E^{3}-E(p_{1}p_{2}+p_{2}p_{3}+p_{3}p_{1})-p_{1}p_{2}p_{3}}{E^{2}p_{1}^{2}p_{2}^{2}p_{3}^{2}}\langle
12\rangle^{2}\langle 2\bar{3}\rangle^{2}\langle\bar{3}1\rangle^{2}.$

$c_{E}$ and $c_{O}$ are the partiy even and parity odd homogeneous
coefficients whereas $c_{T}$ is the parity even nonhomogeneous coefficient.
The remaining helicities can be obtained via exchanges or by the complex
conjugation.

Just like the previous example of $\langle JJJ\rangle$, while the homogeneous
contribution is in accordance with statement $1$ (2.2.1), the nonhomogeneous
part seems to violate statement $2$ (2.2.1) as it is nonzero in the $(---)$
and $(+++)$ helicities. However, we shall show that by adding a suitable
contact term, we can render such contributions zero. Writing an ansatz for the
contact term and fixing the coefficients by demanding that it should cancel
out the nonhomogeneous contribution in the $(---)$ and $(+++)$ helicities we
find,

$\displaystyle\langle TTT\rangle_{\text{contact}}$
$\displaystyle=c_{T}\bigg{[}\frac{8(z_{1}\cdot p_{2})(z_{1}\cdot
z_{2})(z_{2}\cdot z_{3})(z_{3}\cdot p_{1})(E+p_{2})}{3}-\frac{8(z_{1}\cdot
p_{2})(z_{1}\cdot z_{3})(z_{2}\cdot p_{1})(z_{2}\cdot z_{3})}{3}(E+p_{3})$
$\displaystyle-\frac{8(z_{1}\cdot z_{2})(z_{1}\cdot z_{3})(z_{2}\cdot
p_{1})(z_{3}\cdot p_{1})}{3}(E+p_{1})$ $\displaystyle-8(z_{1}\cdot
z_{2})(z_{1}\cdot z_{3})(z_{2}\cdot
z_{3})\bigg{(}p_{1}^{3}+2p_{1}^{2}(p_{2}+p_{3})+2p_{1}(p_{2}^{2}+p_{3}^{2})+(p_{2}+p_{3})(p_{2}^{2}+p_{2}p_{3}+p_{3}^{2})\bigg{)}\bigg{]}.$
(23)

By redefining,

$\displaystyle\langle TTT\rangle=\langle TTT\rangle-\langle
TTT\rangle_{\text{contact}},$ (24)

we find,

$\displaystyle\langle
T^{-}T^{-}T^{-}\rangle=\bigg{(}\frac{(c_{O}+ic_{E})p_{1}p_{2}p_{3}}{E^{6}}\bigg{)}\langle
12\rangle^{2}\langle 23\rangle^{2}\langle 31\rangle^{2},$
$\displaystyle\langle
T^{-}T^{-}T^{+}\rangle=c_{T}\frac{-2}{3p_{1}^{2}p_{2}^{2}p_{3}E^{2}}\bigg{(}5p_{1}^{4}+3p_{1}^{3}(4p_{2}+3p_{3})+p_{1}^{2}(14p_{2}^{2}+11p_{2}p_{3}+9p_{3}^{2})+$
(25)
$\displaystyle\qquad+p_{2}(5p_{2}^{3}+9p_{2}^{2}p_{3}+9p_{2}p_{3}^{2}+5p_{3}^{3})+p_{1}(12p_{2}^{3}+11p_{2}^{2}p_{3}+16p_{2}p_{3}^{2}+5p_{3}^{3}\bigg{)}\langle
12\rangle^{2}\langle 2\bar{3}\rangle^{2}\langle\bar{3}1\rangle^{2}$

proving that the contact term (2), renders the nonhomogeneous $\langle
TTT\rangle$ zero in the $(---)$ and $(+++)$ helicities.

We now turn to an example where there also exists a zero-helicity
configuration.

### $\textbf{Example~{}3}:\langle TJJ\rangle$

The helicity decomposition of this correlator is as follows:161616This
expression can easily be obtained from the free theory computations using the
definitions provided in (2.2.1).

$\displaystyle\langle
T^{-}J^{-}J^{-}\rangle=\frac{(c_{1}+ic_{1}^{\prime})p_{1}}{E^{4}}\langle
12\rangle^{2}\langle 31\rangle^{2}-c_{J}\frac{\langle 12\rangle^{2}\langle
31\rangle^{2}(p_{2}+p_{3})}{p_{1}^{2}p_{2}p_{3}},$ $\displaystyle\langle
T^{+}J^{-}J^{-}\rangle=-c_{J}\frac{\langle
23\rangle^{4}(p_{1}^{2}-(p_{2}-p_{3})^{2})^{2}(p_{2}+p_{3})}{\langle
12\rangle^{2}\langle 31\rangle^{2}p_{1}^{2}p_{2}p_{3}}$ (26)
$\displaystyle\langle T^{-}J^{-}J^{+}\rangle=-c_{J}\frac{\langle
12\rangle^{4}}{3E^{2}\langle
23\rangle^{2}p_{1}^{2}p_{2}p_{3}}(E-2p_{1})^{2}\big{(}3E^{2}(E-p_{1})+8(p_{1}^{2}-E^{2})p_{2}+8(p_{1}+E)p_{2}^{2}\big{)}.$

$c_{1}$ and $c_{1}^{\prime}$ are the parity even and odd homogeneous
coefficients while $c_{J}$ is the coefficient of the nonhomogeneous parity
even contribution.

It seems that not only does $\langle TJJ\rangle$ contain a nonhomogeneous
contribution in the $(---)$ and $(+++)$ helicity configurations but also a
zero helicity configuration, both of which should not exist according to
statements $2$ (2.2.1) and $3$ (17).

By writing an ansatz for contact terms to cancel these contributions, we were
able to find one that simultaneously cancels out both of them:

$\displaystyle\langle TJJ\rangle_{\text{contact}}=-4c_{J}(z_{1}\cdot
z_{2})(z_{1}\cdot z_{3})(p_{2}+p_{3}).$ (27)

By redefining,

$\displaystyle\langle TJJ\rangle_{new}=\langle TJJ\rangle-\langle
TJJ\rangle_{\text{contact}}\,,$ (28)

we find

$\displaystyle\langle
T^{-}J^{-}J^{-}\rangle_{new}=\frac{(c_{1}+ic_{1}^{\prime})p_{1}}{E^{4}}\langle
12\rangle^{2}\langle 31\rangle^{2}$ $\displaystyle\langle
T^{+}J^{-}J^{-}\rangle_{new}=0$ $\displaystyle\langle
T^{-}J^{-}J^{+}\rangle_{new}=\frac{8c_{J}\langle 12\rangle^{4}}{3\langle
23\rangle^{2}p_{1}^{2}E^{2}}(E-2p_{1})^{2}(E+p_{1})\,.$ (29)

Thus, the simple contact term (27) suffices to render the-nonhomogeneous
$\langle TJJ\rangle$ zero in the $(---)$ and $(+++)$ helicities as well as the
zero-helicity configurations.

### $\textbf{Example~{}4}:\langle J_{3}TJ\rangle$

Let us now consider a correlator involving a higher-spin current to show that
our statements also hold in such cases as well. Since the homogeneous part of
this correlator obeys statement $1$ (2.2.1), we focus on the apparent
discrepancy between this correlator’s nonhomogeneous part with our statements
$2$ (2.2.1) and $3$ (17). The nonhomogeneous part of this correlator in the
$(---)$ and $(+--)$ configurations are respectively given by,171717See
footnote 16.

$\displaystyle\langle J_{3}^{-}T^{-}J^{-}\rangle_{nh}$
$\displaystyle=a\frac{-\langle 12\rangle^{4}\langle
31\rangle^{2}}{p_{1}^{3}p_{2}^{2}p_{3}}\big{(}6p_{2}^{3}-p_{1}^{2}p_{3}+4p_{2}(p_{1}+2p_{2})p_{3}-7(p_{1}-2p_{2})p_{3}^{2}+14p_{3}^{3}\big{)},$
$\displaystyle\langle J_{3}^{+}T^{-}J^{-}\rangle_{nh}$
$\displaystyle=a\frac{\langle
23\rangle^{6}(p_{1}+p_{2}-p_{3})^{2}(p_{1}-p_{2}+p_{3})^{4}}{\langle
12\rangle^{2}\langle
31\rangle^{4}p_{1}^{3}p_{2}^{3}p_{3}}\big{(}-6p_{2}^{3}-14p_{3}^{3}+$ (30)
$\displaystyle\qquad+(p_{1}^{2}+4p_{1}p_{2}-8p_{2}^{2})p_{3}-7(p_{1}+2p_{2})p_{3}^{2}\big{)}\,,$

$a$ is the coefficient of the parity even nonhomogeneous contribution.
Although these expressions are quite complicated, we were able to write an
ansatz and determine a contact term that can simultaneously cancel both these
expressions, rendering the nonhomogeneous contribution zero in the $(---)$,
$(+++)$ helicities as well as the two zero-helicity configurations. The
explicit expression for the contact term is as follows:

$\displaystyle\langle J_{3}TJ\rangle_{\text{contact}}$
$\displaystyle=4a(z_{1}\cdot z_{2})\bigg{(}4(z_{1}\cdot
p_{2})\big{(}9(z_{2}\cdot p_{1})(z_{1}\cdot z_{3})-7(z_{1}\cdot
p_{2})(z_{2}\cdot z_{3})\big{)}p_{3}+$ $\displaystyle+(z_{1}\cdot
z_{2})\big{(}-14(z_{3}\cdot p_{1})(z_{1}\cdot p_{2})p_{3}+(z_{1}\cdot
z_{3})(12p_{2}^{3}+9p_{1}^{2}p_{3}+13p_{2}^{2}p_{3}+3p_{3}^{3})\big{)}\bigg{)}.$

By redefining,

$\displaystyle\langle J_{3}TJ\rangle_{nh,new}=\langle
J_{3}TJ\rangle_{nh}-\langle J_{3}TJ\rangle_{\text{contact}},$ (31)

we find,

$\displaystyle\langle J_{3}^{-}T^{-}J^{-}\rangle_{nh,new}=\langle
J_{3}^{+}T^{-}J^{-}\rangle_{nh,new}=0.$ (32)

With this, we conclude our examples and attempt to generalize our findings to
arbitrary correlators inside the triangle.

### Conclusion

In all the examples that we considered, we showed that it is always possible
to add contact terms to the correlators to render the nonhomogeneous
contributions zero in the $(---)$, $(+++)$ helicities as well as the zero-
helicity configuration where applicable. Essentially, what this means is that
the Ward-Takahashi identity for the correlators in these helicity sectors can
be removed via re-defining the correlator by adding particular contact terms.
If we take this as an input, i.e. the Ward-Takahashi identities being trivial
in these helicity configurations, one can prove in general that the
nonhomogeneous contribution to the correlator vanishes in the $(---)$, $(+++)$
and zero-helicity sector. This follows from the statement that there exists no
solution (that is consistent with the operator product expansion Jain:2021whr
) to the conformal Ward identities in these configurations. Of course, in the
$(\pm\pm\pm)$ configuration the homogeneous correlators are solutions. In
other words, there are no additional solutions that are consistent with the
OPE. We also investigated correlators involving half-integer spin currents in
appendix D, where we find a perfect agreement with our statements.

Thus, after all the dust has settled, we see that using statements $1$, $2$
and $3$, (2.2.1), (2.2.1) and (17)), the expressions for three-point functions
inside the triangle for generic CFTs in the eight helicity configurations are,

$\displaystyle\langle
J_{s_{1}}^{-}J_{s_{2}}^{-}J_{s_{3}}^{-}\rangle=(n_{f}-n_{b}-2in_{\text{odd}})\langle
J_{s_{1}}^{-}J_{s_{2}}^{-}J_{s_{3}}^{-}\rangle_{h},~{}\,\langle
J_{s_{1}}^{+}J_{s_{2}}^{+}J_{s_{3}}^{+}\rangle=(n_{f}-n_{b}+2in_{\text{odd}})\langle
J_{s_{1}}^{+}J_{s_{2}}^{+}J_{s_{3}}^{+}\rangle_{h},$ $\displaystyle\langle
J_{s_{1}}^{-}J_{s_{2}}^{-}J_{s_{3}}^{+}\rangle=(n_{b}+n_{f})\langle
J_{s_{1}}^{-}J_{s_{2}}^{-}J_{s_{3}}^{+}\rangle_{nh},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\langle
J_{s_{1}}^{+}J_{s_{2}}^{+}J_{s_{3}}^{-}\rangle=(n_{b}+n_{f})\langle
J_{s_{1}}^{+}J_{s_{2}}^{+}J_{s_{3}}^{-}\rangle_{nh},$ $\displaystyle\langle
J_{s_{1}}^{-}J_{s_{2}}^{+}J_{s_{3}}^{-}\rangle=(n_{b}+n_{f})\langle
J_{s_{1}}^{-}J_{s_{2}}^{+}J_{s_{3}}^{-}\rangle_{nh},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\langle
J_{s_{1}}^{+}J_{s_{2}}^{-}J_{s_{3}}^{+}\rangle=(n_{b}+n_{f})\langle
J_{s_{1}}^{+}J_{s_{2}}^{-}J_{s_{3}}^{+}\rangle_{nh},$ $\displaystyle\langle
J_{s_{1}}^{+}J_{s_{2}}^{-}J_{s_{3}}^{-}\rangle=(n_{b}+n_{f})\langle
J_{s_{1}}^{+}J_{s_{2}}^{-}J_{s_{3}}^{-}\rangle_{nh},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\langle
J_{s_{1}}^{-}J_{s_{2}}^{+}J_{s_{3}}^{+}\rangle=(n_{b}+n_{f})\langle
J_{s_{1}}^{-}J_{s_{2}}^{+}J_{s_{3}}^{+}\rangle_{nh},$ (33)

and if there are any configurations with net zero helicity, the correlators
are identically zero. We now turn to the study of three-point functions that
are outside the triangle.

####  2.2.2 Three-point functions that are outside the triangle

Three-point functions of conserved currents that violate the spin triangle
inequality take the following form Giombi:2011rz :

$\displaystyle\langle J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle=n_{f}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FF}+n_{b}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FB}\,,$ (34)

i.e. in contrast to (8), the parity odd structure is incompatible with current
conservation.181818There can exist parity odd nonhomogeneous contributions but
these are all contact term contributions. For instance, take the FF or FB
Ward-Takahashi identity and replace all the two-point functions appearing
there by their parity odd counterparts. This gives rise to two new structures
that source these parity odd Ward-Takahashi identities. It would be
interesting to see the consequences of the existence of these structures but
we leave such an exercise for the future. It is also interesting to note that
in contrast to the situation inside the triangle (13), there is no homogeneous
correlator outside the triangle. The reason for this is that the free bosonic
and free fermionic Ward-Takahashi identities are different outside the
triangle Jain:2021whr . It is also important to note that the Ward-Takahashi
identity is only due to the highest spin current in the correlator. For
example, consider a correlator with a spin four current and two spin one
currents, i.e., $\langle J_{4}J_{1}J_{1}\rangle$. Here, the only Ward-
Takahashi identity is with respect to the spin-four current.

Just as we have made for three-point functions inside the triangle, we shall
make some statements for their counterparts that are outside the triangle.
Without loss of generality, consider a correlator $\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle$ with $s_{1}>s_{2}+s_{3}$, $s_{2}\geq
s_{3}$. We have then, the following statements for correlators that violate
the spin triangle inequality:

Statement 4: The $FF-FB$ correlators have support iff $J_{s_{2}}$ and
$J_{s_{3}}$ have the same helicities.

$\displaystyle\langle J_{s_{1}}^{h_{1}}J_{s_{2}}^{-}J_{s_{3}}^{-}\rangle_{FF-
FB}\neq 0\,,\qquad\langle
J_{s_{1}}^{h_{1}}J_{s_{2}}^{+}J_{s_{3}}^{+}\rangle_{FF-FB}\neq 0$
$\displaystyle\langle J_{s_{1}}^{h_{1}}J_{s_{2}}^{-}J_{s_{3}}^{+}\rangle_{FF-
FB}=\langle J_{s_{1}}^{h_{1}}J_{s_{2}}^{+}J_{s_{3}}^{-}\rangle_{FF-FB}=0.$
(35)

Statement 5: The $FF+FB$ correlators identically vanish when the helicities of
$J_{s_{2}}$ and $J_{s_{3}}$ coincide.

$\displaystyle\langle
J_{s_{1}}^{h_{1}}J_{s_{2}}^{-}J_{s_{3}}^{-}\rangle_{FF+FB}=0\,,\qquad\langle
J_{s_{1}}^{h_{1}}J_{s_{2}}^{+}J_{s_{3}}^{+}\rangle_{FF+FB}=0$
$\displaystyle\langle
J_{s_{1}}^{h_{1}}J_{s_{2}}^{-}J_{s_{3}}^{+}\rangle_{FF+FB}\neq
0\,,\qquad\langle
J_{s_{1}}^{h_{1}}J_{s_{2}}^{+}J_{s_{3}}^{-}\rangle_{FF+FB}\neq 0\,.$ (36)

We verify statements $4$ and $5$ in appendix F via a few examples. A different
argument for the validity of statement $4$ is presented later.

If we allow the current conservation to be “slightly” broken we instead have
Giombi:2011rz ,

$\displaystyle\langle J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle=n_{f}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FF}+n_{odd}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{odd}+n_{b}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FB}\,,$ (37)

where the odd piece is the one that is not conserved but rather, slightly-
broken. However, it was shown in Jain:2021gwa that the odd part can be given
by an epsilon transform of the difference of the fermionic and bosonic
correlators:

$\displaystyle\langle J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle=n_{f}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FF}+n_{odd}~{}\epsilon\cdot\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FF-FB}+n_{b}\langle
J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{FB}\,.$ (38)

The question now is, epsilon transform with respect to what? The answer is
provided in the below statement:

Statement 6: The epsilon transform required to obtain the odd correlator can
be performed only with respect to the lowest two spins in the correlator. The
proof of statement $6$ can be found in appendix G. For example consider
$\langle J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle$ with $s_{1}>s_{2}$, $s_{2}\geq
s_{3}$. Then,

$\displaystyle\langle J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{\text{odd}}=\langle
J_{s_{1}}\epsilon\cdot J_{s_{2}}J_{s_{3}}\rangle_{\text{FF-CB}}=\langle
J_{s_{1}}J_{s_{2}}\epsilon\cdot J_{s_{3}}\rangle_{\text{FF-CB}}\,.$ (39)

Let us now turn to an explicit example of a CFT with the slightly-broken
higher-spin symmetry, i.e. to Chern-Simons matter theories.

####  2.2.3 Quasi-Fermionic Theory

The theory of interest is $SU(N_{f})$ Chern-Simons+fundamental fermions
(henceforth called the quasi-fermionic (QF) theory). The action of the QF
theory is,

$\displaystyle S_{FF,CS}=\int
d^{3}x\bigg{(}\bar{\psi}\not{D}\psi+i\epsilon^{\mu\nu\rho}\frac{\kappa_{f}}{4\pi}\Tr{A_{\mu}\partial_{\nu}A_{\rho}-\frac{2i}{3}A_{\mu}A_{\nu}A_{\rho}}\bigg{)}.$
(40)

As usual, we take the large $N_{f}\to\infty$ limit. We also take the limit
$\kappa_{f}\to\infty$ keeping the t’Hooft coupling
$\lambda_{f}=\frac{N_{f}}{\kappa_{f}}$ fixed. The spectrum of this theory
includes a parity odd $\Delta=2+\order{\frac{1}{N_{f}}}$ scalar $O_{2}$,
conserved spin-$1$ and spin-$2$ currents $J$ and $T$ and weakly nonconserved
currents $J_{s},s=3,4,\cdots$ with $\Delta_{s}=s+1+\order{\frac{1}{N_{f}}}$.

This theory enjoys a strong-weak duality with the $SU(N_{b})$ critical bosonic
theory, which is the theory obtained by flowing to the Wilson-Fisher fixed
point starting with the following action:

$\displaystyle S_{CB,CS}=\int
d^{3}x\bigg{(}D_{\mu}\bar{\phi}D_{\mu}\phi+\frac{\lambda_{4}}{N_{b}}(\bar{\phi}\phi)^{2}+i\epsilon^{\mu\nu\rho}\frac{\kappa_{b}}{4\pi}\Tr{A_{\mu}\partial_{\nu}A_{\rho}-\frac{2i}{3}A_{\mu}A_{\nu}A_{\rho}}\bigg{)}.$
(41)

We take the limits $N_{b}\to\infty,\lambda_{4}\to\infty$ and
$\kappa_{b}\to\infty$ keeping fixed $\frac{\lambda_{4}}{N_{b}}$ and
$\lambda_{b}=\frac{N_{b}}{\kappa_{b}}$. The spectrum of this theory includes a
parity even $\Delta=2+\order{\frac{1}{N_{b}}}$ scalar $O_{2}$, conserved
spin-$1$ and spin-$2$ currents $J$ and $T$ and weakly nonconserved currents
$J_{s},s=3,4,\cdots$ with $\Delta=s+1+\order{\frac{1}{N_{b}}}$.

The nonconservation is not seen at the level of two-point correlation
functions, which take the form of the exactly conserved case, (49) with
$c_{s,even}=\frac{\tilde{N}}{1+\tilde{\lambda}^{2}}$ and
$c_{s,odd}=\frac{\tilde{N}\tilde{\lambda}}{1+\tilde{\lambda}^{2}}$. Three-
point functions take the form of (37) with the coefficients are given by
Maldacena:2012sf 191919We define $\tilde{N}$ and $\tilde{\lambda}$ in (45).

$\displaystyle n_{f}=\frac{\tilde{N}}{1+\tilde{\lambda}^{2}}\,,\qquad
n_{odd}=\frac{\tilde{N}\tilde{\lambda}}{1+\tilde{\lambda}^{2}}\,,\qquad
n_{b}=\frac{\tilde{N}\tilde{\lambda}^{2}}{1+\tilde{\lambda}^{2}}\,.$ (42)

If the three-point function is inside the triangle, then the correlator is
conserved with respect to all the currents. However, when the spin triangle
inequality is violated, the odd part of the correlator gives rise to the
nonconservation.

Let us now summarize the structure of three-point correlators outside the
triangle while employing (38) together with statements $4,5$ and $6$, (2.2.2),
(2.2.2) and (39) in mind.

### Helicity decomposition of correlators outside the triangle

We see that the expressions for three-point functions outside the triangle for
CFTs with the slightly-broken higher-spin symmetry in the eight helicity
configurations are (we assume $s_{1}>s_{2}$, $s_{2}\geq s_{3}$ in the
expression below)

$\displaystyle\langle
J_{s_{1}}^{-}J_{s_{2}}^{-}J_{s_{3}}^{-}\rangle=\tfrac{n_{f}-n_{b}-2in_{odd}}{2}\langle
J_{s_{1}}^{-}J_{s_{2}}^{-}J_{s_{3}}^{-}\rangle_{FF-FB},$ $\displaystyle\langle
J_{s_{1}}^{+}J_{s_{2}}^{+}J_{s_{3}}^{+}\rangle=\tfrac{n_{f}-n_{b}+2in_{odd}}{2}\langle
J_{s_{1}}^{+}J_{s_{2}}^{+}J_{s_{3}}^{+}\rangle_{FF-FB},$ $\displaystyle\langle
J_{s_{1}}^{-}J_{s_{2}}^{-}J_{s_{3}}^{+}\rangle=\tfrac{n_{f}+n_{b}}{2}\langle
J_{s_{1}}^{-}J_{s_{2}}^{-}J_{s_{3}}^{+}\rangle_{FF+FB},$ $\displaystyle\langle
J_{s_{1}}^{+}J_{s_{2}}^{+}J_{s_{3}}^{-}\rangle=\tfrac{n_{f}+n_{b}}{2}\langle
J_{s_{1}}^{+}J_{s_{2}}^{+}J_{s_{3}}^{-}\rangle_{FF+FB},$ $\displaystyle\langle
J_{s_{1}}^{-}J_{s_{2}}^{+}J_{s_{3}}^{-}\rangle=\tfrac{n_{f}+n_{b}}{2}\langle
J_{s_{1}}^{-}J_{s_{2}}^{+}J_{s_{3}}^{-}\rangle_{FF+FB},$ $\displaystyle\langle
J_{s_{1}}^{+}J_{s_{2}}^{-}J_{s_{3}}^{+}\rangle=\tfrac{n_{f}+n_{b}}{2}\langle
J_{s_{1}}^{+}J_{s_{2}}^{-}J_{s_{3}}^{+}\rangle_{FF+FB},$ $\displaystyle\langle
J_{s_{1}}^{+}J_{s_{2}}^{-}J_{s_{3}}^{-}\rangle=\tfrac{n_{f}-n_{b}-2in_{odd}}{2}\langle
J_{s_{1}}^{+}J_{s_{2}}^{-}J_{s_{3}}^{-}\rangle_{FF-FB},$ $\displaystyle\langle
J_{s_{1}}^{-}J_{s_{2}}^{+}J_{s_{3}}^{+}\rangle=\tfrac{n_{f}-n_{b}+2in_{odd}}{2}\langle
J_{s_{1}}^{-}J_{s_{2}}^{+}J_{s_{3}}^{+}\rangle_{FF-FB}.$ (43)

With the three-point correlators both inside and outside the triangle under
our control, we use the results of this section in the quest to identify the
(anti-)chiral limit in Chern-Simons matter theory.

##  3 (Anti-)chiral limit for Chern-Simons matter theory

Our goal in this section is to find a closed subsector of Chern-Simons matter
theories that is dual to (anti-)Chiral higher-spin gravity in
$\text{AdS}_{4}$. In the main text, we focus on the QF theory. For the case of
Chern-Simons+fundamental bosons, please refer to appendix C.

The limit that we take to reach the (anti-)Chiral theory should be a
nonunitary one as the (anti-)chiral bulk theory is nonunitary. To find out the
correct prescription, we begin by analyzing correlation functions of the
single trace primary operators in the Chern-Simons matter theory.

Before we proceed, let us set the notation straight. In contrast to
Jain:2022ajd , we normalize our single trace operators such that their two-
point functions are $\order{1}$. We also rescale our currents and scalar
operator:

$\displaystyle\tilde{J}_{s}^{\pm}=e^{\mp i\theta}J_{s}^{\pm}$
$\displaystyle\tilde{O}_{2}=\cos{\theta}~{}O_{2},$ (44)

where $\theta=\frac{\pi\lambda}{2}$ and $\lambda=\frac{N}{k}$ is the ’t Hooft
coupling in the Chern-Simons matter theory. Note that the rescaling of the
currents is different for their positive and negative components. For real
$\theta$ this is required as $J_{s}^{+}$ and $J_{s}^{-}$ are complex
conjugates of each other. However, if we complexify $\theta$ this is no longer
true and $J_{s}^{+}$ and $J_{s}^{-}$ become independent quantities. Finally,
we define for convenience,

$\displaystyle\tilde{N}=2N\frac{\sin{2\theta}}{2\theta}\,,$
$\displaystyle\tilde{\lambda}=\tan{\theta}\,.$ (45)

We now state two limits that, as we shall show, project us into the anti-
chiral or chiral sectors.

$\displaystyle\textbf{anti-chiral
limit:~{}}(\tilde{N}\to\infty,\tilde{\lambda}\to-i\equiv\theta\to-i\infty)~{}\textbf{with~{}}\frac{e^{i\theta}}{\sqrt{\tilde{N}}}=g_{ac},$
(46)

and,

$\displaystyle\textbf{chiral limit:~{}}(\tilde{N}\to\infty,\tilde{\lambda}\to
i\equiv\theta\to
i\infty)~{}\textbf{with~{}}\frac{e^{-i\theta}}{\sqrt{\tilde{N}}}=g_{c}.$ (47)

Let us begin by investigating the consequences of these limits for two-point
functions.

###  3.1 Two-point functions

The two-point functions of the spin-$s$ currents in the QF theory are given
by,

$\displaystyle\langle J_{s}J_{s}\rangle_{QF}=\langle
J_{s}J_{s}\rangle_{even}+\tilde{\lambda}\langle J_{s}J_{s}\rangle_{odd}\,,$
(48)

where the odd piece is given in terms of the even one through an epsilon
transformation as we discussed in section 2. In spinor-helicity variables, the
correlator takes the following form in the two nonzero helicity configurations
(note that this and all other expressions in spinor-helicity variables are
after the rescaling (3)):

$\displaystyle\langle\tilde{J}_{s}^{-}\tilde{J}_{s}^{-}\rangle_{QF}=(1-i\tilde{\lambda})\langle
J_{s}^{-}J_{s}^{-}\rangle_{even}\,,\qquad\langle\tilde{J}_{s}^{+}\tilde{J}_{s}^{+}\rangle_{QF}=(1+i\tilde{\lambda})\langle
J_{s}^{+}J_{s}^{+}\rangle_{even}.$ (49)

Let us now take the limit $\tilde{\lambda}\to-i$ in (49). In this limit, only
the $(--)$ configuration survives. We obtain,

$\displaystyle\lim_{\tilde{\lambda}\to-i}\bigg{(}\langle\tilde{J}_{s}^{-}\tilde{J}_{s}^{-}\rangle_{QF}=2\langle
J_{s}^{-}J_{s}^{-}\rangle_{even}\,,\qquad\langle\tilde{J}_{s}^{+}\tilde{J}_{s}^{+}\rangle_{QF}=0\bigg{)}.$
(50)

Similarly, as $\tilde{\lambda}\to i$, we obtain only the $(++)$ helicity
configuration. The story for the scalar two-point function is simple: it
reads,

$\displaystyle\langle\tilde{O}_{2}\tilde{O}_{2}\rangle_{QF}=\langle
O_{2}O_{2}\rangle_{FF}\,.$ (51)

Let us move on to the three-point case where we can test this limit further.

###  3.2 Three-point functions

We will show that (46) and (47) do indeed produce (anti-)chiral three-point
functions. Let us begin with correlators that are inside the triangle.

####  3.2.1 Spinning correlators inside the triangle

###  Example 1: $\langle TJJ\rangle$

This correlator is given by,

$\displaystyle\langle
TJJ\rangle_{QF}=\frac{1}{\sqrt{\tilde{N}}(1+\tilde{\lambda}^{2})}\big{(}\langle
TJJ\rangle_{FF}+\tilde{\lambda}~{}\epsilon\cdot\langle TJJ\rangle_{FF-
FB}+\tilde{\lambda}^{2}\langle TJJ\rangle_{FB}\big{)}.$ (52)

Using our results for correlators inside the triangle (2) with $n_{f},n_{b}$
and $n_{\text{odd}}$ given in (42), we obtain the expression of this
correlator in the eight helicity configurations:

$\displaystyle\langle\tilde{T}^{-}\tilde{J}^{-}\tilde{J}^{-}\rangle=\frac{e^{i\theta}}{\sqrt{\tilde{N}}}\langle
T^{-}J^{-}J^{-}\rangle_{h}\,,\qquad$
$\displaystyle\langle\tilde{T}^{+}\tilde{J}^{+}\tilde{J}^{+}\rangle$
$\displaystyle=\frac{e^{-i\theta}}{\sqrt{\tilde{N}}}\langle
T^{+}J^{+}J^{+}\rangle_{h}\,,$
$\displaystyle\langle\tilde{T}^{-}\tilde{J}^{-}\tilde{J}^{+}\rangle=\frac{e^{i\theta}}{\sqrt{\tilde{N}}}\langle
T^{-}J^{-}J^{+}\rangle_{nh}\,,\qquad$
$\displaystyle\langle\tilde{T}^{+}\tilde{J}^{+}\tilde{J}^{-}\rangle$
$\displaystyle=\frac{e^{-i\theta}}{\sqrt{\tilde{N}}}\langle
T^{+}J^{+}J^{-}\rangle_{nh}\,,$
$\displaystyle\langle\tilde{T}^{-}\tilde{J}^{+}\tilde{J}^{-}\rangle=\frac{e^{i\theta}}{\sqrt{\tilde{N}}}\langle
T^{-}J^{+}J^{-}\rangle_{nh}\,,\qquad$
$\displaystyle\langle\tilde{T}^{+}\tilde{J}^{-}\tilde{J}^{+}\rangle$
$\displaystyle=\frac{e^{-i\theta}}{\sqrt{\tilde{N}}}\langle
T^{+}J^{-}J^{+}\rangle_{nh}\,,$
$\displaystyle\langle\tilde{T}^{-}\tilde{J}^{+}\tilde{J}^{+}\rangle=0\,,$
$\displaystyle\langle\tilde{T}^{+}\tilde{J}^{-}\tilde{J}^{-}\rangle$
$\displaystyle=0\,.$ (53)

Let us now take the anti-chiral limit defined in (46). In this limit we
obtain,

$\displaystyle\langle\tilde{T}^{-}\tilde{J}^{-}\tilde{J}^{-}\rangle=g_{ac}\langle
T^{-}J^{-}J^{-}\rangle_{h}\,,\qquad$
$\displaystyle\langle\tilde{T}^{+}\tilde{J}^{+}\tilde{J}^{+}\rangle$
$\displaystyle=0\,,$
$\displaystyle\langle\tilde{T}^{-}\tilde{J}^{-}\tilde{J}^{+}\rangle=g_{ac}\langle
T^{-}J^{-}J^{+}\rangle_{nh}\,,\qquad$
$\displaystyle\langle\tilde{T}^{+}\tilde{J}^{+}\tilde{J}^{-}\rangle$
$\displaystyle=0\,,$
$\displaystyle\langle\tilde{T}^{-}\tilde{J}^{+}\tilde{J}^{-}\rangle=g_{ac}\langle
T^{-}J^{+}J^{-}\rangle_{nh}\,,\qquad$
$\displaystyle\langle\tilde{T}^{+}\tilde{J}^{-}\tilde{J}^{+}\rangle$
$\displaystyle=0\,,$
$\displaystyle\langle\tilde{T}^{-}\tilde{J}^{+}\tilde{J}^{+}\rangle=0\,,$
$\displaystyle\langle\tilde{T}^{+}\tilde{J}^{-}\tilde{J}^{-}\rangle$
$\displaystyle=0.$ (54)

Equation (3) contains only the net negative helicity configurations of the
correlators and is precisely the definition of the anti-chiral sector, showing
that the limit (46) is as we desired. One point to note is that if we had not
re-scaled the currents as in (3), we would only have been able to retain the
$(---)$ helicity configuration. However, with the rescaling, we are able to
obtain the full anti-chiral sector viz (3). Let us consider another example
now to further illustrate and validate (46):

###  Example 2: $\langle TTT\rangle$

This correlator is given by 202020Following footnote 7, there can, in
principle, exist a parity odd nonhomogeneous contact term. If we allow for
such contributions, $\langle TTT\rangle_{nh,\text{even}}$ can be cancelled out
by $\langle TTT\rangle_{nh,\text{odd}}$ in the $(---)$ and $(+++)$ helicities
as can be seen given the expressions in Jain:2021vrv . However, for
correlators such as $\langle TJJ\rangle$, there is no nonhomogeneous parity
odd contact term possible and thus our parity even contact term analysis of
section 2 is essential.

$\displaystyle\langle
TTT\rangle=\frac{1}{\sqrt{\tilde{N}}(1+\tilde{\lambda}^{2})}\big{(}\langle
TTT\rangle_{FF}+\epsilon\cdot\langle TTT\rangle_{FF-
FB}+\tilde{\lambda}^{2}\langle TTT\rangle_{FB}\big{)}.$ (55)

Using (2) and (42), we see that this correlator in the eight helicity
configurations reads

$\displaystyle\langle\tilde{T}^{-}\tilde{T}^{-}\tilde{T}^{-}\rangle=\frac{e^{i\theta}}{\sqrt{\tilde{N}}}\langle
T^{-}T^{-}T^{-}\rangle_{h}~{}~{},~{}~{}\langle\tilde{T}^{+}\tilde{T}^{+}\tilde{T}^{+}\rangle=\frac{e^{-i\theta}}{\sqrt{\tilde{N}}}\langle
T^{+}T^{+}T^{+}\rangle_{h}$
$\displaystyle\langle\tilde{T}^{-}\tilde{T}^{-}\tilde{T}^{+}\rangle=\frac{e^{i\theta}}{\sqrt{\tilde{N}}}\langle
T^{-}T^{-}T^{+}\rangle_{nh}~{}~{},~{}~{}\langle\tilde{T}^{+}\tilde{T}^{+}\tilde{T}^{-}\rangle=\frac{e^{-i\theta}}{\sqrt{\tilde{N}}}\langle
T^{+}T^{+}T^{-}\rangle_{nh}$
$\displaystyle\langle\tilde{T}^{-}\tilde{T}^{+}\tilde{T}^{-}\rangle=\frac{e^{i\theta}}{\sqrt{\tilde{N}}}\langle
T^{-}T^{+}T^{-}\rangle_{nh}~{}~{},~{}~{}\langle\tilde{T}^{+}\tilde{T}^{-}\tilde{T}^{+}\rangle=\frac{e^{-i\theta}}{\sqrt{\tilde{N}}}\langle
T^{+}T^{-}T^{+}\rangle_{nh}$
$\displaystyle\langle\tilde{T}^{+}\tilde{T}^{-}\tilde{T}^{-}\rangle=\frac{e^{i\theta}}{\sqrt{\tilde{N}}}\langle
T^{+}T^{-}T^{-}\rangle_{nh}~{}~{},~{}~{}\langle\tilde{T}^{-}\tilde{T}^{+}\tilde{T}^{+}\rangle=\frac{e^{-i\theta}}{\sqrt{\tilde{N}}}\langle
T^{-}T^{+}T^{+}\rangle_{nh}.$ (56)

Taking the anti-chiral limit (46) we obtain only the anti-chiral helicity
configurations as desired:

$\displaystyle\langle\tilde{T}^{-}\tilde{T}^{-}\tilde{T}^{-}\rangle=g_{ac}\langle
T^{-}T^{-}T^{-}\rangle_{h}\,,$
$\displaystyle\langle\tilde{T}^{+}\tilde{T}^{+}\tilde{T}^{+}\rangle=0$
$\displaystyle\langle\tilde{T}^{-}\tilde{T}^{-}\tilde{T}^{+}\rangle=g_{ac}\langle
T^{-}T^{-}T^{+}\rangle_{nh}\,,$
$\displaystyle\langle\tilde{T}^{+}\tilde{T}^{+}\tilde{T}^{-}\rangle=0$
$\displaystyle\langle\tilde{T}^{-}\tilde{T}^{+}\tilde{T}^{-}\rangle=g_{ac}\langle
T^{-}T^{+}T^{-}\rangle_{nh}\,,$
$\displaystyle\langle\tilde{T}^{+}\tilde{T}^{-}\tilde{T}^{+}\rangle=0$
$\displaystyle\langle\tilde{T}^{+}\tilde{T}^{-}\tilde{T}^{-}\rangle=g_{ac}\langle
T^{+}T^{-}T^{-}\rangle_{nh}\,,$
$\displaystyle\langle\tilde{T}^{-}\tilde{T}^{+}\tilde{T}^{+}\rangle=0.$ (57)

We note that if the nonhomogeneous correlator was nonzero in the $(---)$
helicity configuration, we would have a contribution of the form
$\frac{e^{3i\theta}}{\sqrt{\tilde{N}}}\langle T^{-}T^{-}T^{-}\rangle_{nh}$
which would be exponentially enhanced compared to all the other helicity
configurations in the anti-chiral limit. It being zero via the addition of
contact terms like (2) is essential to obtain a uniform scaling for all the
anti-chiral helicity configurations as in (3).

Let us now consider a correlator outside the triangle and investigate if our
findings carry over.

####  3.2.2 Spinning correlators outside the triangle

###  Example 3: $\langle J_{4}J_{1}J_{1}\rangle$

Using (2) and (42), we obtain that the eight helicity configurations of this
correlator are

$\displaystyle\langle\tilde{J}_{4}^{-}\tilde{J}^{-}\tilde{J}^{-}\rangle=\frac{e^{i\theta}}{2\sqrt{\tilde{N}}}\langle
J_{4}^{-}J^{-}J^{-}\rangle_{FF-CB}\,,$
$\displaystyle\langle\tilde{J}_{4}^{+}\tilde{J}^{+}\tilde{J}^{+}\rangle=\frac{e^{-i\theta}}{2\sqrt{\tilde{N}}}\langle
J_{4}^{+}J^{+}J^{+}\rangle_{FF-CB}$
$\displaystyle\langle\tilde{J}_{4}^{-}\tilde{J}^{-}\tilde{J}^{+}\rangle=\frac{e^{i\theta}}{2\sqrt{\tilde{N}}}\langle
J_{4}^{-}J^{-}J^{+}\rangle_{FF+CB}\,,$
$\displaystyle\langle\tilde{J}_{4}^{+}\tilde{J}^{+}\tilde{J}^{-}\rangle=\frac{e^{-i\theta}}{2\sqrt{\tilde{N}}}\langle
J_{4}^{+}J^{+}J^{-}\rangle_{FF+CB}$
$\displaystyle\langle\tilde{J}_{4}^{-}\tilde{J}^{+}\tilde{J}^{-}\rangle=\frac{e^{i\theta}}{2\sqrt{\tilde{N}}}\langle
J_{4}^{-}J^{+}J^{-}\rangle_{FF+CB}\,,$
$\displaystyle\langle\tilde{J}_{4}^{+}\tilde{J}^{-}\tilde{J}^{+}\rangle=\frac{e^{-i\theta}}{2\sqrt{\tilde{N}}}\langle
J_{4}^{+}J^{-}J^{+}\rangle_{FF+CB}$
$\displaystyle\langle\tilde{J}_{4}^{-}\tilde{J}^{+}\tilde{J}^{+}\rangle=\frac{e^{i\theta}}{2\sqrt{\tilde{N}}}\langle
J_{4}^{-}J^{+}J^{+}\rangle_{FF-CB}\,,$
$\displaystyle\langle\tilde{J}_{4}^{+}\tilde{J}^{-}\tilde{J}^{-}\rangle=\frac{e^{-i\theta}}{2\sqrt{\tilde{N}}}\langle
J_{4}^{+}J^{-}J^{-}\rangle_{FF-CB}\,.$ (58)

Let us now take the anti-chiral limit defined in (46). In this limit we
obtain,

$\displaystyle\langle\tilde{J}_{4}^{-}\tilde{J}^{-}\tilde{J}^{-}\rangle=\frac{g_{ac}}{2}\langle
J_{4}^{-}J^{-}J^{-}\rangle_{FF-CB}\,,$
$\displaystyle\langle\tilde{J}_{4}^{+}\tilde{J}^{+}\tilde{J}^{+}\rangle=0$
$\displaystyle\langle\tilde{J}_{4}^{-}\tilde{J}^{-}\tilde{J}^{+}\rangle=\frac{g_{ac}}{2}\langle
J_{4}^{-}J^{-}J^{+}\rangle_{FF+CB}\,,$
$\displaystyle\langle\tilde{J}_{4}^{+}\tilde{J}^{+}\tilde{J}^{-}\rangle=0$
$\displaystyle\langle\tilde{J}_{4}^{-}\tilde{J}^{+}\tilde{J}^{-}\rangle=\frac{g_{ac}}{2}\langle
J_{4}^{-}J^{+}J^{-}\rangle_{FF+CB}\,,$
$\displaystyle\langle\tilde{J}_{4}^{+}\tilde{J}^{-}\tilde{J}^{+}\rangle=0$
$\displaystyle\langle\tilde{J}_{4}^{-}\tilde{J}^{+}\tilde{J}^{+}\rangle=\frac{g_{ac}}{2}\langle
J_{4}^{-}J^{+}J^{+}\rangle_{FF-CB}\,,$
$\displaystyle\langle\tilde{J}_{4}^{+}\tilde{J}^{-}\tilde{J}^{-}\rangle=0\,,$
(59)

which indeed is the anti-chiral sector. Note that we used the facts that the
$FF+CB$ contribution to this correlator vanishes in the $(---)$ as well as the
$(-++)$ helicity configurations (as well as their complex conjugate
helicities). We also used the fact that the $FF-CB$ contribution vanishes in
the $(--+)$ helicity as well as its complex conjugate and their
$(2\leftrightarrow 3)$ permutations. These facts were explicitly obtained in
Jain:2021whr . Some comments about parity odd contact terms are also in order.
The FF and CB Ward-Takahashi identities are distinct outside the triangle.
Thus, if we take the $\langle JJ\rangle$ two-point functions appearing on the
RHS of these Ward-Takahashi identities and replace them with the parity odd
two-point function, we will obtain two distinct parity odd correlation
functions that are, of course, contact term contributions. It is not obvious
what role they play but it would be interesting to further explore their
consequences in the future.

####  3.2.3 Correlators involving the scalar operator

### One scalar

Let us first consider a correlator with a single scalar insertion and two
currents. In the QF theory, such correlators are given by

$\displaystyle\langle J_{s_{1}}J_{s_{2}}O_{2}\rangle$
$\displaystyle=\frac{1}{\sqrt{\tilde{N}}}\big{(}\langle
J_{s_{1}}J_{s_{2}}O_{2}\rangle_{FF}+\tilde{\lambda}\langle
J_{s_{1}}J_{s_{2}}O_{2}\rangle_{CB}\big{)}$
$\displaystyle=\frac{1}{\sqrt{\tilde{N}}}\big{(}\langle
J_{s_{1}}J_{s_{2}}O_{2}\rangle_{FF}+\tilde{\lambda}\epsilon\cdot\langle
J_{s_{1}}J_{s_{2}}O_{2}\rangle_{FF}\big{)}\,,$ (60)

where in the second line, the epsilon transform is performed with respect to
the lower amongst the two spins. This correlator in the four helicity
configurations (after the rescaling (3) and assuming $s_{1}\geq s_{2}$) is
given by,

$\displaystyle\langle\tilde{J}_{s_{1}}^{-}\tilde{J}_{s_{2}}^{-}\tilde{O}_{2}\rangle=\frac{e^{i\theta}}{\sqrt{\tilde{N}}}\langle
J_{s_{1}}^{-}J_{s_{2}}^{-}O_{2}\rangle_{FF}\,,$
$\displaystyle\langle\tilde{J}_{s_{1}}^{+}\tilde{J}_{s_{2}}^{+}\tilde{O}_{2}\rangle=\frac{e^{-i\theta}}{\sqrt{\tilde{N}}}\langle
J_{s_{1}}^{+}J_{s_{2}}^{+}O_{2}\rangle_{FF}$
$\displaystyle\langle\tilde{J}_{s_{1}}^{-}\tilde{J}_{s_{2}}^{+}\tilde{O}_{2}\rangle=\frac{e^{i\theta}}{\sqrt{\tilde{N}}}\langle
J_{s_{1}}^{-}J_{s_{2}}^{+}O_{2}\rangle_{FF}\,,$
$\displaystyle\langle\tilde{J}_{s_{1}}^{+}\tilde{J}_{s_{2}}^{-}\tilde{O}_{2}\rangle=\frac{e^{-i\theta}}{\sqrt{\tilde{N}}}\langle
J_{s_{1}}^{+}J_{s_{2}}^{-}O_{2}\rangle_{FF}\,.$ (61)

Taking the anti-chiral limit (46) we obtain,

$\displaystyle\langle\tilde{J}_{s_{1}}^{-}\tilde{J}_{s_{2}}^{-}\tilde{O}_{2}\rangle=g_{ac}\langle
J_{s_{1}}^{-}J_{s_{2}}^{-}O_{2}\rangle_{FF}\,,$
$\displaystyle\langle\tilde{J}_{s_{1}}^{+}\tilde{J}_{s_{2}}^{+}\tilde{O}_{2}\rangle=0$
$\displaystyle\langle\tilde{J}_{s_{1}}^{-}\tilde{J}_{s_{2}}^{+}\tilde{O}_{2}\rangle=g_{ac}\langle
J_{s_{1}}^{-}J_{s_{2}}^{+}O_{2}\rangle_{FF}\,,$
$\displaystyle\langle\tilde{J}_{s_{1}}^{+}\tilde{J}_{s_{2}}^{-}\tilde{O}_{2}\rangle=0\,,$
(62)

which is indeed the desired result.

### Two scalars

The QF theory correlator involving two scalars is given by

$\displaystyle\langle
J_{s}O_{2}O_{2}\rangle=\frac{(1+\tilde{\lambda}^{2})}{\sqrt{\tilde{N}}}\langle
J_{s}O_{2}O_{2}\rangle_{FF}\,.$ (63)

In the two independent helicity configurations, it is given by,

$\displaystyle\langle\tilde{J}_{s}^{-}\tilde{O}_{2}\tilde{O}_{2}\rangle=\frac{e^{i\theta}}{\sqrt{\tilde{N}}}\langle
J_{s}^{-}\tilde{O}_{2}\tilde{O}_{2}\rangle_{FF}\,,$
$\displaystyle\langle\tilde{J}_{s}^{+}\tilde{O}_{2}\tilde{O}_{2}\rangle=\frac{e^{-i\theta}}{\sqrt{\tilde{N}}}\langle
J_{s}^{+}O_{2}O_{2}\rangle_{FF}\,.$ (64)

Taking the anti-chiral limit (46) yet again results in the desired answer:

$\displaystyle\langle\tilde{J}_{s}^{-}\tilde{O}_{2}\tilde{O}_{2}\rangle=g_{ac}\langle
J_{s}^{-}O_{2}O_{2}\rangle_{FF}\,,$
$\displaystyle\langle\tilde{J}_{s}^{+}\tilde{O}_{2}\tilde{O}_{2}\rangle=0\,.$
(65)

### Three-point Summary

Based on the examples considered above, one can easily show that the anti-
chiral limit, (46), works for any three-point correlator that is inside or
outside the triangle, also for the ones involving scalar operators, thus
establishing that what we have uncovered is indeed the subsector that is dual
to anti-Chiral higher-spin gravity. Indeed, at the three-point level, the main
signature of (anti-)Chiral higher-spin gravity is that it features cubic
interactions with the total helicity being (negative) positive.

Let us now explore the consequences of taking this limit at higher points,
starting with the four-point case.

##  4 Beyond three-point

A general comment is that beyond three-point and at the loop level chiral and
anti-chiral interactions can mix in the bulk, see section 5, which is not in
contradiction with having them as closed subsectors on their own. However, we
should not expect the simple limiting procedure we proposed above to work as
it is, but it is instructive to see what happens when we take the limit.

###  4.1 Four-point functions

In this subsection, we investigate what happens to four-point correlators in
the QF theory when we take the anti-chiral limit (46). Let us begin with the
scalar case.

### Example 1: $\langle O_{2}O_{2}O_{2}O_{2}\rangle$

The scalar correlator in the CS+fermionic matter theory is given by
Bedhotiya:2015uga ; Turiaci:2018nua

$\displaystyle\langle\tilde{O}_{2}\tilde{O}_{2}\tilde{O}_{2}\tilde{O}_{2}\rangle=\frac{\cos^{4}(\theta)(1+\tilde{\lambda}^{2})^{2}}{\tilde{N}}\langle
O_{2}O_{2}O_{2}O_{2}\rangle_{FF}=\frac{1}{\tilde{N}}\langle
O_{2}O_{2}O_{2}O_{2}\rangle_{FF}\,,$ (66)

where we used $\tilde{\lambda}=\tan(\theta)$. Therefore, in the chiral limit
and the anti-chiral limits (47) and (46), this correlator goes to
zero.212121Given the fact that the fermionic scalar four-point function is
non-local Sleight:2017pcz ; Jain:2023juk , it is a nice feature that it drops
out in this limit. This is an indication that we are focusing on a local sub-
sector.

### Example 2: $\langle J_{s}O_{2}O_{2}O_{2}\rangle$

In the anti-chiral limit (46), this correlator in the two helicity
configurations is given by Li:2019twz ; Silva:2021ece

$\displaystyle\langle\tilde{J}_{s}^{-}\tilde{O}_{2}\tilde{O}_{2}\tilde{O}_{2}\rangle=\frac{e^{i\theta}}{\tilde{N}}\big{(}\cos(\theta)\langle
J_{s}^{-}O_{2}O_{2}O_{2}\rangle_{FF}+\sin(\theta)\langle
J_{s}^{-}O_{2}O_{2}O_{2}\rangle_{CB}\big{)}\,,$
$\displaystyle\langle\tilde{J}_{s}^{+}\tilde{O}_{2}\tilde{O}_{2}\tilde{O}_{2}\rangle=\frac{e^{-i\theta}}{\tilde{N}}\big{(}\cos(\theta)\langle
J_{s}^{+}O_{2}O_{2}O_{2}\rangle_{FF}+\sin(\theta)\langle
J_{s}^{+}O_{2}O_{2}O_{2}\rangle_{CB}\big{)}\,.$ (67)

If we now take the anti-chiral limit (46) we see that

$\displaystyle\langle\tilde{J}_{s}^{-}\tilde{O}_{2}\tilde{O}_{2}\tilde{O}_{2}\rangle=\frac{g^{2}}{2}\big{(}\langle
J_{s}^{-}O_{2}O_{2}O_{2}\rangle_{FF}-i\langle
J_{s}^{-}O_{2}O_{2}O_{2}\rangle_{CB}\big{)}\,,$
$\displaystyle\langle\tilde{J}_{s}^{+}O_{2}O_{2}O_{2}\rangle=0\,,$ (68)

thus showing that the anti-chiral limit (46) is well defined and indeed picks
out just the anti-chiral sector for these types of four-point functions.

In the above examples, we notice a very nice separation between the chiral
(net negative helicity) and anti-chiral (net positive helicity) sectors.
However, by investigating several more general four-point functions in
appendix I, we see that in addition to taking the (anti-)chiral limit, one
needs to throw away some extra pieces to obtain finite results. However, we
postpone further analysis at the four- and higher-point levels to future work.

###  4.2 Anomalous dimensions in the (anti-)chiral limit

The anomalous dimensions of the spin $s$ currents in the CS+fermionic matter
theory at leading order in $\tilde{N}$ are given by Giombi:2016zwa

$\displaystyle\gamma_{s}=\frac{1}{\tilde{N}}\bigg{(}\frac{a_{s}\tilde{\lambda}^{2}}{(1+\tilde{\lambda}^{2})}+\frac{b_{s}\tilde{\lambda}^{2}}{(1+\tilde{\lambda}^{2})^{2}}\bigg{)}=\frac{\sin^{2}(\theta)}{\tilde{N}}\bigg{(}a_{s}+b_{s}\cos^{2}(\theta)\bigg{)}\,,$
(69)

where we used $\tilde{\lambda}=\tan(\theta)$. Let us now take the anti-chiral
limit (46). In this limit we have

$\displaystyle\gamma_{s}=-\frac{a_{s}}{4}g_{ac}^{2}-\frac{b_{s}}{16}e^{2i\theta}g_{ac}^{2}+\cdots.$
(70)

Note that $a_{s}$ is a rational function of $s$ that approaches constant for
large spins, while $b_{s}\sim\log s$ for large spins. The latter signals that
the CFT is a gauge theory. It seems that the chiral limit should freeze these
gauge degrees of freedom and the $b_{s}$ term needs to be dropped.

As for the scalar operator, its anomalous dimension is given by Jain:2019fja

$\displaystyle\gamma_{0}=-\frac{32}{3\pi^{2}}\frac{\tilde{\lambda}^{2}}{\tilde{N}(1+\tilde{\lambda}^{2})}\,.$
(71)

Using the definition $\tilde{\lambda}=\tan\theta$ we see that this equals

$\displaystyle\gamma_{0}=-\frac{32}{3\pi^{2}}\frac{(\sin\theta)^{2}}{\tilde{N}}\,.$
(72)

In the anti-chiral limit, this quantity is finite and equals,

$\displaystyle\gamma_{0}=\frac{8g_{ac}^{2}}{3\pi^{2}}\,.$ (73)

An important point to note here is that (70) and (73) could receive
corrections from non-planar diagrams. In the usual setting, we take the ’t
Hooft limit $\lambda=\frac{N}{k}$ to be finite while taking $N$ and $k$ to
infinity. However, the anti-chiral limit (46) that we take corresponds to
$\lambda\to-i\infty$ which means we take $N\to\infty$ faster than we take
$k\to\infty$. This can alternatively be interpreted as $N\to\infty$ while
taking $k\to i$ thus indicating that the contributions of the non-planar
diagrams to the anomalous dimensions need to be kept while taking this limit.
Thus, it may be appropriate or even necessary to perform the re-summation of
these contributions. For instance, the divergent part of the anomalous
dimension of the spin $s$ currents (70) could be rendered finite by these
contributions. We leave a detailed investigation of this to the future.

##  5 Holographic predictions for correlators

In this section we change the gears and move into the bulk. It is quite easy
to get a number of predictions by analysing the structure of interactions in
Chiral theory without having to actually compute any holographic correlators.
Some of the statements below go a bit beyond what is known at present on the
CFT side. Several points should be stressed before we proceed.

Firstly, as we show below, certain corrections vanish identically in the bulk
because it is impossible to draw a tree/loop diagram that would contribute.
These are robust predictions. However, we expect that the higher-spin symmetry
implies many cancellations. For example, there have to be no loop corrections
to holographic correlators for the free CFT duals. Therefore, certain
combinations of loop diagrams must vanish, even though it is possible to draw
them. To take this possibility into account to refer to a situation when a
contribution exists, but only an explicit computation can tell us if it
vanishes or not, we will use the sign
$\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny?}}}}{{\neq}}}0$.

Secondly, we will consider the most general case of the bosonic ($U(M)$ or
$O(M)$) gauged Chiral higher-spin gravity. On the CFT side, this corresponds
to vector models with some leftover global symmetries, e.g. $U(M)$ or $O(M)$.
However, there are two standard cases in the literature: (a) the spectrum
contains higher-spin currents of all integer spins; (b) the spectrum has
higher-spin currents with even spins only. These are obtained by simple
reductions of the general statements that are made for the nonabelian Chiral
theory: for (a) the sum of spins meeting at a vertex has to be even; for (b)
the spins themselves have to be even. For example, for case (a) $\langle
TTJ\rangle=0$ and there is no bulk cubic vertex with spins $2-2-1$.
Supersymmetric extensions are also possible, but we do not discuss them in
detail.

Thirdly, one should be careful in comparing the bulk predictions to CFT while
the chiral limit on the CFT side is not yet well-understood. Indeed, the
hypothetical bulk dual of the full Chern-Simons matter theories (see also the
picture in the introduction) has the same spectrum as Chiral theory. Chiral
and anti-chiral interactions cover all possible cubic interactions. Starting
from the quartic order this is not the case and there has to be some
structures (let us call them nonchiral) not belonging to (anti-)Chiral
theories. Starting from four-point functions (and loop corrections)
holographic correlators receive contributions from (anti-)Chiral theories
separately and from various mixtures of (anti-)chiral interactions together
with the nonchiral ones.

Let us begin by giving a schematic form of the Lagrangian of Chiral theory.
The Lagrangian is known in the light-cone gauge in flat space Metsaev:1991nb ;
Metsaev:1991mt ; Ponomarev:2016lrm , up to the cubic order in $AdS_{4}$
Metsaev:2018xip ; Skvortsov:2018uru . With the equations of motion
Skvortsov:2018uru ; Sharapov:2022wpz ; Sharapov:2022awp it is easy to see what
kind of contact interaction vertices are present at higher orders in
$AdS_{4}$, which can be summarized as

$\displaystyle\mathcal{L}$ $\displaystyle=\sum_{s\geq
0}\Phi_{-s}\square\Phi_{s}+\sum_{N\geq 3}g^{N-2}\sum_{|\lambda|\geq
N-2}V^{N}_{\lambda_{1},...,\lambda_{N}}\,.$ (74)

Here, the first term is the sum of kinetic terms for all spins. Note that a
massless spin-$s$, $s>0$ field has two degrees of freedom that can be
associated with helicity $+s$ and $-s$ states, which we denote $\Phi_{\pm
s}$.222222The Lagrangian written as if we use the light-cone gauge
Metsaev:2018xip where the helicity structure is manifest. Note that the
kinetic terms do reduce to simple $\square=\partial_{m}\partial^{m}$ in the
light-cone gauge after an appropriate rescaling despite $AdS_{4}$ is curved,
which is due to the fact that massless higher-spin fields are also conformally
invariant in certain field descriptions. In particular, the bulk-to-bulk
propagator connects helicity $\lambda$ to helicity $-\lambda$. The most
important information about the interactions is that at order $N$ the total
helicity $|\lambda|=\sum_{i}\lambda_{i}$ entering the vertex
$V^{N}_{\lambda_{1},...,\lambda_{N}}$ must be greater or equal to $N-2$ for
chiral theory or $-|\lambda|>N-2$ for the anti-Chiral one. For example, all
cubic interactions with $\lambda_{1}+\lambda_{2}+\lambda_{3}>0$ are present in
Chiral theory and those with $\lambda_{1}+\lambda_{2}+\lambda_{3}<0$ are
present in anti-Chiral one. In reality, the helicity is integer in the bosonic
theory and we have $\lambda_{1}+\lambda_{2}+\lambda_{3}\geq 1$ (or $\leq-1$).
For simplicity, we confine ourselves to anti-Chiral theory as in the main text
above.232323Sketchy arguments to explain the possibility of the chiral
truncation from the bulk vantage point can be found in appendix A.

##### Three-point functions.

The main constraint for the three-point functions is the aforementioned
$\lambda_{1}+\lambda_{2}+\lambda_{3}<0$. In particular, $0-0-0$ vanishes.
Therefore, $\langle OOO\rangle=0$ and all three-point correlators where the
sum of the helicities greater or equal to zero vanish. This is exactly what we
observed above. It is important that Chiral theory can be shown to be a unique
theory under certain assumptions.242424Assumptions can vary, for example, one
can assume that there is at least one higher-spin field that has nontrivial
self-interactions. The latter forces one to introduce all other spins and all
other (anti-)chiral interactions, see Metsaev:1991nb ; Metsaev:1991mt ;
Ponomarev:2016lrm . It is important to note that there are two contractions of
Chiral theory Ponomarev:2017nrr that can be thought of as higher-spin
extensions of selfdual Yang-Mills and of self-dual gravity (separately). They
contain higher-spin fields but exhibit only one-derivative Yang-Mills-type
interactions or two-derivative gravitational-type interactions, but no genuine
higher-spin interactions. These two theories have simple actions
Krasnov:2021nsq .

##### Four-point functions.

There are two types of contributions to the four-point functions at tree
level: exchanges and contact, but they satisfy the same constraint
$-|\lambda|\geq N-2$ as one can easily see. Therefore, for a four-point
correlator to be nonzero we should have
$\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}\leq 2$.

For example, for $0-0-0-0$ the only vertex that can contribute to the exchange
is $0-0-(-s)$ and the bulk-to-bulk propagator connects $+s$ to $-s$ and
$0-0-(\pm s)$ cannot be both in the same Chiral theory. Therefore, $\langle
OOOO\rangle=0$ and, in general, we expect

$\displaystyle\langle J_{-s}OOO\rangle$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny?}}}}{{\neq}}}0\,,\quad
s\geq 2\,,$ $\displaystyle\begin{aligned} \langle J_{-s}OOO\rangle&=0\,,\quad
s=0,1\,,\\\ \langle J_{+s}OOO\rangle&=0\,,\quad s\geq 0\,,\end{aligned}$ (75)

where the correlator on the left can still be zero as a result of some
cancellation, but in general we expect that it does not vanish.

##### Higher-point functions.

For higher point functions we have $-|\lambda|\geq N-2$. In particular, we
expect that

$\displaystyle\langle J_{-s}OOOO\rangle$
$\displaystyle\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny?}}}}{{\neq}}}0\,,\quad
s\geq 3\,,$ $\displaystyle\begin{aligned} \langle J_{-s}OOOO\rangle&=0\,,\quad
s=0,1,2\,,\\\ \langle J_{+s}OOOO\rangle&=0\,,\quad s\geq 0\end{aligned}$ (76)

For example, all $\langle OO...O\rangle=0$.

##### Loop corrections.

Flat space computations Skvortsov:2018jea ; Skvortsov:2020gpn show that there
are no UV divergences in the theory at least at one-loop. Any $n$-point one-
loop diagram can be obtained by connecting two lines in some $(n+2)$-point
tree-level diagram. Therefore, for a one-loop correction to an $n$-point
correlator we get $-|\lambda|\geq n$.

For example, for two-point functions, we have $\lambda_{1}+\lambda_{2}\leq-2$.
Therefore, we do not expect any one-loop correction to $\langle OO\rangle$,
but there has to be one to $\langle J_{-s}J_{-s}\rangle$, $s\geq 1$. For
three-point functions at one-loop, we have
$\lambda_{1}+\lambda_{2}+\lambda_{3}\leq-3$ and so on. In particular, there
should not be any loop corrections to $\langle OO...O\rangle$.

Assuming Chiral theory does not have any UV-divergences to all orders and,
hence, no new counterterms need to be introduced that could change the simple
conclusions here, we can estimate that an $l$-loop correction to an $n$-point
function comes by gluing $2l$ lines in a $(2l+n)$-point tree level diagram.
Therefore, we have to have $-|\lambda|\geq 2l+n-2$. As a consequence, we see
that for any given correlation function $\langle
J_{\lambda_{1}}...J_{\lambda_{n}}\rangle$ loop corrections stop at a certain
finite loop order, $2l\leq-|\lambda|-n+2$. The latter means that the $1/N$
expansion is convergent (we can say nonuniformly because the order depends on
the observable being computed).

##  6 Discussion and Conclusions

In the paper, we made the first step towards identifying the hidden closed
subsector of Chern-Simons matter theories that is dual to Chiral higher-spin
gravity. It is quite easy to do so at the level of three-point functions by
taking a certain limit of the coupling constant. The main selection rule here
is to keep the correlators with total helicity positive/negative. At higher
orders the picture is less clear since chiral and anti-chiral sectors can mix.
Indeed, this is what we observed in appendix I. However, if we naively attempt
to bootstrap higher-point correlators of the (anti-)chiral theory using the
three-point functions as an input, we obtain simple but different results from
directly taking the (anti-)chiral limit of the correlators as we showed in
appendix H. The latter procedure should be consistent with the bulk picture.

One should not rush to compare some of our bulk conclusions to the CFT results
above because the existence of a closed subsector is not the same as existence
of a limiting subsector. We see that at higher orders chiral and anti-chiral
theories/subsectors mix with each other and with the nonchiral ones and a
further refinement of our limiting procedure may be needed. For example, one
may need to drop certain parts of correlation functions to land on the chiral
subsector.

It is worth stressing that the perturbation series in the bulk seems to
converge (nonuniformly in spins), which is dual to the convergence of the
$1/N$ expansion in the chiral subsector. The chiral subsector can be
bootstrapped with the help of the slightly-broken higher-spin symmetry
starting from the three-point functions we have. Therefore, there is a good
chance of establishing a pair of AdS/CFT dual theories where both sides are
nontrivial, computable and are defined independently of each other.

In general, one might expect the slightly-broken higher-spin symmetry to be
useful even at small $N$, given that the anomalous dimensions of higher-spin
currents are small and get even smaller with spin. It would be interesting to
find out if the existence of chiral subsectors can also be established
directly at small $N$ via the numerical bootstrap techniques, where the main
challenge seems to be to implement the notion of helicity into the structure
of correlators and OPE coefficients and to be able to target nonunitary
theories.

On the way towards exact models of AdS/CFT correspondence, it is interesting
to identify the AdS/CFT duals of self-dual Yang-Mills and of self-dual
gravity. In flat space all tree-level amplitudes in these theories vanish and
the only nontrivial ones are all-helicity-plus one-loop amplitudes. However,
it is not so in (anti-)de Sitter and it would be interesting to compute the
corresponding AdS/CFT correlators. Already tree-level amplitudes of type
$(++...+-)$ do not vanish. The existence of these theories implies that there
are closed subsectors of the current $\langle J...J\rangle$ and stress tensor
$\langle T...T\rangle$ correlators. However, these two subsectors are quite
small. Nevertheless, they are universal since every (local) CFT has a stress
tensor and many have global symmetry currents, while the existence of the
slightly-broken higher-spin currents is special to large-$N$ vector models. It
also should be taken into account that both SDYM and SDGR are UV finite
theories Krasnov:2016emc .

SDYM and SDGR are contained inside (anti-)Chiral theory. SDYM has only the
$(--+)$ cubic vertex and SDGR also has the $(--+)$ vertex and stops there at
least in the light-cone gauge Lipstein:2023pih ; Neiman:2023bkq . Therefore,
our expressions for $\langle T^{-}T^{-}T^{+}\rangle_{nh}$ and $\langle
J^{-}J^{-}J^{+}\rangle_{nh}$ immediately give the three-point holographic
functions in SDGR and SDYM, respectively. As for the four-point ones, it easy
to see that $\langle J^{-}J^{-}J^{-}J^{+}\rangle$ in SDYM coincides with the
one in Chiral theory with all integer spins. Similarly, $\langle
T^{-}T^{-}T^{-}T^{+}\rangle$ in SDGR coincides with the one on Chiral theory
restricted to even spins. In fact the same should be true for all $\langle
J^{-}J^{-}...J^{-}J^{+}\rangle$ and $\langle T^{-}T^{-}...T^{-}T^{+}\rangle$
since exchanges via higher-spin fields are suppressed in these correlators.

In the same vein, it would be interesting to identify the chiral subsectors in
ABJ theories since via triality Chang:2012kt one can argue that
$\mathcal{N}=6$ $U(M)$-gauged Chiral theory should be dual to the subsector of
the vector-like limit of ABJ theories. The latter indicates the existence of a
closed subsector in tensionless strings on $AdS_{4}\times\mathbb{CP}^{3}$.

Our (anti-)chiral limit can be understood as taking $N\to\infty$ while keeping
the Chern-Simons level fixed at $k=\pm i$. The analysis of Chern-Simons theory
with a complex level has been performed by Witten Witten:1989ip ;
Witten:2010cx . It would be interesting to see if one can perform an analogous
analysis when we also add matter into the mix and see for instance, if the
bosonization dualities persist. In the literature, supersymmetric Chern-Simons
matter theories have been analyzed in the so-called M-theory limit which
corresponds to $N\to\infty$ with $k$ finite Kapustin:2009kz ; Marino:2011eh ,
which is quite similar in spirit to our (anti-)chiral limit. By localization,
the partition function on $S^{3}$ reduces to a matrix model which is then
interpreted as the partition function of an ideal Fermi gas of non-interacting
particles. The level $k$ is mapped to Planck’s constant and, hence, one can
study the theory at finite $k$ by employing the WKB approximation. This
formalism also enables one to compute non-perturbative (instanton) effects. It
would thus be interesting to understand the implications of the (anti-)chiral
limits in this vein.

The works mentioned above were in the context of ABJM or ABJ theory with
matter in the adjoint or bi-fundamental representations. There has also been
some work on the supersymmetric versions of Chern-Simons theory with
fundamental matter Inbasekar:2015tsa ; Gur-Ari:2015pca ; Aharony:2019mbc . It
would be interesting to see if we can define (anti-)chiral limits analogous to
(47) and (46) to obtain a sub-sector that is dual to a supersymmetric
(anti-)chiral higher-spin gravity. The formalism of super spinor-helicity
variables developed in Jain:2023idr for $\mathcal{N}=1$ and $\mathcal{N}=2$
theories would provide a perfect stage for the start of such exploits.

Another avenue of interest is the connection between
$\text{AdS}_{4}/\text{CFT}_{3}$ and flat space holography. $\text{CFT}_{3}$
correlators/ $\text{AdS}_{4}$ amplitudes are related to S-matrix in four
dimensions. The framework of Celestial holography relates four-dimensional
S-matrices to $\text{CFT}_{2}$ correlators. It would thus be interesting to
see the implications of taking the (anti-)chiral limit on the celestial
correlators.

## Acknowledgments

We would like to thank Ofer Aharony, Trivko Kukolj and Rohit.R Kalloor for
useful discussions and for coordinating the submission of this manuscript with
their upcoming paper. We are grateful to Simone Giombi, Saurabh Pant and Sasha
Zhiboedov for very useful discussions. SJ would also like to thank the
organizers and participants of conferences organized in BITS Goa and IIT
Kanpur where part of this work was presented. This project has received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement No 101002551).
E.S. is a Research Associate of the Fund for Scientific Research – FNRS,
Belgium. S.J and D.K.S would like to acknowledge their debt to the people of
India for their steady support of research in basic sciences.

##  Appendix A Holographic Chiral limit

Let us discuss the chiral limit in the bulk, in the holographic dual of Chern-
Simons vector models. This theory is understood at present only at the level
of the underlying $L_{\infty}$-algebra. This algebra determines, in principle,
formally consistent classical equations of motion in the form of a free
differential algebra Vasiliev:1990en , see also Sharapov:2019vyd for the
general solution to the problem of constructing $L_{\infty}$-extension
starting from free higher spin fields. We will not need any concrete technical
details of the construction to identify the chiral limit. There are two
fields: a one-form $\omega$ and a zero-form $C$, which are generating
functions containing fields of all spins together with certain auxiliary
fields. Based on the form degree counting, the general structure has to be

$\displaystyle d\omega$
$\displaystyle=\mathcal{V}(\omega,\omega)+\mathcal{V}(\omega,\omega,C)+\mathcal{V}(\omega,\omega,C,C)+\mathcal{O}(C^{3})\,,$
(77) $\displaystyle dC$
$\displaystyle=\mathcal{U}(\omega,C)+\mathcal{U}(\omega,C,C)+\mathcal{O}(C^{3})\,,$
(78)

where $\mathcal{V}$ and $\mathcal{U}$ are certain multilinear maps (the
structure maps of some $L_{\infty}$-algebra). These multilinear maps have the
following general structure

$\displaystyle\mathcal{V}(\omega,\omega,C)$
$\displaystyle=g\,\mathcal{V}_{10}(\omega,\omega,C)+\bar{g}\,\mathcal{V}_{01}(\omega,\omega,C)\,,$
(79) $\displaystyle\mathcal{V}(\omega,\omega,C,C)$
$\displaystyle=g^{2}\,\mathcal{V}(\omega,\omega,C,C)_{20}+\bar{g}g\,\mathcal{V}_{11}(\omega,\omega,C,C)+\bar{g}^{2}\,\mathcal{V}_{02}(\omega,\omega,C,C)\,,$
(80) $\displaystyle\mathcal{U}(\omega,C,C)$
$\displaystyle=g\,\mathcal{U}_{10}(\omega,C,C)+\bar{g}\,\mathcal{U}_{01}(\omega,C,C)\,,$
(81)

where $g$ and $\bar{g}$ can be taken to be independent complex coupling
constants. More generally, we have

$\displaystyle\mathcal{V}(\omega,\omega,\overbrace{C,...,C}^{n})$
$\displaystyle=\sum_{i+j=n}g^{i}\bar{g}^{j}\,\mathcal{V}_{i,j}(\omega,\omega,\overbrace{C,...,C}^{n})$
(82) $\displaystyle\mathcal{U}(\omega,\overbrace{C,...,C}^{n+1})$
$\displaystyle=\sum_{i+j=n}g^{i}\bar{g}^{j}\,\mathcal{U}_{i,j}(\omega,\overbrace{C,...,C}^{n+1})$
(83)

The general structures above can be either extracted from Vasiliev’s equations
Vasiliev:1990en , or defined directly Sharapov:2019vyd . The
$L_{\infty}$-algebra itself is defined only modulo canonical transformations,
which correspond to field redefinitions

$\displaystyle\omega$ $\displaystyle\rightarrow\omega+f(\omega,C)$
$\displaystyle C$ $\displaystyle\rightarrow C+g(C)$ (84)

from the field theory point of view. Such redefinitions are in general very
nonlocal from the field theory point of view and it is still an open problem
to define the vertices $\mathcal{V}$ and $\mathcal{U}$ in such a way that
systematic holographic calculations are possible. Assuming that the resulting
theory can be defined and the locality considerations do not modify the
general structure of the equations,252525Note that $g$ and $\bar{g}$
correspond to two nontrivial deformations at the $L_{\infty}$-level
Sharapov:2020quq . In fact, provided locality is not taken into account, there
are infinitely many other deformations that enter at each odd order in $C$,
i.e. in $\mathcal{V}_{i,j}$ with $i+j=\text{odd}$. They correspond to the
phase function $\theta_{\star}(X)=-\theta_{\star}(-X)$ in Vasiliev’s
equations. These deformations do not seem to find any natural explanation on
the CFT side, see e.g. Maldacena:2012sf ; Boulanger:2015ova . It may well be
that the holographic dual of Chern-Simons vector models may need to activate
(some of) these parameters, see e.g. Giombi:2011kc . This will complicate the
matters discussed here. Another possibility is that the careful treatment of
locality can change the cohomological analysis, i.e. the structure of the
deformation, by introducing some new/eliminating some old deformation
parameters. We assume that none of this will happen.  it is easy to see that
the (anti-)chiral limits should correspond simply to $g=0$ or $\bar{g}=0$.
Imposing unitarity one has to have $g=|g|e^{i\theta}$ and
$\bar{g}=|g|e^{-i\theta}$, where $|g|\sim 1/\sqrt{N}$ is the bulk coupling.
Therefore, the chiral limit is nonunitary.

From the bulk perspective, the chiral limit has improved locality properties,
and the perturbatively local equations of motion are known explicitly
Sharapov:2022wpz ; Sharapov:2022awp and are a smooth uplift of the Chiral
theory from flat space Skvortsov:2018uru to anti-de Sitter.

##  Appendix B Notation and three dimensional spinor-helicity variables

In this appendix, we set straight the notation used in the paper. The
abbreviations we use are provided in the table below.

Abbreviation | Full Form
---|---
$\langle...\rangle_{\text{QF}}$ | In quasi-fermionic theory
$\langle...\rangle_{\text{FF/FB}}$ | In free fermionic/bosonic theory
$\langle...\rangle_{\text{CB}}$ | In critical bosonic theory
$\langle...\rangle_{\text{odd}}$ | Parity odd correlator
$\langle...\rangle_{\text{FF+FB}}$ | $\langle...\rangle_{\text{FF}}+\langle...\rangle_{\text{FB}}$
$\langle...\rangle_{\text{FF-FB}}$ | $\langle...\rangle_{\text{FF}}-\langle...\rangle_{\text{FB}}$
$\langle...\rangle_{\text{nh}}$ | nonhomogeneous
$\langle...\rangle_{\text{h}}$ | homogeneous
Table 1: Some notation

While writing correlators in momentum space/spinor-helicity variables we
suppress the momentum-conserving delta functions. In spinor-helicity
variables, we denote the negative (positive) helicity component of a spin $s$
current $J_{s}^{\mu_{1}\cdots\mu_{s}}$ that one obtains after contraction with
the transverse null polarizations $z_{\mu_{1}}^{\pm}\cdots z_{\mu_{s}}^{\pm}$
as $J_{s}^{-}$($J_{s}^{+}$). This is shorthand for $J_{s}^{-\cdots-}$
($J_{s}^{+\cdots+}$) where there are $s$ minuses (pluses). We now briefly
review the formalism of three-dimensional spinor-helicity variables.

We work with the usual flat Euclidean metric,

$\displaystyle\delta_{\mu\nu}=\textbf{diag}(1,1,1),$ (85)

with which vector indices are raised and lowered. Since upper and lower
indices are identical in this case, we do not need to distinguish between
them. Our convention for the three-dimensional Levi-Civita symbol is,

$\displaystyle\epsilon_{123}$ $\displaystyle=\epsilon^{123}=1,$
$\displaystyle\epsilon_{\mu\nu\rho}\epsilon_{\alpha\beta\rho}$
$\displaystyle=\delta_{\mu\alpha}\delta_{\nu\beta}-\delta_{\mu\beta}\delta_{\nu\alpha}.$
(86)

Spinor indices on the other hand are raised and lowered using the two-
dimensional Levi-Civita symbol $\epsilon_{ab}$ which is given by,

$\displaystyle\epsilon_{12}$ $\displaystyle=\epsilon^{12}=1,$
$\displaystyle\epsilon_{ab}\epsilon^{ac}$ $\displaystyle=\delta_{b}^{c}.$ (87)

Our convention for raising and lowering indices is as follows: given a spinor
$A_{a}$ we have,

$\displaystyle A^{a}=\epsilon^{ab}A_{b}\iff A_{a}=\epsilon_{ba}A^{b}.$ (88)

We choose the following representations of the Pauli matrices

$\displaystyle\sigma^{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\,,\qquad\sigma^{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}\,,\qquad\sigma^{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\,,$ (89)

which satisfy,

$\displaystyle(\sigma^{\mu})^{a}_{b}(\sigma^{\nu})^{c}_{a}=\delta^{\mu\nu}\delta^{c}_{b}+i\epsilon^{\mu\nu\rho}(\sigma^{\rho})^{c}_{b}.$
(90)

Given a three-vector $p_{\mu}$, we can trade it for a matrix
$(\not{p})^{a}_{b}$ in the following way:

$\displaystyle(\not{p})^{a}_{b}=p_{\mu}(\sigma^{\mu})^{a}_{b}=\lambda_{b}\bar{\lambda}^{a}+p\delta^{a}_{b}.$
(91)

We can extract the “energy” through the following bracket:

$p=-\frac{1}{2}\langle\lambda\bar{\lambda}\rangle.$ (92)

By contracting spinors associated to different momenta, we can form the angle
bracket:

$\displaystyle\langle ij\rangle=\lambda_{ia}\lambda_{j}^{a}.$ (93)

The polarization vectors associated to the currents are given by,

$\displaystyle
z_{\mu}^{-}(\sigma^{\mu})^{a}_{b}=(\not{z}^{-})^{a}_{b}=\frac{\lambda_{b}\lambda^{a}}{p},$
(94) $\displaystyle
z_{\mu}^{+}(\sigma^{\mu})^{a}_{b}=(\not{z}^{+})^{a}_{b}=\frac{\bar{\lambda}_{b}\bar{\lambda}^{a}}{p}.$
(95)

Polarization vectors of higher-spin operators can be formed by taking products
of these spin-one polarization vectors. Momentum conservation in these
variables is given by,

$\displaystyle\sum_{i=1}^{n}\lambda_{ib}\bar{\lambda}_{i}^{a}=-E\delta^{a}_{b}.$
(96)

$E=\sum_{i=1}^{n}p_{i}$ is the total energy. The three-dimensional dot product
of two three-vectors $x$ and $y$ can be written in spinor notation using,

$\displaystyle x\cdot y=\frac{1}{2}(\not{x})^{a}_{b}(\not{y})^{b}_{a}.$ (97)

Since we work with parity odd correlation functions as well, we will require
the following formula:

$\displaystyle\epsilon^{\mu\nu\rho}=\frac{1}{2i}(\sigma^{\mu})^{a}_{b}(\sigma^{\nu})^{c}_{a}(\sigma^{\rho})^{b}_{c}.$
(98)

Finally, we have used the following notation:

$\epsilon^{v_{1}v_{2}v_{3}}=v_{1\mu}v_{2\nu}v_{3\rho}\epsilon^{\mu\nu\rho},$
(99)

for any three vectors $v_{1},v_{2}$ and $v_{3}$. More details as well as some
useful formulae can be found in appendix A of Jain:2023idr .

##  Appendix C A Chiral limit for the Quasi Bosonic theory

In this appendix, we shall define a chiral limit for correlation functions in
the Chern-Simons$+$fundamental bosons theory (henceforth called the quasi-
bosonic (QB) theory). Let us first present the Lagrangian description of the
theory.

The $U(N)$ Regular bosonic theory is described by the following action:

$\displaystyle S=\int
d^{3}x~{}\bigg{(}D_{\mu}\bar{\phi}D^{\mu}\phi+i\epsilon^{\mu\nu\rho}\frac{\kappa_{b}}{4\pi}\Tr{A_{\mu}\partial_{\nu}A_{\rho}-\frac{2i}{3}A_{\mu}A_{\nu}A_{\rho}}+(\frac{2\pi}{k_{b}})^{2}(\lambda_{6}+1)(\bar{\phi}\phi)^{3}\bigg{)}.$
(100)

We take the limits $\kappa_{b}\to\infty,N\to\infty$ while keeping fixed
$\lambda=\frac{N}{k_{b}}$.

This theory enjoys a strong-weak duality with the critical fermionic theory,
the action of which is provided in equation $(1.5)$ in Aharony:2018pjn .

In terms of correlators, the main difference between the QF and QB theories is
the fact that the three-point function of scalars is nonzero in the latter
case. It is given by Aharony:2012nh 262626Our normalization for the operator
in contrast to section 5.1 in Aharony:2012nh is as follows:
$O_{1,ours}=\frac{O_{1,theirs}}{\sqrt{N(1+\tilde{\lambda}^{2})}}$

$\displaystyle\langle
O_{1}O_{1}O_{1}\rangle_{QB}=\frac{1}{\sqrt{N(1+\tilde{\lambda}^{2})}}\bigg{(}1+\frac{\tilde{\lambda}^{2}}{4}\big{(}3-\frac{\lambda_{6}}{8\pi^{2}\lambda^{2}}\big{)}\bigg{)}\langle
O_{1}O_{1}O_{1}\rangle_{FB}.$ (101)

We see that we can make this three-point function identically zero by choosing
the coefficient of the $\phi^{6}$ term, $\lambda_{6}$ as follows:

$\displaystyle\lambda_{6}=32(3+\frac{4}{\tilde{\lambda}^{2}})\tan^{-1}(\tilde{\lambda})^{2}.$
(102)

We then define chiral and anti-chiral limits much in the same way as (47) and
(46). The analysis of the spinning correlators is then identical to that of
the quasi-fermionic theory presented in section 3 since the $\lambda_{6}$
coefficient only contributes to the scalar three-point function.

##  Appendix D Generalization of statements in section 2.2.1 to correlators
involving fermionic operators

In this appendix, we shall generalize our statements about correlation
functions inside the triangle in section 2 ((2.2.1),(2.2.1) and (17), to be
precise) to those that include half-integer spin operators. The reason for the
same is to show that our statements are general enough. For brevity, we shall
only focus on the nonhomogeneous parts of the correlators. This is because the
homogeneous contributions all obey statement $1$ (2.2.1). Let us begin with a
simple example.272727The correlators that appear in this section can be
computed from the free bosonic and free fermionic theories and using the
definition of the nonhomogeneous piece (13). Schematically, for the Abelian
examples, $O_{1/2}\sim\psi\bar{\phi}+cc$,
$J_{3/2}\sim\psi\partial\bar{\phi}+cc$ and $J$ and $T$ are the $U(1)$ current
and the stress tensor respectively.

###  Example 1: $\langle O_{1/2}O_{1/2}J\rangle$

The nonhomogeneous contribution to this correlator is nonzero in the $(---)$
and the zero-helicity sector $(--+)$ and their complex conjugate
ones.282828These expressions can be obtained by solving the conformal Ward
identities in spinor-helicity variables like in Jain:2021vrv . The expressions
are respectively given by,

$\displaystyle\langle
O_{1/2}^{-A}O_{1/2}^{-B}J^{-B}\rangle_{nh}=f^{ABC}\frac{\langle
23\rangle\langle 31\rangle}{\sqrt{p_{1}p_{2}}p_{3}^{2}},$
$\displaystyle\langle
O_{1/2}^{-A}O_{1/2}^{-B}J^{+C}\rangle_{nh}=f^{ABC}\frac{\langle
12\rangle^{2}((p_{1}-p_{2})^{2}-p_{3}^{2})}{\langle 23\rangle\langle
31\rangle\sqrt{p_{1}p_{2}}p_{3}^{2}}.$ (103)

Consider now, the following contact term:

$\displaystyle\langle
O_{1/2}^{A}O_{1/2}^{B}J^{C}\rangle_{contact}=-f^{ABC}\frac{\zeta_{1a}\zeta_{2b}(\not{z}_{3})^{ab}}{p_{3}},$
(104)

and redefine,

$\displaystyle\langle O_{1/2}^{A}O_{1/2}^{B}J^{C}\rangle_{new,nh}=\langle
O_{1/2}^{A}O_{1/2}^{B}J^{C}\rangle_{nh}-\langle
O_{1/2}^{A}O_{1/2}^{B}J^{C}\rangle_{contact}.$ (105)

The redefined nonhomogeneous correlator satisfies,

$\displaystyle\langle O_{1/2}^{-A}O_{1/2}^{-B}J^{-B}\rangle_{new,nh}=\langle
O_{1/2}^{-A}O_{1/2}^{-B}J^{+B}\rangle_{new,nh}=0,$ (106)

that is, it vanishes in the $(---)$ as well as zero helicity sector (and of
course, their complex conjugate configurations).

### Example 2:$~{}\langle J_{3/2}J_{3/2}J\rangle$

Let us now focus on a correlator that involves two spin three-half currents.
The nonhomogeneous part of this correlator in the $(---)$ helicity is given
by,

$\displaystyle\langle J_{3/2}^{-}J_{3/2}^{-}J^{-}\rangle_{nh}=\frac{\langle
12\rangle^{2}\langle 23\rangle\langle
31\rangle(2E+p_{3})}{(p_{1}p_{2})^{3/2}p_{3}}.$ (107)

We can, however, cancel this contribution by adding the following simple
contact term:

$\displaystyle\zeta_{1a}\zeta_{1b}\zeta_{1c}\zeta_{2e}\zeta_{2f}\zeta_{2g}z_{3\mu}\langle
J_{3/2}^{abc}J_{3/2}^{efg}J^{\mu}\rangle_{contact}=-\zeta_{1a}\zeta_{1b}\zeta_{1c}\zeta_{2e}\zeta_{2f}\zeta_{2g}z_{3}^{gc}\epsilon^{af}\epsilon^{be}(E+p_{3}).$
(108)

Redefining

$\displaystyle\langle J_{3/2}J_{3/2}J\rangle_{new,nh}=\langle
J_{3/2}J_{3/2}J\rangle_{nh}-\langle J_{3/2}J_{3/2}J\rangle_{contact},$ (109)

we find

$\displaystyle\langle J_{3/2}^{-}J_{3/2}^{-}J^{-}\rangle_{new,nh}=0,$ (110)

showing that the nonhomogeneous part of this correlator is zero when the
helicities of all the insertions coincide.

###  Example 3: $\langle J_{3/2}O_{1/2}J\rangle$

The nonhomogeneous part of this correlator is nonzero in both the $(---)$ and
$(+--)$ helicities which should not be the case according to statements $2$
and $3$ ((2.2.1) and (17)):

$\displaystyle\langle J_{3/2}^{-}O_{1/2}^{-}J^{-}\rangle=-\frac{\langle
12\rangle\langle 31\rangle^{2}}{p_{1}^{3/2}\sqrt{p_{2}}p_{3}}$
$\displaystyle\langle J_{3/2}^{+}O_{1/2}^{-}J^{-}\rangle=\frac{\langle
23\rangle^{3}(E-2p_{3})^{2}(E-2p_{2})}{\langle 12\rangle^{2}\langle 31\rangle
p_{1}^{3/2}\sqrt{p_{2}}p_{3}}.$ (111)

However, these contributions can be removed via the addition of the following
contact term:

$\displaystyle z_{1abc}\zeta_{2b}z_{3\mu}\langle
J^{abc}_{3/2}O_{1/2}^{b}J^{\mu}\rangle_{contact}=\frac{2\zeta_{1a}\zeta_{2b}}{p_{2}}\big{(}i\epsilon^{ab}\epsilon^{z_{1}z_{3}p_{2}}-(z_{1}\cdot
p_{2})\not{z}_{3}^{ab}\big{)}.$ (112)

After redefining

$\displaystyle\langle J_{3/2}O_{1/2}J\rangle_{new,nh}=\langle
J_{3/2}O_{1/2}J\rangle_{nh}-\langle J_{3/2}O_{1/2}J\rangle_{contact},$ (113)

we find,

$\displaystyle\langle J_{3/2}^{-}O_{1/2}^{-}J^{-}\rangle_{new,nh}=\langle
J_{3/2}^{+}O_{1/2}^{-}J^{-}\rangle_{new,nh}=0,$ (114)

which validates both statements ((2.2.1) and (17)).

###  Example 4: $\langle TJ_{3/2}O_{1/2}\rangle$

Let us now consider an example that involves the stress tensor. The
nonhomogeneous part of this correlator is not zero in both the $(---)$ and
$(+--)$ helicity sectors:

$\displaystyle\langle T^{-}J_{3/2}^{-}O_{1/2}^{-}\rangle_{nh}=-\frac{\langle
12\rangle^{3}\langle
31\rangle(p_{2}+p_{3})}{p_{1}^{2}p_{2}^{3/2}\sqrt{p_{3}}}\,,$
$\displaystyle\langle T^{+}J_{3/2}^{-}O_{1/2}^{-}\rangle_{nh}=\frac{\langle
23\rangle^{3}(E-2p_{3})(E-2p_{2})^{3}(p_{2}+p_{3})}{\langle 12\rangle\langle
31\rangle^{3}p_{1}^{2}p_{2}^{3/2}\sqrt{p_{3}}}\,.$ (115)

Consider the following contact term:

$\displaystyle z_{1\mu}z_{1\nu}z_{2abc}\zeta_{3e}\langle
T^{\mu\nu}J_{3/2}^{abc}O_{1/2}^{e}\rangle_{contact}=(p_{2}+p_{3})\not{z}_{1}^{ab}\not{z}_{1}^{ce}z_{2abc}\zeta_{3e}.$
(116)

By redefining

$\displaystyle\langle TJ_{3/2}O_{1/2}\rangle_{new,nh}=\langle
TJ_{3/2}O_{1/2}\rangle_{nh}-\langle TJ_{3/2}O_{1/2}\rangle_{contact},$ (117)

we obtain

$\displaystyle\langle T^{-}J_{3/2}^{-}O_{1/2}^{-}\rangle_{new,nh}=\langle
T^{+}J_{3/2}^{-}O_{1/2}^{-}\rangle_{new,nh}=0,$ (118)

thereby validating statements $2$ and $3$ ((2.2.1) and (17)) once again.

To summarize, we have established in this appendix that our statements about
correlators inside the triangle ((2.2.1), (2.2.1) and (17)) are robust and
generic.

##  Appendix E An argument for statements 2 and 3 in section 2.2.1 via the
CFT-S matrix correspondence

In this appendix, we shall present an argument for statements $2$ and $3$
((2.2.1) and (17)) by employing the connection between CFT correlators and
scattering amplitudes in one higher dimension.

By the AdS/CFT duality, CFT correlators are dual to AdS scattering amplitudes.
Thus, we can take a flat space limit to obtain the flat space scattering
amplitude from the CFT correlator Maldacena:2011nz ; Raju:2012zr . This limit
corresponds to taking the total energy $E=\sum_{i=1}^{n}p_{i}\to 0$. We shall
now use this correspondence to motivate the results we obtained above.

Let us first consider $\langle TTT\rangle_{nh}$. This nonhomogeneous stress
tensor three-point function is dual to the Einstein gravity graviton three-
point amplitude Jain:2021qcl . By taking the flat space limit of this
correlator, the correspondence tells us that we should obtain the flat space
Einstein gravity three-point amplitude. In flat space, the Einstein gravity
three-point function has no contribution in the $(---)$ and $(+++)$ helicity
configurations. For higher spins and amplitudes involving different spins as
well, non-higher-derivative interactions are zero in the $(---)$ and $(+++)$
helicities, a fact that is most easily seen while attempting to write down
such interactions in light-cone variables and noticing that such terms in the
Lagrangian cannot simultaneously satisfy dimensional analysis, helicity
neutrality, and locality. Therefore, the corresponding AdS amplitude/CFT
correlators should be free of total energy ($E$) poles. It is thus plausible
that such contributions can be removed via the addition of contact terms,
which is indeed what we showed in section 2 and appendix D for not only
$\langle TTT\rangle$ but for many more examples.

Let us now discuss the zero-helicity sector. In flat space, three-point
amplitudes with net zero-helicity have been shown to be inconsistent with
unitarity and locality McGady:2013sga . This was proved by showing that the
zero-helicity three-point amplitudes cannot be glued together to produce a
valid (local, consistent with unitary) four-point amplitude and hence such
three-point seeds should vanish. While we do not have such an argument in CFT,
we can make use of the CFT/S-matrix correspondence again. Assume that the CFT
correlator exists in the zero-helicity sector. By taking a flat space limit,
we recover a zero-helicity flat space amplitude. However, as argued above,
such amplitudes do not exist and hence the CFT correlator should vanish in the
flat space limit. Thus, our conclusion is that zero-helicity configurations of
correlators cannot have total energy singularities. Therefore, it is plausible
to remove such contributions via adding contact terms to the correlators and
is exactly what we found in section 2 and appendix D.

##  Appendix F Correlators outside the triangle: Examples

In this appendix, we shall illustrate via three examples, the validity of
statements $4$ and $5$ ((2.2.2) and (2.2.2)) presented in section 2. Let us
begin with the example of $\langle J_{4}J_{1}J_{1}\rangle$ following
Jain:2021whr .

### Example 1: $\langle J_{4}J_{1}J_{1}\rangle$

The $FF-FB$ correlator in the independent helicity configurations takes the
following forms Jain:2021whr :

$\displaystyle\langle J_{4}^{-}J_{1}^{-}J_{1}^{-}\rangle_{FF-FB}$
$\displaystyle=-\langle 12\rangle^{4}\langle
31\rangle^{4}\langle\bar{2}\bar{3}\rangle^{2}\frac{3E^{5}+5E^{4}k_{1}+8E^{3}k_{1}^{2}+12E^{2}k_{1}^{3}+16Ek_{1}^{4}+16k_{1}^{5}}{524288E^{8}k_{1}^{4}},$
$\displaystyle\langle J_{4}^{+}J_{1}^{-}J_{1}^{-}\rangle_{FF-FB}$
$\displaystyle=-\langle
23\rangle^{6}\langle\bar{1}\bar{2}\rangle^{4}\langle\bar{3}\bar{1}\rangle^{4}\frac{3E+k_{1}}{524288E^{8}k_{1}^{4}},~{}~{}~{}\langle
J_{4}^{-}J_{1}^{-}J_{1}^{+}\rangle_{FF-FB}=0.$ (119)

This statement is perfectly consistent with statement $4$ (2.2.2), i.e. the
vanishing of $FF-FB$ when the helicities of the spin one currents are
different. Let us now examine the $FF+FB$ correlator. In spinor-helicity
variables we have Jain:2021whr

$\displaystyle\langle J_{4}^{-}J_{1}^{-}J_{1}^{-}\rangle_{FB+FF}$
$\displaystyle=0,~{}~{}~{}\langle
J_{4}^{+}J_{1}^{-}J_{1}^{-}\rangle_{FB+FF}=0\,,$ $\displaystyle\langle
J_{4}^{-}J_{1}^{-}J_{1}^{+}\rangle_{FB+FF}$ $\displaystyle=\langle
12\rangle^{6}\langle
31\rangle^{2}\langle\bar{2}\bar{3}\rangle^{4}\frac{5E^{3}+5E^{2}k_{1}+4Ek_{1}^{2}+2k_{1}^{3}}{262144E^{8}k_{1}^{4}}\,,$
(120)

which is in perfect agreement with statement 5 (2.2.2) as the correlator
vanishes when the helicities of the spin one currents coincide.

Let us now proceed to an example which involves half-integer spin insertions.

###  Example 2: $\langle O_{1/2}O_{1/2}T\rangle$

This correlator can be computed via Wick contractions. For the free bosonic
stress tensor we have

$\displaystyle\zeta_{1a}\zeta_{2b}z_{3\alpha}z_{3\beta}\langle
O^{a}_{1/2}(p_{1})O^{b}_{1/2}(p_{2})T^{\alpha\beta}_{\textbf{FB}}(p_{3})\rangle=2\zeta_{1a}\zeta_{2\alpha}z_{3\alpha}z_{3\beta}(\sigma^{\rho})^{ab}\int\frac{d^{3}l}{(2\pi)^{3}}\frac{l^{\rho}(l^{\alpha}-p_{1}^{\alpha})(l^{\beta}-p_{1}^{\beta})}{l^{2}(l-p_{1})^{2}(l+p_{2})^{2}}$
$\displaystyle=-\frac{\zeta_{1a}\zeta_{2b}(p_{1}\cdot
z_{3})}{32p_{1}p_{2}E^{3}}\bigg{(}(p_{1}\cdot
z_{3})(E+2p_{2})\bigg{(}-p_{2}(\not{p_{1}}^{ab})+p_{1}(\not{p_{2}}^{ab})\bigg{)}+2p_{1}p_{2}E(E+p_{3})(\not{z_{3}})^{ab}\bigg{)}.$
(121)

A similar computation for the free fermionic stress tensor yields,

$\displaystyle\zeta_{1a}\zeta_{2b}z_{3\alpha}z_{3\beta}\langle
O^{a}_{1/2}(p_{1})O^{b}_{1/2}(p_{2})T^{\alpha\beta}_{\textbf{FF}}(p_{3})\rangle=\zeta_{1a}\zeta_{2b}z_{3\alpha}z_{3\beta}(\sigma^{\beta})^{d}_{c}(\sigma^{\rho})^{cb}(\sigma^{\chi})^{a}_{d}\int\frac{d^{3}l}{(2\pi)^{3}}\frac{(l^{\alpha}-p_{1}^{\alpha})(l^{\rho}+p_{2}^{\rho})(l^{\chi}-p_{1}^{\chi})}{l^{2}(l-p_{1})^{2}(l+p_{2})^{2}}$
$\displaystyle=\frac{\zeta_{1a}\zeta_{2b}}{32p_{1}p_{2}p_{3}E^{3}}\bigg{(}-(p_{1}\cdot
z_{3})^{2}\bigg{(}(4(p_{1}+p_{2})^{3}+(p_{1}+p_{2})(10p_{1}+11p_{2})p_{3}+3(2p_{1}+3p_{2})p_{3}^{2})\not{p_{1}}^{ab}-(1\leftrightarrow
2)\bigg{)}$ $\displaystyle+(p_{1}\cdot
z_{3})(p_{1}+p_{2})(2p_{1}+2p_{2}+3p_{3})(p_{1}^{2}+p_{2}^{2}-p_{3}^{2})(\not{z_{3}})^{ab}-2i(p_{1}\cdot
z_{3})(p_{1}+p_{2})(2p_{1}+2p_{2}+3p_{3})\epsilon^{ab}\epsilon^{z_{3}p_{1}p_{2}}\bigg{)}.$
(122)

We now consider the difference and sum of the bosonic and fermionic
contributions along with the addition of a contact term:

$\displaystyle\langle O_{1/2}O_{1/2}T\rangle_{\textbf{FF-FB}}=\langle
O_{1/2}O_{1/2}T_{\textbf{FF}}\rangle-\langle
O_{1/2}O_{1/2}T_{\textbf{FB}}\rangle-\mathcal{C}_{O_{\frac{1}{2}}O_{\frac{1}{2}}T}\,,$
$\displaystyle\langle O_{1/2}O_{1/2}T\rangle_{\textbf{FB+FF}}=\langle
O_{1/2}O_{1/2}T_{\textbf{FB}}\rangle+\langle
O_{1/2}O_{1/2}T_{\textbf{FF}}\rangle+\mathcal{C}_{O_{\frac{1}{2}}O_{\frac{1}{2}}T}\,,$
(123)

where the contact term is given by

$\mathcal{C}_{O_{\frac{1}{2}}O_{\frac{1}{2}}T}=\frac{1}{8p_{3}}\zeta_{1a}\zeta_{2b}(p_{1}\cdot
z_{3})(\not{z_{3}})^{ab}\,.$ (124)

We now find in accordance with statement $4$ (2.2.2) that the $FF-FB$
correlator vanishes when the $O_{1/2}$ operators have different helicities.

$\displaystyle\langle
O_{1/2}^{-}(p_{1})O_{1/2}^{-}(p_{2})T^{-}(p_{3})\rangle_{\textbf{FF-
FB}}=-\frac{\langle 23\rangle^{2}\langle
31\rangle^{2}\bigg{(}4(p_{1}+p_{2})^{4}+7(p_{1}+p_{2})^{3}p_{3}-3(p_{1}+p_{2})^{2}p_{3}^{2}-11(p_{1}+p_{2})p_{3}^{3}+3p_{3}^{4}\bigg{)}}{128\langle
12\rangle\sqrt{p_{1}}\sqrt{p_{2}}p_{3}^{3}E^{3}},$ $\displaystyle\langle
O_{1/2}^{-}(p_{1})O_{1/2}^{+}(p_{2})T^{-}(p_{3})\rangle_{\textbf{FF-FB}}=0,$
$\displaystyle\langle
O_{1/2}^{+}(p_{1})O_{1/2}^{-}(p_{2})T^{-}(p_{3})\rangle_{\textbf{FF-FB}}=0,$
$\displaystyle\langle
O_{1/2}^{-}(p_{1})O_{1/2}^{-}(p_{2})T^{+}(p_{3})\rangle_{\textbf{FF-
FB}}=-\frac{\langle
12\rangle^{3}(4(p_{1}+p_{2})+3p_{3})((p_{1}-p_{2})^{2}-p_{3}^{2})^{2}}{128\langle
23\rangle^{2}\langle 31\rangle^{2}\sqrt{p_{1}}\sqrt{p_{2}}p_{3}^{3}}.$ (125)

We also see that the FF+FB correlator obeys statement $5$ (2.2.2):

$\displaystyle\langle
O_{1/2}^{-}(p_{1})O_{1/2}^{-}(p_{2})T^{-}(p_{3})\rangle_{\textbf{FB+FF}}=0,$
$\displaystyle\langle
O_{1/2}^{-}(p_{1})O_{1/2}^{+}(p_{2})T^{-}(p_{3})\rangle_{\textbf{FB-
FF}}=-\frac{\langle 23\rangle\langle
31\rangle^{3}(p_{1}+p_{2})(p_{1}+p_{2}-p_{3})^{2}(2p_{1}+2p_{2}+3p_{3})}{64\langle
12\rangle^{2}\sqrt{p_{1}}\sqrt{p_{2}}p_{3}^{3}E^{2}},$ $\displaystyle\langle
O_{1/2}^{+}(p_{1})O_{1/2}^{-}(p_{2})T^{-}(p_{3})\rangle_{\textbf{FB-
FF}}=\frac{\langle 23\rangle^{3}\langle
31\rangle(p_{1}+p_{2})(p_{1}+p_{2}-p_{3})^{2}(2p_{1}+2p_{2}+3p_{3})}{64\langle
12\rangle^{2}\sqrt{p_{1}}\sqrt{p_{2}}p_{3}^{3}E^{2}},$ $\displaystyle\langle
O_{1/2}^{-}(p_{1})O_{1/2}^{-}(p_{2})T^{+}(p_{3})\rangle_{\textbf{FB-FF}}=0.$
(126)

To conclude this appendix, let us consider one more example, this time with a
correlator involving a spin $4$ current and two spin half operators.

### Example 3$:\langle J_{4}O_{1/2}O_{1/2}\rangle$

Computing this correlator via Wick contractions, we see that the $FF-FB$ and
$FF+FB$ combinations automatically obey statements $5$ and $6$ ((2.2.2) and
(39)). The expressions in the independent helicities are as follows:

$\displaystyle\langle J_{4}^{-}O_{1/2}^{-}O_{1/2}^{-}\rangle_{FF-
FB}=\frac{\langle 12\rangle^{4}\langle 31\rangle^{4}}{4096\langle
23\rangle^{3}E^{5}p_{1}^{4}\sqrt{p_{2}p_{3}}}\bigg{(}15E^{7}-70E^{6}p_{1}+84E^{5}p_{1}^{2}-128p_{1}^{7}\bigg{)}\,,$
$\displaystyle\langle J_{4}^{-}O_{1/2}^{+}O_{1/2}^{+}\rangle_{FF-
FB}=-\frac{\langle 12\rangle^{4}\langle 31\rangle^{4}}{4096\langle
23\rangle^{5}E^{3}p_{1}^{4}\sqrt{p_{2}p_{3}}}(E-2p_{1})^{5}\bigg{(}15E^{2}+10Ep_{1}+4p_{1}^{2}\bigg{)}\,,$
$\displaystyle\langle J_{4}^{-}O_{1/2}^{\pm}O_{1/2}^{\mp}\rangle_{FF-FB}=0\,,$
(127)

and

$\displaystyle\langle
J_{4}^{-}O_{1/2}^{-}O_{1/2}^{+}\rangle_{FF+FB}=\frac{\langle
12\rangle^{5}\langle 31\rangle^{3}}{1024\langle
23\rangle^{4}E^{4}p_{1}^{4}\sqrt{p_{2}p_{3}}}(E-2p_{1})^{4}\bigg{(}5E^{3}+5E^{2}p_{1}+4p_{1}^{2}+2p_{1}^{3}\bigg{)}\,,$
$\displaystyle\langle J_{4}^{-}O_{1/2}^{+}O_{1/2}^{+}\rangle=0\,,$
$\displaystyle\langle J_{4}^{-}O_{1/2}^{-}O_{1/2}^{-}\rangle=0\,,$ (128)

thereby verifying the robustness of our statements.

##  Appendix G The correct parity even vs. parity odd relation outside the
triangle

In section 2, we made a statement (39) about the correct prescription for the
epsilon transform for correlators in slightly-broken higher-spin theories
whose insertions violate the spin triangle inequality. In this appendix, we
prove said statement by employing slightly-broken higher-spin equations.

Without loss of generality assume that $s_{1}>s_{2}+s_{3}$, $s_{2}\geq s_{3}$
for a correlator $\langle J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle$ that is outside
the triangle. It is easy to show that in this correlator,
$J_{s_{2}},J_{s_{3}}$ behave like exactly conserved currents and the
nonconservation is entirely due to $J_{s_{1}}$.292929This is easy to see as
follows: Assume on the contrary that $\partial\cdot J_{s_{2}}\sim
J_{s_{1}}\partial^{n}J_{s_{3}}$ (schematically). Then the LHS has scaling
dimension $s_{2}+2+\order{\frac{1}{N}}$ while the RHS has $s_{1}+s_{3}+n+2$.
Matching these we obtain $s_{2}=s_{1}+s_{3}+n$. Since $n\geq 0$, this is a
contradiction and hence $\partial\cdot J_{s_{2}}$ does not contain
$J_{s_{1}}\partial^{n}J_{s_{3}}$. Note that this is similar to the statement
that the only Ward-Takahashi identity for correlators outside the triangle is
due to the highest spin.  An example of such a correlator is $\langle
J_{4}J_{1}J_{1}\rangle$. It was observed in Jain:2021whr that for $\langle
J_{4}J_{1}J_{1}\rangle$, the epsilon transformation to obtain the odd piece
should be performed with respect to the $J_{1}$ operators, i.e. the currents
that are exactly conserved in this correlator. Does this pattern extend to any
correlator outside the triangle? We shall answer this question via the
analysis of the slightly-broken higher-spin equations. Before we present the
detailed analysis, we summarize our results: the parity odd part of the
correlator $\langle J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle$ (with
$s_{1}>s_{2}+s_{3}$, $s_{2}\geq s_{3}$) can be obtained in the following
equivalent ways:

$\displaystyle\langle J_{s_{1}}J_{s_{2}}J_{s_{3}}\rangle_{odd}=\langle
J_{s_{1}}\epsilon\cdot J_{s_{2}}J_{s_{3}}\rangle_{FF-CB}=\langle
J_{s_{1}}J_{s_{2}}\epsilon\cdot J_{s_{3}}\rangle_{FF-CB}\,,$ (129)

which are epsilon transforms with respect to $J_{s_{2}}$ or $J_{s_{3}}$, the
currents that are conserved in this correlator. An alternative quick way to
arrive at the same conclusion is to use the fact that the epsilon transform is
conformally invariant if and only if it is performed on a conserved current
(see Caron-Huot:2021kjy ) and in the above case, this picks out the ones with
respect to $J_{s_{2}}$ and $J_{s_{3}}$.

Further validation of (129) comes from the fact that it has the following
implications on the $FF-FB$ correlators ($h_{1}$ is the helicity of
$J_{s_{1}}$ below):

$\displaystyle\langle
J_{s_{1}}^{h_{1}}J_{s_{2}}^{\pm}J_{s_{3}}^{\mp}\rangle_{FF-FB}=0\,,$ (130)

which is nothing but statement $4$ (2.2.2). Let us now first illustrate how to
obtain (129) via an example:

###  An Example: $\langle J_{5}TJ\rangle$

Consider the equation:

$\displaystyle\langle[Q_{3}^{\mu\nu},J_{5}^{\rho\sigma\gamma\omega\phi}(x_{1})]T^{\alpha\beta}(x_{2})O_{2}(x_{3})\rangle+\langle
J_{5}^{\rho\sigma\gamma\omega\phi}(x_{1})[Q_{3}^{\mu\nu},T^{\alpha\beta}(x_{2})]O_{2}(x_{3})\rangle+\langle
J_{5}^{\rho\sigma\gamma\omega\phi}(x_{1})T^{\alpha\beta}(x_{2})[Q_{3}^{\mu\nu},O_{2}(x_{3})]\rangle$
$\displaystyle=\int
d^{3}x\langle\partial_{\psi}J_{3}^{\mu\nu\psi}(x)J_{5}^{\rho\sigma\gamma\omega\phi}(x_{1})T^{\alpha\beta}(x_{2})O_{2}(x_{3})\rangle\,,$
(131)

where $Q_{5}$ is a higher-spin charge corresponding to the current $J_{5}$ and
the other operators have their usual meaning. We then use the current algebra,
the nonconservation of the spin-$3$ current, convert it to the momentum space,
and substitute the expansion of the correlators appearing in the resulting
equation.303030The procedure for the same follows the analysis of Jain:2022ajd
. The nonconservation operator of the $J_{3}$ current is also given in (B.15)
in Jain:2022ajd . By expanding the higher-spin equation about
$\tilde{\lambda}=i$ we obtain the following equation that has to be satisfied:

$\displaystyle\langle
J_{5}^{\rho\sigma\gamma\omega\phi}T^{\alpha\beta}J^{\mu}\rangle_{odd}=\frac{\epsilon^{\mu
ab}p_{3a}}{p_{3}}\big{(}\langle
J_{5}^{\rho\sigma\omega\gamma\phi}T^{\alpha\beta}J^{b}\rangle_{FF}-\langle
J_{5}^{\rho\sigma\omega\gamma\phi}T^{\alpha\beta}J^{b}\rangle_{FB}\big{)}\,,$
(132)

which is the epsilon transform with respect to $J$.

Let us now show that by solving a different higher-spin equation we obtain an
epsilon transform with respect to $T$. Consider,

$\displaystyle\langle[Q_{4}^{\nu\alpha\beta},J_{5}^{\rho\sigma\gamma\omega\phi}(x_{1})]O_{2}(x_{2})J^{\mu}(x_{3})\rangle+\langle
J_{5}^{\rho\sigma\gamma\omega\phi}(x_{1})[Q_{4}^{\nu\alpha\beta},O_{2}(x_{2})]J^{\mu}(x_{3})\rangle+\langle
J_{5}^{\rho\sigma\gamma\omega\phi}(x_{1})O_{2}(x_{2})[Q_{4}^{\nu\alpha\beta},J^{\mu}(x_{3})]\rangle$
$\displaystyle=\int
d^{3}x\langle\partial_{\psi}J_{4}^{\psi\nu\alpha\beta}(x)J_{5}^{\rho\sigma\gamma\omega\phi}(x_{1})O_{2}(x_{2})J^{\mu}(x_{3})\rangle\,.$
(133)

Similar to the previous exercise, we use the higher-spin algebra, the
nonconservation equation, convert to Fourier space and use the expansion of
the correlators. We then expand the resulting equation about
$\tilde{\lambda}=i$ which yields the following equation:

$\displaystyle\langle
J_{5}^{\rho\sigma\gamma\omega\phi}T^{\alpha\beta}J^{\mu}\rangle_{odd}=\frac{\epsilon^{\alpha
ab}p_{2a}}{p_{2}}\big{(}\langle
J_{5}^{\rho\sigma\omega\gamma\phi}T^{b\beta}J^{\mu}\rangle_{FF}-\langle
J_{5}^{\rho\sigma\omega\gamma\phi}T^{b\beta}J^{\mu}\rangle_{FB}\big{)}\,,$
(134)

which is an epsilon transform with respect to $T$.

Since we now have two different expressions for the odd part of the
correlator, viz. (132) and (134), let us contract these expressions with the
appropriate polarization vectors and then equate them to derive the
consequences. In the $(\pm--)$ and $(\pm++)$ helicity configurations, that is,
when $T$ and $J$ have the same helicities, it is easy to see that (132) and
(134) agree. However, when $T$ and $J$ have opposite helicities, (132) yields

$\displaystyle\langle J_{5}^{\pm}T^{-}J^{+}\rangle_{odd}=i\langle
J_{5}^{\pm}T^{-}J^{+}\rangle_{FF-FB}\,,$ (135)

whereas (134) gives