aainstitutetext: Department of Particle Physics and Astrophysics, Weizmann
Institute of Science, Rehovot 7610001, Israelbbinstitutetext: School of
Natural Sciences, Institute for Advanced Study, Princeton 08540, NJ,
USAccinstitutetext: Department of Theoretical Physics, Tata Institute of
Fundamental Research, Homi Bhabha road, Mumbai 400005, India

# A chiral limit for Chern-Simons-matter theories

Ofer Aharony c Rohit R. Kalloor a Trivko Kukolj<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Large $N$ quasi-fermionic Chern-Simons-matter theories have an approximate
higher-spin symmetry that strongly constrains their correlation functions. In
particular, the 3-point functions for generic spins are combinations of 3
structures (with specific dependence on the positions and helicities), and the
coupling-dependence of the coefficient of each structure is uniquely
determined. In the past few years, several relations between different
structures were found. In this paper we show that all the relations between
the structures follow from (or, conversely, they imply) a specific form
written by Skvortsov for the vertices of the dual higher-spin gravity theory
on four-dimensional anti-de Sitter space, when written in spinor-helicity
variables. The dual bulk theory has a specific limit where it simplifies and
becomes a “chiral higher-spin gravity theory”, and we discuss what can be said
about this limit in the dual Chern-Simons-matter theories, where it involves
an analytic continuation to complex couplings.

##  1 Introduction

An interesting example of the AdS/CFT correspondence is the duality between
Chern-Simons-matter theories and higher-spin gravity theories on four
dimensional anti-de Sitter (AdS) space. This duality was first formulated for
the $O(N)$-invariant sector of the theory of $N$ free massless real scalars in
$2+1$ dimensions, and for the critical $O(N)$ model related to this theory
Klebanov:2002ja (1). These are dual to minimal higher-spin gravity theories
which only have fields of even spin, and there is a straightforward
generalization to $U(N)$-invariant theories of $N$ complex scalars, dual to
higher-spin gravity theories that have fields of all spins. A similar duality
was then discovered Sezgin:2003pt (2) for $O(N)$ (or $U(N)$)-invariant
theories of $N$ real (complex) free fermions, and for the critical Gross-
Neveu-type fixed points associated with them. It was then discovered
Aharony:2011jz (3, 4, 5) that these two dualities are the end-points of a
family of dualities (which becomes continuous in the large $N$ limit), between
Chern-Simons-matter theories and parity-breaking versions of higher-spin
gravity. In this paper we will focus on the quasi-fermionic family of
theories, whose scalar operator has dimension $\Delta_{0}=2+O(1/N)$. This
family has three different descriptions. At large $N$, these are (1) A
$U(N_{f})_{k_{f}}$ Chern-Simons (CS) theory coupled to a fermion in the
fundamental representation of $U(N_{f})$; (2) A $U(N_{b})_{k_{b}}$ Chern-
Simons theory coupled to the $U(N_{b})$ global symmetry of the critical
$U(N_{b})$ model (which is the non-trivial fixed point of $N_{b}$ complex
scalars deformed by a $|\phi^{2}|^{2}$ operator); (3) A (quantum version of a)
higher-spin gravity theory Fradkin:1987ks (6, 7, 8, 9, 10) on $AdS_{4}$, which
has a coupling $g$ (such that $n$-point vertices scale as $g^{n-2}$) and a
parity-breaking-parameter $\theta$. Each of these theories also has an
$O(N)$-version, which we will not discuss here. At finite $N$ all the
parameters become discrete, each precise duality is slightly more complicated
Aharony:2015mjs (11, 12), and each of these theories has several different
versions, but that will not affect our discussion here.

In the large $N$ limit, the local operators in these theories include one
operator for every integer spin (these are called “single-trace operators”),
and the products of these operators. The theories in this class have an
approximate higher-spin symmetry at large $N$, meaning that the divergence of
the spin $s$ currents ($s>2$) is given by a linear combination of terms which
are products of two other currents111By a slight abuse of notations, we will
call all the “single-trace operators” currents, including the operator of spin
zero. and derivatives acting on them, with coefficients proportional to $1/N$.
It was proven in Maldacena:2012sf (13) that the 3-point functions in theories
with this property are uniquely fixed (at leading order in the large $N$
limit) up to two parameters; in the field theory descriptions these may be
identified with $N$ and with the ’t Hooft coupling $N/k$, and on the gravity
side with $g$ and $\theta$. It is believed (but not yet proven) that all
higher-point functions are also determined by the same parameters; a proof of
this would imply that all three theories described in the previous paragraph
are indeed equivalent at large $N$.

In Maldacena:2012sf (13) the 3-point functions were written as a linear
combination of three different structures with a different dependence on the
positions and indices of the operators, with coefficients that were different
functions of the parameters (when one or more of the operators have spin zero,
less structures appear). However, by writing the 3-point functions in momentum
space and in spinor-helicity variables, it was noticed in a series of papers
Jain:2021vrv (14, 15, 16, 17, 18) that the different structures are related,
and in particular that in spinor-helicity variables some of them are
proportional to each other. Independently of this, the cubic vertices of the
dual higher-spin gravity theories were analyzed in Skvortsov:2018uru (19)
using light-cone coordinates and using a parameterization of the bulk fields
that is closely related to the spinor-helicity basis of operators. It was
suggested there that the 3-point vertices for fields with helicities
$s_{1},s_{2},s_{3}$ (which can be $\pm s$ for the spin $s$ gauge fields, or
$0$ for the scalar) have coefficients whose coupling dependence is
$ge^{in\theta}$ with $n={\rm sign}(s_{1})+{\rm sign}(s_{2})+{\rm
sign}(s_{3})-{\rm sign}(s_{1}+s_{2}+s_{3})$ (and they vanish when
$s_{1}+s_{2}+s_{3}=0$). When translated into 3-point functions in the field
theory, obtaining this coupling dependence requires very specific relations
between the three structures described above.

In this paper we show that the relations between different 3-point functions
structures take precisely the correct form to be consistent with the higher-
spin gravity vertices of Skvortsov:2018uru (19). This includes the relations
analyzed in Jain:2021vrv (14, 15, 16, 17, 18, 19) and a few additional
relations, and there are no additional relations that do not follow from this
form.

It was noted in Skvortsov:2018uru (19) that this form of the 3-point vertices
is consistent with having a specific non-unitary limit of the higher-spin
gravity theories, in which only 3-point vertices where the sum of the
helicities is positive are non-zero. This limit was called a “chiral higher-
spin gravity”222Such theories were constructed in flat space in Metsaev:1991mt
(20, 21, 22), and further analyzed in Skvortsov:2018jea (23, 24, 25, 26, 27,
28, 29) (among others)., and it was argued to have special properties such as
bulk locality Ponomarev:2016lrm (22) and absence of bulk UV divergences
Skvortsov:2020wtf (30). The duality of the higher-spin gravity theories to
Chern-Simons-matter theories implies that the latter should also have such a
non-unitary chiral limit, in which only 3-point functions with positive total
helicity are non-zero. By translating the gravity limit to the field theory,
we describe the corresponding limit on the field theory side, involving a
continuation of the Chern-Simons level to complex values. The limit required
to obtain the chiral theories is not strictly consistent with the ’t Hooft
large $N$ limit (in which the relation between the 3-point functions of
higher-spin gravity and of CS-matter theories is proven), and it would be
interesting to understand better the properties of this limit in the field
theory.

We begin in section 2 by reviewing the CS-matter and higher-spin gravity
theories mentioned above, and what is known about their 3-point functions. In
section 3 we analyze the 3-point functions for different values of the spins
and helicities, and show that their different structures are related precisely
in the correct way to reproduce the higher-spin gravity action in the form
written in Skvortsov:2018uru (19). In section 4 we analyze the “chiral limit”
of these theories, and the subtleties with understanding it in the Chern-
Simons-matter theories.

The generalization of our work to quasi-bosonic theories, where in the bulk
the scalar operator obeys different boundary conditions corresponding to
$\Delta=1+O(1/N)$, is straightforward. These theories have an extra coupling
constant that at large $N$ only affects the 3-point function of the scalar
(and higher-point functions), and which in the holographic description appears
only in the boundary conditions for the scalar. The only difference between
the two types of theories in the 3-point functions that do not involve 3
scalars is a Legendre transform with respect to the source of the scalar
operator; in momentum space this is the same as multiplying each correlation
function of the scalar at momentum $p$ by $1/|p|$.

Note added: During the work on this paper we became aware of Sachin (31),
which contains some overlapping results.

##  2 A Review of quasi-fermionic Chern-Simons-Matter Theory

Quasi-fermionic (QF) Chern-Simons-Matter theories are a special case of a
broad class of three dimensional CFTs involving matter coupled to Chern-Simons
gauge fields. The field theories in this class are believed to have two dual
descriptions, both of which can be thought of as low-energy fixed points of
matter coupled to Yang-Mills-Chern-Simons gauge fields. For simplicity we
describe here only the case of a single, fundamental matter field, though
there is a straightforward generalization to multiple matter fields. And, we
will describe the field theories in the large $N$ limit, ignoring order one
shifts in the ranks or levels as well as the distinction between $U(N)$ and
$SU(N)$ groups, which show up at higher orders in $1/N$. In this article, we
will be working exclusively in the Euclidean signature.

As a fermionic theory, we have a Dirac fermion ${\psi}$ in the fundamental
representation of the gauge group $U(N_{f})$, coupled to gluons $A^{a}_{\mu}$
with Chern-Simons interactions at level $k_{f}$:

$\displaystyle S_{FF+CS}=-\frac{ik_{f}}{8\pi}\int
d^{3}x{\epsilon}^{\mu\nu\rho}\bigg{(}A_{\mu}^{a}\partial_{\nu}A_{\rho}^{a}+\frac{1}{3}f^{abc}A^{a}_{\mu}A^{b}_{\nu}A^{c}_{\rho}\bigg{)}+\int
d^{3}x{\overline{\psi}}{\gamma}^{\mu}D_{\mu}{\psi}.$ (1)

In the large $N_{f}$ limit this theory has an ’t Hooft coupling
$\lambda_{f}\equiv N_{f}/k_{f}$ 333The convention we will use in this paper is
that the Chern-Simons levels are the ones appearing when the theory is
regularized using dimensional regularization. They are related to the levels
$\kappa_{f}$ appearing when flowing from a Yang-Mills-Chern-Simons theory by
$k_{f}={\rm sign}(\kappa_{f})(|\kappa_{f}|+N_{f})$. This implies that
$|k_{f}|\geq N_{f}$ and that $|\lambda_{f}|\leq 1$; and similarly for the
bosonic theory. . Alternatively, one may describe the same QF theory as a
bosonic theory by starting from the $U(N_{b})$ critical scalar model of
$N_{b}$ complex scalars $\phi$ and coupling it to Chern-Simons interactions at
level $k_{b}$:

$\displaystyle S_{CB+CS}=-\frac{ik_{b}}{8\pi}\int
d^{3}x{\epsilon}^{\mu\nu\rho}\bigg{(}{\tilde{A}}_{\mu}^{a}\partial_{\nu}{\tilde{A}}_{\rho}^{a}+\frac{1}{3}f^{abc}{\tilde{A}}^{a}_{\mu}{\tilde{A}}^{b}_{\nu}{\tilde{A}}^{c}_{\rho}\bigg{)}+\int
d^{3}x\bigg{(}|{\tilde{D}}_{\mu}\phi|^{2}+\sigma(\phi^{\dagger}\phi)\bigg{)}.$
(2)

This theory has an ’t Hooft coupling $\lambda_{b}\equiv N_{b}/k_{b}$, and
$\sigma(x)$ is an auxiliary field.

The fermionic description (1) and the bosonic description (2) are believed to
be related by a strong-weak duality, and describe the same theory, with the
identifications Aharony:2012nh (5):

$\displaystyle k_{b}=-N_{f}\ \text{sign}(k_{f}),\qquad\qquad N_{b}=|k_{f}|.$
(3)

This statement goes by the name of 3D Bosonization.444 More general versions
of the duality were formulated outside the large $N$ ’t Hooft limit
Aharony:2015mjs (11, 32, 33), and there are also extensions to other gauge
groups or matter content (see, for instance, Jain:2013gza (34, 35, 36, 37,
38)), and similar dualities for supersymmetric theories Giveon:2008zn (39,
40).

At large $N_{f}$ (equivalently large $N_{b}$) it was shown that the QF
theories with different couplings form a line of unitary conformal fixed
points $0\leq\lambda_{f}\leq 1$ Giombi:2011kc (4) (by using a parity
transformation we can choose $\lambda_{f}\geq 0$ without loss of generality).
In this limit the duality relates $\lambda_{b}=\lambda_{f}-{\rm
sign}(\lambda_{f})$. The special endpoints of this interval are the singlet
sector of the free fermion $U(N_{f})$ model, reached by setting
$\lambda_{f}\to 0$ (or equivalently $\lambda_{b}\to 1$), and the singlet
sector of the critical $U(N_{b})$ scalar model, reached by setting
$\lambda_{f}\to 1$ (or equivalently $\lambda_{b}\to 0$). These two endpoints
are referred to as the free fermion ($FF$) limit, and the critical boson
($CB$) limit, respectively.

The addition of Chern-Simons interactions does not introduce new local
operators, but just modifies the field bilinears already present in free and
critical vector models Giombi:2016zwa (41). Therefore, the spectrum of
“single-trace” operators of these theories consists of a tower555In the
present case, we have a unique spin-$s$ current for each $s\geq 0$, however by
choosing a different gauge group, different truncations/extensions of the
spectrum are admissible Maldacena:2012sf (13). of integer spin primaries
$(j^{(s\geq 0)}(x))^{\mu_{1}\mu_{2}...\mu_{s}}$, each transforming in the
spin-$s$ representation of $SO(3)$. Here we encounter a defining feature of
the QF Chern-Simons-Matter theory, which is the presence of an approximate
global higher-spin symmetry Maldacena:2012sf (13). While initially argued from
considering higher-spin gravity duals, it has also been explicitly verified
Giombi:2011kc (4, 41) that the higher-spin ($s>2$) currents of the QF theories
develop divergences only at subleading orders in $N$, in the form of multi-
trace operators. Ignoring Lorentz indices, derivatives and numerical factors
(depending on $\lambda$) that can be found, for instance, in Giombi:2016zwa
(41), we write this schematically as: $(\partial\cdot j^{(s\geq
3)})\simeq\frac{1}{N_{f,b}}\sum_{s_{1},s_{2}}j^{(s_{1})}j^{(s_{2})}+\mathcal{O}(1/N_{f,b}^{2})$,
where the sum goes over $s_{1}+s_{2}\leq s-2$.

It was shown by Maldacena and Zhiboedov in Maldacena:2011jn (42) that the only
theories satisfying exact higher-spin symmetry are free theories (assuming a
single conserved spin-2 current). Nevertheless, weakly broken higher-spin
symmetry, as described above for QF theories (1), (2) still imposes strong
constraints on current correlators, with all 3-point functions belonging to a
two-parameter family of correlators with a tractable amount of different
kinematic structures Maldacena:2012sf (13). These parameters can be related to
the microscopic parameters in the fermionic and bosonic descriptions. The
correlators of the QF theory are controlled Maldacena:2012sf (13) by a large
parameter $\tilde{N}$ which is of order $N_{f,b}$, and which plays the role of
the effective number of degrees of freedom. The other parameter
$\tilde{\lambda}$ controls the degree of higher-spin symmetry breaking. These
parameters were identified in Aharony:2012nh (5, 43) as:

$\displaystyle\tilde{\lambda}=\tan\bigg(\frac{\pi\lambda_{f}}{2}\bigg{missing})=-\cot\bigg(\frac{\pi\lambda_{b}}{2}\bigg{missing});\qquad\tilde{N}=\frac{2N_{f}\sin(\pi\lambda_{f})}{\pi\lambda_{f}}=\frac{2N_{b}\sin(\pi\lambda_{b})}{\pi\lambda_{b}}.$
(4)

With this identification, $\tilde{\lambda}\to 0$ corresponds to the FF limit,
while $\tilde{\lambda}\to\infty$ corresponds to the CB limit. Alternatively,
we can describe the correlators in the $CB+CS$ description by the bosonic
$\tilde{\lambda}_{b}=\tan(\frac{\pi\lambda_{b}}{2})$ parameter, satisfying
$\tilde{\lambda}_{b}=-\tilde{\lambda}^{-1}$, so that taking
$\tilde{\lambda}_{b}\to 0$ corresponds to the CB theory.

The holographic dual of large $N$ Chern-Simons-Matter theory is a higher-spin
gravity theory, whose classical equations of motion were written down in
Vasiliev:1990en (7, 8, 9, 10). There are two parity-preserving versions of
these theories on $AdS_{4}$, and a parity-breaking parameter $\theta$
affecting 3-point and higher correlation functions that interpolates between
these two versions. The 3-point vertices of these theories were constructed in
Skvortsov:2018uru (19), in light-cone gauge and in a particular gauge-fixing
of the bulk fields. These interaction vertices match the dual CFT three-point
functions of single-trace operators. Written in light-cone gauge, each
spin-$s$ field ($s>0$) in the bulk is represented in terms of two helicity
eigenstates666We denote the bulk radial coordinate by $z$.
$\Phi_{\lambda}(p;z)$, where $\lambda=\pm s$ is the helicity, and $p$ is the
momentum in the three dimensions of the CFT (for $s=0$ there is a single bulk
field $\Phi_{0}(p;z)$). In the dual CFT, these fields are related to the two
physical components of the corresponding spin-$s$ current $j^{(s)}_{\pm}(p)$,
written in the spinor-helicity basis777Recall that symmetric traceless
conserved currents in $2+1$ dimensions have only two independent components.
Their components parallel to the direction of the momentum vanish by
conservation, and tracelessness leaves just two independent components in the
other directions. We review the spinor-helicity basis for these operators in
appendix A.. The final expression for the terms with up to three fields in the
classical bulk action is:

$\displaystyle S$ $\displaystyle=S_{2}+S_{3}+\bar{S}_{3},$ (5) $\displaystyle
S_{2}$ $\displaystyle=\frac{1}{2}\sum_{\lambda}\int
dzd^{3}p\left(\Phi_{\lambda}(p;z)\right)^{\dagger}\left(\partial_{z}^{2}-p^{2}\right)\Phi_{\lambda}(p;z),$
(6) $\displaystyle S_{3}$
$\displaystyle=g\left(e^{+2\imath\theta}\mathbf{V}^{+++}+e^{+\imath\theta}\mathbf{V}^{++0}+\left(\mathbf{V}^{++-}+\mathbf{V}^{+00}\right)+e^{-\imath\theta}\mathbf{V}^{+-0}+e^{-2\imath\theta}\mathbf{V}^{+--}\right),$
(7) $\displaystyle\bar{S}_{3}$
$\displaystyle=g\left(e^{-2\imath\theta}\bar{\mathbf{V}}^{---}+e^{-\imath\theta}\bar{\mathbf{V}}^{--0}+\left(\bar{\mathbf{V}}^{--+}+\bar{\mathbf{V}}^{-00}\right)+e^{+\imath\theta}\bar{\mathbf{V}}^{-+0}+e^{+2\imath\theta}\bar{\mathbf{V}}^{-++}\right).$
(8)

The quadratic part of the action $S_{2}$ simply represents the action of free
fields on $AdS_{4}$. The factors $\mathbf{V}$ and $\bar{\mathbf{V}}$ denote
three-field interaction vertices explicitly given in Skvortsov:2018uru (19),
whose precise form will not be important for us; their indices refer to the
signs of the helicities of the participating fields. The $\mathbf{V}$’s are
non-zero only when the sum of the three helicities of the participating fields
is positive, and the $\bar{\mathbf{V}}$’s only when it is negative.

This description of the bulk theory involves two parameters $(g,\theta)$,
where $\theta\in[0,\tfrac{\pi}{2}]$ takes one from the gravity dual of the
critical bosonic theory to the free fermionic one. As was pointed out in
Skvortsov:2018uru (19), it is natural to consider two specific limits. By
dropping either the chiral vertices $S_{3}$ (which all have a positive sum of
spinor-helicities) or the anti-chiral vertices $\bar{S}_{3}$ (which all have a
negative sum of spinor-helicities) and setting $\theta=0$, one obtains the
action of (anti-)Chiral Higher-Spin gravity on $AdS_{4}$. We will discuss the
interpretation of this limit in the Chern-Simons-matter theories in Section 4.

###  2.1 Two-point functions

As all the current operators are primaries of specific spin $s$ and scaling
dimension $\Delta_{s}$, their two-point functions at separated points are
fixed by conformal invariance, to a single, parity-even structure. We will
normalize the currents such that the coefficients of this structure are
independent of $\lambda$ and scale like $\tilde{N}$. At finite, non-zero
$\tilde{\lambda}$, the two-point functions of spinning currents in the
theories (1), (2) develop also parity-odd contact terms. It will be useful to
define the momentum space epsilon-transform of a current operator
$j^{(s)}_{\mu_{1}...\mu_{s}}(p)$ Jain:2021gwa (16):

$\displaystyle({\epsilon}\cdot
j^{(s)})_{\mu_{1}...\mu_{s}}(p)\equiv{\epsilon}_{\nu\rho(\mu_{1}}\frac{p^{\nu}}{|\vec{p}|}(j^{(s)})^{\rho}_{\mu_{2}\cdots\mu_{s})}(p).$
(9)

The brackets in the subscript denote symmetrization with the appropriate
$1/s!$ factor. The epsilon-transform appears frequently when relating
different structures of two- and three-point functions of QF and QB theories.
Conveniently, in the spinor-helicity basis, this operation takes the form of a
helicity-dependent phase (see (51)). The two-point functions at leading order
in the large $N$ limit may now be written as:

$\displaystyle\langle j^{(0)}j^{(0)}\rangle_{QF}$
$\displaystyle=\tilde{N}\langle j^{(0)}j^{(0)}\rangle_{FF},$ (10)
$\displaystyle\langle j^{(s)}j^{(s)}\rangle_{QF}$
$\displaystyle=\tilde{N}\bigg{(}\langle
j^{(s)}j^{(s)}\rangle_{FF}+g_{s}(\tilde{\lambda})\langle
j^{(s)}j^{(s)}\rangle_{odd}\bigg{)},\text{ for }s\geq 1,$ (11)
$\displaystyle\langle j^{(s)}j^{(s)}\rangle_{odd}$
$\displaystyle=\langle({\epsilon}\cdot j^{(s)})j^{(s)}\rangle_{FF}.$ (12)

Here the subscript $odd$ denotes contact terms, given explicitly by (12),
whose coefficient can be a general odd function $g_{s}(\tilde{\lambda})$ of
the coupling. Interestingly, perturbative computations of the vector current
two-point functions in the bosonic and fermionic descriptions give different
answers for these factors Aharony:2012nh (5, 43), specifically:

$\displaystyle g_{1}(\tilde{\lambda})=\begin{cases}\tilde{\lambda},&\text{in
the FF+CS description }\eqref{FF+CS},\\\
\tilde{\lambda}_{b}=-\tilde{\lambda}^{-1},&\text{in the CB+CS description
}\eqref{CB+CS}.\end{cases}$ (13)

So far, as far as we know, only one higher-spin contact term was computed; the
stress-tensor two-point function was computed in Aharony:2012nh (5) in the
$CB+CS$ description, to be $g_{2}(\tilde{\lambda})=\tilde{\lambda}_{b}$. Based
on this example and on the general structure of the perturbative computations
leading to it, it is natural to assume that $g_{s}$ is independent of the spin
and equal to (13) for all $s$, and we will assume this from here on.

In general one would expect the correlation functions in dual descriptions of
the same theory to agree at separate positions (and this is indeed true), but
they can differ by contact terms (which can be interpreted as a change of
variables between the sources in one description and in another). However, the
contact terms appearing in $\eqref{2:2ptS}$ are not entirely arbitrary. When
coupling the spin-$s$ currents to background gauge fields $a_{s}$, the contact
terms (12) arise from background Chern-Simons terms. For the vector current
$j^{(1)}$, such a term looks like
$F[a]\propto...+\frac{i\kappa_{1}}{4\pi}\epsilon^{\mu\nu\rho}\int
d^{3}xa_{1\mu}\partial_{\nu}a_{1\rho}$ (related to a momentum-space two-point
function $\langle
j^{(1)}_{\mu}(p)j^{(1)}_{\nu}(-p)\rangle\propto\epsilon_{\mu\nu\rho}p^{\rho}$),
and similarly one may have gravitational Chern-Simons terms and higher-spin
analogs. The quantization of the Chern-Simons levels implies that the
fractional coefficients of these terms $(\kappa_{s}\mod 1)$ cannot be shifted
by introducing local counterterms, and hence represent universal parameters of
the theory Closset:2012vp (44). 888Note that (13) is not enough to imply that
the bulk duals of (1) and (2) differ in the value of $\theta$-terms, as the
functions $\tilde{N}g_{1}(\tilde{\lambda})$ are just the leading order
contributions to $(\kappa_{1}\mod 1)$ in the large $\tilde{N}$ expansion. The
large $N$ bosonic and fermionic descriptions have different contact terms, but
their fractional parts should agree at any finite value of $N$.

In the bulk theory, these contact-term coefficients correspond to
$\theta$-angles. For an example of the $U(1)$ case, where the bulk term is
proportional to $F\wedge F$, see Closset:2012vp (44), and the bulk dual theory
is expected to also contain $R\wedge R$ and higher-spin $\theta$-terms. The
choice of prefactors999Up to a shift by integer multiples, reflecting the
periodicity of $\theta$-angles. for contact terms in (11) then reflects
different possibilities for choosing the bulk $\theta$-terms, and will play an
interesting role when discussing the chiral limits of Chern-Simons-Matter
theory below.

Written in spinor-helicity variables, the $QF$ two-point functions (11) take
on a simple form for general values of spin Jain:2021gwa (16). The only non-
vanishing components are those with non-zero helicity:

$\displaystyle\langle
j^{(s)}_{+}(q)j^{(s)}_{+}(-q)\rangle=\tilde{N}\bigg{(}1-ig_{s}(\tilde{\lambda})\bigg{)}\frac{\langle\bar{1}\bar{2}\rangle^{2s}}{|\vec{q}|};\qquad\langle
j^{(s)}_{-}(q)j^{(s)}_{-}(-q)\rangle=\tilde{N}\bigg{(}1+ig_{s}(\tilde{\lambda})\bigg{)}\frac{\langle
12\rangle^{2s}}{|\vec{q}|}.$ (14)

Note that these correlators come with momentum-conserving delta functions,
which we will choose to omit throughout this article.

###  2.2 Three-point functions

It was shown in Maldacena:2012sf (13) that the correlation functions of the
currents in QF theories may be expanded in up to three distinct structures.
Assuming that the two-point functions are normalized as in (10)-(11), one
finds that the three-point function of three scalars vanishes (up to contact
terms), while the other correlators at separated points are given by (where
$s_{i}>0$):

$\displaystyle\begin{aligned} \langle
j^{(s)}j^{(0)}j^{(0)}\rangle_{QF}&=\tilde{N}\langle
j^{(s)}j^{(0)}j^{(0)}\rangle_{FF}=\tilde{N}\langle
j^{(s)}j^{(0)}j^{(0)}\rangle_{CB},\\\ \langle
j^{(s_{1})}j^{(s_{2})}j^{(0)}\rangle_{QF}&=\tilde{N}\bigg{(}\frac{1}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s_{1})}j^{(s_{2})}j^{(0)}\rangle_{FF}+\frac{\tilde{\lambda}}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s_{1})}j^{(s_{2})}j^{(0)}\rangle_{CB}\bigg{)},\\\ \langle
j^{(s_{1})}j^{(s_{2})}j^{(s_{3})}\rangle_{QF}&=\tilde{N}\bigg{(}\frac{1}{1+\tilde{\lambda}^{2}}\langle
j^{(s_{1})}j^{(s_{2})}j^{(s_{3})}\rangle_{FF}+\frac{\tilde{\lambda}}{1+\tilde{\lambda}^{2}}\langle
j^{(s_{1})}j^{(s_{2})}j^{(s_{3})}\rangle_{odd}\\\
&\qquad\qquad+\frac{\tilde{\lambda}^{2}}{1+\tilde{\lambda}^{2}}\langle
j^{(s_{1})}j^{(s_{2})}j^{(s_{3})}\rangle_{CB}\bigg{)}.\end{aligned}$ (15)

The subscripts $FF$ and $CB$ denote correlators in the free fermion and
critical boson theories with a single field, and correspond to the
$\tilde{\lambda}\to 0,\infty$ limits, respectively 101010Note that in the
large $N$ limit, three-point functions of the CB theory not containing scalar
currents are identical to the free boson theory three-point functions, while
the ones with scalars are related to free boson correlators by multiplying
them by the scalar momentum. For a complete discussion, see Aharony:2012nh
(5).. The additional $odd$ pieces do not appear in free theories, and in
Section 3 we will discuss their relations to the $FF$ and $CB$ pieces.

In this article, we will write the three-point functions in the normalization
common in the field theory literature, with the two-point functions scaling
like $\tilde{N}$, as in (10),(11) (note that the precise normalization we are
using for the scalar operator is different from the one used in
Maldacena:2012sf (13)). However, in the higher-spin gravity action, the
kinetic terms of the fields (giving their two-point functions at separated
points) were normalized to one, so in order to compare the three-point
functions to the bulk three-point vertices, we must use a normalization where
the two-point functions scale like $\tilde{N}^{0}$. We will call this the
gravity normalization.

The three-point functions (15) are written neglecting any contact terms, which
may appear with general $\tilde{\lambda}$-dependences and may differ across
fermionic (1) and bosonic (2) descriptions, similarly to the two-point
function case. For simplicity, we will ignore the three-point contact terms in
this paper.

The form of the three-point functions obtained from the bulk theory (5) is
similar to (15). The vertices of the $CB$ and $FF$ theories may be obtained by
setting $\theta=0,\tfrac{\pi}{2}$, respectively. In order to obtain the
correct normalization for the 3-point functions we need to set
$g=\tfrac{1}{\sqrt{\tilde{N}}}$, and then one finds that the gravity action
gives (in our normalization):

$\displaystyle\begin{aligned} \langle
j^{(s)}j^{(0)}j^{(0)}\rangle_{QF}&=\tilde{N}\langle
j^{(s)}j^{(0)}j^{(0)}\rangle_{FF}=\tilde{N}\langle
j^{(s)}j^{(0)}j^{(0)}\rangle_{CB},\\\ \langle
j^{(s_{1})}j^{(s_{2})}j^{(0)}\rangle_{QF}&=\tilde{N}\bigg{(}\cos(\theta)\langle
j^{(s_{1})}j^{(s_{2})}j^{(0)}\rangle_{CB}+\sin(\theta)\langle
j^{(s_{1})}j^{(s_{2})}j^{(0)}\rangle_{FF}\bigg{)},\\\ \langle
j^{(s_{1})}j^{(s_{2})}j^{(s_{3})}\rangle_{QF}&=\tilde{N}\bigg{(}\cos^{2}(\theta)\langle
j^{(s_{1})}j^{(s_{2})}j^{(s_{3})}\rangle_{CB}+\cos(\theta)\sin(\theta)\langle
j^{(s_{1})}j^{(s_{2})}j^{(s_{3})}\rangle_{odd}\\\
&\qquad+\sin^{2}(\theta)\langle
j^{(s_{1})}j^{(s_{2})}j^{(s_{3})}\rangle_{FF}\bigg{)}.\end{aligned}$ (16)

Comparing with (15), we arrive at:

$\displaystyle\tan(\theta)\leftrightarrow-\tilde{\lambda}_{b}=\tilde{\lambda}^{-1},\qquad\cot(\theta)\leftrightarrow\tilde{\lambda}.$
(17)

In Section 3, we will show that the three-point functions in the spinor-
helicity basis reproduce the form of the bulk action (5), with the above map
between parameters.

##  3 Consistency with the bulk action

The bulk action (5) has a simple form in the light-cone gauge, which does not
just imply (16) but also implies relations between the different structures
such that each correlator with fixed helicities comes with a fixed power of
$e^{i\theta}$. In this section we discuss relations (some of which were first
discussed in Jain:2021vrv (14, 15, 16, 17, 18)) between various tensor-
structures of the three-point functions, and show that they match precisely
with what is expected from the form of the bulk action (5) (up to local
contact terms). We present the relations that don’t follow from these works as
conjectures, and refer to explicit computations of correlation functions that
support them.

We split our analysis into three cases – with one, two, and three spinning
operators, respectively.111111The scalar three-point function at leading order
in large $N$ is a contact term that may be set to zero.

###  3.1 $\langle j^{(s)}j^{(0)}j^{(0)}\rangle_{QF}$

As was discussed in Section 2:

$\displaystyle\langle j^{(s)}_{\pm}j^{(0)}j^{(0)}\rangle_{QF}=\tilde{N}\langle
j^{(s)}_{\pm}j^{(0)}j^{(0)}\rangle_{FF}=\tilde{N}\langle
j^{(s)}_{\pm}j^{(0)}j^{(0)}\rangle_{CB}$ (18)

for all $s$, up to contact terms. The vertices from (5) that contribute to
this three-point function are:

$\displaystyle S\sim g(\mathbf{V}^{+00}+\bar{\mathbf{V}}^{-00}).$ (19)

We see that the three-point function in the gravity normalisation is precisely
of the expected form, given the mapping between $g$ and $\tilde{N}$ discussed
above.

###  3.2 $\langle j^{(s)}j^{(s^{\prime})}j^{(0)}\rangle_{QF}$

We set $s\geq s^{\prime}$ without loss of generality. Excluding contact terms,
the correlator is determined by two pieces Maldacena:2012sf (13):

$\displaystyle\langle j^{(s)}j^{(s^{\prime})}j^{(0)}\rangle_{QF}$
$\displaystyle=\tilde{N}\left(\frac{1}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s)}j^{(s^{\prime})}j^{(0)}\rangle_{FF}+\frac{\tilde{\lambda}}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s)}j^{(s^{\prime})}j^{(0)}\rangle_{CB}\right).$ (20)

It was noted in Jain:2021whr (17) (and supported by an explicit calculation of
$\langle j^{(3)}j^{(1)}j^{(0)}\rangle$ there) that for various helicities, the
two structures satisfy:

$\displaystyle\langle j^{(s)}_{a}j^{(s^{\prime})}_{\pm}j_{0}\rangle_{FF}$
$\displaystyle=\pm\imath\langle
j^{(s)}_{a}j^{(s^{\prime})}_{\pm}j_{0}\rangle_{CB},$ (21)

independent of the helicity $a=\pm$ of $j^{(s)}$ (but note that this is not
necessarily true if we exchange $s$ and $s^{\prime}$).121212The relation was
written down in Jain:2021whr (17) using an epsilon-transform, and simplified
to (21) using (51). The signs on the right-hand side of (21) may differ from
Jain:2021whr (17) due to a mismatch in conventions (see Appendix A for our
conventions). The invariant part of the claim is the form of the result, and
that the same sign works for every $s,s^{\prime}$. We have verified that the
explicit calculation of $\langle j^{(1)}j^{(1)}j^{(0)}\rangle$ in Gur-
Ari:2012lgt (43) gives (21) in our conventions. Using (17), we now have:

$\displaystyle\langle
j^{(s)}_{a}j^{(s^{\prime})}_{\pm}j^{(0)}\rangle_{QF}=\tilde{N}\left(\frac{\pm\imath+\tilde{\lambda}}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s)}_{a}j^{(s^{\prime})}_{\pm}j^{(0)}\rangle_{CB}\right)=\tilde{N}e^{\pm\imath\theta}\langle
j^{(s)}_{a}j^{(s^{\prime})}_{\pm}j^{(0)}\rangle_{CB}.$ (22)

The result (22) is perfectly compatible with the form of the relevant bulk
vertices:

$\displaystyle S\sim
g\left(e^{\imath\theta}\mathbf{V}^{++0}+e^{-\imath\theta}\mathbf{V}^{+-0}\right)+(cc),$
(23)

where $(cc)$ stands for complex conjugation, and $\mathbf{V}$ is non-zero only
when the sum of the helicities is positive.

Note that for (21) to be consistent when $s=s^{\prime}$, all zero-total-
helicity correlators in this class must vanish (up to contact terms):

$\displaystyle\langle j^{(s)}_{+}j^{(s)}_{-}j_{0}\rangle_{FF,CB}=0.$ (24)

This was indeed observed in Jain:2021vrv (14). We have also verified this in
the $CB$ theory for $s\leq 6$. This agrees with the absence of zero-total-
helicity vertices in the bulk action (5).

###  3.3 $\langle j^{(s)}j^{(s^{\prime})}j^{(s^{\prime\prime})}\rangle_{QF}$

We order $s\geq s^{\prime}\geq s^{\prime\prime}$ without loss of generality.
The relevant correlator is:

$\displaystyle\langle
j^{(s)}j^{(s^{\prime})}j^{(s^{\prime\prime})}\rangle_{QF}$
$\displaystyle=\tilde{N}\left(\frac{1}{1+\tilde{\lambda}^{2}}\langle
j^{(s)}j^{(s^{\prime})}j^{(s^{\prime\prime})}\rangle_{FF}+\frac{\tilde{\lambda}}{1+\tilde{\lambda}^{2}}\langle
j^{(s)}j^{(s^{\prime})}j^{(s^{\prime\prime})}\rangle_{odd}\right.$
$\displaystyle\left.\qquad\qquad+\frac{\tilde{\lambda}^{2}}{1+\tilde{\lambda}^{2}}\langle
j^{(s)}j^{(s^{\prime})}j^{(s^{\prime\prime})}\rangle_{CB}\right).$ (25)

The odd pieces were dealt with in the analyses by Jain et. al. Jain:2021whr
(17). At least in the case where $s<s^{\prime}+s^{\prime\prime}$, they found
the following relation between the $odd$ and $FF-CB$ (which they called
$homogeneous$) tensor structures:

$\displaystyle\langle
j^{(s)}_{a}j^{(s^{\prime})}_{b}j^{(s^{\prime\prime})}_{c}\rangle_{odd}$
$\displaystyle=\begin{cases}-\imath\langle
j^{(s)}_{a}j^{(s^{\prime})}_{+}j^{(s^{\prime\prime})}_{+}\rangle_{FF-
CB}&b=c=+\\\ \imath\langle
j^{(s)}_{a}j^{(s^{\prime})}_{b}j^{(s^{\prime\prime})}_{c}\rangle_{FF-
CB}&\text{otherwise}\end{cases}$ (26)

irrespective of $a=\pm$.131313Once again, the exact sign on the right-hand
side of (26) – in our conventions – may be confirmed by checking it against
the collinear result for $\langle j^{(2)}j^{(1)}j^{(1)}\rangle$ from
Aharony:2012nh (5). For consistency with the bulk action, this relation must
be true in general (the authors of Jain:2021whr (17) expect this to be true as
well, but do not prove it). Using this in (25), we get:

$\displaystyle\langle
j^{(s)}_{a}j^{(s^{\prime})}_{b}j^{(s^{\prime\prime})}_{c}\rangle_{QF}$
$\displaystyle=\begin{cases}\tilde{N}e^{-\imath(\tfrac{\pi}{2}-\theta)}\left(\frac{1}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s)}_{a}j^{(s^{\prime})}_{+}j^{(s^{\prime\prime})}_{+}\rangle_{FF}+\frac{\imath\tilde{\lambda}}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s)}_{a}j^{(s^{\prime})}_{+}j^{(s^{\prime\prime})}_{+}\rangle_{CB}\right)&b=c=+\\\
\tilde{N}e^{\imath(\tfrac{\pi}{2}-\theta)}\left(\frac{1}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s)}_{a}j^{(s^{\prime})}_{b}j^{(s^{\prime\prime})}_{c}\rangle_{FF}-\frac{\imath\tilde{\lambda}}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s)}_{a}j^{(s^{\prime})}_{b}j^{(s^{\prime\prime})}_{c}\rangle_{CB}\right)&\text{otherwise}\end{cases}$
(27)

The relevant bulk vertices are:

$\displaystyle S\sim
g\left(e^{2\imath\theta}\mathbf{V}^{+++}+\mathbf{V}^{++-}+e^{-2\imath\theta}\mathbf{V}^{+--}\right)+(cc).$
(28)

To proceed, we must deal with either side of the triangle inequality
$s\lessgtr s^{\prime}+s^{\prime\prime}$ separately.

####  3.3.1 Inside the triangle: $s<s^{\prime}+s^{\prime\prime}$

In this case, since the non-conservation of the spin $s$ current only contains
(at leading order in large $N$) other currents $j^{(s_{1})}j^{(s_{2})}$ with
$s_{1}+s_{2}<s$, all the currents in the correlator are conserved at leading
order in $1/\tilde{N}$, namely141414If the divergence is applied to
$s^{\prime}$ or $s^{\prime\prime}$, this statement holds even outside the
triangle. The difference is the effective conservation of $j^{(s)}$.

$\displaystyle\langle\partial.j^{(s)}j^{(s^{\prime})}j^{(s^{\prime\prime})}\rangle_{QF}=\langle
j^{(s)}\partial.j^{(s^{\prime})}j^{(s^{\prime\prime})}\rangle_{QF}=\langle
j^{(s)}j^{(s^{\prime})}\partial.j^{(s^{\prime\prime})}\rangle_{QF}=0.$ (29)

Consistency with the bulk action requires the following:

$\displaystyle\langle
j_{a}^{(s)}j_{b}^{(s^{\prime})}j_{c}^{(s^{\prime\prime})}\rangle_{FF}$
$\displaystyle=\begin{cases}-\langle
j_{a}^{(s)}j_{b}^{(s^{\prime})}j_{c}^{(s^{\prime\prime})}\rangle_{CB}&a=b=c\\\
\langle
j_{a}^{(s)}j_{b}^{(s^{\prime})}j_{c}^{(s^{\prime\prime})}\rangle_{CB}&\text{otherwise}\end{cases}$
(30)

The second equality was observed in several cases by the Ward identity
analysis of Jain et. al. (for instance, Jain:2021vrv (14)), and also by the
explicit computations of $\langle j^{(2)}j^{(2)}j^{(2)}\rangle_{FF}$ and
$\langle j^{(2)}j^{(2)}j^{(2)}\rangle_{CB}$ (for instance, Maldacena:2011nz
(45)). The latter also supports the first equality. Assuming this result, (27)
becomes:

$\displaystyle\langle
j^{(s)}_{+}j^{(s^{\prime})}_{b}j^{(s^{\prime\prime})}_{c}\rangle_{QF}$
$\displaystyle=\begin{cases}\tilde{N}e^{-\imath(\tfrac{\pi}{2}-\theta)}\left(\frac{-1+\imath\tilde{\lambda}}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s)}_{+}j^{(s^{\prime})}_{+}j^{(s^{\prime\prime})}_{+}\rangle_{CB}\right)&b=c=+\\\
\tilde{N}e^{\imath(\tfrac{\pi}{2}-\theta)}\left(\frac{1-\imath\tilde{\lambda}}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s)}_{+}j^{(s^{\prime})}_{b}j^{(s^{\prime\prime})}_{c}\rangle_{CB}\right)&\text{otherwise}\end{cases}$
$\displaystyle=\begin{cases}\tilde{N}e^{2\imath\theta}\langle
j^{(s)}_{+}j^{(s^{\prime})}_{+}j^{(s^{\prime\prime})}_{+}\rangle_{CB}&b=c=+\\\
\tilde{N}\langle
j^{(s)}_{+}j^{(s^{\prime})}_{b}j^{(s^{\prime\prime})}_{c}\rangle_{CB}&\text{otherwise}\end{cases}$
(31)

as expected from the bulk action (28) (noting that in this case
$s-s^{\prime}-s^{\prime\prime}<0$, so we should use ${\bar{\mathbf{V}}}^{+--}$
for $b=c=-$). The other cases may be obtained by complex conjugation.

####  3.3.2 Outside the triangle: $s\geq s^{\prime}+s^{\prime\prime}$

In this case, $j^{(s)}$ is not always conserved. An explicit computation of
$\langle j^{(4)}j^{(1)}j^{(1)}\rangle_{FF}$ and $\langle
j^{(4)}j^{(1)}j^{(1)}\rangle_{CB}$ (see Jain:2021whr (17)) points to the
following relation:

$\displaystyle\langle
j_{a}^{(s)}j_{b}^{(s^{\prime})}j_{c}^{(s^{\prime\prime})}\rangle_{FF}$
$\displaystyle=\begin{cases}-\langle
j_{a}^{(s)}j_{b}^{(s^{\prime})}j_{c}^{(s^{\prime\prime})}\rangle_{CB}&b=c\\\
\langle
j_{a}^{(s)}j_{b}^{(s^{\prime})}j_{c}^{(s^{\prime\prime})}\rangle_{CB}&\text{otherwise}\end{cases}$
(32)

which we conjecture to be true for all $s,s^{\prime},s^{\prime\prime}$ outside
the triangle. Our hypothesis is also supported by explicit computations of
$\langle j^{(2)}j^{(1)}j^{(1)}\rangle$ (see, for instance, Jain:2021vrv (14)
or Aharony:2012nh (5, 46)). Assuming this, (27) becomes:

$\displaystyle\langle
j^{(s)}_{+}j^{(s^{\prime})}_{b}j^{(s^{\prime\prime})}_{c}\rangle_{QF}$
$\displaystyle=\begin{cases}\tilde{N}e^{-\imath(\tfrac{\pi}{2}-\theta)}\left(\frac{-1+\imath\tilde{\lambda}}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s)}_{+}j^{(s^{\prime})}_{+}j^{(s^{\prime\prime})}_{+}\rangle_{CB}\right)&b=c=+\\\
\tilde{N}e^{\imath(\tfrac{\pi}{2}-\theta)}\left(\frac{-1-\imath\tilde{\lambda}}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s)}_{+}j^{(s^{\prime})}_{-}j^{(s^{\prime\prime})}_{-}\rangle_{CB}\right)&b=c=-\\\
\tilde{N}e^{\imath(\tfrac{\pi}{2}-\theta)}\left(\frac{1-\imath\tilde{\lambda}}{\sqrt{1+\tilde{\lambda}^{2}}}\langle
j^{(s)}_{+}j^{(s^{\prime})}_{b}j^{(s^{\prime\prime})}_{c}\rangle_{CB}\right)&\text{otherwise}\end{cases}$
$\displaystyle=\begin{cases}\tilde{N}e^{2\imath\theta}\langle
j^{(s)}_{+}j^{(s^{\prime})}_{+}j^{(s^{\prime\prime})}_{+}\rangle_{CB}&b=c=+\\\
\tilde{N}e^{-2\imath\theta}\langle
j^{(s)}_{+}j^{(s^{\prime})}_{-}j^{(s^{\prime\prime})}_{-}\rangle_{CB}&b=c=-\\\
\tilde{N}\langle
j^{(s)}_{+}j^{(s^{\prime})}_{b}j^{(s^{\prime\prime})}_{c}\rangle_{CB}&\text{otherwise}\end{cases}$
(33)

The other cases may be obtained by complex conjugation. Only the unbarred
vertices of (28) are relevant and the match is easy to see.

To complete the comparison of three-point functions we discuss the case where
the spins saturate the triangle inequality $s=s^{\prime}+s^{\prime\prime}$.
Here the $(+--)$ correlators have zero total helicity, and must vanish for our
conjecture (and for the matching to the gravity action) to be consistent. We
have verified this in the $CB$ theory for low values of spin.

##  4 A chiral limit for Chern-Simons-matter theories

It was noted in Skvortsov:2018uru (19) that the chiral (i.e. positive total
helicity) three-point vertices of the bulk action (the ones appearing in (7))
were sufficient to construct a representation of the $AdS_{4}$ algebra, and
this truncation was called a chiral higher-spin gravity, generalizing similar
constructions in flat space Ponomarev:2016lrm (22, 23, 30). One way to obtain
this theory is to drop by hand the terms in (8), but the field theory
interpretation of this is not clear. We will argue that one can construct a
limit of the full higher-spin gravity theory that has the same effect. For
simplicity we will discuss this limit in the dual Chern-Simons-matter
theories, where it involves taking a specific (non-unitary) limit of $N$ and
$k$ and a specific rescaling of the currents. The translation to the bulk,
where it involves taking a limit of $(g,\theta)$ and rescaling the bulk
fields, is straightforward given the relation between them described in
section 2.

We would like to find a limit of the parameters
$({\tilde{N}},\tilde{\lambda})$ and a rescaling of the currents that keeps
only the 3-point functions related to (7) and not those of (8). In general the
rescaling could depend on the value of the spin, but it seems that the
simplest option is to perform a universal rescaling of all $j_{+}^{(s)}$
operators and of all $j_{-}^{(s)}$ operators (independent of $s$). Then, by
comparing the coefficients of the $(++-)$ vertices in (7) and (8), it is clear
that we must take a limit where $e^{i\theta}\to 0$. In the unitary higher-spin
gravity theories $\theta$ is real, and correspondingly in the dual CFTs
$\tilde{\lambda}$ (related to it by (17)) is real; however in order to obtain
the chiral limit we must continue these parameters to complex values (in
particular $\theta\to i\infty$), leading to non-unitary theories.

Let us denote $e^{i\theta}=\epsilon$ (it can have any phase), and assume that
we take the $\epsilon\to 0$ limit while doing a rescaling such that

$\displaystyle\tilde{N}$ $\displaystyle={\tilde{c}}\epsilon^{a},$ (34)
$\displaystyle j^{(s)}_{+}$ $\displaystyle\to
c^{\prime}\epsilon^{b}j^{(s)}_{+},$ (35) $\displaystyle j^{(s)}_{-}$
$\displaystyle\to c^{\prime}\epsilon^{c}j^{(s)}_{-},$ (36) $\displaystyle
j_{0}$ $\displaystyle\to c^{\prime}\epsilon^{d}j_{0}.$ (37)

By requiring that the correlation functions related to the six vertices in (7)
remain finite, and recalling that in the 3-point functions their pre-factor is
$\tilde{N}$ rather than $g$, we find that we need to have

$\displaystyle a+2+3b$ $\displaystyle=0,\qquad a+1+2b+d=0,\qquad a+2b+c=0,$
(38) $\displaystyle a+b+2d$ $\displaystyle=0,\qquad a-1+b+c+d=0,\qquad
a-2+b+2c=0.$ (39)

There are only three linearly independent equations here, so we need one
additional constraint, and requiring that the 2-point function of $j_{0}$ (10)
remains finite tells us that $a+2d=0$, so that we have

$\displaystyle a=-2,\qquad b=0,\qquad c=2,\qquad d=1.$ (40)

With this scaling all 3-point functions related to (7) have a finite
$\epsilon\to 0$ limit, while all 3-point functions related to (8) go to zero,
as desired. We can further choose the finite prefactors such that the 2-point
function of $j_{0}$ is normalized to one, while all remaining 3-point
functions are proportional to some new bulk chiral-higher-spin-gravity
coupling constant $g^{\prime}$; this will happen if we take
${\tilde{c}}=1/g^{\prime 2}$, $c^{\prime}=g^{\prime}$. We will discuss the
2-point functions of $j_{+}$ and $j_{-}$ below.

In the Chern-Simons-matter theories, the limit we need to take on the
parameters is then:

$\displaystyle\tilde{N}$ $\displaystyle=-\frac{1}{g^{\prime
2}}\frac{1+\imath\tilde{\lambda}}{1-\imath\tilde{\lambda}}\simeq\frac{1}{g^{\prime
2}\epsilon^{2}}\rightarrow\infty,$ (41) $\displaystyle\tilde{\lambda}$
$\displaystyle\rightarrow-\imath.$ (42)

Translating this via (4) to the couplings of the fermionic Chern-Simons-matter
theories, we obtain

$\displaystyle\lambda_{f}=\frac{N_{f}}{k_{f}}$ $\displaystyle\rightarrow
1-\imath\infty,$ (43) $\displaystyle k_{f}$
$\displaystyle\rightarrow\frac{-\imath\pi}{g^{\prime 2}}.$ (44)

Depending on the choice of phase of $g^{\prime}$, at least one of
$(N_{f},k_{f})$ must be complex, such that the CFT is no longer unitary. For
instance, we can choose to keep $N_{f}$ real and positive. In this case
$g^{\prime}$ is imaginary, and we need to take the limit :

$\displaystyle N_{f}\rightarrow\infty$ $\displaystyle
k_{f}\in\imath\mathbbm{R}_{>0}$ (45)

where large $|k_{f}|$ corresponds to having weak coupling in the gravity
theory (small $|g^{\prime}|$). The limit in the scalar variables takes a
similar form. Alternatively, by choosing a different phase for $g^{\prime}$,
we can keep the Chern-Simons level real, and take the rank of the gauge group
to be large and imaginary. Note that in various applications of Chern-Simons
theory (such as the relation to the Jones polynomial) one uses a parameter $q$
which in our notation is $q=\exp(2\pi i/k_{f})$, and the limit we are taking
is $N_{f}\to\infty$, $q^{-N_{f}}\to 0$ with $q$ fixed (and close to $1$ if we
want the dual gravity theory to be weakly coupled). In this limit, we obtain a
“chiral Chern-Simons-matter” theory, in which (after rescaling the operators
as above) only positive-total-helicity 3-point functions are non-zero.

Similarly, there is an antichiral limit where we keep only the negative-total-
helicity vertices. This can be obtained by taking $\theta\to-\theta$, and
exchanging the rescaling of $j^{(s)}_{+}$ with that of $j^{(s)}_{-}$. In the
field theory this means $\lambda_{f}\to i\infty$, with $k_{f}$ having an
opposite phase.

Recall that we defined $k_{f}$ as the Chern-Simons level when regularized by a
dimensional reduction scheme; it is not so clear what $k_{f}$ behaves like
when flowing from a Yang-Mills-Chern-Simons theory, since the relation between
the two $k$’s involves ${\rm sign}(k_{f})$ and it is not clear how to continue
this to complex $k$’s.

###  4.1 Two-point functions and contact terms

The only remaining correlators to consider are the two-point functions of
$j^{(s)}_{+}$ and of $j^{(s)}_{-}$, which are more subtle since as discussed
in section 2 they depend on the choice of contact terms.

Since in the limit we are taking ${\tilde{N}}\propto 1/\epsilon^{2}$,
obtaining a finite limit for the 2-point function (14) of $j^{(s)}_{+}$
requires having $(1-ig_{s})\propto\epsilon^{2}$. Clearly this requires the
contact term $g_{s}$ to become complex, but since we are already taking the
physical couplings to be complex, it is not surprising that the contact term
coefficients (related to the levels of Chern-Simons terms for background
fields) also become complex. Indeed, it is easy to see that if we choose the
contact terms to take their values (13) computed from perturbation theory,
either in the fermionic description of the Chern-Simons-matter theories,
$g_{s}(\tilde{\lambda})=\tilde{\lambda}$, or in the bosonic description,
$g_{s}(\tilde{\lambda})=-1/\tilde{\lambda}$, then we have
$(1-ig_{s})\propto\epsilon^{2}$, so that with either one of these natural
choices the 2-point function of $j^{(s)}_{+}$ remains finite.

Before our rescaling, with this choice of contact terms the two-point function
of $j^{(s)}_{-}$ diverges, but after the rescaling we discussed above (needed
to obtain finite 3-point functions) we find that this two-point function goes
to $0$. Since we are in a non-unitary theory, there is nothing inconsistent
about having operators with vanishing 2-point functions and finite 3-point
functions, and this is what we obtain.

Note that since we are taking different rescalings for different spinor-
helicity components of the fields, it is not so obvious how to define
operators in position space in our theories. The two-point functions at
separated points, which are proportional to ${\tilde{N}}$, diverge in our
limit (for all spins) before we do any rescaling, and it is not clear how to
interpret the rescaled operators directly in position space.

Our discussion so far is not completely self-consistent, since we considered
contact terms in the two-point functions (which play an important role in
making them finite in our limit), but we ignored all contact terms in the
three-point functions. Indeed, not only do contact terms appear in
perturabtive computations of 3-point functions (see, for instance
Aharony:2012nh (5, 43)), but the higher-spin Ward identities even relate some
contact terms in two-point functions to contact terms in three-point functions
Kukolj:2024 (46), so that we should include at least these contact terms, if
we want our limit to be consistent with the approximate higher-spin symmetry.
We believe that including these contact terms will not modify the consistency
of the limit we described here, but we leave their full analysis to future
work.

In this paper we only considered two-point and three-point functions; it would
be interesting to study higher-point functions, and to see if they are well-
defined in the chiral limit discussed here, and if they agree with the
expectations from the chiral higher-spin gravity theory. It would also be
interesting to study how correlation functions of non-local operators as in
Gabai:2022vri (47, 48, 49) behave in our limit.

###  4.2 Defining the limit of Chern-Simons-matter theories

Our discussion above led us to consider a specific complex limit of Chern-
Simons-matter theories. There are two major subtleties in understanding the
behavior of these theories in this limit.

One issue is that $SU(N)$ Chern-Simons theories are well-defined for positive
integers $N$ and for integer $k$, while our limit requires an analytic
continuation to complex values (for one or both of these). Clearly, already
for pure Chern-Simons theories, such a limit is not uniquely defined, and it
is not clear if there is any preferred limit. Another place where a similar
continuation appears (for pure Chern-Simons theories) is in the duality of
Chern-Simons theories to topological string theories Gopakumar:1998ki (50,
51), where the string coupling $g_{s}$ is mapped (in our convention) to $i/k$,
and the ’t Hooft coupling $N/k$ is related to a complex parameter (related to
a combination of the size of a 2-cycle and the $B$ field on it). It would be
interesting to understand if the analytic continuation required in that
context can be used also for the chiral higher-spin gravity theories;
unfortunately it is subtle to describe the coupling of the Chern-Simons theory
to matter fields in the topological string language Aganagic:2017tvx (52, 53)
so it is not clear how to check this.

The second issue is that even though the correlation functions we discussed
were computed only in the ’t Hooft limit of large $N$, $k$ with fixed $N/k$,
the limit we eventually need to consider for the chiral higher-spin gravity is
different, with large $N$ and fixed $k$. This limit is less well-understood,
with no good methods to perform computations, and there is no reason why non-
planar diagrams could not lead to significant corrections to the correlation
functions we used, that may modify the results. From the bulk point of view
the chiral higher-spin gravity theory seems to have special properties which
may make it well-defined (and weakly coupled in the small $g^{\prime}$ / large
$k$ limit), and clearly also in the CS-matter theories such a limit (with only
positive-helicity correlators) should have many simplifying properties. The
known leading non-planar corrections to correlators, such as the corrections
to 2-point functions of spin $s>2$ operators related to their anomalous
dimensions computed in Giombi:2016zwa (41), naively diverge in the limit
discussed above. However, since our limit treats different-helicity-components
differently, and since contact terms play an important role in defining it, it
is not clear if this naive computation of the non-planar corrections is
relevant in our limit or not (recall that the 2-point function at separated
points diverges also at the leading planar order, as discussed above, but this
is fixed by including the contact terms and the rescaling). It is also
possible that a resummation of the non-planar corrections could behave
differently. It would be interesting to understand how to think about the full
correlation functions in our limit, and to see if they are well-defined.

## Acknowledgments

We would like to thank Sachin Jain, Dhruva K. S., and Evgeny Skvortsov for
useful discussions and for coordinating the submission of this manuscript with
their upcoming paper. RRK and TK would also like to thank Erez Urbach for
discussions. This work was supported in part by Israel Science Foundation
grant no. 2159/22, by Simons Foundation grant 994296 (Simons Collaboration on
Confinement and QCD Strings), by the Minerva foundation with funding from the
Federal German Ministry for Education and Research, by the German Research
Foundation through a German-Israeli Project Cooperation (DIP) grant
“Holography and the Swampland”, and by a research grant from Martin
Eisenstein. OA is the Samuel Sebba Professorial Chair of Pure and Applied
Physics. The work of RRK was also supported by the Infosys Endowment for the
study of the Quantum Structure of Spacetime. He would also like to acknowledge
his debt to the people of India for their generous and steady support to
research in the basic sciences.

##  Appendix A The spinor-helicity basis

We briefly review the basics of the spinor-helicity formalism in three
dimensional flat space. For a more comprehensive overview of spinor-helicity
variables in three and four dimensions, we direct the reader to
Baumann:2020dch (54, 16, 14) and referenes therein.

We start by embedding a general Euclidean three-vector $\vec{p}$ as a null
four-vector $\vec{p}\to p_{\mu}=\big{(}|\vec{p}|,\vec{p}\big{)}$ in four
dimensional Minkowski space. The corresponding Lorentz group may be decomposed
as $SO(1,3)=SL(2,\mathbb{C})_{undot}\otimes SL(2,\mathbb{C})_{dot}$ with an
additional reality condition imposed. The undotted and dotted subgroups are
generated by151515We use the notation $\sigma^{\mu}=(\mathds{1},\sigma^{i})$,
where $\sigma^{i}$ are Pauli matrices. $(\sigma^{\mu})^{\beta}_{\alpha}$ and
$(\sigma^{\nu})^{\dot{\beta}}_{\dot{\alpha}}$, respectively. The spinor
indices can be raised and lowered by contracting with the invariant epsilon
tensors ${\epsilon}^{\alpha\beta}\in SL(2,\mathbb{C})_{undot}$ and
${\epsilon}^{\dot{\alpha}\dot{\beta}}\in SL(2,\mathbb{C})_{dot}$. We can now
identify each four-vector as the outer product of two spinors:

$\displaystyle p_{\mu}\to
p_{\alpha\dot{\alpha}}=p_{\mu}(\sigma^{\mu})_{\alpha\dot{\alpha}}=\lambda_{\alpha}\bar{\lambda}_{\dot{\alpha}}.$
(46)

When restricting ourselves to three-dimensional vectors of $SO(3)$, we can
introduce an orthogonal timelike vector $\tau_{\mu}=(1,\vec{0})$, giving us a
way to identify dotted and undotted indices via
$\mathds{1}_{\alpha\dot{\alpha}}=-\tau_{\mu}(\sigma^{\mu})_{\alpha\dot{\alpha}}$.
This reflects the fact that $SO(3)$ only preserves a single
$SL(2,\mathbb{C})_{undot}=SL(2,\mathbb{C})$ subgroup. Thus to each spinor
$\lambda_{\alpha}\in SL(2,\mathbb{C})$, we associate a barred spinor
$\bar{\lambda}_{\alpha}=-\lambda_{\dot{\alpha}}{\epsilon}^{\dot{\alpha}\dot{\beta}}\mathds{1}_{\dot{\beta}\alpha}$,
so that:

$\displaystyle\vec{p}=\frac{1}{2}(\vec{\sigma})^{\alpha\beta}\lambda_{\alpha}\bar{\lambda}_{\beta}.$
(47)

For each three-vector $\vec{p}$, we can introduce two polarization vectors
$\vec{z}^{\pm}$, satisfying $(\vec{z}^{\pm})^{2}=0$ and
$\vec{z}^{\pm}\crossproduct\vec{p}=\mp i|\vec{p}|\vec{z}^{\pm}$. In spinor-
helicity variables, the polarization vectors take the form:

$\displaystyle
z^{-}_{\alpha\beta}=\frac{\lambda_{\alpha}\lambda_{\beta}}{2|\vec{p}|};\qquad\qquad
z^{-}_{\alpha\beta}=\frac{\bar{\lambda}_{\alpha}\bar{\lambda}_{\beta}}{2|\vec{p}|}.$
(48)

When dealing with multiple vectors $\vec{p}_{i}$, to each we associate a pair
of spinors $\lambda_{i}^{\alpha}$ and $\bar{\lambda}_{i}^{\beta}$. To contract
spinors in an $SL(2,\mathbb{C})$ invariant way, we define the spinor dot-
products as:

$\displaystyle\langle
ij\rangle={\epsilon}_{\alpha\beta}\lambda_{i}^{\alpha}\lambda_{j}^{\beta};\qquad\langle
i\bar{j}\rangle={\epsilon}_{\alpha\beta}\lambda_{i}^{\alpha}\bar{\lambda}_{j}^{\beta};\qquad\langle\bar{i}\bar{j}\rangle={\epsilon}_{\alpha\beta}\bar{\lambda}_{i}^{\alpha}\bar{\lambda}_{j}^{\beta}.$
(49)

We may now use (49) to translate expressions between momentum space and
spinor-helicity variables. In practice, to obtain a specific helicity
component of a correlator from its momentum space expression, we dot the
$SO(3)$ indices161616For the remainder of the section we denote the vector
indices of $SO(3)$ by greek letters, disregarding the embedding in four
dimensions. of the currents in the following fashion:

$\displaystyle\langle
j_{a}^{(s_{1})}...j_{b}^{(s_{2})}\rangle=\langle\big{(}(z_{1}^{a})^{\mu_{1}..\mu_{s_{1}}}\cdot
j^{(s_{1})}_{\mu_{1}..\mu_{s_{1}}}(p_{1})\big{)})...\big{(}(z_{2}^{b})^{\nu_{1}..\nu_{s_{2}}}\cdot
j^{(s_{2})}_{\nu_{1}..\nu_{s_{2}}}(p_{2})\big{)}\rangle$ (50)

Where
$(z_{i}^{a})^{\mu_{1}..\mu_{s_{i}}}=(z_{i}^{a})^{\mu_{1}}...(z_{i}^{a})^{\mu_{s_{i}}}$
are the polarization vectors corresponding to $\vec{p_{i}}$, and $a=\pm$ is
the helicity sign. One can now use the expressions given in Appendix C.2 of
Baumann:2020dch (54), to rewrite the correlators as products of spinors. The
expressions for two- and three-point functions simplify considerably,
highlighting the elegance of spinor-helicity variables for massless fields.
The epsilon-transform relation (9) takes a particularly simple form:

$\displaystyle\begin{aligned} (z^{a})^{\mu_{1}..\mu_{s}}\cdot({\epsilon}\cdot
j^{(s)})_{\mu_{1}..\mu_{s}}(p)&=-{\epsilon}_{\rho\nu(\mu_{1}}\frac{p^{\nu}(z^{a})^{\mu_{1}}...(z^{a})^{\mu_{s}}}{|\vec{p}|}\big{(}j^{(s)}\big{)}^{\rho}_{\mu_{2}..\mu_{s})}(p)\\\
&=-\frac{ia|\vec{p}|(z^{a})_{\rho}(z^{a})^{\mu_{2}}...(z^{a})^{\mu_{s}}}{|\vec{p}|}\big{(}j^{(s)}\big{)}^{\rho}_{\mu_{2}..\mu_{s})}(p)=-iaj^{(s)}_{a}.\end{aligned}$
(51)

Hence, the epsilon-transform in spinor-helicity variables acts as a phase.
Note that, regardless of helicity, applying the epsilon transform two times
consecutively just produces a minus sign. As a simple illustration, we present
the different helicity components of the spin-1 current two-point function:

$\displaystyle\langle
j^{(1)}_{\mu}(p_{1})j^{(1)}_{\nu}(p_{2})\rangle_{FF}=\tilde{N}\frac{p_{1\mu}p_{1\nu}-g_{\mu\nu}p_{1}^{2}}{16p_{1}}$
(52) $\displaystyle\langle
j^{(1)}_{a}j^{(1)}_{b}\rangle_{FF}=\frac{\tilde{N}}{16p_{1}}\bigg{(}(z_{1}^{a}\cdot
p_{1})(z_{2}^{b}\cdot p_{1})-p_{1}^{2}(z_{1}^{a}\cdot z_{2}^{b})\bigg{)}$ (53)
$\displaystyle\langle
j^{(1)}_{a}j^{(1)}_{b}\rangle_{odd}=(z_{1}^{a})^{\mu}(z_{2}^{b})^{\nu}\langle({\epsilon}\cdot
j^{(1)})_{\mu}j^{(1)}_{\nu}\rangle_{FF}=-ia\langle
j^{(1)}_{a}j^{(1)}_{b}\rangle_{FF}$ (54)

On shell, $\vec{p}_{2}=-\vec{p}_{1}$, and therefore
$\vec{z}_{2}^{\pm*}=\vec{z}_{1}^{\pm}$. We use the identities found in
Baumann:2020dch (54) and replace the scalar products in (53) by spinor
products. It is straightforward to see that mixed helicity components vanish.
Putting together the QF two-point function (11), we find the two non-zero
components:

$\displaystyle\langle
j^{(1)}_{+}j^{(1)}_{+}\rangle=\frac{\tilde{N}}{256}\bigg{(}1-ig_{1}(\tilde{\lambda})\bigg{)}\frac{\langle\bar{1}\bar{2}\rangle^{2}}{p_{1}};\qquad\langle
j^{(1)}_{-}j^{(1)}_{-}\rangle=\frac{\tilde{N}}{256}\bigg{(}1+ig_{1}(\tilde{\lambda})\bigg{)}\frac{\langle
12\rangle^{2}}{p_{1}}$ (55)

The result can be generalized to spin-$s$ currents, leading to (14).

Finally, let’s lay out our convention for the spinor-helicity currents by
obtaining the sign of the two-point contact term from the results of Gur-
Ari:2012lgt (43). There, the momentum was chosen to be in the $x^{3}$
direction, and light-cone variables were used, with light-cone
currents171717We purposefully write light-cone indices as $m,p$ to distinguish
them from the spinor-helicity indices $\mp$.
$j^{(1)}_{m,p}\equiv\tfrac{1}{\sqrt{2}}\left(j^{(1)}_{1}\pm\imath
j^{(1)}_{2}\right)$. In these variables the two-point function of spin-$1$
currents was computed explicitly:

$\displaystyle\langle j_{m}(-q)j_{p}(q)\rangle$
$\displaystyle\propto-\tilde{N}|q|-\imath\tilde{N}\tilde{\lambda}q,$ (56)

where the second term is a contact term. Given the direction of the momentum,
we can identify the current in spinor-helicity variables:

$\displaystyle j_{+}(q)=\begin{cases}j_{p}(-q),&q>0\\\
j_{m}(-q),&q<0\end{cases}$ (57)

This gives (say for $q<0$)

$\displaystyle\langle j_{+}(q)j_{+}(-q)\rangle$ $\displaystyle=\langle
j_{m}(-q)j_{p}\rangle\propto\tilde{N}\left(1-\imath\tilde{\lambda}\right)q,$
(58)

as in section 2.

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