CERN-TH-2023-067
ZMP-HH/23-6

On Higher-Spin Points and Infinite Distances
in Conformal Manifolds

Florent Baume1,2 and José Calderón-Infante3
1 Department of Physics and Astronomy, University of Pennsylvania

Philadelphia, PA 19104, U.S.A.

2 II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee
149,

22607 Hamburg, Germany

3 Theoretical Physics Department, CERN,

CH-1211 Geneva 23, Switzerland

<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

Distances in the conformal manifold, the space of CFTs related by marginal
deformations, can be measured in terms of the Zamolodchikov metric. Part of
the CFT Distance Conjecture posits that points in this manifold where part of
the spectrum becomes free, called higher-spin points, can only be at infinite
distance from the interior. There, an infinite tower of operators become
conserved currents, and the conformal symmetry is enhanced to a higher-spin
algebra. This proposal was initially motivated by the Swampland Distance
Conjecture, one of pillars of the Swampland Program. In this work, we show
that the conjecture can be tackled using only methods from the conformal
toolkit, and without relying on the existence of a weakly-coupled gravity
dual. Via conformal perturbation theory combined with properties of
correlators and of the higher-spin algebra, we establish that higher-spin
points are indeed at infinite distance in the conformal manifold. We make no
assumptions besides the usual properties of local CFTs, such as unitarity and
the existence of an energy-momentum tensor. In particular, we do not rely on a
specific dimension of spacetime (although we assume $d>2$), nor do we require
the presence of supersymmetry.

###### Contents

  1. 1 Introduction
    1. 1.1 Sketch of the Proof
  2. 2 Distances from Conformal Perturbation Theory
    1. 2.1 Spinning Conformal Correlators
    2. 2.2 Conformal Perturbation Theory
    3. 2.3 A Criterion for Finite Versus Infinite Distance
  3. 3 Higher-Spin Points are at Infinite Distance
    1. 3.1 Conservation Laws and Higher-Spin Correlators
    2. 3.2 Higher-Spin Conserved Currents and Marginal Operators
    3. 3.3 Weakly-broken Higher-Spin Symmetries
    4. 3.4 Infinite Distance and Exponentially Conserved Currents
  4. 4 Flavor Enhancements at (In)Finite Distances
  5. 5 Conclusions
  6. A Current-Conservation Conditions
  7. B Surface Integrals and Ward Identities

##  1 Introduction

Exactly marginal operators play a special rôle in local Conformal Field Theory
(CFT): one can use them to deform the original theory without breaking
spacetime symmetries and reach infinite families of CFTs related by the tuning
of associated coupling constants. These span the so-called conformal manifold,
and are not uncommon in four dimensions and lower. Supersymmetry is however
usually required in order to protect marginal operators against unwanted
quantum corrections and maintain a vanishing beta-function, see e.g.
references [1, 2, 3, 4] for early works.

While there are examples of non-supersymmetric conformal manifolds in two
dimensions related to the compact free boson and its toroidal orbifold
generalizations, to the best of our knowledge all well-established examples in
higher dimensions are supersymmetric. Furthermore, the possible types of
supersymmetry-preserving exactly marginal deformations being severely
restricted by representation theory, supersymmetric conformal manifolds cannot
exist in dimensions greater than four, see reference [5] for an exhaustive
analysis. Recently, there has been a growing effort to study conformal
manifolds via various techniques in supersymmetric theories [6, 7, 8, 9, 10,
11, 12, 13], as well as with methods not relying on such additional structures
[14, 15, 16, 17, 18, 19, 20].

Conformal manifolds can allow for special points where a sector of the CFT
decouples from the rest of the spectrum and becomes free. At these points, an
infinite tower of higher-spin operators become conserved currents, and the
conformal symmetry enlarges to an infinite-dimensional higher-spin
algebra.111By higher-spin operators, we will always mean operators
transforming in the $\ell$-traceless-symmetric representation of the Lorentz
group. These symmetry-enhanced points, dubbed higher-spin (HS) points, can
tell us much about the underlying physics, as it was shown that the converse
is also true: the presence of a higher-spin symmetry always implies that at
least part of the theory is free. First shown in $d=3$ by Maldacena and
Zhiboedov [21], this theorem was swiftly generalized to arbitrary dimensions
[22, 23, 24, 25, 26, 27]. Moreover, a direct corollary is that the presence of
a single higher-spin conserved current is a sufficient condition to obtain
infinite tower of such operators, as it is essentially a consequence of the
closure of the commutation relations of the algebra generators. Hunting for an
operator becoming a conserved current in a certain limit can therefore help us
find dual descriptions in terms of a weakly-coupled sector.

A conformal manifold comes furthermore equipped with the Zamolodchikov metric,
defined from the two-point function of the marginal operators. The properties
of correlators in unitary theories then ensure that it has the expected
properties of a metric, and enables one to define a notion of distance between
different CFTs related by marginal deformations.

Given both the potential presence of symmetry-enhanced points and the ability
to measure distances between theories, it is worthwhile to find universal
features of conformal manifolds, perhaps relating them. For instance, one can
ask about those CFTs that lie at infinite distance from generic points. It is
then natural to expect that they correspond to physically-distinguished
theories. Such special cases occur, for example, at points where there is a
symmetry enhancement. In particular, one may wonder whether there is a
connection between infinite distances and higher-spin enhancements in the
conformal manifold.

Through the AdS/CFT correspondence, questions about universal properties of
CFTs are very much in the spirit of the Swampland Program [28], see e.g.
references [29, 30, 31, 32] for reviews. In the bulk, the conformal manifold
maps to the space of vacuum expectation values of massless scalar fields, i.e.
the moduli space of the gravity theory. Within the scope of the Swampland
Program, considerable effort has been recently spent with the goal of probing
and studying the structure of these spaces, and a number of conjectures
pointing to universal behaviors have been formulated. On the field-theory
side, this can serve as a guide to find universal properties of conformal
manifolds.

For instance, with the advent of the AdS/CFT correspondence the importance of
HS points was promptly recognized, for instance in the duality between vector
models and gravitational theories of higher-spin fields [33] and its
generalizations, see e.g. reference [34] for a review. Moreover, the breaking
of this symmetry in $\mathcal{N}=4$ super-Yang–Mills can be understood
holographically as realizing the Pantagruelic Higgs mechanism (“La Grande
Bouffe”) responsible for giving a mass to higher-spin fields in the
$\text{AdS}_{5}\times S^{5}$ bulk theory [35, 36, 37].

In the same vein, there is a prominent Swampland conjecture pointing to
universal behaviors of infinite-distance points in the moduli space: the
Swampland Distance Conjecture [38]. It states that as one approaches a point
that is at infinite distance from the interior, an infinite tower of states
become massless, and does so exponentially fast with the distance, i.e.
$M_{\text{tower}}\sim e^{-\alpha_{G}\cdot\text{dist}}$ in Planck units. These
points are further expected to occur as special limits, and are related either
to a decompactification, where the tower corresponds Kaluza–Klein modes, or a
limit for which a fundamental string becomes weakly coupled, a proposal called
the Emergent String Conjecture [39].

The Swampland Distance Conjecture enjoys a large body of evidence in string
theory compactifications leading to flat space backgrounds, in particular in
supersymmetric setups [38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 39, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60]. Despite the amount of support in favor of
the conjecture, one could worry that it is merely coming from two lamppost
effects: string theory and supersymmetry. It is however challenging to go
beyond these regimes due to the lack of a general framework describing any
theory of quantum gravity, and the lack of technical tools in non-
supersymmetric setups.

A promising way out is the holographic approach. This framework is _a priori_
independent of string theory, and there are powerful, non-perturbative,
approaches in the conformal toolkit to go beyond supersymmetry. In this way,
not only can the Swampland Program serve as guidance for questions related to
conformal field theories, but methods such as the conformal bootstrap can help
in the gathering of evidence for Swampland conjectures going beyond lamppost
effects. This “holographic Swampland Program” has in fact already bore fruits
when applied to the absence of global symmetries in quantum gravity [61, 62]
the Weak Gravity Conjecture [63, 64, 65, 66, 67, 68, 69, 70, 71], moduli
stabilization, and the scale-separation problem [72, 73, 74, 75, 76, 77, 78].

For the Distance Conjecture this approach originated in references [79, 80].
There, it was found that in large classes of supersymmetric CFTs (SCFTs) the
Swampland Distance Conjecture is naturally realized in the conformal manifold
as one approaches a higher-spin point. The prototypical example is that of
four-dimensional $\mathcal{N}=2$ SCFTs, for which marginal couplings
correspond to complexified gauge parameters. Around the weak-coupling point,
$\tau\to i\infty$, the Zamolodchikov metric turns out to be approximately
hyperbolic, $\chi_{\tau\bar{\tau}}\sim(\text{Im}\tau)^{-2}$ [81]. The geodesic
distance growing logarithmically with $\tau$, the free point is at infinite
distance. Combined with the fact that HS currents have an anomalous dimension
given at one loop by
$\gamma=\Delta-\Delta_{\text{free}}\sim(\text{Im}\tau)^{-1}$, they become
conserved exponentially fast with the Zamolodchikov distance.

This process is natural in supersymmetric gauge theories and leads to the
intriguing possibility that it is in fact the general mechanism behind the
Swampland Distance Conjecture from an holographic perspective. In reference
[80], this expectation was encapsulated in the CFT Distance Conjecture.

##### CFT Distance Conjecture:

given the conformal manifold of a local CFT in $d>2$,

  * Conjecture I: all higher-spin (HS) points are at infinite distance;

  * Conjecture II: all CFTs at infinite distance are HS points;

  * Conjecture III: the anomalous dimensions of the higher-spin operators becoming conserved currents at the HS point go to zero exponentially fast with the geodesic distance:

$\gamma=\Delta-\Delta_{\text{free}}\sim
e^{-\alpha_{\chi}\text{dist}_{\chi}}\,.$ (1.1)

The distances are computed with respect to the Zamolodchikov metric, and the
coefficient $\alpha_{\chi}$ is expected to be related to the central charge of
the theory, $C_{T}$. In particular, $\sqrt{C_{T}}\,\alpha_{\chi}$ should at
least of order one [79, 80]. By local CFT, we further mean theories whose
spectrum include the energy-momentum tensor. This condition is very relevant
from the Swampland perspective, since it translates to dynamical gravity in
the bulk. Moreover, this conjecture relies on the conformal manifold being
parameterized by exactly marginal operators, which is required to have a well-
defined notion of distance in terms of the Zamolodchikov metric. For instance,
there are theories without local energy-momentum tensor that preserve
conformal invariance for any value of a continuous parameter, but that do not
have an associated local marginal operator, see e.g. references [82, 83].

Furthermore, note that the conjecture does not claim that any CFT should have
a conformal manifold. For instance, in five and six dimensions, there are no
supersymmetry-preserving marginal deformations, and it is believed that all 5d
and 6d CFTs are supersymmetric isolated points. In addition, it is not
required that all conformal manifolds must contain infinite-distance points.
There are indeed known cases of compact conformal manifolds [10], and
superpotential deformations in $\mathcal{N}=1$ four-dimensional SCFTs are
expected to always lead to finite distances [3, 8, 80]. Due to the close
resemblance between 4d $\mathcal{N}=1$ and 3d $\mathcal{N}=2$ SCFTs, it is
further predicted that conformal manifold of three-dimensional theories should
always be compact [80].

In two dimensions the status of the conjecture is unclear, as both the global
conformal groups and the Virasoro algebra always admits higher-spin currents.
Moreover, there also exists HS algebras that are finitely generated. A
possible extension of the CFT Distance Conjecture in two dimensions could come
from the fact that limiting points where the scalar gap vanishes, $\Delta\to
0$, are at infinite distance [80]. Indeed, it was similarly conjectured that
the converse is also true [84, 85]. Due to these complications, we will
however restrict ourselves to cases where the spacetime dimension is $d>2$.

As discussed above, the evidence for the conjecture mainly comes from SCFTs.
Despite this, as stated the CFT Distance Conjecture does not make any
assumption on the existence of supersymmetry nor does it differentiate between
dimensions. This suggests that the mechanism behind this feature of conformal
manifolds should not rely on supersymmetry, regardless of whether there are
conformal manifold associated with non-supersymmetric CFTs. If this is to be
the case, there should exist a way to prove at least part of the conjecture in
a way that is independent of the dimension and without assuming additional
ingredients. This is quite reminiscent of many of the tools used in the study
of CFTs. A celebrated example is the set of equations constraining the form of
the conformal block expansion of four-point functions—the keystone of the
modern conformal bootstrap—which vary continuously with the dimension [86,
87]. Supersymmetry then merely imposes additional selection rules between the
blocks. In the context of the holographic Swampland program, a general result
in this direction can be found in references [88, 89], where infinite distance
limits have been linked to the factorization of correlators in the theory. For
instance, this happens at HS points that contain a scalar operator saturating
the unitarity bound, $\Delta=(d-2)/2$, and thus corresponding to a free scalar
field.

In this work, we will show that it is indeed possible to prove for Conjecture
I in such a way, i.e., only assuming that the CFT is local and possesses a
conformal manifold. The crux of our argument relies on the constraints imposed
by HS symmetry on correlators involving higher-spin currents.

We will use that, away from a given reference CFT, the changes in the physical
data can be tracked in terms of conformal perturbation theory [90]. For
instance, scaling dimensions can be obtained from three-point functions
involving a relevant operator deforming the theory. Such techniques have been
for instance utilized to explore the space of theories beyond the conformal
regime and (re)discover new fixed points, see e.g. references [91, 92, 93, 94,
95, 96] for recent advances in that direction.

Here, we will focus on exactly marginal deformations to travel in the
conformal manifold, where changes in the conformal dimensions away from a
reference point are encoded in three-point functions involving marginal
operators. Deforming away from the HS point we can then compute anomalous
dimensions at leading order and relate them to the Zamolodchikov distance in a
way that does not require a Lagrangian description, or a precise microscopic
description of the marginal operators, only their existence.

###  1.1 Sketch of the Proof

As our proof will invoke conformal correlators involving higher-spin
operators, as well as sometimes technical properties of higher-spin algebras
and the induced conservation relations, we now summarize the steps needed to
prove the first part of the CFT Distance conjecture, namely that all higher-
spin (HS) points are at infinite distance in the conformal manifold.

The relevant data, the conformal dimension $\Delta_{\ell}$ of spin-$\ell$
operators $J_{\ell}$, is encoded in their two-point functions. Choosing a
trajectory corresponding to a coordinate $t$ in the conformal manifold
$\mathcal{M}$, one can use conformal perturbation theory to show that
$\Delta_{\ell}(t)$ is controlled by the following differential equation [17]:

$\frac{d\Delta_{\ell}}{dt}=-C_{JJ\mathcal{O}}(t)\,,$ (1.2)

where $C_{JJ\mathcal{O}}$ is a specific combination of the coefficients
appearing in the correlator $\left<J_{\ell}J_{\ell}\mathcal{O}\right>$. Since
the coefficients of the three-point functions are related to those appearing
in Operator Product Expansions (OPEs), we will often use a shorthand, and
refer to $C_{JJ\mathcal{O}}$ as the “OPE combination”.

The behavior of the anomalous dimensions near a reference point of
$\mathcal{M}$ therefore depends on how $C_{JJ\mathcal{O}}$ varies in that
neighborhood. In particular, if for any marginal operator $\mathcal{O}$ with
properly normalized two-point function, we have
$C_{JJ\mathcal{O}}\sim\gamma^{1+a}$ with $a\geq 0$ close to a HS point, then
the above differential equation implies the anomalous dimension vanishes at
$\text{dist}_{\chi}\sim t\to\infty$. That is, the HS point is at infinite
distance from any other point.

To show this we introduce an expansion parameter $g$, related to the
coordinate $t$, measuring the breaking of the conservation equation for the HS
currents—that is, the divergence no longer vanishes:

$\partial\cdot J_{\ell}=g\,K_{\ell-1}\,,$ (1.3)

where $K_{\ell-1}$ is a spin-$(\ell-1)$ operator that depend on the
microscopic details of theory, and has conformal dimension $\Delta=d-1+\ell$
at the higher-spin point. This will allow us to compute the OPE combination as
a perturbative series in $g$ of the form

$C_{JJ\mathcal{O}}=C_{JJ\mathcal{O}}^{(0)}+C_{JJ\mathcal{O}}^{(1)}\,g+C_{JJ\mathcal{O}}^{(2)}\,g^{2}+\cdots\,.$
(1.4)

Most importantly, this can be rewritten in terms of quantities evaluated at
the HS point using _Anselmi’s trick_ [97]: by taking different numbers of
divergences on $\left<J_{\ell}J_{\ell}\mathcal{O}\right>$, and employing the
broken Ward identity given in equation (1.3), each coefficient in the
expansion above gets naturally related to three-point functions involving
different numbers of the operator $K_{\ell}$, _evaluated at the higher-spin
point_ :

$C_{JJ\mathcal{O}}=C_{JJ\mathcal{O}}^{\text{HS}}+C_{JK\mathcal{O}}^{\text{HS}}\,g+C_{KK\mathcal{O}}^{\text{HS}}\,g^{2}+\cdots\,.$
(1.5)

Here $C_{JK\mathcal{O}}$ and $C_{KK\mathcal{O}}$ denote certain linear
combinations of the OPE coefficients appearing in the correlators
$\left<J_{\ell}K_{\ell-1}\mathcal{O}\right>$ and
$\left<K_{\ell-1}K_{\ell-1}\mathcal{O}\right>$, respectively.

Using the constraints imposed by HS symmetry on these correlators, we will
show that $C_{JJ\mathcal{O}}^{\text{HS}}=0=C_{JK\mathcal{O}}^{\text{HS}}$ for
any marginal operator. The first non-trivial contribution to
$C_{JJ\mathcal{O}}$ therefore comes at order $g^{2}$ or higher. An application
of Anselmi’s trick on the two-point functions of $J_{\ell}$ is furthermore
well known to show that at leading order we have the relation $\gamma\sim
g^{2}$, and we finally obtain that in the neighborhood of a HS point, we must
indeed have:

$C_{JJ\mathcal{O}}\sim\gamma^{1+a}\quad\text{ with }a\geq 0\,,$ (1.6)

for any marginal operator. We then conclude that any point of the conformal
manifold with a higher-spin symmetry must be at infinite distance.

The rest of this work fills in the details and is organized as follow: in
Section 2, we review the structure of two- and three-point correlators
involving higher-spin operators as well as conformal perturbation theory, and
give a criterion to decide whether a point of the conformal manifold is at
finite or infinite distance. In Section 3 we show that HS points are at
infinite distance following the procedure sketched above, relegating the more
technical computations to the appendices. In Section 4 we briefly discuss the
fate of points where a new flavor conserved current appear in the spectrum. We
give our conclusions in Section 5.

##  2 Distances from Conformal Perturbation Theory

To characterize limiting behavior in the conformal manifold, we need to
understand how the relevant conformal data changes under marginal
deformations. For our purpose this is the conformal dimension $\Delta_{\ell}$
of higher-spin operators, as HS points are characterized by a tower of
operators saturating unitarity bounds. As we will review in this section, the
changes are encoded in an evolution equation relating the conformal dimension
to the coefficients of a particular set of three-point functions. This is
possible due to the severe constraints imposed by conformal symmetry, which
completely fixes the kinematic structure of the three-point functions up to
these coefficients.

In this section, we review those constraints, as well as how correlators vary
under conformal perturbation theory. Working in the neighborhood of a
reference point of the conformal manifold, this will then enable us to find a
criterion allowing us to decide whether a point is at finite or infinite
distance.

###  2.1 Spinning Conformal Correlators

The central objects of this work are higher-spin (HS) operators $J_{\ell}$
becoming conserved currents at the HS point. Away from this special point they
develop an anomalous dimension and are no longer conserved, but remain
conformal primaries transforming in the $\ell$-traceless-symmetric
representation of the $d$-dimensional Lorentz group, i.e. spin-$\ell$
operators. When studying such operators, it is convenient to use an index-free
notation by contracting all Lorentz indices with polarization vectors,
$\xi^{\mu}$:222For the sake of clarity, we will often drop the dependence of
the operators on their polarization vectors when there are no ambiguities,
leaving it implicit, e.g. $J_{\ell}(x)=J_{\ell}(x,\xi)$.

$J_{\ell}(x,\xi)=J_{\ell,\,\mu_{1}\dots\mu_{\ell}}\,\xi^{\mu_{1}}\cdots\xi^{\mu_{\ell}}\,.$
(2.1)

The structure of correlators involving higher-spin operators was studied in
detail in references [98, 99], where it was shown that the tensor structures
allowed by conformal invariance can be constructed out of simple building
blocks. For instance, the two-point function of two spinning operators is
constrained to have the following form:

$\left<J_{\ell}(x_{1},\xi_{1})J_{\ell}(x_{2},\xi_{2})\right>=\frac{H_{12}(x_{12};\xi_{1},\xi_{2})^{\ell}}{|x_{12}|^{2\tau}}\,,\quad$
(2.2)

where we defined $x_{ij}=x_{i}-x_{j}$, and we use a convention where the
denominator is given in terms of the twist:

$\tau=\Delta_{\ell}-\ell\,.$ (2.3)

In this case, there is only a single kinematic function, $H_{12}$, respecting
conformal covariance. It is related to the well-known spin-one projector
appearing e.g. in scattering amplitudes involving gauge bosons, and its
explicit expression can be found in equation (B.1). Note that we will be
working in an orthonormal basis for a given spin $\ell$, thereby fixing the
prefactor of the two-point functions to one.

On the other hand, three-point functions involving _a priori_ different spins,
which we will refer to as a type-$(\ell_{1},\ell_{2},\ell_{3})$ correlators,
have more than a single kinematic structure allowed by symmetry, and can be
constructed out of six different building blocks [99]. In this work, we will
only deal with type-$(\ell,\ell,0)$ and type-$(\ell,\ell-1,0)$ correlators,
leading to a simplification of the allowed structures, as they can be
constructed out of only on three building blocks. These include the
aforementioned kinematic function $H_{12}$, as well two additional blocks,
denoted $V_{1}\,,V_{2}$. Their precise form in terms of spacetime coordinates
and polarization vectors will be relevant only in a few technical steps
performed in the appendices. In the main text, we will only need to know that
they allow us to construct a basis of conformally-covariant structures, and we
have therefore relegated their definitions to Appendix B.

As we will see shortly, a key rôle will be played by the three-point functions
of two higher-spin operators and a marginal operator $\mathcal{O}$. This is a
type-$(\ell,\ell,0)$ correlator, generically taking the following form:

$\left<J_{\ell}(x_{1},\xi_{1})J_{\ell}(x_{2},\xi_{2})\mathcal{O}(x_{3})\right>=\sum_{n=0}^{\ell}C_{JJ\mathcal{O}}^{\,n}\,\Theta_{n}(x_{i};\xi_{1},\xi_{2}).$
(2.4)

There are $\ell+1$ independent conformal structures $\Theta_{n}$ that depend
both on Lorentz coordinates and the polarization vectors of the two higher-
spin operators. In terms of the building blocks, these conformally-covariant
functions are given by

$\Theta_{n}(x_{i};\xi_{1},\xi_{2})=\frac{H_{12}^{\ell-n}\big{(}V_{1}V_{2}\big{)}^{n}}{|x_{12}|^{\tau_{123}}|x_{13}|^{\tau_{132}}|x_{23}|^{\tau_{231}}}\,,\qquad\tau_{ijk}=\tau_{i}+\tau_{j}-\tau_{k}\,.$
(2.5)

The numbers $C_{JJ\mathcal{O}}^{\,n}\in\mathbb{R}$ are related to the Operator
Product Expansion (OPE) and therefore depend on the microscopic details of the
theory under consideration, and we often refer to them as the OPE
coefficients.

We will also be led to consider three-point functions of a marginal operator
and two different operators $J_{\ell}$ and $K_{\ell-1}$ of adjacent spins. For
those correlators of type $(\ell,\ell-1,0)$, one finds a very similar
expression to that of equation (2.4):

$\left<J_{\ell}(x_{1},\xi_{1})K_{\ell-1}(x_{2},\xi_{2})\mathcal{O}(x_{3})\right>=\sum_{n=0}^{\ell-1}C_{JK\mathcal{O}}^{\,n}\,\widetilde{\Theta}_{n}(x_{i};\xi_{1},\xi_{2})\,,$
(2.6)

where there are now $(\ell-1)$ independent conformal structures given in terms
of the building blocks by:

$\widetilde{\Theta}_{n}(x_{i};\xi_{1},\xi_{2})=\frac{H_{12}^{\ell-n-1}V_{1}^{n+1}V_{2}^{n}}{|x_{12}|^{\tau_{123}}|x_{13}|^{\tau_{132}}|x_{23}|^{\tau_{231}}}\,,\qquad\tau_{ijk}=\tau_{i}+\tau_{j}-\tau_{k}\,.$
(2.7)

Note that as these two cases illustrate, contrary to the usual intuition
coming from three-point functions of scalar operators where everything is
fixed up to a single coefficient, there can be more than one when dealing with
spinning correlators. As we will see below, there can however be relations
between them in a given theory, and this will be one of the ingredients of our
proof. For instance, conservation conditions reduce the number of independent
conformal structures, leading to constraints on the OPE coefficients

We close by commenting on the special case where $d=3$. The conformal
structures that we have reviewed are parity even. For $d\geq 4$ these are all
the structures that can appear in these correlators, but in three dimensions
there are $\ell$ and $\ell-1$ extra parity-odd structures allowed to appear in
$\left<J_{\ell}J_{\ell}\mathcal{O}\right>$ and
$\left<J_{\ell}K_{\ell-1}\mathcal{O}\right>$, respectively. Nevertheless, they
will not play any rôle in our analysis. First, parity-odd structures will not
appear in conformal perturbation theory applied to conformal dimensions, and
second, there is no mixing between parity-odd and parity-even structures in
the constraints imposed by HS symmetry. We will come back to this point in
sections 2.2 and 3.1.

###  2.2 Conformal Perturbation Theory

As we will now review, the usefulness of two- and three-point functions
reveals itself in the study of conformal perturbations [90], allowing one to
track the evolution of the conformal data as the parameters of the theory are
tuned. In particular, since exactly marginal deformations do not break
conformal invariance and span the conformal manifold $\mathcal{M}$, we can use
conformal perturbation theory around a reference point of $\mathcal{M}$ to, at
least formally, follow the modifications of the conformal data as a function
of the couplings.

Placing ourselves at a given point, $t^{i}\in\mathcal{M}$ we can ask how the
conformal data behaves under small variations. If a Lagrangian description is
available, this amounts to consider the deformation

$\mathcal{L}_{t+\delta t}=\mathcal{L}_{t}+\delta t^{i}\mathcal{O}_{i}\,,$
(2.8)

and a standard perturbation analysis can be performed. More generally, we need
not make any special assumptions about the reference theory—in particular, it
is not necessary to assume deformations around a Gaussian fixed point produced
from a free theory. We can also consider non-Lagrangian cases by promoting
coupling constants to background sources and work directly with the effective
action $W[t^{i}(x)]$.

In a $d$-dimensional CFT, given a set of exactly marginal operators
$\mathcal{O}_{i}$ and their associated coupling constants $t^{i}$, the
conformal manifold is endowed with the Zamolodchikov metric, defined as the
coefficient of the two-point function of the marginal operators:333More
precisely, the marginal operators $\mathcal{O}_{i}$ give a basis of the
conformal manifold tangent bundle, and $\chi_{ij}$ are the components of the
metric in such a basis. Notice that this might not be a coordinate basis. For
instance, it can be taken to be an orthonormal basis in which
$\chi_{ij}=\delta_{ij}$.

$\left<\mathcal{O}_{i}(x_{1})\mathcal{O}_{j}(x_{2})\right>_{t}=\frac{\chi_{ij}(t)}{|x_{12}|^{2d}}\,,\qquad\Delta_{i}(t)=d\,.$
(2.9)

where $\left<\cdots\right>_{t}$ indicates that the correlator is calculated at
a given point of the conformal manifold. We stress that by marginal operator,
we mean an operator whose conformal dimension is strictly fixed to that of
spacetime _throughout the conformal manifold_ and never becomes (ir)relevant
at any point of $\mathcal{M}$, regardless of the mechanism protecting it
against quantum corrections.

By unitarity, $\chi_{ij}$ has the properties required to be a _bona fide_
metric, and the Zamolodchikov metric allows us to define distances between two
points along a trajectory $\Gamma\subset\mathcal{M}$:

$\text{dist}_{\chi}=\int_{\Gamma}ds\,,\qquad
ds^{2}=\chi_{ij}\,dt^{i}dt^{j}\,.$ (2.10)

This is the distance used in the CFT Distance Conjecture. Denoting the
parameter of a trajectory by $t$ with a slight abuse of notation, the
coordinates are given by $t^{i}(t)$, and under small variations we have:

$\delta t^{i}\,\mathcal{O}_{i}=\delta
t\,\mathcal{O}\,,\quad\mathcal{O}=\frac{dt^{i}}{dt}\,\mathcal{O}_{i}\,.$
(2.11)

This makes clear that each trajectory is related to a specific combination of
marginal operators, understood as a smooth vector field in the tangent bundle
of the conformal manifold. Reversing the logic, a result that is valid for any
marginal operator holds for any trajectory in the conformal manifold. This is
relevant for our analysis, as we aim to obtain arguments that do not depend on
the details of a given family of CFTs and that are independent of the
trajectory that we consider. Furthermore, from the previous equation we see
that a reparametrization of the trajectory amounts to a change in the
normalization of $\mathcal{O}$. In particular, this means that if
$\mathcal{O}$ is chosen to be properly normalized, then $t$ is nothing but the
distance traveled along the trajectory in terms of the Zamolodchikov metric.

As previously alluded to, we are interested in the conformal dimensions of
spin-$\ell$ operators and their behavior along a trajectory in the conformal
manifold as a function of the distance. In other words, we want to find the
conformal dimension as a function of $t$: $\Delta_{\ell}=\Delta_{\ell}(t)$.
This dependence is encoded in the two-point function
$\left<J_{\ell}J_{\ell}\right>$. Given a reference point, at first order in
perturbation theory the variation induced by a change $\delta t$ on this
correlator is given by:

$\left\langle J_{\ell}(x_{1})J_{\ell}(x_{2})\right\rangle_{t+\delta
t}=\left\langle J_{\ell}(x_{1})J_{\ell}(x_{2})\right\rangle_{t}+\delta
t\left.\int d^{d}y\left\langle
J_{\ell}(x_{1})J_{\ell}(x_{2})\mathcal{O}(y)\right\rangle_{t}\right|_{\text{reg.}}\,.$
(2.12)

In a Lagrangian theory, equation (2.12) can be understood from the path
integral formulation, but can also be obtained directly from a variation of
the effective action $W[t^{i}(x)]$. Note that the integral must be regularized
to take care of the divergences appearing as $y\to x_{1},x_{2}$, which induces
a renormalization of the operators $J_{\ell}$ to obtain a well-defined
quantity. This procedure is well defined, see e.g. reference [100], and can
also be understood holographically [101, 102].

Up to this point we have only used perturbation theory, but not that the
deformation is exactly marginal and therefore that the resulting two-point
functions remain constrained by conformal invariance. The variation of the
correlator must thus take the generic form:

$\left\langle J_{\ell}(x_{1})J_{\ell}(x_{2})\right\rangle_{t+\delta
t}-\left\langle
J_{\ell}(x_{1})J_{\ell}(x_{2})\right\rangle_{t}=-\frac{H(x_{12})^{\ell}}{|x_{12}|^{2\tau}}\,2\,\delta\Delta_{\ell}\log|x_{12}|\,,$
(2.13)

with $\delta\Delta_{\ell}$ the change in the conformal dimension induced by
$\delta t$.444In general there could also be another term related to the
change of the prefactor in front of the two-point function, i.e., the
normalization of the spin $\ell$ operator. We however take this operator to be
in an orthonormal basis at any point in the conformal manifold. This
corresponds to a choice of renormalization scheme, and will not influence out
computations at the order we consider, see e.g. reference [20] for a detailed
discussion. This shift is therefore encoded in the logarithmic divergence
obtained after integration of the three-point function given in equation
(2.12).

The integration over the three-point function in terms of the conformal
structure of type-$(\ell,\ell,0)$ correlators defined in equation (2.4) must
be consistent with (2.12), and must lead to the same type of divergences. This
means that the shift of $\Delta_{\ell}$ is controlled by a certain linear
combination of the OPE coefficients, $C^{\,}_{JJ\mathcal{O}}$:

$\delta\Delta_{\ell}=-\delta t\,C_{JJ\mathcal{O}}+\mathcal{O}(\delta
t^{2})\,,\qquad
C_{JJ\mathcal{O}}=\sum_{n=0}^{\ell}w_{n}C_{JJ\mathcal{O}}^{\,n}\,.$ (2.14)

This was shown in reference [17], where the renormalization procedure was
carefully performed by combining the embedding formalism with dimensional
regularization. The coefficients $w_{n}$—which we recall here for
completeness—were found to be given by:

$w_{n}=2\pi^{\frac{d}{2}}\,\frac{\Gamma(n+1)}{\Gamma(n+\frac{d}{2})}\,.$
(2.15)

Their precise values will not be relevant to us beyond the fact that, at first
order in conformal perturbation theory, the change in $\Delta_{\ell}$ is
encoded in a particular linear combination of the type-$(\ell,\ell,0)$ OPE
coefficients, which we refer to as the OPE combination $C_{JJ\mathcal{O}}$.

Importantly, equation (2.14) is true at any point in the conformal manifold,
which allows one to exponentiate it to its differential version [19]. This
gives us an evolution equation for the conformal dimensions in the conformal
manifold as we change $t$:

$\frac{d\Delta_{\ell}}{dt}=-\,C_{JJ\mathcal{O}}(t)\,.$ (2.16)

A possible cross-check for the above equation is that it transforms
covariantly under reparameterization of the marginal coupling $t$ discussed in
equation (2.11), and consequently takes into account changes in the
normalization for the marginal operator $\mathcal{O}$.

As commented in Section 2.1, the parity-odd structures that can appear in
$d=3$ do not contribute to equation (2.16). After integrating over the
position of the marginal operator, the leftover function of $x_{1}$ and
$x_{2}$ is again parity odd. This is inconsistent with (2.13), which is parity
even, and they therefore cannot contribute to the variation of
$\Delta_{\ell}$.

Knowing the OPE combination $C_{JJ\mathcal{O}}(t)$, we can thus in principle
reconstruct $\Delta_{\ell}$ at every point of the trajectory specified by
$\mathcal{O}$ via the evolution equation (2.16), and measure the distance
traveled with respect to the Zamolodchikov metric. However, this information
of course is not readily available. Usually, one may try to extract
information about $C_{JJ\mathcal{O}}(t)$ by also using conformal perturbation
theory. For instance, at first order the evolution of the three-point function
$\left\langle J_{\ell}J_{\ell}\mathcal{O}\right\rangle$ is encoded into the
four-point function $\left\langle
J_{\ell}J_{\ell}\mathcal{O}\mathcal{O}\right\rangle$, leading to a similar
differential equation for $C_{JJ\mathcal{O}}(t)$ [16, 19, 20]. However, due to
presence of this four-point function, after imposing the usual associativity
constraints, one is required to have knowledge about the conformal data of
operators appearing in the various OPEs involving $\mathcal{O}$ and
$J_{\ell}$, substantially complicating the analysis. Furthermore, when
applying this approach one should worry about the radius of convergence of
conformal perturbation theory, since it amounts to reconstruct information of
the whole conformal manifold starting from a single point by using conformal
perturbation theory. Here we take a different approach, one more suited to
distinguishing between finite and infinite distance points in the conformal
manifold.

###  2.3 A Criterion for Finite Versus Infinite Distance

From here on, we will focus our attention on the neighborhood of a point where
certain spin-$\ell$ operators have dimension $\Delta_{\ell}^{\ast}$—or
similarly for a locus defined by
$\\{\Delta_{\ell}(t)=\Delta_{\ell}^{\ast}\\}$. As in the previous section, we
also consider a trajectory specified by a marginal operator $\mathcal{O}$,
understood as a smooth vector field in the neighborhood of the point under
consideration. One can think of different choices of $\mathcal{O}$ as
different ways of approaching this point, as depicted in Figure 2.1. In
particular, drawing conclusions about any marginal operator in this
neighborhood is equivalent to finding properties of any trajectory approaching
this point in any possible direction.

Figure 2.1: Different trajectories, corresponding to distinct marginal
operators $\mathcal{O}$ and $\mathcal{O}^{\prime}$, in a neighborhood of the
conformal manifold $\mathcal{M}$ leading to the same point with conformal
dimension $\Delta^{\ast}$.

The idea behind our criterion to characterize whether a point is at (in)finite
distance from a generic point in the interior of $\mathcal{M}$ is then the
following: if one is able to obtain $C_{JJ\mathcal{O}}$ as a function of
$\Delta_{\ell}$ close to the point under consideration, one can recover the
behavior of the conformal dimension as a function of $t$ in that neighborhood
using the evolution equation (2.16). As argued above, when the two-point
function of $\mathcal{O}$ is chosen in an orthonormal basis, $t$ is then
interpreted as the Zamolodchikov distance traveled along the trajectory
specified by $\mathcal{O}$ itself. We can then use the behavior of
$\Delta_{\ell}(t)$ as it approaches $\Delta_{\ell}^{\ast}$ to learn whether
this point is at finite or infinite distance along the given trajectory.

Notice however that one can only obtain non-trivial information about the
traveled distance when the conformal dimension varies along the trajectory. We
hence need to pick a spin-$\ell$ operator whose conformal dimension is not
fixed to a specific value over the whole conformal manifold. For instance, the
energy-momentum tensor, conserved flavor currents, and protected BPS operators
in supersymmetric theories have this property, and are therefore excluded from
this analysis. We will thus assume that the higher-spin operators are
conserved only at specific points or loci of the conformal manifold.

To illustrate this idea let us start with the most trivial—but also most
generic— example, namely the case where the OPE combination can be considered
constant in the neighborhood defined by $\Delta^{\ast}_{\ell}$:
$C_{JJ\mathcal{O}}\to C_{JJ\mathcal{O}}^{*}\neq 0$ as
$\Delta_{\ell}\to\Delta_{\ell}^{*}$. Defining the change in conformal
dimension as $\gamma_{\ell}$,555By abuse of notation, in this section we use
$\gamma_{\ell}=\Delta_{\ell}-\Delta_{\ell}^{\ast}$ for the deviation of the
conformal dimension with respect to an arbitrary point of the conformal
manifold in analogy to the usual anomalous dimension
$\gamma=\Delta-\Delta_{\text{free}}$. When the reference point corresponds to
a free theory, $\Delta^{\ast}=\Delta_{\text{free}}$, as will be the case in
the next sections, the two quantities coincide. the evolution equation (2.16)
reduces to:

$\frac{d\gamma_{\ell}}{dt}=-\,C_{JJ\mathcal{O}}^{\,*}+\mathcal{O}(\gamma_{\ell})\,,\qquad
C_{JJ\mathcal{O}}^{\,*}=\text{const}\,,\qquad\gamma_{\ell}=\Delta_{\ell}-\Delta_{\ell}^{\ast}\,.$
(2.17)

Therefore, the leading behavior in the change of conformal dimension is given
simply as a linear approximation

$\gamma_{\ell}(t)\simeq\gamma_{\ell}^{0}-C_{JJ\mathcal{O}}^{\,*}\,t\,,$ (2.18)

where we have fixed the integration constant in terms of the deviation at the
point where we put the origin of the trajectory, namely
${\gamma}_{\ell}(t=0)=\gamma_{\ell}^{0}$. Of course, this approximation is
only valid when the deviation $\gamma_{\ell}(t)$ is sufficiently small.

We have therefore obtained the solution to equation (2.16) close to
${\gamma}_{\ell}=0$ when $C_{JJ\mathcal{O}}$ takes a non-zero value at this
point. Importantly, the solution given in equation (2.18) reaches
$\gamma_{\ell}=0$ for finite values of $t$. In other words, this point is at
finite distance from the initial point $t=0$ along the trajectory defined by
$\mathcal{O}$.

To summarize, we see that if at some point in the conformal manifold we have
$C_{JJ\mathcal{O}}^{*}\neq 0$ for a given marginal operator $\mathcal{O}$,
then we have found a trajectory reaching this point within finite distance.
Furthermore, finding a single trajectory for which this is true is enough to
imply that the point under consideration is a finite-distance point in the
conformal manifold.

Let us now give an example for which the point is reached at infinite
distance. If the OPE coefficients grows linearly with the deviation,

$C_{JJ\mathcal{O}}({\gamma}_{\ell})\simeq\alpha_{\chi}\,{\gamma}_{\ell}\,,\qquad\gamma_{\ell}=\Delta_{\ell}-\Delta^{\ast}_{\ell}\,,$
(2.19)

for some constant $\alpha_{\chi}>0$, the evolution equation discussed above is
solved by an exponential. The behavior of the deviation as it reaches zero is
of the form:

${\gamma}_{\ell}={\gamma}_{\ell}^{0}\,e^{-\alpha_{\chi}\,t}\,,$ (2.20)

with the integration constant again fixed by
${\gamma}_{\ell}(t=0)={\gamma}_{\ell}^{0}$. As $t$ is again interpreted as the
distance in the conformal manifold, we have the case of a trajectory with the
exponential behavior typical of the CFT Distance Conjecture, see equation
(1.1). Note that ${\gamma}_{\ell}\to 0$ for $t\to\infty$ regardless of the
point where we place $t=0$, which means that this point is at infinite
distance along the trajectory defined by the marginal operator $\mathcal{O}$.
However, this does not mean that we have an infinite-distance point. Indeed,
it could be that $\mathcal{O}$ is associated with a highly-non-geodesic
trajectory, but there exists another trajectory for which the point can be
reached within a finite distance.

To characterize whether a point where the conformal dimension is given by
$\Delta_{\ell}^{\ast}$ is reached at finite distance or not along certain
trajectory, let us consider a general case where $C_{JJ\mathcal{O}}$ is
parameterized by:666This parametrization is motivated by a perturbative
description valid around the point under consideration, in which both
$C_{JJ\mathcal{O}}$ and ${\gamma}_{\ell}$ enjoy an expansion in powers of a
small parameter. For instance, this will be the case when we discuss HS points
in later sections.

$C_{JJ\mathcal{O}}({\gamma}_{\ell})\sim{\gamma}_{\ell}^{1+a}\qquad\text{ as
}{\gamma}_{\ell}=\Delta_{\ell}-\Delta^{\ast}_{\ell}\to 0\,.$ (2.21)

Close to the reference point, and up to subleading corrections, the deviation
from $\Delta_{\ell}^{*}$ then behaves as:

$t\sim\,\left\\{\,\begin{aligned} \log\,{\gamma}_{\ell}\,,\quad&\text{if
}a=0\,,\\\ {\gamma}_{\ell}^{-a}\,,\quad&\text{if }a\neq 0\,.\\\
\end{aligned}\right.$ (2.22)

For the same reasons as above, $t$ measures the distance from the distance
from an arbitrary point encoded in the integration constant. The case $a<0$
corresponds to a finite distance, while $a\geq 0$ is at infinite distance. For
$a=0$ we have not only the critical case between finite and infinite distance,
but also the one where the deviation ${\gamma}_{\ell}$ decreases exponentially
fast with the distance, that is the one associated with the third part of the
CFT Distance Conjecture explained in the introduction. This is depicted in
Figure 2.2.

Once again, we stress that if $a\geq 0$, this does not mean that the point at
$t=0$ is an infinite-distance point, but could be associated with a non-
geodesic trajectory and reach the point in an seemingly infinite-distance
fashion. In order to ensure that it is truly an infinite-distance point, we
need to show that it is the case for _any_ trajectory. Conversely, a marginal
operator inducing a trajectory where a point is at finite distance is
sufficient to imply it is a finite-distance point, albeit it is maybe not
attained via the shortest path. We have therefore found the following
criterion.

##### CFT Distance Criterion:

In a neighborhood of a point of a conformal manifold $\mathcal{M}$ where
higher-spin operators have conformal dimension $\Delta_{\ell}^{\ast}$, the
behavior of the OPE combination $C_{JJ\mathcal{O}}$ defined in equation (2.14)
as $\gamma_{\ell}=\Delta_{\ell}-\Delta_{\ell}^{\ast}\to 0$ is sufficient to
decide whether it is a:777Even though the criterion is tailored to the
behavior given in equation (2.21) close to the reference point, a similar
analysis can be performed for other parametrization.

  * -

Finite-distance point: $\exists\,\mathcal{O}\text{ such that
}C_{JJ\mathcal{O}}\sim{\gamma}_{\ell}^{1+a}\text{ with }a<0\,,$

  * -

Infinite-distance point: $C_{JJ\mathcal{O}}\sim{\gamma}_{\ell}^{1+a}\text{
with }a\geq 0\ \forall\mathcal{O}\,.$

(a) Finite distance

(b) Infinite distance

Figure 2.2: Two different trajectories associated with a marginal operator in
the neighborhood of a point with dimension $\Delta^{*}$. If the OPE
combination $C_{JJ\mathcal{O}}(t)$ remains approximately constant in the
neighbourhood of the point, it is a finite-distance point. On the other hand,
if $C_{JJ\mathcal{O}}\sim(\Delta-\Delta^{*})^{1+a}$ with $a\geq 0$, it is at
infinite distance from any other point.

Note that this criterion does not depend on the precise nature of the operator
$J_{\ell}$, and at no point did we assume a specific value of $\ell$, and
should be valid for an conformal primary. To apply it, we only need an
operator whose conformal dimension is not constant along the whole trajectory.
Therefore, any operator for which this reasoning is applicable should give us
the same answer. In particular, there cannot be an operator placing a point at
finite distance while another predicts an infinite-distance behavior. This
implies non-trivial relations between three-point functions of the form
$\langle J_{\ell}J_{\ell}\mathcal{O}\rangle$ and the possible marginal
operators for the conformal manifold to make sense as a metric space.

In this section, we have discussed how to differentiate between finite- and
infinite-distance points of the conformal manifold, which typically requires
understanding the behavior of $C_{JJ\mathcal{O}}$ close to that point for one
or potentially even all marginal operators of the theory, an information that
may be challenging to access in general. Despite this hurdle, we can draw an
interesting observation: any point where $C_{JJ\mathcal{O}}$ is non-vanishing
for any marginal operator is a finite-distance point. Conversely, a vanishing
OPE combination is a necessary, albeit not sufficient, condition to have an
infinite-distance point.

##  3 Higher-Spin Points are at Infinite Distance

Having a criterion to decide whether a point of the conformal manifold
$\mathcal{M}$ is at infinite distance, we now apply the procedure described in
the previous section to higher-spin (HS) points. There, (part) of the HS
operators become conserved currents and the conformal group enhances to a
larger, infinite-dimensional higher-spin symmetry.

In order to apply the CFT Distance Criterion to show that HS points are at
infinite distance, we will assume that a given HS operator is never conserved
everywhere in the conformal manifold, as this can happen only if it is part of
a free subsector that is completely decoupled from the rest of the theory at
all points of $\mathcal{M}$. Moreover, following the discussion at the end of
the previous section, it is in principle enough to show that HS points are at
infinite distance for an operator with e.g. spin $\ell=4$; there is indeed
always a spin-four conserved current at any HS point [21, 22, 23, 24, 25].
While we could in principle set $\ell=4$ from now on, we will keep the spin
arbitrary and specialize to the spin-four case for illustrative purposes.

###  3.1 Conservation Laws and Higher-Spin Correlators

At the HS point, the anomalous dimension of the higher-spin operator vanish
and its conformal dimension saturates the unitarity bound for spin-$\ell$
operators: $\Delta_{\ell}=d-2+\ell$. As is well known in Conformal Field
Theory, when this happens the associated conformal multiplet develops a null
state corresponding to a conservation condition:

$\partial\cdot
J_{\ell}=\partial^{\mu}J_{\ell,\,\mu\nu_{1}\dots\mu_{\ell}}\,\xi^{\nu_{2}}\dots\xi^{\nu_{\ell}}=0\,.$
(3.1)

This identity can be used to constrain correlators involving these conserved
currents, as their divergences have to vanish up to contact terms. We will
often refer to this as the _current-conservation condition_ , and it will
allow us to reduce the number of conformal structures that can appear in a
given correlator. Starting with general type-$(\ell_{1},\ell_{2},\ell_{3})$
correlators and taking the divergence with respect to a HS operator, the
current conservation conditions will impose constraints on its OPE
coefficients. An important observation is that the null states associated with
the divergence of conserved currents transform as spin-$(\ell-1)$ conformal
primaries, see e.g. reference [99]. The current-conservation condition of a
type-$(\ell_{1},\ell_{2},\ell_{3})$ correlator with respect to the first
current is therefore equivalent to the vanishing of a
type-$(\ell_{1}-1,\ell_{2},\ell_{3})$ three-point function. As was already
realized in reference [21], since both the type-$(\ell_{1},\ell_{2},\ell_{3})$
and type-$(\ell_{1}-1,\ell_{2},\ell_{3})$ correlators can be written in terms
of the same building blocks, see Section 2.1, we can instead write the
divergence as a differential operator acting on the space of these building
blocks. This considerably simplifies an otherwise rather cumbersome
computation. The precise form of this operator can be found in reference
[103], and we use it for cases of interest in Appendix A.

As already mentioned in Section 2.1, we note that parity-even and -odd
structures do not mix in the current-conservation condition, as divergences
preserve parity. To verify this, one can observe that the spin changes by one
when taking the divergence, and that spin-$\ell$ operators acquires a sign
$(-1)^{\ell}$ under parity. This—together with the fact that parity-odd
structures do not contribute to equation (2.16)—allows us to ignore all
allowed parity-odd $(d=3)$-structures for our purposes.

By Noether’s theorem, the HS conserved currents also have associated conserved
charges by integrating them over a codimension-one surface $\Sigma$. Note
that, contrary to the usual flavor charges, higher-spin charges have spin and
we again use polarization vectors to contract the extra-indices. We then
obtain a formal one-form that can be used to construct the charge:888 A more
general construction involves contracting the $(\ell-1)$ remaining indices
with a conformal Killing tensor $\zeta$ satisfying
$\partial^{(\mu}\zeta^{\nu_{1}\dots\nu_{\ell-1})}=0$. For our purpose however,
it will be sufficient to consider constant polarization vectors, which forms a
subclass of those tensors. We defer to reference [23] for a review of the
construction of the space of conformal Killing tensors and the associated
charges.

$Q_{\ell}(\xi)=\int_{\Sigma}\check{J}_{\ell}(\xi)\,,\qquad\check{J}_{\ell}(\xi)=J_{\ell,\,\mu\nu_{1}\dots\nu_{\ell-1}}\,\xi^{\nu_{1}}\dots\xi^{\nu_{\ell-1}}dx^{\mu}\,.$
(3.2)

While the textbook approach to conserved charges involves integrating over the
$(d-1)$ space coordinates, the usefulness of more general surfaces and induced
topological properties of the charges have long been recognized, and have
recently come in full force in the context of higher-form symmetries [104].
Note however that the charges $Q_{\ell}(\Sigma,\xi)$ have spin $\ell-1$ and
act on (possibly-spinning) local operators and the currents transform in
$\ell$-symmetric-traceless representations of the Lorentz group. While they do
generalize the notion of ordinary spin-one flavor charges, they do not
correspond to the type of generalized symmetries discussed in reference [104],
which act on extended objects and the involved currents transform in _anti_
-symmetric spacetime representations.

The conserved charges can be also used to constrain correlators involving HS
conserved currents. Indeed, when inserted in a correlator, we must take into
account contact terms—that is, contributions at coincident points:

$\left<(\partial\cdot
J_{\ell})(y,\xi)\Phi_{1}(x_{1})\cdots\Phi_{n}(x_{n})\right>\propto\sum_{k=1}^{n}\delta(y-x_{k})\left<\Phi_{1}(x_{1})\cdots[Q(\xi),\Phi_{k}(x_{k})]\cdots\Phi_{n}(x_{n})\right>\,,$
(3.3)

up to a constant related to normalization, and the operators $\Phi_{n}$ are
possibly spinning. Integrating over a $d$-dimensional submanifold of spacetime
containing only a single position $x_{k}$, we can therefore select a given
commutator and reduce the number of operators appearing in the correlator. On
the other hand, using Stokes’ theorem on the LHS of equation (3.3), we can
integrate the current, understood as a formal one-form as in equation (3.2),
over the codimension-one surface $\Sigma$ bounding the submanifold. The
charges are of course topological and we can deform the surface as long we are
not crossing over another point, in which case we must take into account the
appropriate commutator. We will refer to this as the _integrated Ward
identity_ and it will allow us to relate the OPE coefficients in a given
correlator through properties of the HS algebra we review below.

Given the conserved charges, there is a further set of identities that one can
impose on correlators to constrain not only their OPE coefficients but also
how the charges can act on different operators. These are particularly
interesting, since they allow us to impose the coexistence of HS symmetry with
other ingredients of the theory like, e.g. the presence of the energy-momentum
tensor. In fact, these constraints were crucial for the original derivation of
the theorems relating HS points to free theories [21, 22, 23, 24, 25].
Following reference [21], we will call them _charge-conservation identities_.
Given a set of operators, the idea is to consider the correlator of their
commutator with $Q_{\ell}$. Since the conserved charge must annihilate the
vacuum, this correlator must vanish. This is:

$0=\left<[Q_{\ell}(\xi),\Phi_{1}(x_{1})\cdots\Phi(x_{n})]\right>=\sum_{k=1}^{n}\left<\Phi_{1}(x_{1})\cdots[Q_{\ell}(\xi),\Phi_{k}(x_{k})]\cdots\Phi_{n}(x_{n})\right>\,.$
(3.4)

This can also be obtained from equation (3.3) by integrating over the entire
of spacetime or any other submanifold that does not have a boundary, and
containing all the positions $x_{k}$. Using Stokes’ theorem, the result must
vanish.

Using the HS algebra to further expand each commutator in equation (3.4) as a
sum over operators appearing in the various HS multiplets, one can infer a set
of constraints involving both the parameters controlling the action of
$Q_{\ell}$ on the operators and their OPE coefficients. In what follows, we
review some of the properties of the HS algebra, give an example of such a
charge-conservation identity, and its consequences.

The commutation relations of the charges must close and generate the full
higher-spin algebra. In general, the action of the charges on an operator of
twist $\tau=\Delta-\ell$ will involve derivatives of the various fields
falling inside the same higher-spin multiplet so that the total twist is
conserved. Tracking the index structure of such commutation relations can
however be cumbersome, and it is instead very convenient to focus on a
particular spacetime direction in order to simplify the index structure.
Following reference [21], we will work in light-cone coordinates:

$ds^{2}=dx^{+}dx^{-}+\delta_{ij}dx^{i}dx^{j}\,,\qquad i,j=2,\dots,d-1\,,$
(3.5)

and use polarization vectors forcing the indices to be in the
$(x^{-})$-direction: $\xi^{\mu}=\delta^{\mu}_{-}$.999In terms of group theory,
this corresponds to consider the celebrated non-compact $\mathfrak{sl}_{2}$
derivative subsector, where the higher-weight states are the $\ell$-traceless-
symmetric conformal primaries. The only relevant generator actions are then
associated with derivatives of the fields in the $x^{-}$direction. To
differentiate HS operators in this regime from those with arbitrary
polarizations, written with uppercase letters, e.g. $J_{\ell}(x,\xi)$, we will
use lowercase letters to indicate spinning operators whose indices all point
in that direction. For instance:

$j_{\ell}=J_{\ell,-\dots-}\,.$ (3.6)

The charges are then defined in terms of the currents with the indices in that
direction

$Q_{\ell}=\int_{\Sigma}dx^{-}d^{d-2}x\;\;j_{\ell}(x)\,,$ (3.7)

where now $\Sigma$ is taken as a “slab” of spacetime in the directions
perpendicular to $x^{+}$. For simplicity, we will henceforth drop the
dependence on the polarization of the charges whenever the context is clear:
$Q_{\ell}(\delta^{\mu}_{-})=Q_{\ell}$.

First, since the charges have twist zero, each term appearing in commutators
involving $Q_{\ell}$ and an operator $k_{\ell^{\prime}}$ of twist $\tau$ must
also have twist $\tau$. Moreover, these can only involve twist-$\tau$
operators $k_{\ell^{\prime\prime}}$ with all indices in the minus direction as
well as $x^{-}$-derivatives, $\partial=\frac{\partial}{\partial x^{-}}$. Due
to this index structure, each derivative increases the spin by one but the
twist remains unchanged. The most general formula for such a commutator is
then:

$[Q_{\ell},k_{\ell^{\prime}}]=\sum_{\ell^{\prime\prime}=\text{max}(\ell^{\prime}-\ell+1,0)}^{\ell+\ell^{\prime}-1}\eta_{\ell^{\prime\prime}}\,\partial^{\ell+\ell^{\prime}-1-\ell^{\prime\prime}}k_{\ell^{\prime\prime}}\,.\\\
$ (3.8)

As discussed above, the coefficients $\eta_{\ell}$ can be constrained—together
with correlators—using the charge-conservation identities, and lead to a
number of useful relations [21, 22, 23, 24, 25]. The simplest example is
obtained by acting with $Q_{\ell}$ on the two-point function
$\left<k_{\ell^{\prime}}k_{\ell^{\prime\prime}}\right>$. From this, one can
show that if $k_{\ell^{\prime\prime}}$ appears in
$[Q_{\ell},k_{\ell^{\prime}}]$ then $k_{\ell^{\prime}}$ must appear in
$[Q_{\ell},k_{\ell^{\prime\prime}}]$.101010One might worry that, for $\tau\geq
d$, there may be extra terms in the commutator in equation (3.8) of the form
$\Box^{n}\mathcal{V}_{\ell^{\prime\prime}}$ with
$\mathcal{V}_{\ell^{\prime\prime}}$ a twist-$(\tau-2n)$ operator. However,
through a similar argument applied to the correlator
$\left<k_{\ell^{\prime}}\mathcal{V}_{\ell^{\prime\prime}}\right>$, one can
show that such terms cannot appear.

Moreover, since we consider local theories, there is a spin-two conserved
current that generates conformal transformations, the energy-momentum tensor
$T_{\mu\nu}$. Additional constraints can be obtained by specifically
considering charge-conservation identities on correlators involving
$t_{2}=T_{--}$, as they enable us to further constrain the appearance of a
given operator in commutators given in equation (3.8). For instance, since
$t_{2}$ is related to the dilatation operator and charges have conformal
dimension $\ell-1$, $j_{\ell}$ must always appear in $[Q_{\ell},t_{2}]$. Then,
using this fact and charge-conservation identities, one can further show that
not all _a priori_ possible conserved currents appear in $[Q_{\ell},t_{2}]$.
When $\ell=4$, one finds that the most general form of the commutator is given
by:

$[Q_{4},t_{2}]=\alpha_{0}\,\partial^{5}j_{0}+\alpha_{1}\,\partial^{4}j_{1}+\alpha_{2}\,\partial^{3}j_{2}+\alpha_{3}\,\partial^{2}j_{3}+\alpha_{4}\,\partial
j_{4}\,.$ (3.9)

By the argument above, $j_{4}$ must appear in the commutator and thus
$\alpha_{4}\neq 0$. Notice that we do not need to assume that the energy-
momentum tensor $t_{2}$ is the unique spin-two conserved current. This is,
$j_{2}$ is a _generic_ spin-two conserved current, that can include
contributions from both $t_{2}$, and possibly any additional spin-two
conserved current. When $t_{2}$ is the unique spin-two conserved current, not
all coefficients $\alpha_{\ell}$ with $\ell=0,\cdots,5$ are non-zero, and it
is possible to show that $\alpha_{1}=0=\alpha_{3}$ [21].

Restricting the possible operators appearing the various commutators will
prove very powerful. It will allow us to show that some of the three-point
functions involving marginal operators must vanish, and consequently will be
crucial to prove that HS point are at infinite distance.

###  3.2 Higher-Spin Conserved Currents and Marginal Operators

Following the CFT Distance Criterion discussed in Section 2.3, a necessary
condition for a HS point to be at infinite distance is that the OPE
combination must vanish for any marginal operator at this point:
$C_{JJ\mathcal{O}}^{\text{HS}}=0$. We will thus now shift our focus to three-
point functions $\left<J_{\ell}J_{\ell}\mathcal{O}\right>$. Our goal is
therefore to show that these correlators are always trivial, regardless of the
chosen marginal operator.

To do so, we will first use the current-conservation condition on this
correlator. This will impose constraints on the various OPE coefficients,
reducing the number of allowed conformal structure to a single one. By further
using an integrated Ward identity, we will then show that the remaining _a
priori_ independent parameter must vanish.

Let us start by imposing conservation of the higher-spin currents, see
equation (3.1), on this three-point function. Up to contact terms, we have:

$\partial_{2}\cdot\left<J_{\ell}(x_{1})J_{\ell}(x_{2})\mathcal{O}(x_{3})\right>=\left<J_{\ell}(x_{1})(\partial\cdot
J_{\ell})(x_{2})\mathcal{O}(x_{3})\right>=0\,,$ (3.10)

where $\partial_{i}$ is understood as taking the divergence with respect to
the operator located at $x_{i}$. As discussed in Section 3.1, the LHS can be
conveniently evaluated using a differential operator in the space of building
blocks $H$, $V_{1}$, and $V_{2}$. Its expression, as well as the result for
this current-conservation condition, can be found in Appendix A. Here, we will
simply give a qualitative description of the procedure needed to show that at
the higher-spin point, this three-point function and the related OPE
combination vanish.

As we have seen in Section 2.1, for a fixed value of $\ell$ there are $\ell+1$
independent type-$(\ell,\ell,0)$ conformal structures, as defined in equation
(2.5). Taking the divergence as in equation (3.10) we then obtain a linear
combination of conformal structures of type $(\ell,\ell-1,0)$. Since there are
only $\ell$ such structures, demanding that the resulting three-point function
vanish leads to a set of $\ell$ linear constraints on the $\ell+1$ OPE
coefficients $C_{JJ\mathcal{O}}^{\,n}$. If these $\ell$ constraints are
independent, we can express all the OPE coefficients in terms of a single one,
and the solution takes the form:

$C_{JJ\mathcal{O}}^{\,n}=v^{n}\,C_{JJ\mathcal{O}}^{\,\ell}\,,\qquad
n=0,\dots,\ell\,,$ (3.11)

where the components of the vector, denoted as $v^{n}\in\mathbb{R}$, are
normalized such that $v^{\ell}=1$. This is found by showing that the current-
conservation constraint in equation (3.10) can be viewed as a recursion
relation in the space of OPE coefficients $C_{JJ\mathcal{O}}^{\,n}$. In
general, solving this for arbitrary values of $\ell$ is difficult, but not
particularly illuminating. It is on the other hand straightforward to do it on
a case-by-case basis. For instance, when $\ell=4$ the coefficients are given
by:

$v^{n}=\left(\frac{-24}{(d-2)d(d+2)(d+6)}\,,\,\frac{48(d+3)}{(d-2)d(d+2)(d+6)}\,,\,\frac{72}{(d-2)d(d+6)}\,,\,\frac{-8(d+7)}{(d-2)(d+6)}\,,\,1\right)\,.$
(3.12)

However, the specific values of $v^{n}$ will not be relevant for our analysis,
only that the current-conservation constraint reduces the number of allowed
structure from $(\ell+1)$ to only one. The details of this computation can be
found in Appendix A.

To see the impact of this constraint on the conformal-perturbation-theory
equations, let us write the OPE combination at the HS point after imposing the
Ward identity (3.10):

$C^{\text{HS}}_{JJ\mathcal{O}}=C_{JJ\mathcal{O}}^{\ell}\,\sum_{n=0}^{\ell}w_{n}v^{n}\,.$
(3.13)

While one could hope that $\sum_{n=0}^{\ell}w_{n}v^{n}=0$, directly showing
that $C_{JJ\mathcal{O}}^{\text{HS}}=0$, this is unfortunately generically not
the case. We have checked this explicitly for $4\leq\ell\leq 10$ in any
dimension by using the expression of coefficients $w_{n}$, given in equation
(2.15), and those of $v^{n}$, obtained by solving the recursive relation
obtained in Appendix A.111111For $\ell=2,$ and $\ell=3$, it turns out that
this quantity does vanish for $d=4$ and $d=14$, respectively. In fact, it is
hard to believe that there is a given spin $\ell$ for which it vanishes for
any dimension. The strategy is then to further show that the yet-undetermined
OPE coefficient $C_{JJ\mathcal{O}}^{\,\ell}$ has to vanish. Doing so will
establish that $C^{\text{HS}}_{JJ\mathcal{O}}=0$—as expected—in a way that
does not depend on the precise values of $v^{n}$ and $w_{n}$.

To show that this is the case, we now turn to the integrated Ward identity,
which allows us to reduce the correlator to a two-point function. Fixing the
coordinates $x_{2}$ and $x_{3}$, we can use a version of the Ward identity
(3.3) applied to our case and integrate the current over a submanifold only
containing $x_{2}$, and we obtain:

$\int_{\Sigma}\left\langle\check{J}_{\ell}(x_{1},\xi_{1})J_{\ell}(x_{2},\xi_{2})\mathcal{O}(x_{3})\right\rangle\propto\left\langle\left[Q_{\ell}(\xi_{1}),J_{\ell}(x_{2},\xi_{2})\right]\mathcal{O}(x_{3})\right\rangle\,.$
(3.14)

The codimension one surface $\Sigma$ is the boundary of the submanifold
containing $x_{2}$. The integral is then performed with respect to the first
HS conserved current $\check{J}_{\ell}$, where the first index is in direction
normal to $\Sigma$, see equation (3.2).

As discussed above, the commutator of an operator with a conserved charge must
involve only operators of the same twist, and in this case they must therefore
all have $\tau=d-2$. However marginal operators have twist $\tau=d$, and as a
consequence there cannot be any operator appearing in the commutator so that
the two-point function on the RHS of equation (3.14) does not vanish, and we
conclude that:

$\left\langle\left[Q_{\ell},J_{\ell}(x_{2})\right]\mathcal{O}(x_{3})\right\rangle=0\,.$
(3.15)

By consistency, the integral on the LHS of equation (3.14) must also vanish.
If we choose $\Sigma$ such that it picks the component of $\check{J}$ in the
direction of $\xi_{1}$, we can then expand the correlator in terms of the
conformal structures, and using equation (3.11), we can factor out the so-far-
undetermined coefficient $C_{JJ\mathcal{O}}^{\,\ell}$ to obtain:

$\int_{\Sigma}\left\langle
J_{\ell}(x_{1},\xi_{1})J_{\ell}(x_{2},\xi_{2})\mathcal{O}(x_{3})\right\rangle=C_{JJ\mathcal{O}}^{\,\ell}\int_{\Sigma}\left(\sum_{n=0}^{\ell}v^{n}\,\Theta_{n}(x_{i};\xi_{1},\xi_{2})\right)\overset{!}{=}0\,.$
(3.16)

If we can find polarization vectors $\xi_{1},\xi_{2}$ so that the integral
gives a non-zero result, the constraint coming from the integrated Ward
identity can only be satisfied if $C_{JJ\mathcal{O}}^{\,\ell}=0$, as desired.
In Appendix B, we show explicitly that the integral indeed does not vanish for
a particular choice of polarization vectors. We therefore conclude that at any
HS point and for any marginal operator $\mathcal{O}$, the three-point function
$\left\langle J_{\ell}J_{\ell}\mathcal{O}\right\rangle$ is trivial, and by
extension so is the OPE combination relevant to the CFT Distance Criterion:

$C^{\text{HS}}_{JJ\mathcal{O}}=0\,.$ (3.17)

This result is in the end not too surprising. In fact, it is known that when
the two spins are different, the OPE coefficients vanish, see e.g. references
[98, 105]. For equal spin however, the conservation condition is not enough to
come to this conclusion, as one must use the integrated Ward identity.

For a theory of 4d free vectors, which include $\mathcal{N}=4$ super-
Yang–Mills in the free-field limit, this result can be obtained from a direct
computation by decomposing the field-strength into (anti-)self dual components
$F=F_{+}+F_{-}$. The only non-vanishing two-point function of the field-
strength is $\langle F_{+}F_{-}\rangle$, and the HS currents and marginal
operators can be written as [106]:

$J_{\ell}\sim F_{+}\partial^{\ell}F_{-}\,,\qquad\mathcal{O}\sim
F_{+}F_{+}+F_{-}F_{-}\,,$ (3.18)

where we have been very cavalier with the index structure. By Wick’s theorem,
the relevant three-point functions then directly vanish:121212We thank E.
Skvortsov for pointing out this very concise argument to us.

$\langle(F_{+}\partial^{\ell}F_{-})\,(F_{+}\partial^{\ell}F_{-})\,(F_{+}F_{+})\rangle=0=\langle(F_{+}\partial^{\ell}F_{-})\,(F_{+}\partial^{\ell}F_{-})\,(F_{-}F_{-})\rangle\,.$
(3.19)

As advertised, we have therefore found that the OPE combination always
vanishes at any HS point, $C^{\text{HS}}_{JJ\mathcal{O}}=0$. We stress that
this result is not sensible to the actual numerology of the coefficients
$w_{n}$ and $v^{n}$, but is rather a consequence of the properties of the
higher-spin symmetry.

This result shows that higher-spin points are not in the regular finite-
distance class for which $C_{JJ\mathcal{O}}\neq 0$. However, the fact that the
OPE combination must vanish is a necessary but not sufficient condition to
show that these points are at infinite distance. To complete the proof, we
require information about their neighborhood, and what happens when the
higher-spin symmetry is weakly broken.

###  3.3 Weakly-broken Higher-Spin Symmetries

In this section we move slightly away from the higher-spin point. This is, we
assume that the higher-spin symmetry is weakly broken by a small parameter $g$
(related, but not to be confused with the marginal coupling $t$ used in
conformal perturbation theory). This is, for the divergence of the higher-spin
operators we have:

$\partial\cdot J_{\ell}=g\,K_{\ell-1}\,.$ (3.20)

Here, we have introduced an operator $K_{\ell-1}$ controlling the higher-spin
symmetry breaking. This operator is a primary of twist $d$ and spin $\ell-1$
only at the HS point, i.e., at leading order in $g\to 0$ [97, 107, 108]. In
fact, the two multiplets recombine into a larger multiplet with primary
$J_{\ell}$—which is no longer conserved due to the anomalous dimension—and
$K_{\ell-1}$ is a mere descendant of the would-be current away from the
higher-spin point [109]. We take this operator to be in an orthonormal basis,
also at leading order in this limit. This in turn fixes the definition of the
higher-spin symmetry breaking parameter $g$ via equation (3.20).

Breaking the HS symmetry, the decoupled sector is not free anymore, and
$K_{\ell-1}$ can be thought of as specifying the type of interactions that are
turned on. However, we will not assume that the CFT data enjoys an expansion
in powers of $g$ around the HS point. The reason is that $g$ is not
necessarily the coupling constant of those interactions. For instance, it
could _a priori_ happen that $\partial\cdot J_{\ell}$ vanishes at tree level,
and the breaking occurs only at higher-loop orders. In this case, the CFT data
would generically have an expansion in fractional powers of $g$. Here, we will
only assume that the CFT data remains finite as the HS point is approached.
This is closely related to the normalization of operators such as $J_{\ell}$,
$K_{\ell-1}$, and $\mathcal{O}$. Normalizations in which we allow the two-
point functions to diverge as $g\to 0$ can lead to similar, unphysical,
divergences in the three-point functions. Nevertheless, this cannot happen
when operators are properly normalized. In this case the three-point-function
coefficients are physical and should remain finite in any _bona fide_ CFT like
those associated with HS points of the conformal manifold.

As pioneered in reference [97], let us now take the double divergence of the
two-point function of two higher-spin operators, and use the broken Ward
identity defined in equation (3.20):

$\partial_{1}\partial_{2}\cdot\left\langle
J_{\ell}(x_{1})J_{\ell}(x_{2})\right\rangle=g^{2}\left\langle
K_{\ell-1}(x_{1})K_{\ell-1}(x_{2})\right\rangle\,.$ (3.21)

The LHS of this expression can be directly computed using the form of the two-
point function as an expansion in terms of the conformal structures, see
equation (2.2). For small anomalous dimensions,
$\gamma_{\ell}=\tau_{\ell}-(d-2)\ll 1$, it is at leading order proportional to
a spin-$(\ell-1)$ two-point function [105, 110]. Since $K_{\ell-1}$ is a
conformal primary for $g\to 0$, the RHS is also proportional to the two-point
function of a spin-$(\ell-1)$ conformal primary in that limit. We then
conclude that, at leading order, the anomalous dimension is given by:

$\gamma_{\ell}\sim g^{2}\,,\qquad\text{as }g\to 0\,,$ (3.22)

where we are ignoring numerical prefactors whose precise values can be found
in references [105, 110]. In this way we see that the first contribution to
the anomalous dimension of the higher-spin operators always appears at order
$g^{2}$ in the higher-spin symmetry-breaking parameter. This result is
sometimes referred to as _Anselmi’s trick_ , who first used the broken Ward
identity to simplify the computations of the anomalous dimensions of the
Konishi current and other spinning operators for four-dimensional
$\mathcal{N}=4$ super-Yang–Mills theory[97].

We can apply similar arguments to the three-point function $\langle
J_{\ell}J_{\ell}\mathcal{O}\rangle$. Let us first take the divergence with
respect to the second higher-spin operator and use equation (3.20):

$\partial_{2}\cdot\left\langle
J_{\ell}(x_{1})J_{\ell}(x_{2})\mathcal{O}(x_{3})\right\rangle=g\left\langle
J_{\ell}(x_{1})K_{\ell-1}(x_{2})\mathcal{O}(x_{3})\right\rangle\,.$ (3.23)

As for the two-point function, at leading order the divergence of the currents
matches with the correlator of spinning primaries. Something similar happens
for the three-point function in the RHS, and we see that the leading-order
corrections to the current-conservation condition, which led to (3.11), are
controlled by the behavior of $\left\langle
J_{\ell}K_{\ell-1}\mathcal{O}\right\rangle$ close to the HS point.

Similarly, a weakly-broken version of the integrated Ward identity goes as
follows: because of equation (3.20), the charges defined in equation (3.2) are
no longer conserved, and as a result their correlators depend on the
codimension-one surface $\Sigma$. One can however define the action of quasi-
conserved charges on a given operator as the result of the integral when the
surface $\Sigma$ enclosing its position shrinks to zero size [108]. This
action is constrained by twist conservation as it was the case when the
symmetry was exactly preserved. By virtue of Stokes’ theorem, one then gets
integrated Ward identities similar to the conserved case, but with an extra
contribution from the correlator where $J_{\ell}$ has been replaced by
$\partial\cdot J_{\ell}=gK_{\ell-1}$, integrated over the volume enclosed by
$\Sigma$. As happens for the current-conservation condition, the leading-order
correction to the integrated Ward identity for
$\left<J_{\ell}J_{\ell}\mathcal{O}\right>$ is again controlled by the behavior
of $\left\langle J_{\ell}K_{\ell-1}\mathcal{O}\right\rangle$ close to the HS
point.

Putting these two results together and taking into account equation (3.17), we
get that for a weakly-broken HS symmetry we have

$C_{JJ\mathcal{O}}\sim g\,C_{JK\mathcal{O}}\qquad\text{ as }g\to 0\,,$ (3.24)

where $C_{JK\mathcal{O}}$ is a certain linear combination of the OPE
coefficients appearing in $\left<J_{\ell}K_{\ell-1}\mathcal{O}\right>$ as
defined in equation (2.6).

If $C_{JK\mathcal{O}}$ is non-vanishing at the HS point, we see that the OPE
combination behaves as $C_{JJ\mathcal{O}}\sim g$ when $g\to 0$. Using equation
(3.22), this would mean that at leading order, we would have
$C_{JJ\mathcal{O}}\sim\gamma^{1/2}$. Following the CFT Distance Criterion
discussed in Section 2.3, this would result in HS points being at finite
distance with respect to the trajectory defined by the marginal operator
$\mathcal{O}$. For the first part of the CFT Distance conjecture to be true,
the OPE combination $C_{JK\mathcal{O}}$ must always be trivial at the HS
point. We now show this is indeed the case.

####  3.3.1 The Correlator $\langle J_{\ell}K_{\ell-1}\mathcal{O}\rangle$ at
the HS Point

To show that at the higher-spin point the correlator $\langle
J_{\ell}K_{\ell-1}\mathcal{O}\rangle$ vanishes, we will go through a similar
procedure to the one we used for the three-point function $\langle
J_{\ell}J_{\ell}\mathcal{O}\rangle$, and begin with the current-conservation
condition. We now have to take the divergence of a type-$(\ell,\ell-1,0)$
correlator with respect to the first operator, from which we obtain a three-
point function of type-$(\ell-1,\ell-1,0)$. Using that $K_{\ell-1}$ is a
conformal primary at this point, and as explained in Section 2.1, both
correlators involve $\ell$ independent conformal structures. As before, this
enable us to find an homogeneous system of $\ell$ linear constraints for
$\ell$ variables $C_{JK\mathcal{O}}^{n}$.

If all the equations were independent, one would automatically get that
$C_{JK\mathcal{O}}^{n}=0$. Unfortunately, we have checked that the system is
not closed for $2\leq\ell\leq 10$ and for any $d$. In general, we therefore
expect that this happens for any spin and any number of spacetime dimensions.
Despite this, we show in the Appendix A that the constraints can always be
used to fix most of the coefficients, and we get a similar result as for
$C_{JJ\mathcal{O}}^{n}$:

$C_{JK\mathcal{O}}^{n}=\tilde{v}^{n}\,C_{JK\mathcal{O}}^{\ell-1}\,,\qquad
n=0,\dots,\ell-1\,,$ (3.25)

where the components of the vector, denoted by $\tilde{v}^{n}\in\mathbb{R}$,
are normalized such that $\tilde{v}^{\ell-1}=1$. In appendix A, we argue that,
similarly to what we have done for the case of
$\left<J_{\ell}J_{\ell}\mathcal{O}\right>$ above, the current-conservation
constraint imposes a recursion relation in the space of OPE coefficients
$C_{JK\mathcal{O}}^{n}$. We have not found a closed-form expression for the
coefficients $\tilde{v}^{n}$, but they can easily be solved for a given spin
programmatically. For instance, for $\ell=4$ it is straightforward to check
that the solution is given by:

$\tilde{v}^{n}=\left(\frac{-24}{(d-2)d(d+2)}\,,\,\frac{36}{(d-2)d}\,,\,\frac{-12}{d-2}\,,\,1\right)\,.$
(3.26)

Once again, the particular values of $\tilde{v}^{n}$ is not important, only
that that the current-conservation condition fixes all the coefficients but
one.

To show that this remaining coefficient is in fact trivial, we again need to
impose the integrated Ward identity. Starting with the analogue of equation
(3.3), we integrate the current over a manifold enclosing the point $x_{2}$
with boundary $\Sigma$, and we obtain:

$\int_{\Sigma}\left\langle\check{J}_{\ell}(x_{1},\xi_{1})K_{\ell-1}(x_{2},\xi_{2})\mathcal{O}(x_{3})\right\rangle\propto\left\langle\left[Q_{\ell}(\xi_{1}),K_{\ell-1}(x_{2},\xi_{2})\right]\mathcal{O}(x_{3})\right\rangle\,,$
(3.27)

and $\check{J}_{\ell}$ is the current with the first index in the direction
normal to the surface. As we did for the three-point function $\left\langle
J_{\ell}J_{\ell}\mathcal{O}\right\rangle$, taking $\Sigma$ to pick the
$\xi_{1}$-direction, the LHS can be expanded in terms of the conformal
structures. Taking into account equation (3.25), we then have:

$\int_{\Sigma}\left\langle
J_{\ell}(x_{1},\xi_{1})K_{\ell-1}(x_{2},\xi_{2})\mathcal{O}(x_{3})\right\rangle=C_{JK\mathcal{O}}^{\ell-1}\int_{\Sigma}\left(\sum_{n=0}^{\ell}\tilde{v}^{n}\widetilde{\Theta}_{n}(x_{i};\xi_{1},\xi_{2})\right)\,.$
(3.28)

As discussed in Appendix B, the same choice of polarization vector and surface
we used for $\left\langle J_{\ell}J_{\ell}\mathcal{O}\right\rangle$ again lead
to a non-vanishing integral. However, a crucial difference is that the
commutator in equation (3.27) is not trivial in this case. Indeed, due to
twist conservation, only twist $\tau=d$ operators can appear in the commutator
from the relation given in equation (3.8). Since marginal operators also have
twist $d$, it can in principle be contained in the commutator, and we cannot
conclude that the three-point function $\left\langle
J_{\ell}K_{\ell-1}\mathcal{O}\right\rangle$ vanishes as in the previous
subsection.

To do so, we need to invoke an ingredient that we have not used so far, namely
the presence of the energy-momentum tensor. Indeed, assuming a local CFT is
what makes the CFT distance conjecture part of the holographic Swampland
Program, as it implies the presence of dynamical gravity in the bulk. Note
that here we do not assume a weakly-coupled gravity dual, and will only use
the interplay of the energy-momentum tensor with the higher-spin algebra. As
alluded to in Section 3.1, it can be used to constrain the operators appearing
in the commutators involving higher-spin charges [21].

In particular, our goal is to show that
$\left<J_{\ell}K_{\ell-1}\mathcal{O}\right>\neq 0$ is inconsistent with a
charge-conservation identity. We will achieve this by considering the action
of $Q_{\ell}$ on the three-point function $\left\langle
T_{2}K_{\ell-1}\mathcal{O}\right\rangle$:

$\left<\left[Q_{\ell},t_{2}k_{\ell-1}\mathcal{O}\right]\right>=\left<\left[Q_{\ell},t_{2}\right]k_{\ell-1}\mathcal{O}\right>+\left<t_{2}\left[Q_{\ell},k_{\ell-1}\right]\mathcal{O}\right>+\left<t_{2}k_{\ell-1}\left[Q_{\ell},\mathcal{O}\right]\right>=0\,,$
(3.29)

which can be understood as integrating equation (3.3) over all of spacetime.
We recall that we use the notation—explained in Section 3.1—where lowercase
operators indicate that the indices are all taken in the $x^{-}$-direction,
e.g. $t_{2}=T_{--}$ for the energy-momentum tensor, and the HS charges are
defined as in equation (3.7).

Let us first discuss the action of one of these charges on the energy-momentum
tensor:

$\left[Q_{\ell},t_{2}\right]=\sum_{s=0}^{\ell}\alpha_{s}\,\partial^{\ell+1-s}j_{s}\,,$
(3.30)

Recall that $\partial$ denotes the partial derivative with respect to the
$x^{-}$-direction. Even though it is allowed by twist and spin conservation,
notice that we have not included $j_{\ell+1}$. The reason is that, were it to
appear in the commutator then $Q_{\ell}$ would not be translation invariant
[21]. Note that we are not assuming uniqueness of the spin two conserved
current, in general $j_{2}$ could contain contributions from both the energy-
momentum tensor $t_{2}$, and any other spin-two current.

The action of $Q_{\ell}$ on $k_{\ell-1}$ and $\mathcal{O}$ leads to similar
expressions, see equation (3.8):

$\begin{split}\left[Q_{\ell},k_{\ell-1}\right]&=\sum_{s=0}^{2\ell-2}\eta_{s}\,\partial^{2\ell-2-s}k_{s}^{\prime}\,,\\\
\left[Q_{\ell},\mathcal{O}\right]&=\sum_{s=0}^{\ell-1}\beta_{s}\,\partial^{\ell-1-s}k_{s}^{\prime}\,.\end{split}$
(3.31)

The primed operators $k_{s}^{\prime}$ are there to remind us that they might
be different from unprimed operators $k_{\ell-1}$ we have considered so far.
Furthermore, the twist-$d$ scalar operators $k_{0}^{\prime}$ might be
different from the marginal operator $\mathcal{O}$ we started with.

Plugging these commutators into equation (3.29), we then obtain:

$\sum_{s=0}^{\ell}\alpha_{s}\,\partial^{\ell+1-s}_{1}\left<j_{s}k_{\ell-1}\mathcal{O}\right>+\sum_{s=0}^{2\ell-2}\eta_{s}\,\partial^{2\ell-2-s}_{2}\left<t_{2}k^{\prime}_{s}\mathcal{O}\right>+\sum_{s=0}^{\ell-1}\beta_{s}\,\partial^{\ell-1-s}_{3}\left<t_{2}k_{\ell-1}k_{s}^{\prime}\right>=0\,,$
(3.32)

where $\partial_{i}$ denotes the partial $x^{-}$-derivative with respect to
the position of the $i$-th operator. Notice that the first sum contains a term
with $\alpha_{\ell}\neq 0$ and $\left<j_{\ell}k_{\ell-1}\mathcal{O}\right>$,
if the charge-conservation identity imposes that this term must vanish, then
we obtain the desired result.

Performing this analysis for general spin $\ell$ is quite involved, but it
becomes more manageable for $\ell=4$. In this case we have:

$\sum_{s=0}^{4}\alpha_{s}\,\partial^{5-s}_{1}\left<j_{s}k_{3}\mathcal{O}\right>+\sum_{s=0}^{6}\eta_{s}\,\partial^{6-s}_{2}\left<t_{2}k^{\prime}_{s}\mathcal{O}\right>+\sum_{s=0}^{3}\beta_{s}\,\partial^{3-s}_{3}\left<t_{2}k_{3}k_{s}^{\prime}\right>=0\,.$
(3.33)

We then have to analyze correlators of type
$\left<J_{s}K_{3}\mathcal{O}\right>$, $\left<T_{2}K_{s}\mathcal{O}\right>$ and
$\left<T_{2}K_{3}K_{s}\right>$. Using the current-conservation condition and
integrated Ward identities, we conclude that:

$\begin{split}\left<J_{s}K_{3}\mathcal{O}\right>&=0\qquad\text{for
}s=2,3\,;\\\ \left<T_{2}K_{s}\mathcal{O}\right>&=0\qquad\text{for
}s=1,\dots,6\,;\\\ \left<T_{2}K_{3}K_{s}\right>&=0\qquad\text{for
}s=0\,.\end{split}$ (3.34)

This can be shown as we did for $\left<J_{\ell}J_{\ell}\mathcal{O}\right>$:
the current conservation condition reduces the number of independent OPE
coefficients to one, and the integrated Ward identity then imposes that the
remaining OPE coefficient must vanish. For the case
$\left<T_{2}K_{1}\mathcal{O}\right>$ it is crucial to use that $T_{2}$ is the
energy-momentum tensor. Only in this case $Q_{2}$ is the charge generating
translations in the minus direction, $Q_{2}=P_{-}$, which guarantees that even
though allowed by twist conservation, $K_{1}$ does not appear in
$[Q_{2},\mathcal{O}]$. The results for the current-conservation conditions and
the integrals for the integrated Ward identities can be found in appendices A
and B.

Using this, the charge-conservation condition reduces to

$\begin{split}0=\,&\alpha_{0}\,\partial^{5}_{1}\left<j_{s}k_{0}\mathcal{O}\right>+\alpha_{1}\,\partial^{4}_{1}\left<j_{1}k_{3}\mathcal{O}\right>+\alpha_{4}\,\partial_{1}\left<j_{4}k_{3}\mathcal{O}\right>+\,\eta_{0}\,\partial^{6}_{2}\left<t_{2}k^{\prime}_{0}\mathcal{O}\right>\\\
&+\beta_{1}\,\partial^{2}_{3}\left<t_{2}k_{3}k_{1}^{\prime}\right>+\beta_{2}\,\partial_{3}\left<t_{2}k_{3}k_{2}^{\prime}\right>+\beta_{3}\,\left<t_{2}k_{3}k_{3}^{\prime}\right>\,.\end{split}$
(3.35)

The last three terms are more complicated to analyze, since the current-
conservation condition allows for more than one conformal structure. However,
including them in the charge-conservation condition becomes more manageable
when coordinates are taken in the light-cone plane,
$x_{i}=(x_{i}^{-},x_{i}^{+},\mathbf{0})$, since only one linear combination of
the conformal structures of these correlators contributes in this
case.131313It is not guaranteed that this linear combination survives the
current-conservation condition. However, that would not change our results,
but make them easier to derive. Moreover, the spacetime dependence of all the
conformal structures simplify drastically. Imposing this charge-conservation
condition for any $x_{i}^{-}$ and $x_{i}^{+}$, we find that there is no
solution if $\left<j_{4}k_{3}\mathcal{O}\right>\neq 0$, thus arriving to the
desired result:

$\left<J_{4}K_{3}\mathcal{O}\right>=0\,.$ (3.36)

This concludes the analysis of the correlator
$\left<J_{\ell}K_{\ell-1}\mathcal{O}\right>$ at the HS point. At least for the
case $\ell=4$, we have been able to show that this correlator must vanish for
any marginal operator $\mathcal{O}$. This in turn implies that

$C_{JK\mathcal{O}}^{\text{HS}}=0\,,$ (3.37)

as we wanted. Following the same procedure as for $\ell=4$, we expect the same
result to hold for $\ell>4$. In fact, this has to be the case for any other HS
conserved current coexisting with $J_{4}$ to generate the higher-spin algebra,
otherwise the conformal manifold would stop making sense as a metric space. We
come back to this point in the next section.

###  3.4 Infinite Distance and Exponentially Conserved Currents

In this section, we have so far shown that by using various properties of the
higher-spin algebra, two of the three-point functions involving marginal
operators and a HS current must always be trivial at the HS point:

$\text{HS point:}\qquad\langle J_{\ell}J_{\ell}\mathcal{O}\rangle=0=\langle
J_{\ell}K_{\ell-1}\mathcal{O}\rangle\,.$ (3.38)

Following the CFT Distance Criterion defined in Section 2.3, the first term is
a necessary condition for HS points to be at infinite distance in the
conformal manifold. On the other hand, from the discussion around equation
(3.23), the fact that second vanishes forbids contributions to the OPE
combination $C_{JJ\mathcal{O}}$ at linear order in the parameter $g$ for a
weakly-broken HS symmetry.

By the same token, one can reach a similar conclusion for
$C_{JK\mathcal{O}}(g)$ away from the HS point. Taking the divergence with
respect to $J_{\ell}$ and using equation (3.20) we have

$\partial_{1}\cdot\left<J_{\ell}(x_{1})K_{\ell-1}(x_{2})\mathcal{O}(x_{3})\right>=g\left<K_{\ell-1}(x_{1})K_{\ell-1}(x_{2})\mathcal{O}(x_{3})\right>\,.$
(3.39)

The leading order correction to the current-conservation condition for
$\left<J_{\ell}K_{\ell-1}\mathcal{O}\right>$ is controlled by
$\left<K_{\ell-1}K_{\ell-1}\mathcal{O}\right>$ at the HS point. The same
observation can be reached for the leading correction to the integrated Ward
identity and the charge-conservation condition [108]. Similarly to what we
obtained in the previous section, and taking into account equation (3.37), we
now have

$C_{JK\mathcal{O}}\sim g\,C_{KK\mathcal{O}}\qquad\text{ as }g\to 0\,,$ (3.40)

where $C_{KK\mathcal{O}}$ is again a certain linear combination of the OPE
coefficients appearing in $\left<K_{\ell-1}K_{\ell-1}\mathcal{O}\right>$.

Since both $K_{\ell}$ and $\mathcal{O}$ are assumed to be in an orthonormal
basis, the correlator $\left<K_{\ell-1}K_{\ell-1}\mathcal{O}\right>$ cannot
blow up as $g\to 0$—see the discussion below equation (3.20). In general, we
therefore have

$\left<K_{\ell-1}K_{\ell-1}\mathcal{O}\right>\sim g^{2a}\qquad\text{ with
}a\geq 0\,\text{ as }g\to 0\,.$ (3.41)

By this equation, we mean that the $\ell$ OPE coefficients of this correlator,
$C_{KKO}^{n}$, have this behavior as $g\to 0$. Combining this with equations
(3.24) and (3.40), we have

$C_{JJ\mathcal{O}}\sim g^{2(1+a)}\qquad\text{ with }a\geq 0\,\text{ as }g\to
0\,$ (3.42)

for the OPE combination controlling the conformal-perturbation-theory equation
(2.16).141414Strictly speaking, it could happen that the coefficients
$C_{JJ\mathcal{O}}^{\,n}$ conspire in such a way that leading correction
vanishes for the linear combination controlling conformal perturbation theory,
$C_{JJ\mathcal{O}}$. Equation (3.42) would then depend on another exponent
$a^{\prime}>a$, but the conclusion would be unchanged. Given the behavior of
the anomalous dimension in equation (3.22), we finally reach the conclusion

$C_{JJ\mathcal{O}}\sim\gamma_{\ell}^{1+a}\qquad\text{ with }a\geq 0\,\text{ as
}\gamma_{\ell}\to 0\,.$ (3.43)

Since we have made no assumption on the marginal operator $\mathcal{O}$, this
behavior is independent of the deformation away from the higher-spin point,
and is precisely the CFT Distance Criterion for an infinite-distance point in
the conformal manifold, as explained in Section 2.3. Furthermore, we have not
made any assumptions on the operators $J_{\ell}$ either, apart from the fact
that they become conserved at the HS point. This then shows that for local
CFTs in any dimension, the first part of the CFT Distance Conjecture is
correct:

_all higher-spin points are at infinite distance in the conformal manifold_.

Strictly speaking, in Section 3.3 we have only proven this for spin-four
currents. However, as discussed below the CFT Distance Criterion in Section
2.3, it is enough to show that a single higher-spin current places the HS
point at infinite distance in the conformal manifold. The rest of the tower
should behave accordingly, and we must also have
$\left<J_{\ell}K_{\ell-1}\mathcal{O}\right>=0$ at the HS point for any other
current. Furthermore, since there is always a spin-four conserved current
generating the higher-spin algebra [21, 22, 23, 24, 25], the case $\ell=4$
suffices to establish that any HS point is at infinite distance.

It is also interesting to elaborate more on the condition for the higher-spin
currents to become conserved exponentially with the distance. This would be
the case if $a=0$ in equation (3.43). In other words, having an exponentially
conserved HS current with the distance as the HS point is approached is linked
to having:

$\left<K_{\ell-1}K_{\ell-1}\mathcal{O}\right>_{\text{HS}}\neq 0\,.$ (3.44)

In particular, this does not mean that all the OPE coefficients
$C_{KK\mathcal{O}}^{n}$ are non-vanishing as the HS point. We only need to
require this for the OPE combination $C_{KK\mathcal{O}}$ controlling the order
$g^{2}$ correction to $C_{JJ\mathcal{O}}$. It is intriguing that the condition
for the exponential behavior precisely involves the last correction to
$C_{JJ\mathcal{O}}$ that can be addressed using Anselmi’s trick, i.e. the
weakly-broken Ward identity (3.20).

Having $\left<K_{\ell-1}K_{\ell-1}\mathcal{O}\right>\neq 0$ at the infinite
distance HS point might seem a bit puzzling. Indeed, one of the corollaries of
the CFT Distance Criterion is that having an operator with non-vanishing three
point function with $\mathcal{O}$ is generically enough to imply that the
point is at finite distance. However, let us recall that $K_{\ell-1}$ is not a
conformal primary outside the HS point, while the CFT Distance Criterion
assumes this for the operator under consideration, and equation (2.16)
therefore does not apply to the descendant $K_{\ell-1}$. The analogous
equation, obtainable through conformal perturbation theory applied to this
non-primary operator, should be such that the behavior of $C_{KK\mathcal{O}}$
close to the HS point must place it at infinite distance. In fact, in our case
this is guaranteed by the fact that, when going away from the HS point, the
current and $K_{\ell-1}$ recombine into a larger multiplet, and their behavior
as we approach the free point must be related.

If equation (3.44) is satisfied, then the exponential decay rate
$\alpha_{\chi}^{(\ell)}$ appearing in equation (1.1) is controlled by the OPE
combination $C_{KKO}$ at the HS point. Schematically, we have:

$\alpha_{\chi}^{(\ell)}\sim\left<K_{\ell-1}K_{\ell-1}\mathcal{O}\right>_{\text{HS}}\,,$
(3.45)

where the numerical coefficients and the OPE combination, $C_{KK\mathcal{O}}$,
can be fixed by the weakly-broken HS symmetry constraints. In fact, this
result can be brought into a more suggestive form. Taking an orthonormal basis
for the set of marginal operators $\mathcal{O}_{i}$ and generalizing the
evolution equation (2.16) we have

$\frac{\partial_{i}\gamma_{\ell}}{\gamma_{\ell}}\sim\left<K_{\ell-1}K_{\ell-1}\mathcal{O}_{i}\right>_{\text{HS}}\,.$
(3.46)

An orthonormal basis for the marginal operators corresponds to an orthonormal
basis for the tangent bundle of the conformal manifold. Therefore, equation
(3.46) gives the components of what is defined as the scalar charge-to-mass
ratio vector used for the convex-hull formulation of the Swampland Distance
Conjecture [111]. This equation is then telling us that, close to the HS
point, this information is encoded in this particular three-point function.

Showing the condition given in equation (3.44) for a single spin, e.g.
$\ell=4$, is not enough to show an exponential behavior for all off them.
There is _a priori_ nothing wrong with having an infinite-distance point for
which the different HS currents become conserved at different rates along a
given trajectory in the conformal manifold. The only thing that is required by
consistency of the conformal manifold as a metric space is that
$\gamma_{\ell}(t)\to 0$ as $t\to\infty$ for all of them.

Despite this, there might be a way of translating the result for a single spin
to the rest of the tower. The idea is to exploit a relation that we have not
discussed yet. Even though the HS symmetry-breaking parameter $g$ is not the
same as the distance $t$ in the conformal manifold, they are related. Indeed,
having shown the exponential behavior for the anomalous dimension of a given
spin $\ell$, and using equation (3.22) we get the relation:

$g_{\ell}^{2}\sim e^{-\alpha_{\chi}^{(\ell)}\,t}\qquad\text{ as
}t\to\infty\,.$ (3.47)

Here we have added a subscript $\ell$ to the HS symmetry breaking parameter to
make clear that _a priori_ one could expect the parameters to be different for
the breaking of the conservation condition of different HS operators. However,
coming back to the underlying picture of introducing an interaction with a
small coupling, we indeed expect them to be related in a power-like fashion.
This is, we expect each of these conservation conditions to be broken at a
given loop order. Note however that this could be avoided if the conservation
condition for some HS currents is broken only by non-perturbative effects.

Following this reasoning, the exponential behavior for a given spin would
carry over to the others, even though the exponential decay rate might still
depend on $\ell$. It would be interesting to show that, as it happens for
super-Yang–Mills theories, all the HS conservation conditions are broken at
the same order, i.e, $g_{\ell}=g$. In fact, it seems plausible that this is
imposed by consistency of the HS symmetry breaking, since this symmetry
relates the different HS currents. If this is case, one would not only
conclude that it is enough to show the exponential behavior for a single spin,
but also that the exponential rate $\alpha_{\chi}$ does not depend on $\ell$
and one has a full tower of HS currents becoming exponentially conserved at
the same rate.

##  4 Flavor Enhancements at (In)Finite Distances

In the previous section, we have used properties of the higher-spin algebra to
show that HS points are at infinite distance. Although we focused on higher-
spin currents, many aspect of this machinery turns out to also be valid for
$\ell=1$. In the conformal manifold, this corresponds to regimes where a new
conserved flavor current appears in the spectrum. These special points need
not automatically be associated with higher-spin points, nor with infinite
distances. A classical family of examples is $(\mathcal{N}=1)$-preserving
deformations of four-dimensional $\mathcal{N}=4$ super-Yang–Mills [3], where
new R-symmetry currents are accompanied with extra supercharges, see e.g.
reference [112] for an analysis of the possible multiplet recombination rules.
These enhancements are associated with superpotential deformations, and are at
finite distance [80].

On the other hand, not all flavor-enhancement point are at finite distance. In
$\mathcal{N}=2$ four-dimensional theories, representation-theoretic arguments
show that spin-one would-be-conserved current are in the same long multiplet
as would-be higher-spin currents. At threshold, the long multiplet then splits
into (at least) two short multiplets and both the flavor and HS currents
become conserved [113, 114, 112]. The flavor-enhancement points are therefore
also higher-spin points, and always at infinite-distance in the conformal
manifold from the results of the previous section. This is related to the
decomposability conjecture [114], stating that non-trivial $\mathcal{N}=2$
conformal manifolds can only be obtained by performing a conformal gauging of
the flavor symmetry of isolated theories. The decoupled sector is then made
out of the spectrum of free vector multiplets.

One can then ask whether a similar fate awaits more general cases, where we
only assume that there exist a neighborhood of the flavor-enhancement point
such that the broken Ward identity is valid:

$\partial_{\mu}J^{\mu}=g\,K_{0}\,.$ (4.1)

At the point where $g=0$, $K_{0}$ is a scalar conformal primary with
$\Delta=d$, but develops an anomalous dimension elsewhere. The two operators
$J_{1}$ and $K_{0}$ recombine into a larger multiplet away from that point,
where $K_{0}$ is a conformal descendant [109].

The first step of our argument required us to show that, when $g=0$, the OPE
combination vanishes: $C_{JJ\mathcal{O}}=0$. The current-conservation
condition applied to the correlator $\langle J_{1}J_{1}\mathcal{O}\rangle$
again fixes the two OPE coefficents to be proportional to one another:

$C_{JJ\mathcal{O}}^{0}=\frac{1}{d}C_{JJ\mathcal{O}}^{1}\,.$ (4.2)

But, contrary to cases where $\ell>1$, we cannot use the integrated Ward
identity to conclude that the remaining coefficient must vanish. Indeed, as
shown in Appendix A for the choice of surface $\Sigma$ discussed in Section
3.2, the integral vanishes and we cannot conclude that $C_{JJ\mathcal{O}}=0$
in this case.

It is nevertheless tempting to assume that the OPE combination should vanish
at the flavor-enhancement point. Were it not the case, taking
$\gamma^{0}_{\ell=1}=0$ as the reference point in equation (2.18), the
anomalous dimension would become negative, $\gamma\simeq-
C_{JJ\mathcal{O}}\,t<0$, and violate unitarity bounds. Since it is at least in
principle possible to deform away from the flavor-enhancement point in the
directions associated with both $\pm\mathcal{O}$, corresponding to the two
directions away from the point along the line parameterized by $t$, one of
them always lead to a violation of unitarity. A possible escape could be that
somehow conformal perturbation theory breaks down in that direction, leaving
us with only a semi-infinite critical line.

For now, let us thus assume that the OPE combination indeed vanishes:
$C_{JJ\mathcal{O}}=0$. We are then led to consider the correlator $\langle
J_{1}K_{0}\mathcal{O}\rangle$ at the point $g=0$. The current-conservation
condition is automatically satisfied, and using the integrated Ward identity
we obtain:

$\int_{\Sigma}dx^{\mu}\,\langle
J_{1,\mu}(x_{1})K_{0}(x_{2})\mathcal{O}(x_{3})\rangle\propto\langle[Q_{1},K_{0}(x_{2})]\mathcal{O}(x_{3})\rangle\,.$
(4.3)

The integral on the LHS can be expanded in terms of the single type-$(1,0,0)$
conformal structure, and, as shown in Appendix B, the result is non-vanishing
in any dimension.

Whether the OPE combination $C_{JK\mathcal{O}}$ is trivial—and by extension
whether the flavor-enhancement point is at infinite distance via the CFT
Distance Criterion—is therefore decided by the commutator $[Q_{1},K_{0}]$. If
any of the marginal operators belong to the same multiplet as the twist-$d$
operator, $[Q_{1},K_{0}]\sim\mathcal{O}$, then the two-point function on the
RHS of equation (4.3) is non-vanishing and $C_{JK\mathcal{O}}\neq 0$. The
associated limiting point is then at finite distance. If on the other hand
none of the marginal operators share a multiplet with $K_{0}$, the RHS of
equation (4.3) is trivial and the flavor-enhancement point is at infinite
distance—provided that $C_{JJ\mathcal{O}}=0$ at that point, as discussed
above.

Let us note that the charge-conservation identity applied in Section 3.3 for
the case of HS symmetry does not help in this case. Precisely for $\ell=1$,
$j_{1}$ cannot appear in $[Q_{1},j_{2}]$, since $Q_{1}$ has vanishing
conformal dimension. This is the usual statement that the energy-momentum
tensor should be invariant under any global symmetry. We thus have to set
$\alpha_{1}=0$ in equation (3.32) and the correlator of interest
$\left<j_{1}K_{0}\mathcal{O}\right>$ does not appear in the charge-
conservation condition.

In addition, the unitarity argument that we presented in favor of
$C_{JJ\mathcal{O}}=0$ at the flavor-enhancement point does not work either. If
we have $C_{JK\mathcal{O}}\neq 0$, then the behavior of the OPE combination is
given by $C_{JJ\mathcal{O}}\sim g\sim\sqrt{\gamma_{1}}$, which leads to
$\gamma_{1}\sim t^{2}$ upon using equation (2.16). This is at finite distance,
but the anomalous dimension is not linear with the Zamolodchikov distance $t$,
as it is the case for $C_{JJ\mathcal{O}}\neq 0$ at the flavor-enhancement
point. Therefore, this would not lead to a violation of the unitarity bound in
any of the two directions departing from the flavor-enhancement point along
the line parametrized by $t$.

The condition we have found for the flavor-enhancement point to be at finite
distance is of course very intuitive. It is equivalent to the statement that
the marginal operators are charged under the symmetry. We could have
alternatively chosen another integration domain so that equation (4.3) is
written in terms of the charges acting on $\mathcal{O}$, which would have led
us to the condition $[Q_{1},\mathcal{O}]\sim K_{0}$. For a
deformation—marginal or not—to preserve a symmetry the associated operator
must be in the singlet representation. In the context of R-symmetry, this was
used to classify possible supersymmetry-preserving deformations in various
dimensions [3, 8, 5]. In the example of $\mathcal{N}=4$ super-Yang Mills
described above, the marginal operators are indeed charged under the extra
symmetry, and this explains why they are at finite distance.

Conversely, if none of the marginal operators are charged under the flavor
symmetry, it cannot be explicitly broken or restored. One would then expect
that in the conformal manifold this point can only be reached asymptotically
and is therefore at infinite distance. This is consistent with a flavor
symmetry emerging as a sector becomes free, namely a higher-spin point. It
then begs the question of the possible existence of other flavor-enhancement
points at infinite distance. According to the second part of the CFT Distance
Conjecture, this cannot be the case, and an emerging flavor symmetry must be
accompanied by a free sector. While in 4d $\mathcal{N}=2$ SCFTs this is
enforced by the superconformal algebra [113, 114, 112], we are not aware of
similar arguments in more general cases. The study of spin-one operators and
their behavior in the conformal manifold therefore seem like an avenue to test
the rest of the CFT Distance Conjecture.

##  5 Conclusions

A weakly-broken higher-spin symmetry imposes remarkably strong constraints on
the form of the correlators. By studying conformal perturbations close to
higher-spin points, we have harnessed it to show that these points must always
be at infinite Zamolodchikov distance in the conformal manifold, establishing
part of the CFT Distance Conjecture.

In particular, we have combined two different types of perturbations. First,
we have used conformal perturbation theory to find a criterion which can be
used to decide whether a point is at infinite distance, given in terms of the
behavior of certain OPE coefficients near the point of the conformal manifold
under consideration. Furthermore, we have also utilized the broken Ward
identities appearing close to points where there is a symmetry enhancement.
Through Anselmi’s trick, this enabled us to find the leading behavior of the
these coefficients without needing to involve the details of a particular
microscopic description.

Combining these two techniques have shown to be powerful as they were
sufficient to show that the first part of the CFT Distance Conjecture is
indeed correct. One can then wonder about the possibility of proving the
converse, namely that infinite-distance points can only be accompanied with a
higher-spin enhancement, and whether the machinery discussed here can be
deployed to prove the rest of the conjecture. One would need to show that the
absence of the HS current at such points lead to an inconsistency, perhaps
violating the assumptions of local unitary CFTs. We expect that the CFT
Distance Criterion could help in this endeavour. We note that efforts have
already been spent in that direction, as it was recently shown using inversion
formulae that singularities of the curvature of the Zamolodchikov metric can
only occur in the presence of higher-spin currents [20], or if the CFT
develops a continuum.

Concerning the third part of the conjecture—stating that the anomalous
dimension vanishes exponentially fast with the distance at higher-spin
points—an analysis connected to our results using conservation identities
seems promising. We have indeed shown that this exponential behavior happens
along a given trajectory if $C_{KK\mathcal{O}}^{\text{HS}}\neq 0$. Therefore,
the third part of the conjecture holds if there is always a marginal operator
$\mathcal{O}$ such that this OPE combination is non-vanishing at the HS point.
Note that the presence of at least one such operators with this property is
enough. The presence of other marginal operators for which
$C_{KK\mathcal{O}}\to 0$ as the HS point is approached would correspond to
non-geodesic trajectories that avoid an exponential behavior, and that are
known to exist in the context of the Swampland Distance Conjecture [45, 111].
It is furthermore remarkable that this condition only requires information
about the HS point, and one could hope that this follows from the higher-spin
algebra. However, the operators $K_{\ell-1}$ are not as constrained as the
related conserved currents $J_{\ell}$. A study of the associated correlators
is then more difficult without considering specific models, and we leave such
analyses for future works.

We stress again that, while the evidence for the conjecture has originated
from limiting behaviors of SCFTs, our arguments have not relied on
supersymmetry, and are valid across dimensions. It would however be
interesting to investigate if we can uncover new properties of conformal
manifolds by imposing this additional structure. Part of our analysis might be
simplified, and connect constraints of different spins. There are also
examples of conformal manifolds that are compact [10], and all conformal
manifolds of three-dimensional CFTs have been conjectured to be so as well
[80]. It would be interesting to check _ad absurdum_ if assuming the presence
of a broken Ward identity for the higher-spin symmetry in those cases leads to
inconsistencies. In the same vein, while we have not assumed the presence of
supercharges, we have not tackled the question of non-supersymmetric conformal
manifolds, and only required the presence of marginal operators regardless of
the mechanism protecting them against quantum corrections. While there have
been hints of possible cases [115], their existence is not yet settled.

We have furthermore seen that for spin-one cases, some of the arguments we
have used fail, particularly to show that $C_{JJ\mathcal{O}}^{\text{HS}}=0$, a
key ingredient of our analysis. Similarly, we could not reach the conclusion
$C_{JK\mathcal{O}}^{\text{HS}}=0$. This is not a flaw but a feature, since
flavor enhancements are known to happen at both finite and infinite distance
in the conformal manifold. Our analysis leads to simple and intuitive
conditions diagnosing which case occurs. One can then ask whether the
condition for having a flavor-enhancement point at infinite distance would
imply that it comes with additional HS conserved currents. This seems to be a
promising scenario for making progress towards the second part of the CFT
Distance Conjecture.

Our work falls into the broader context of the holographic Swampland Program,
and provides a step forward in the understanding of the constraints imposed by
quantum gravity through the AdS/CFT correspondence. While we have not
discussed the gravity duals of the local CFTs, our results have implications
for theories of quantum gravity in AdS backgrounds [79, 80]. A striking
feature of our analysis is that it does not assume that the families of CFTs
are holographic, but only that they have an energy-momentum tensor. This
signals that the Swampland Distance Conjecture could hold beyond weakly-
coupled Einstein gravity. On more general grounds, one may wonder what
constraints must be imposed to conformal theories to address other Swampland
conjectures.

Furthermore, our results lend more validity to the physical intuition that
Vasiliev-type gravity—with infinitely many massless HS fields—should be
infinitely far away from Einstein gravity, i.e. they should not be connected
by small deformations [80]. They are moreover in agreement with the Emergence
proposal—see e.g. references [116, 117, 44, 46, 30, 118, 119, 120, 121] for
works in this direction—which implies that points for which an infinite number
of weakly-coupled fields become massless should always be at infinite
distance. It would also be interesting to connect with the results that were
obtained in references [88, 89], by further studying how the correlators
factorize, particularly when only a subsector of the theory becomes free. In
addition, further exploring conformal manifolds of local CFTs (possibly with a
weakly-coupled gravity dual) and their possible HS points might shed some
light on the fate of the Emergent String Conjecture [39] beyond flat
backgrounds, see reference [79] for an analysis of this conjecture applied to
examples of AdS/CFT dual pairs in string theory.

Finally, this work is but an example of possible synergies between conformal
theories and the Swampland Program. The Distance Conjectures have been
developed by studying the constraints on field theories imposed by quantum
gravity, but have been applied to conformal manifold independently of this
origin. Other Swampland conjectures might therefore serve as a guide for other
questions on the field-theory side, and motivate particular answers. On the
other hand, conformal techniques have proven to lead to very rigorous results,
and could allow one to establish more thoroughly—or disprove—some of the more
speculative part of the program, and go beyond the current lamppost.

### Acknowledgements

We thank A. Antunes, S. Komatsu, M. Montero, K. Papadodimas, F. Tellander, A.
Uranga, I. Valenzuela, T. Weigand and A. Zhiboedov for helpful discussions and
comments on early versions of the manuscript, as well as E. Skvortsov for
helpful correspondence. J.C. would like to thank the Department of Physics at
Harvard University and the Instituto de Física Teórica in Madrid for
hospitality during early stages of this work. F.B. is supported by the German
Research Foundation through a German-Israeli Project Cooperation (DIP) grant
“Holography and the Swampland”, by the Swiss National Science Foundation
(SNSF), grant number P400P2_194341, and in part by the Deutsche
Forschungsgemeinschaft under Germany’s Excellence Strategy EXC 2121 Quantum
Universe 390833306. The work by J.C. is partially supported by the Spanish
Agencia Estatal de Investigacion through the grant “IFT Centro de Excelencia
Severo Ochoa CEX2020-001007-S, the grants PGC2018-095976-B-C21 and
PID2021-123017NB-I00, funded by MCIN/AEI/10.13039/ 501100011033, by ERDF A way
of making Europe, and the FPU grant no. FPU17/04181 from the Spanish Ministry
of Education.

##  Appendix A Current-Conservation Conditions

In Section 3, we discuss the various constraints the higher-spin algebra
imposes on correlators. In particular, the Ward identity $\partial\cdot
J_{\ell}=0$ can be used to relate some of the OPE coefficients of a given
three-point function involving $J_{\ell}$. As pointed out in references [21,
103], taking the divergence of the three-point functions can be rephrased in
terms of a differential operator acting in the space of the building blocks
constituting the conformal structures. For the correlators we are considering,
there are only three building blocks for the conformal structures defined in
equations (2.5) and (2.7): $H_{12}$, $V_{1}$ and $V_{2}$ (their expressions as
function of the coordinates and polarization vectors are shown in equation
(B.1)).

In all generality, the expression of the differential operator for
type-$(\ell_{1},\ell_{2},\ell_{3})$ correlators depends on all six building
blocks, and was given in reference [103], whose conventions we follow. For our
purpose, we will always consider three-point functions involving a marginal
operator, $\ell_{3}=0$, which simplifies considerably the expressions we will
have to deal with. We will successively treat the two cases at hand, namely
$\langle J_{\ell}J_{\ell}\mathcal{O}\rangle$ and $\langle
J_{\ell}K_{\ell-1}\mathcal{O}\rangle$. For clarity, we reproduce here the
conformal structure in both cases, using that $J_{\ell}$ has twist $d-2$,
while $K_{\ell-1}$ and $\mathcal{O}$ have twist $d$:

$\displaystyle\Theta_{n}$
$\displaystyle=\frac{\mathcal{P}_{n}}{|x_{12}|^{d-4}|x_{13}|^{d}|x_{23}|^{d}}\,,\qquad$
$\displaystyle\mathcal{P}_{n}$
$\displaystyle=H_{12}^{\ell-n}(V_{1}V_{2})^{n}\,,$ (A.1)
$\displaystyle\widetilde{\Theta}_{n}$
$\displaystyle=\frac{\widetilde{\mathcal{P}}_{n}}{|x_{12}|^{d-2}|x_{13}|^{d-2}|x_{23}|^{d+2}}\,,\qquad$
$\displaystyle\widetilde{\mathcal{P}}_{n}$
$\displaystyle=H_{12}^{\ell-n-1}V_{1}^{n+1}V_{2}^{n}\,.$

We have denoted the numerators $\mathcal{P}$ and $\tilde{\mathcal{P}}$, as
these are the quantities on which the differential operator discussed above is
applied.

### Current Conservation for $\langle J_{\ell}J_{\ell}\mathcal{O}\rangle$

In this case, the conservation equation leads to the following constraint:

$\langle J_{\ell}(x_{1},\xi_{1})(\partial\cdot
J_{\ell})(x_{2},\xi_{2})\mathcal{O}(x_{3})\rangle=0\quad\Leftrightarrow\quad\mathcal{D}_{2}\,\langle
J_{\ell}(x_{1},\xi_{1})J_{\ell}(x_{2},\xi_{2})\mathcal{O}(x_{3})\rangle=0\,.$
(A.2)

where the differential operator, understood as acting on the second current,
reduces from the original expression [103] to:

$\mathcal{D}_{2}=\frac{\mathcal{D}_{2}^{(3)}}{d-2}+\mathcal{D}_{2}^{(2)}+(d-2)\mathcal{D}_{2}^{(1)}-2\left(\frac{\delta\mathcal{D}_{2}^{(2)}}{d-2}+\delta\mathcal{D}_{2}^{(1)}\right)\,,$
(A.3)

with

$\begin{split}\mathcal{D}_{2}^{(1)}&=V_{1}\,\partial_{H_{12}}\,,\\\
\mathcal{D}_{2}^{(2)}&=2H_{12}V_{1}\,\partial_{H_{12}}^{2}+V_{1}\,\partial_{V_{1}}\partial_{V_{2}}+\left(H_{12}+2V_{1}V_{2}\right)\,\partial_{V_{2}}\partial_{H_{12}}\,,\\\
\mathcal{D}_{2}^{(3)}&=2H_{12}\left(H_{12}+2V_{1}V_{2}\right)\,\partial_{V_{2}}\partial_{H_{12}}^{2}+\left(H_{12}+2V_{1}V_{2}\right)\,\partial_{V_{1}}\partial_{V_{2}}^{2}\,,\\\
\delta\mathcal{D}_{2}^{(1)}&=\partial_{V_{2}}-V_{1}\,\partial_{H_{12}}\,,\\\
\delta\mathcal{D}_{2}^{(2)}&=V_{2}\,\partial_{V_{2}}^{2}+2H_{12}\,\partial_{V_{2}}\partial_{H_{12}}-2V_{1}H_{12}\,\partial_{H_{12}}^{2}\,.\\\
\end{split}$ (A.4)

Note that we can safely ignore all terms involving building blocks that do not
appear in the type-$(\ell,\ell,0)$ conformal structures. Furthermore, in
reference [103], the operator is defined as acting on the first operator. To
get to the above equation, we have exchanged the indices $1\leftrightarrow 2$.

As explained in Section 3, this operator can then be applied to each of the
conformal structures, $\Theta_{n}$. The differential operator is taking care
of the contribution from the denominator by design, and we find the following
relation for the numerator of a given structure:

$\begin{split}(d-2)\mathcal{D}_{2}\mathcal{P}_{n}&=n^{2}(n-1)\,\widetilde{\mathcal{P}}_{n-2}\\\
&+n\left[d(\ell-2)+2(3+\ell^{2}-2\ell(n+2)+2n^{2}+n)\right]\,\widetilde{\mathcal{P}}_{n-1}\\\
&+(\ell-n)(d+2n)(d-4+2\ell-2n)\,\widetilde{\mathcal{P}}_{n}\,.\end{split}$
(A.5)

As expected, all dependence on the building blocks can be written in terms of
the numerator of the type-$(\ell,\ell-1,0)$ conformal structures
$\widetilde{\Theta}_{n}$, see equation (A.1). We can then plug in those
relations in the numerator of the three-point function $\langle
J_{\ell}J_{\ell}\mathcal{O}\rangle$:

$\mathcal{P}=\sum_{n=0}^{\ell}C_{JJ\mathcal{O}}^{\,n}\mathcal{P}_{n}\,.$ (A.6)

After shifting the sums and a bit of algebra to simplify the expression,
everything can be rewritten as a single sum:

$\begin{split}(d-2)\mathcal{D}_{2}\mathcal{P}&=\sum_{n=0}^{\ell-1}\bigg{[}(n+2)^{2}(n+1)\,C_{JJ\mathcal{O}}^{n+2}\\\
&+(n+1)\left[d(\ell-2)+2(6+\ell^{2}-2\ell(n+3)+2n^{2}+5n)\right]\,C_{JJ\mathcal{O}}^{n+1}\\\
&+(\ell-n)(d+2n)(d-4+2\ell-2n)\,C_{JJ\mathcal{O}}^{n}\bigg{]}\,\widetilde{\mathcal{P}}_{n}\,,\end{split}$
(A.7)

where we have defined $C_{JJ\mathcal{O}}^{\ell+1}=0$. Note that this is a
boundary condition, as the OPE coefficient $C_{JJ\mathcal{O}}^{n}$ have a
physical meaning only for $0\leq n\leq\ell$. The current conservation
condition then imposes that the prefactors in front of each
$\widetilde{\mathcal{P}}_{n}$ must all vanish, leading to a system of
constraints:

$A(n)\,C_{JJ\mathcal{O}}^{n+2}+B(n)\,C_{JJ\mathcal{O}}^{n+1}+D(n)\,C_{JJ\mathcal{O}}^{n}=0\,,\quad
n=0,\dots,\ell-1\,,$ (A.8)

where we have defined

$\begin{split}A(n)&=(n+2)^{2}(n+1)\,,\\\
B(n)&=(n+1)\left[d(\ell-2)+2(6+\ell^{2}-2\ell(n+3)+2n^{2}+5n)\right]\,,\\\
D(n)&=(\ell-n)(d+2n)(d-4+2\ell-2n)\,,\end{split}$ (A.9)

and one has to keep in mind the boundary condition:

$C_{JJ\mathcal{O}}^{\ell+1}=0\,.$ (A.10)

As discussed in the main text, this is a homogeneous system of $\ell$
equations for $\ell+1$ variables. For given $d>2$ and $\ell\geq 0$, one can
check that $D(n)\neq 0$ for $0\leq n\leq\ell-1$, and we can rewrite this as:

$C_{JJ\mathcal{O}}^{n}=-\frac{1}{D(n)}\left(A(n)\,C_{JJ\mathcal{O}}^{n+2}+B(n)\,C_{JJ\mathcal{O}}^{n+1}\right)\,,\quad
n=0,\dots,\ell-1\,.$ (A.11)

In this form, we clearly see that this is a recursion relation in the space of
OPE coefficients $C_{JJ\mathcal{O}}^{n}$. Using $C^{\ell}_{JJ\mathcal{O}}$ and
$C_{JJ\mathcal{O}}^{\ell+1}=0$ as initial conditions and starting from
$n=\ell-1$, we can find any of the OPE coefficients as being proportional to
$C^{\ell}_{JJ\mathcal{O}}$, and we indeed find the result given in equation
(3.11).

Notice that since $A(n)\neq 0$ for $0\leq n\leq\ell-1$, we could instead have
started with $n=0$ and solved the system in ascending order. However, in this
way $C_{JJ\mathcal{O}}^{\ell+1}=0$ cannot be used as an initial condition.
This will lead to solutions in terms of $C_{JJ\mathcal{O}}^{0}$ and
$C_{JJ\mathcal{O}}^{1}$, plus the constraint imposed by the boundary condition
$C_{JJ\mathcal{O}}^{\ell+1}=0$. As it is not guaranteed that the last
constraint is non-trivial, we find it more convenient to write the solution in
terms of $C_{JJ\mathcal{O}}^{n}$.

As advertised in Section 3, given the current conservation condition for
$\langle J_{\ell}J_{\ell}\mathcal{O}\rangle$, we have shown that we can always
write all the OPE coefficients $C_{JJ\mathcal{O}}^{n}$ in terms of
$C_{JJ\mathcal{O}}^{\ell}$ as in equation (3.11). In other words, this
condition reduces the number of allowed conformal structures from $\ell+1$ to
only one. For arbitrary dimension and spin, we have not found a closed-form
solution of the recursion relation. However, it can be easily solved for fixed
values programmatically, see equations (4.2) and (3.12) for $\ell=1,4$.

### Current Conservation for $\langle J_{\ell}K_{\ell-1}\mathcal{O}\rangle$

Imposing the divergence of the higher-spin current $J_{\ell}$ in the case of
the three-point function $\left<J_{\ell}K_{\ell-1}\mathcal{O}\right>$ proceeds
_mutatis mutandis_ from that of $\langle J_{\ell}J_{\ell}\mathcal{O}\rangle$.
The divergence is again translated into a differentiation in the space of
building blocks:

$\langle(\partial\cdot
J_{\ell})(x_{1},\xi_{1})K_{\ell-1}(x_{2},\xi_{2})\mathcal{O}(x_{3})\rangle=0\quad\Leftrightarrow\quad\mathcal{D}_{1}\,\langle
J_{\ell}(x_{1},\xi_{1})K_{\ell-1}(x_{2},\xi_{2})\mathcal{O}(x_{3})\rangle=0$
(A.12)

The differential operator applied to the first operator is now simpler than in
the previous case, and given by [103]:

$\mathcal{D}_{1}=\frac{\mathcal{D}_{1}^{(3)}}{d-2}+\mathcal{D}_{1}^{(2)}+(d-2)\mathcal{D}_{1}^{(1)}\,,$
(A.13)

where we have defined each component as:

$\begin{split}\mathcal{D}_{1}^{(1)}&=V_{2}\,\partial_{H_{12}}\,,\\\
\mathcal{D}_{1}^{(2)}&=2H_{12}V_{2}\,\partial_{H_{12}}^{2}+V_{2}\,\partial_{V_{1}}\partial_{V_{2}}+\left(H_{12}+2V_{1}V_{2}\right)\,\partial_{V_{1}}\partial_{H_{12}}\,,\\\
\mathcal{D}_{1}^{(3)}&=2H_{12}\left(H_{12}+2V_{1}V_{2}\right)\,\partial_{V_{1}}\partial_{H_{12}}^{2}+\left(H_{12}+2V_{1}V_{2}\right)\,\partial_{V_{1}}^{2}\partial_{V_{2}}\,.\\\
\end{split}$ (A.14)

Once again, following the procedure explained in Section 3, this operator is
applied to the numerator of the correlator. The numerators
$\widetilde{\mathcal{P}}_{n}$ of the type-$(\ell,\ell-1,0)$ conformal
structure $\widetilde{\Theta}_{n}$ then satisfies the following relation:

$\begin{split}(d-2)\mathcal{D}_{1}\widetilde{\mathcal{P}}_{n}&=(n+1)n^{2}\,\mathcal{P}_{n-1}\\\
&+(n+1)\left[(\ell-1)(d+2\ell-6)-(4\ell-6)n+4n^{2}\right]\,\mathcal{P}_{n}\\\
&+(\ell-1-n)(d+2n)(d-6+2\ell-2n)\,\mathcal{P}_{n+1}\,.\end{split}$ (A.15)

As expected, all dependence on the building blocks can be written in terms of
the numerator of type-$(\ell-1,\ell-1,0)$ conformal structures $P_{n}$, see
equation (A.1).151515Note that in equation (A.1) the conformal structures are
given for spin $\ell$, and one therefore needs to substitute $\ell\to\ell-1$.
Plugging these relations into the numerator of the full three-point function
$\left<J_{\ell}K_{\ell-1}\mathcal{O}\right>$:

$\widetilde{\mathcal{P}}=\sum_{n=0}^{\ell-1}C_{JK\mathcal{O}}^{n}\widetilde{\mathcal{P}}_{n}\,,$
(A.16)

we then have

$\begin{split}(d-2)\mathcal{D}_{1}\widetilde{\mathcal{P}}&=\sum_{n=0}^{\ell-1}(n+1)n^{2}\,C_{JK\mathcal{O}}^{n}\mathcal{P}_{n-1}\\\
&+\sum_{n=0}^{\ell-1}(n+1)\left[(\ell-1)(d+2\ell-6)-(4\ell-6)n+4n^{2}\right]\,C_{JK\mathcal{O}}^{n}\mathcal{P}_{n}\\\
&+\sum_{n=0}^{\ell-1}(\ell-1-n)(d+2n)(d-6+2\ell-2n)\,C_{JK\mathcal{O}}^{n}\mathcal{P}_{n+1}\,.\end{split}$
(A.17)

Shifting the sums, we can rewrite the sums in terms of summands containing
only $\mathcal{P}_{n}$ if we impose
$C_{JK\mathcal{O}}^{-1}=0=C_{JK\mathcal{O}}^{\ell}$. Note that this is not a
constraint on the OPE coefficients, as for type-$(\ell,\ell-1,0)$ correlators
$C_{JK\mathcal{O}}^{n}$ has a physical meaning only for $0\leq n\leq\ell-1$.
After a few algebraic manipulations, we find:

$\begin{split}(d-2)\mathcal{D}_{1}\widetilde{\mathcal{P}}&=\sum_{n=0}^{\ell-1}\bigg{[}(n+2)(n+1)^{2}\,C_{JK\mathcal{O}}^{n+1}\\\
&+(n+1)\left[(\ell-1)(d+2\ell-6)-(4\ell-6)n+4n^{2}\right]\,C_{JK\mathcal{O}}^{n}\\\