††institutetext: Department of Mathematics, King’s College London, Strand,
London, WC2R 2LS, UK

# A Walk Through $Spin(1,d+1)$

Vladimir Schaub<EMAIL_ADDRESS>
###### Abstract

We construct unitary irreducible representation of the de Sitter group, that
forms the basis for the study of $dS_{d+1}$ QFT. Using the intertwining kernel
analysis, we discuss bosonic symmetric tensor, and fermionic higher-spinors.
Particular care is given to the structure and construction of exceptional
series and fermionic operators. We discuss the discrete series, and explain
how and why the exceptional series give rise to seemingly non-invariant
correlators in de Sitter. Using tools from Clifford analysis, we show that for
$d>3$, there are no exceptional fermionic representations, and so no unitary
(higher)-gravitino fields. We also consider the structure of representations
for $d=3$ and $d=2$, as relevant for the study of $dS_{4}$, the only space
allowing for unitary gravitino and its generalisation, and of $dS_{3}$.

##  1 Introduction

What we can say about a physical system revolves around its symmetries. Much
of what we know of quantum field theories springs out from Wigner’s program,
and its identification between unitary irreducible representation and
particles Wigner:1939cj ; Weinberg:1995mt . As it has become apparent from
observation that our space-time is expanding Hubble:1929ig , the leading
incorporation of gravity to QFT at large scale is that of a background rigid
de Sitter spacetime structure. The study of de Sitter QFT is a vibrant area of
research, which touches on experiments, trying to understand the physics of
the CMB Bousso:2007gp ; Baumann:2020ksv ; Baumann:2020dch ; Baumann:2019oyu ;
Baumann2018 , and grapples with fundamental questions aiming to understand its
physical properties at the level of correlation functions Bros:1990cu ;
Weinberg2005 ; Bros:2010rku ; Sleight:2019mgd ; Sleight:2019hfp ;
Gorbenko:2019rza ; Sengor:2019mbz ; Sleight:2020obc ; Sengor:2021zlc ;
Sleight:2021iix ; DiPietro:2021sjt ; Hogervorst:2021uvp ; Goodhew:2021oqg ;
Melville:2021lst ; Heckelbacher:2022hbq ; Schaub:2023scu ; DiPietro:2023inn ;
Anninos:2024fty ; Loparco:2023rug ; Loparco:2023akg ; Galante:2023uyf ;
Loparco:2024ibp , of the wavefunction Anninos:2014lwa ; Goodhew:2020hob ;
Salcedo:2022aal ; Anninos2021 , or in a holographic perspective
Strominger:2001pn ; Deser:2003gw ; Anninos:2011ui ; Anninos:2017eib ;
Hertog:2019uhy ; Pethybridge:2024qci  .

It seems crucial then, to have a good physical understanding of the Hilbert
space structure consistent with the de Sitter isometries, meaning
representation theory of $SO(1,d+1)$, the conformal group, and its double
cover, which is the one actually relevant for a quantum Hilbert space. The
present work is a step in that direction. This is very clearly not the first
work on the de Sitter group : the study of $SO(1,d+1)$ is a central element of
the theory of non-compact Lie groups, with (non-exhaustive) references dating
back more almost 80 years and spanning decades Thomas:1941aa ; Bargmann:1946me
; Gelfand:1947aa ; Harish-Chandra:1947aa ; Newton:1950aa ; Dixmier:1961aa ;
Hirai:1962aa ; Takahashi:1963aa ; Ottoson:1968aa ; Gavrilik:1975aa . From a
physicist perspective however, the most crucial early reference seems to be
Dobrev:1977qv , since it places the construction of the unitary
representations in a footing that is close to how quantum field are
manipulated ; working with correlation functions, quotienting out null states,
and making harmonic decompositions. This account, though extremely complete,
only investigated a specific class of representation, the totally symmetric
traceless tensors. A modern treatment along these lines is given in
Sun:2021thf .

This creates a problem, since the existing mathematical literature, most
notably the very exhaustive Hirai:1962aa ; Ottoson:1968aa ; Gavrilik:1975aa ,
are not formulated in a language that is anywhere close to how we think of
building fields, using instead Gelfand-Tsetlin patterns and recursively
building unitary generators. These tools are of course extremely powerful and
interesting in and of themselves, but they hide a lot of the physical content
beyond sophistication. So, even though they have been used successfully to
discuss more general representation like antisymmetric tensors etc.
Basile:2017aa , this method is not transparent and an alternative, following
the lines of Dobrev:1977qv , is ultimately preferable.

An apt illustration of the present situation, and the motivation for this
work, is the case of the (partially)-conserved (massless in the bulk) tensor-
spinors Deser:2001us ; Bonifacio:2023prb ; Bonifacio:2018zex ;
Bonifacio:2021mrf ; Blauvelt:2022wwa . These are not unitary irreducible
representation in $dS_{d+1}$, except in $d=3$. This specific dimension
dependent feature seems to have been mostly overlooked in the literature,
hidden away in the list of indices and weights, until it was pointed out by
Letsios:2023qzq ; Letsios:2023awz ; Letsios:2024nmf , which rediscovered it by
studying the mode functions of a few of these (partially)-massless higher-
spinors across dimension.111The method used in Letsios:2023qzq ;
Letsios:2023awz ; Letsios:2024nmf  does not construct the representation from
scratch at the level of the Hilbert space itself, but rather shows that the
explicit field mode solutions transform in a unitary representation. Since
this family of representation includes the gravitino and our space is well
described at large distance by $dS_{4}$, this rather exceptional fact must be
thoroughly understood, especially given supersymmetry’s difficult cohabitation
with expanding space Pilch:1984aw ; Deser:2003gw ; Anous:2014lia ;
Bergshoeff:2015tra ; Hertog:2019uhy ; Anninos:2023exn . Beyond these
exceptional representations, more mundane questions such as why there are no
complementary series spinors do not seem to have received a simple physical
explanation so far.

The goal of the present work is to give a self-contained treatment, along the
lines of Dobrev:1976vr ; Sun:2021thf , of the symmetric tensor-spinors unitary
irreducible representations, leveraging the modern framework of conformal
field theories, drawing notably Kravchuk:2016qvl ; Karateev:2018oml ;
Karateev:2018uk ; Iliesiu:2015qra ; Iliesiu:2017nrv . This might make the
endeavour sound more restrictive than it really is. Precisely because
supersymmetry is hard to come by in de Sitter, one cannot avoid studying the
fermions by themselves and for their own sakes, even more so since fermions
are plentiful in our universe. While spin-$1/2$ fermions have received some
attention in the literature, see for example Pethybridge:2021rwf ;
Schaub:2023scu , it would be wrong to discard higher-spinors. After all, there
are plenty of known resonances, see fig. 1, which are of this type in our
universe.

Figure 1: Some baryon with spin $3/2$ generated by QCD, organised in a
decuplet Gell-Mann:1961omu . Dressed multi-quark states are expected to
overlap with excitations of all spin and in all families of unitary
representations.

Because of the nature of these puzzling spinorial representations, which are
in the exceptional (and actually discrete) series, one must confront the
associated intricacies of these families of representations. On of the goal is
to give here an exposition that makes the structure and construction of these
representations clear, using the bosonic case as a review. Although
essentially contained in the analysis of Dobrev:1977qv ; Sun:2021thf , this is
still worthwhile from the standpoint of pedagogy and self-containment. This is
not the only draw, since we tried to give a more detailed account of the
peculiar, logarithmic yet conformally invariant, inner product of these
representations and their quotient space structure, and relating it to the
recent progress on the study of the discrete series in $dS_{2}$ and $dS_{4}$
Anninos:2023lin ; Sengor:2023buj ; RiosFukelman:2023mgq ; Anninos:2024fty . An
important take away from our discussion, summarised in the appendix A, is that
the interesting features of these special representations is that they are
exceptional, not whether they are discrete, since their features persist
whether or not the group element have finite square-average.

Studying spinorial representations is an ambitious project by itself, and so
this work will not address all of the associated issues. Notably, we will not
consider the question of the characters of the representations, which can be
understood in principle by applying the results of Hirai:1965aa ; Sun:2021thf
, nor discuss the tensor-product decomposition Dobrev:1976vr ;
Penedones:2023uqc . We leave these matters to future work. We will also not
tackle the case $d=1$, relevant for the study of $dS_{2}$, simply because it
was felt to be adequately studied in the present literature, see e.g.
Kitaev:2017hnr ; Sun:2021thf ; Kravchuk:2021akc .

The paper proceeds as follows. In sec. 2, we review the construction of
unitary irreducible representations, using the symmetric traceleless tensor
representations as an explicit case study. We explain the general strategy,
detail the harmonic decomposition of the inner product. We investigate closely
the exceptional points, first discussing scalars and then spinning
representations. In sec. 3, we perform the analogous analysis for tensor-
spinor representations. The technique for the harmonic decomposition is not
standard, and so we carefully derive them. The output is a proof that for
$d>3$, there are no spinorial representations beyond the principal series
because of a pairing of eigenvalues. What differentiates lower-dimension is
clear : the Clifford algebra contracts, and one must study it case by case.
This is done in sec. 4, for $d=3$ and $d=2$, where one can treat bosonic and
fermionic representations at once. It is found that for $d=3$ only, at the
exceptional points one can define a positive inner product, and so there are
indeed some unitary partially-massless higher-spinors. We conclude with a
discussion of the results and future research directions.

##  2 Story of Bose : Symmetric Tensor Representations

The goal of this section is to review the construction of the unitary
irreducible representations (UIR) for the symmetric traceless tensors (STT).
The central tool is Knapp-Stein intertwining kernel Knapp:1971aa . Induced
representations generate “elementary representations”, on which one tries to
find an invariant, unitary inner product. The $L^{2}$ inner product, defines
the Principal Series $\mathcal{P}_{\Delta}$. Another non-trivial inner product
exists, which is a conformal 2pt. function. Through a partial wave analysis,
we find bounds on the allowed ranges of $\Delta$, defining the complementary
series $\mathcal{C}_{\Delta}$. For exceptional points, this analysis
breakdowns, giving rise to a peculiar inner-product and the associated
exceptional (sometimes discrete) series, $\mathcal{V}_{n}$,
$\mathcal{U}_{\ell,t}$. We give a self-contained, conceptual discussion and
try to give interpretation for this last procedure, which explains some
features of fields in de Sitter.

Our exposition is geared for the two other cases later considered ; fermionic
representations, and lower-dimensions. Hence, care is given to the crucial
role played by the harmonic decomposition to investigate the positive
definiteness, and to the methodology for the investigation of exceptional
points.

###  2.1 Bruhat Decomposition And Intertwining Kernel

The method of induced representations for semi-simple non-compact group
follows by using a particular factorisation into subgroups Wigner:1939cj ;
Dobrev:1977qv ; Knapp:1982aa ; Knapp:1986aa  To consider unitary irreducible
representations (UIR) of the group $G=SO(1,d+1)$, we start from the Bruhat
decomposition into factors

$\displaystyle N$ $\displaystyle=\left\\{e^{b\cdot K}\right\\},$
$\displaystyle A$ $\displaystyle=\left\\{e^{\lambda D}\right\\},$
$\displaystyle M$ $\displaystyle=\left\\{e^{\frac{1}{2}\omega\cdot
M}\right\\},$ $\displaystyle\widetilde{N}$ $\displaystyle=\left\\{e^{x\cdot
P}\right\\}\,.$ (1)

With $K$ the generator of special conformal transformations, $D$ of dilation,
$M$ of rotations, and $P$ of translations. A representation of the subgroup
$NAM$ lifts to a representation of the whole group. More precisely, starting
from representations in the kernel of the nilpotent $N$, transforming as
irreducible representations $(\rho,\Delta)$ of $M$ and $A$, one can act with
an element of $\widetilde{N}$ to generate a full “elementary” representation;
modulo some conditions regarding smoothness. The associated states can be
associated to point in the homogeneous space $G/NAM\sim\mathbb{R}^{d}$, which
define a group action.222This construction is called the “non-compact”
picture. It renders the $M$ subgroup evident. The “compact” or $K$-picture,
makes the maximal compact subgroup $K=SO(d+1)$ manifest, and can be obtained
from an inverse-stereographic projection. It is associated with the Iwasawa
decomposition, whereby one writes a group element as $KAN$.

We start with bosonic representation of the subgroup $SO(d)$, and we limit
ourselves to fields in a representation $\rho$, with associated abstract
indices $\alpha,\beta,\ldots$, given by a young tableau with a single row of
$\ell$ boxes, i.e. symmetric traceless tensors of spin-$\ell$.333In the
classic work Dobrev:1977qv , these are called type I representations. We use
anti-hermitian generators, $Q^{\dagger}=-Q$. This defines complex conjugation
and gives a framework to compute the norm of states, defining a notion of
unitarity. The transformation properties of the elementary representation is
conveniently encoded in braket notationHogervorst:2021uvp ; Sun:2021thf , by
defining a primary state

$\displaystyle D\ket{\Delta,0}_{\alpha}$
$\displaystyle=\Delta\ket{\Delta,0}_{\alpha}\,,$ (2) $\displaystyle
K_{i}\ket{\Delta,0}_{\alpha}$ $\displaystyle=0\,,$ $\displaystyle
M_{kl}\ket{\Delta,0}_{\alpha}$
$\displaystyle=-(\Sigma_{kl})_{\alpha}{}^{\beta}\ket{\Delta,0}_{\beta}\,,$

where the matrices $\Sigma_{kl}$ are the matrix representation of the $M_{kl}$
in the representation $\rho$ of $SO(d)$, i.e. $D_{\rho}(M_{kl})=\Sigma_{kl}$.
This state is irreducible under $A\sim D$, $N\sim K$, and $M$. The leftover
$\widetilde{N}\sim P$ are translations that produce the whole conformal family

$\displaystyle\ket{\Delta,x}_{\alpha}$ $\displaystyle=e^{x\cdot
P}\ket{\Delta,0}_{\alpha}\,,$ (3)

Which span the whole elementary representation. These states for finite $x$
transform as one would expect a primary with scaling dimension $\Delta$ to. We
define the elementary representation as the functional space
$\mathcal{F}_{\Delta,\ell}$, generically not unitary nor irreducible, of
states created by wavefunctions of the form

$\displaystyle\ket{\psi}=\int\mathrm{d}^{d}x\
\psi^{\alpha}(x)\ket{\overline{\Delta},x}_{\alpha}\,.$ (4)

where we defined $\overline{\Delta}=d-\Delta$. The transformation properties
of the states induces a transformation of the wavefunctions $\psi$, which is
that of a primary operator with scaling dimension $\Delta$ in an Euclidean
CFTd, transforming under the conjugate representation $\rho_{C}$ of $SO(d)$.
We now would like to stress the following important fact : the base-states are
a useful tool to induce the transformation properties of the wavefunctions
$\psi$, which should ultimately be promoted to distribution-valued operators.
All along however, they will be treated as functions to define the
representations. It is the properly smeared states $\ket{\psi}$ which are part
of the elementary representations, not the underlying
$\ket{\overline{\Delta},x}$. Although in most instances one can work with
ultra-local wavefunctions, this fails to capture some of the physics in the
degenerate instances we will encounter, where one needs to quotient out null-
subspaces. This distinction is crucial to understand the peculiar nature of
the exceptional points later on.

We consider wavefunctions which are smooth and on which the generator of
special conformal transformation is well behaved.444This can be understood as
an artefact of the non-compact picture. In the compact picture, this simply
translates into smoothness over $K$, the compactification of $\mathbb{R}^{d}$.
This implies an asymptotic condition, which is the mapping of the local
continuity unto infinity,

$\displaystyle\psi(x)\overset{x\to\infty}{\approx}\frac{1}{(x^{2})^{\Delta}}\sum_{n}h_{n}\left(\frac{x^{\mu}}{x\cdot
x}\right)\quad\mid\quad\tilde{x}\cdot\partial
h_{n}(\tilde{x})=n\,h_{n}(\tilde{x})\,,$ (5)

where $h_{n}$ is an homogeneous tensor-polynomial of order $n$, with scaling
dimension $-n$ in $x$. This fixes the functional space
$\mathcal{F}_{\Delta,\ell}$, and the elementary representation.

It is a general result known as subrepresentation theorem Knapp:1986aa ;
Casselman:1989aa , that any (unitary) irreducible representation can be found
as an invariant subspace or quotient of elementary representation. Our goal is
then to study the norm of states to find these spaces. It turns out that,
except for isolated points, the representations are irreducible, making the
analysis more streamlined.

To study unitarity and irreducibility, we need to define an inner-product
$\bra{\cdot}\ket{\cdot}$ on $\mathcal{F}_{\Delta,\ell}$, which has to to
satisfy multiple desiderata : positive definiteness, normalisability and
irreducibility. It must take the generic form

$\displaystyle\bra{\psi_{1}}\ket{\psi_{2}}$
$\displaystyle=\int\mathrm{d}^{d}x\ \mathrm{d}^{d}y\
\psi_{1}{}^{\dagger}_{\alpha}(x)G_{\overline{\Delta},\ell}(x,y)^{\alpha}{}_{\beta}\psi_{2}^{\beta}(y)=\big{(}\psi_{1},G_{\overline{\Delta},\ell}[\psi_{2}]\big{)}\,.$
(6)

The operation $(\cdot,\cdot)$ is the $L^{2}$ pairing of states transforming in
conjugate representations.555Conjugation acts on representations of $A$,
dilations, by sending $\Delta$ to $\overline{\Delta}=d-\Delta$ The function
$G_{\Delta}$ is called the intertwining kernel, as convoluting a state of
$\mathcal{F}_{d-\Delta,\ell}$ with $G_{\Delta,\ell}$ gives (barring
exceptional cases) a state in $\mathcal{F}_{\Delta,\ell}$.666The operator
$G_{\Delta}$ is also known as the Knapp-Stein intertwining operator
Knapp:1971aa , the intertwiner, and the shadow transform Ferrara:1972kab ;
Dolan:2004up ; Simmons-Duffin:2014wb ; Karateev:2018oml .

$\displaystyle G_{\Delta,\ell}:\quad\mathcal{F}_{d-\Delta,\ell}$
$\displaystyle\to\mathcal{F}_{\Delta,\ell}\,,$ (7)
$\displaystyle\psi^{\alpha}(x)$ $\displaystyle\mapsto\int
G_{\Delta,\ell}(x,y)^{\alpha}{}_{\beta}\psi^{\beta}(y)\mathrm{d}^{d}y\,.$

Conformal invariance implies Ward identities on the kernel, that in turn
implies that $G$ transforms like a 2-point function of conformal primary
fields with scaling dimension $\Delta$. Intuitively, this follows from
$G_{\overline{\Delta},\ell}(x,y)=\bra{\overline{\Delta},x}\ket{\overline{\Delta},y}$.777This
convention for the kernel differs from that of Sun:2021thf , in which
$G_{\Delta}=\bra{\overline{\Delta},x}\ket{\overline{\Delta},y}$. This implies
two qualitatively different cases :

  1. 1.

$\imaginary(\Delta)\neq 0$. Conformal invariance imposes
$\Delta+\Delta^{\dagger}=d$, giving $\Delta=\frac{d}{2}+i\nu$. The only
allowed kernel is ultra-local, $G_{\frac{d}{2}-i\nu}(x,y)=\delta^{d}(x-y)$ and
the fields transform in the irreducible888The intersection of the principal
series with the real axis, $\nu=0$, asks for more care. For some choices of
$\rho$, this point can belong to a reducible intertwiner, and so to a
reducible elementary representation. This does not arise for STT
representations, but it happens for spinors. This reduction simply amounts to
quotienting by the null descendants of the differential equation satisfied by
the state. This reducibility is not of an interesting type in general. and
unitary Principal Series $\mathcal{P}_{\frac{d}{2}+i\nu,\ell}$, which is
equipped with the usual $L^{2}$ inner product $(\cdot,\cdot)$.

  2. 2.

$\Delta\in\mathbb{R}$, where the kernel is a non-trivial function of the two
separation, and requires further analysis both for unitarity and
irreducibility.

    1. i.

For almost all values of $\Delta$, the representation will be irreducible. For
some range of $\Delta$, it will moreover be positive, giving rise to the
complementary series $\mathcal{C}_{\Delta}$

    2. ii.

For some values of $\Delta$, the representation will be reducible, giving the
exceptional (and discrete) series.

The second case is the more involved one, to which we now specialise. To study
the kernel, we use the index-free notation Costa:2011wa , whereby we contract
all free tensorial indices with null-vectors $z_{i}$,

$\displaystyle G_{\Delta,\ell}(x,y;z_{1},z_{2})$
$\displaystyle=z_{1}^{\mu_{1}}\ldots
z_{1}^{\mu_{\ell}}G_{{\Delta},\ell}(x,y){}_{\mu_{1}\ldots\mu_{\ell}}{}^{\nu_{1}\ldots\nu_{\ell}}z_{2,\nu_{1}}\ldots
z_{2,\nu_{\ell}}$ (8) $\displaystyle=C_{\Delta,\ell}\frac{\left(z_{1}\cdot
z_{2}-2\frac{z_{1}\cdot(x-y)z_{2}\cdot(x-y)}{(x-y)^{2}}\right)^{\ell}}{(x-y)^{2\Delta}}$
$\displaystyle=C_{\Delta,\ell}\frac{\left(z_{1}\cdot I_{(x-y)}\cdot
z_{2}\right)^{\ell}}{(x-y)^{2\Delta}}\,,$

Where $I_{(x-y)}^{\mu\nu}$ is the inversion tensor and the constant
$C_{\Delta,\ell}$ is a normalisation choice which will be fixed later on. For
generic values of $\Delta$, the intertwining kernel is invertible. One can
pick choices of normalisation for $C_{\Delta,\ell}$ to try to make these two
map the inverse of each other. In this work, we take the simpler convention
that this overall constant will be left mostly as is. The reason is to avoid
obfuscating when the harmonic decomposition of the kernel becomes singular.

To investigate positivity and irreducibility, we use a version of the partial
wave analysis, writing the kernel in a form manifesting the covariance under
translation, scaling, and rotations.999This is an instance of a systematic
procedure. The general idea is to perform a harmonic decomposition of the
intertwiner over the parabolic subgroup $\widetilde{N}AM$. From translation
invariance, we rewrite the inner product in momentum space

$\displaystyle\bra{\psi_{1}}\ket{\psi_{2}}$
$\displaystyle=\int\frac{\mathrm{d}^{d}p}{(2\pi)^{d}}\
\widehat{\psi_{1}}^{\dagger}(p)\cdot\widehat{G}_{\overline{\Delta},\ell}(p)\cdot\widehat{\psi_{2}}(p)\,,$
(9)

defining the momentum space intertwiner

$\displaystyle\widehat{G}_{\Delta,\ell}(p)$
$\displaystyle=\int\mathrm{d}^{d}x\,e^{ip\cdot x}\,G_{\Delta,\ell}(x,0)$ (10)
$\displaystyle=C_{\Delta,\ell}(z_{1}\cdot
z_{2})^{l}\sum_{k=0}^{\ell}\binom{\ell}{k}\left(\frac{2z_{1}\cdot\partial_{p}z_{2}\cdot\partial_{p}}{z_{1}\cdot
z_{2}}\right)^{k}\int\mathrm{d}^{d}x\,\frac{e^{ip\cdot
x}}{(x^{2})^{\Delta+k}}\,.$

The last expression can be evaluated using Schwinger integrals leading to the
identity

$\displaystyle\int\mathrm{d}^{d}x\,\frac{e^{ip\cdot
x}}{(x^{2})^{\Delta}}=\frac{2^{d-2\Delta}\sqrt{\pi}^{d}\Gamma(\frac{d}{2}-\Delta)}{\Gamma(\Delta)}\absolutevalue{p}^{2\Delta-d}\,,$
(11)

and one can proceeds with the resummations. For convenience, we set the
shorthand $h=d/2$, $\nu=\Delta-h$, which are standardly used. Using Leibniz
rules twice and reordering the sums, we find

$\displaystyle(z_{1}\cdot
z_{2})^{\ell}\sum_{m=0}^{\ell}\binom{\ell}{m}\frac{\Gamma(\nu-k)}{4^{m}\Gamma(\Delta+m)}\left(\frac{2z_{1}\cdot\partial_{p}z_{2}\cdot\partial_{p}}{z_{1}\cdot
z_{2}}\right)^{m}\absolutevalue{p}^{-2\nu+2k}$ (12) $\displaystyle=$
$\displaystyle\absolutevalue{p}^{2\Delta-d}\frac{(-2)^{\ell}\ \ell!\
\Gamma(\ell-\nu)}{\Gamma(\ell+\Delta)}\left(\frac{z_{1}\cdot pz_{2}\cdot
p}{p\cdot p}\right)^{\ell}P_{\ell}^{(h-2,\nu-\ell)}\left(\frac{p\cdot
pz_{1}\cdot z_{2}}{z_{1}\cdot pz_{2}\cdot p}-1\right)\,.$

The function $P_{n}^{a,b}(t)$ is a Jacobi Polynomial. It depends on a single
parameter

$\displaystyle t=\frac{p\cdot pz_{1}\cdot z_{2}}{z_{1}\cdot pz_{2}\cdot
p}-1=\left(z_{1}\cdot z_{2}-\frac{z_{1}\cdot pz_{2}\cdot p}{p\cdot
p}\right)\frac{p\cdot p}{z_{1}\cdot pz_{2}\cdot p}\,,$ (13)

which measures the angle between $z_{1}$ and $z_{2}$ in the plane
perpendicular to the vector $p$. To make this effect transparent, we can
without loss of generality consider the configuration

$\displaystyle p$ $\displaystyle=\begin{pmatrix}\absolutevalue{p}\\\
0\end{pmatrix}=\absolutevalue{p}\widehat{p},$ $\displaystyle z_{1}$
$\displaystyle=\begin{pmatrix}+i\sqrt{z^{2}}\\\ z\end{pmatrix},$
$\displaystyle z_{2}$
$\displaystyle=\begin{pmatrix}-i\sqrt{\overline{z}^{2}}\\\
\overline{z}\end{pmatrix},$ $\displaystyle z\cdot\overline{z}$
$\displaystyle=t\sqrt{z^{2}\overline{z}^{2}}\,.$ (14)

This parametrisation is tuned so that $z_{1}^{\dagger}\sim z_{2}$, as happens
in the inner product we consider. We have determined

$\displaystyle\widehat{G}_{\Delta,\ell}(p)$
$\displaystyle\equiv\absolutevalue{p}^{2\Delta-d}C_{\Delta,\ell}\frac{(-2)^{\ell}\pi^{h}\ell!\
\Gamma(\ell-\nu)}{\Gamma(\ell+\Delta)}\sqrt{z^{2}\overline{z}^{2}}^{\ell}P_{\ell}^{(h-2,\nu-\ell)}\left(t\right)\,.$
(15)

The kernel now makes the invariance under translation ($\widetilde{N}$),
dilation ($A$), and rotations ($M$). Taking out the overall factor of
$\absolutevalue{p}$ which diagonalises the action of dilation, the remaining
pieces are defined by the value of the intertwiner for given $\hat{p}^{\mu}$
on $S^{d}$. It must then be decomposable into irreducible representations of
the stabiliser group of $p$, which is $SO(d-1)$. This implies a further
decomposition of $\widehat{G}_{\Delta,l}(z,\overline{z})$ in a basis of
special orthogonal functions associated to the group $SO(d-1)$, known as the
zonal harmonics. The statements of positivity and of irreducibility are then
statements regarding the harmonic expansion of the intertwining kernel.

Although the topic of harmonic decomposition is well understood and standard,
it is pedagogically worthwhile to review it to setup the logic for when we
tackle the spinor case, which is similar but more intricate. We now take a
detour to representation theory, before continuing on to the construction of
the complementary and exceptional series.

###  2.2 Mathematical Interlude : Harmonics Polynomials And $SO(d)$

We have taken a generic function of three vector variables $p$, $z_{1}$ and
$z_{2}$, and have found it can be recast as a scaling piece and a function of
the vectors $z^{\mu}$ and $\overline{z}^{\mu}$ living in $\mathbb{R}^{d-1}$,
which are no more null. The resulting homogeneous polynomial in
$z,\overline{z}$ is an encoding of a specific bi-tensor . Such an object can
be decomposed on a basis of projectors on definite representations
Bargmann:1977gy ; Dolan:2001wg ; Dolan:2004up ; Dolan:2012wt ; Costa:2014rya .

An irreducible representations of $SO(d-1)$ can be represented by polynomials
in the following manner. As any (tensorial) representation lies in the
repeated tensor product of the vector representation, we can identify a
representation by specifying its projector over a space of tensors of a given
number $m$ of indices, $\mathcal{T}_{m}^{(d-1)}$. In this section, we deal
with symmetric traceless tensors, whose projectors satisfy

$\displaystyle\pi_{m}{}_{\mu_{1}\ldots\mu_{m}}{}^{\nu_{1}\ldots\nu_{m}}$
$\displaystyle=\pi_{m}{}_{(\mu_{1}\ldots\mu_{m})}{}^{(\nu_{1}\ldots\nu_{m})}\,,$
(16) $\displaystyle
h^{\mu_{i}\mu_{j}}\pi_{m}{}_{\mu_{1}\ldots\mu_{m}}{}^{\nu_{1}\ldots\nu_{m}}$
$\displaystyle=0\,,$
$\displaystyle\pi_{m\mu_{1}\ldots\mu_{m}}{}^{\sigma_{1}\ldots\sigma_{m}}\pi_{m\sigma_{1}\ldots\sigma_{m}}{}^{\nu_{1}\ldots\nu_{m}}$
$\displaystyle=\pi_{m}{}_{\mu_{1}\ldots\mu_{m}}{}^{\mu_{1}\ldots\mu_{m}}\,.$

In what follows, we will take all the indices to be perpendicular to $p$. One
can simply take $h$ to stand in for the induced metric on the orthogonal plane
to $p$, instead of the identity matrix. We will denote the inner product with
$h$ using the symbol $a\circ b$, $a^{2}=a\circ a$. This projector can be
converted into a polynomial, by contracting this projector with generic
complex vectors. Using generic vectors $z$ and $\overline{z}$ in
$\mathbb{C}^{d-1}$, the resulting object satisfies the properties

$\displaystyle\pi_{m}(z,\overline{z})$ $\displaystyle=z^{\mu_{1}}\ldots
z^{\mu_{m}}\pi_{m}{}_{\mu_{1}\ldots\mu_{m}}{}^{\nu_{1}\ldots\nu_{m}}\overline{z}_{\nu_{1}}\ldots\overline{z}_{\nu_{m}}\,,$
(17) $\displaystyle\pi_{m}(\lambda z,\rho\overline{z})$
$\displaystyle=(\lambda\rho)^{m}\pi_{m}(z,\overline{z})\,,$
$\displaystyle\partialderivative{z}\circ\partialderivative{z}\pi_{m}(z,\overline{z})$
$\displaystyle=0\,,$

Which defines $\pi$ as an homogeneous polynomial, where tracelessness
translates into the projector being a harmonic function.

The upshot is that a given (STT) representation of $SO(d-1)$ is equivalent to
a given harmonic homogeneous polynomial. These are called spherical harmonics.
The adjective spherical is associated to the homogeneity property, which
allows us to fix $\pi$ given its value on the sphere. These conditions
uniquely define the spherical harmonics. Start from the ansatz

$\displaystyle\pi_{m}(z,\overline{z})=(z^{2}\overline{z}^{2})^{\frac{m}{2}}f\left(t=\frac{z\circ\overline{z}}{\sqrt{z^{2}\overline{z}^{2}}}\right)\,,$
(18)

where $f$ is a polynomial of order $m$ in the variable $t$, harmonicity
translates into a differential equation for $f$

$\displaystyle(t^{2}-1)f^{\prime\prime}(t)+(d-2)tf^{\prime}(t)$
$\displaystyle=l(l+d-3)f(t),$ $\displaystyle f(t)$
$\displaystyle=c_{(m)}C^{h-\frac{3}{2}}_{m}(t)\,.$ (19)

Where the factor of $d$ is the dimension of the full-space, matching the rest
of this work, not the $p$-orthogonal one, and $C_{m}$ is the $m$-th Gegenbauer
polynomial. We will sometime use $\alpha=h-\frac{3}{2}$ as a shorthand. The
normalisation can be fixed by considering the leading order
$(z\circ\overline{z})^{m}$ terms which must be set to one, giving

$\displaystyle c_{(m)}$
$\displaystyle=\frac{m!}{2^{m}\left(h-\frac{3}{2}\right)_{m}}\,,$ (20)

Which then fully defines the spherical harmonics.

Spherical harmonics are adapted to decompose a tensor of $SO(d-1)$ of fixed
rank into its irreducible components. The problem at hand for us is however
slightly different, it is the decomposition of a tensor of $SO(d)$ into its
$SO(d-1)$ irreducible pieces. Such a decomposition must encode the branching
of representations, and the associated basis are called the zonal harmonics.
Indeed, consider a spherical harmonic of $SO(d)$ pulled-back to the space
$\mathbb{R}^{d-1}$. The resulting polynomial, although originally traceless,
will not be traceless with respect to induced metric. It is then reducible,
allowing it to be decomposed into a sum of harmonics of $SO(d-1)$ associated
to spins $m\leq l$, multiplied by appropriate trace terms.101010For $d=3$, the
projectors over $SO(3)$ are a special form of the Gegenbauer polynomial given
by Legendre Polynomials. There, the decomposition into irreducible
representations of the subgroup $SO(2)$ is done by Fourier decomposition and
one has $-l\leq m\leq l$ instead. For $d=2$, the zonal decomposition is
singular since the representations are all one-dimensional.

To make this explicit, we introduce an external vector $p$ into the
discussion, and a bigger inner-product $\cdot$, such that $p\cdot p=1$ and
$a\circ b=a\cdot b-a\cdot p\ b\cdot p$. The vector $p$ encodes the breaking of
$SO(d)$ into $SO(d-1)$. The breaking of the vector representation gives a
breaking of the tensor representation living in its tensor-product, giving
rise to projectors $\pi^{\ell}_{m}$ from $\mathcal{T}^{(d)}_{\ell}$ into STT
representations with spin-$m$ of $SO(d-1)$, $\ell\geq m\geq 0$, which are
called the zonal projectors. By definition, they satisfy

$\displaystyle\pi^{\ell}_{\ell\
}{}_{\mu_{1}\ldots\mu_{\ell}}{}^{\nu_{1}\ldots\nu_{\ell}}$
$\displaystyle=\pi_{\ell\
}{}_{\mu_{1}\ldots\mu_{\ell}}{}^{\nu_{1}\ldots\nu_{\ell}}\,,$ (21)
$\displaystyle p^{\mu_{1}}\ldots p^{\mu_{\ell-m+1}}\pi^{\ell}_{m\
}{}_{\mu_{1}\ldots\mu_{\ell}}{}^{\nu_{1}\ldots\nu_{\ell}}$
$\displaystyle=0\,,$ $\displaystyle\pi^{\ell}_{m\
}{}_{\mu_{1}\ldots\mu_{\ell}}{}^{\sigma_{1}\ldots\sigma_{\ell}}\pi^{\ell}_{m^{\prime}\
}{}_{\sigma_{1}\ldots\sigma_{\ell}}{}^{\nu_{1}\ldots\nu_{\ell}}$
$\displaystyle=\delta_{m,m^{\prime}}\pi^{\ell}_{m\
}{}_{\mu_{1}\ldots\mu_{\ell}}{}^{\mu_{1}\ldots\mu_{\ell}}\,,$

and acting on a given spin-$\ell$ tensor of $SO(d)$, outputs its spin-$m$
symmetric traceless tensor of $SO(d-1)$ part.

Using the polynomial encoding, it is straightforward to find an expression for
the zonal harmonics. As previously alluded, these projectors must involve the
spherical harmonics with some “trace term” encoding the non-zero contractions
in the $p$-direction. We extend the vectors $z$, $\overline{z}$ in the $p$
direction such, allowing us to write objects such as $z\cdot p$, while $z\circ
p=0$ still, as the $\circ$ product is $p$ orthogonal.111111We derive the zonal
projectors first as functions of $\mathbb{C}^{d}\times\mathbb{C}^{d}$
initially, which are then restricted to live on the subset of null vectors,
$\mathbb{K}^{d}\times\mathbb{K}^{d}$ with
$\mathbb{K}^{d}=\\{z\in\mathbb{C}^{d}\mid z\cdot z=0\\}$. The zonal harmonics
are harmonic functions of the reduced laplacian
$\partial_{z}\circ\partial_{z}$, but not of the full laplacian
$\partial_{z}\cdot\partial_{z}$ which includes the $p$ direction. Hence, they
must be proportional to spherical harmonics for $SO(d-1)$, up to polynomial
functions of $z\cdot p$ and $\overline{z}\cdot p$. The following ansatz
follows,

$\displaystyle\pi_{m}^{\ell}(z,\overline{z})$ $\displaystyle=a_{\ell,m}(z\cdot
p\ \overline{z}\cdot p)^{\ell-m}\pi_{m}(z,\overline{z}),\quad\quad\ell\geq
m\geq 0\,,$ (22)

for some normalisation choice $a_{\ell,m}$. The normalisation is fixed by
requiring that the harmonics offer a partition of unity on the space of
symmetric tensors of $SO(d)$ with $l$ indices $\mathcal{ST}^{(d)}_{\ell}$,

$\displaystyle\sum_{m=0}^{\ell}\pi^{\ell}_{m}(z,\overline{z})$
$\displaystyle=\left(z\cdot z\
\overline{z}\cdot\overline{z}\right)^{\ell/2}\frac{\ell!}{2^{\ell}(\alpha+\frac{1}{2})_{\ell}}C_{\ell}^{\alpha+\frac{1}{2}}\left(\frac{z\cdot\overline{z}}{\sqrt{z\cdot
z\ \overline{z}\cdot\overline{z}}}\right)\,.$ (23)

By specialising to the configuration for which writing $z\cdot z=(z\cdot
p)^{2}$, one can rewrite this as a decomposition of a function of $t$ into a
sum of Gegenbauer polynomial, which can be inverted using an integral
identity. The output of this computation is the normalisation

$\displaystyle
a_{\ell,m}=\binom{\ell}{m}\frac{2^{\ell-m}\left(m+\alpha+\frac{1}{2}\right)_{\ell-m}}{\left(2m+2\alpha+1\right)_{\ell-m}}\,.$
(24)

By construction, the zonal harmonics give an orthogonal decomposition of the
set $\mathcal{ST}_{\ell}$. Any map $\tau$ from $\mathcal{ST}_{\ell}$ to itself
can then be written as a weighted sum of zonal harmonics

$\displaystyle\tau\in\mathcal{ST}_{\ell}^{(d)}\Rightarrow\tau=\sum_{m=0}^{\ell}\kappa_{m}\,\pi^{\ell}_{m}\,.$
(25)

Such a map is positive definite provided $\kappa_{m}\geq 0$. If some
projectors have negative coefficient, although it is possible to define a
positive map by restricting to a subspace orthogonal to the faulty projectors,
restricting to this subspace defines a covariant equations of motions
Weinberg:1995mt . This is not conformally invariant in general. The correct
procedure is to write the most general conformally invariant kernel and then
consider whether it is positive. One is free to restrict to a subspace only if
it is invariant. This can happen if some projectors are absent (simply
quotienting by null states), or if there exists a conformally invariant
splitting. The latter must be associated to a freedom in the choice of the
kernel.

###  2.3 Complementary Series

We have the tools setup to decompose the intertwiner unto the basis of zonal
harmonics. To do so, we consider a specialisation of the previously derived
partial wave expansion. If we pick $z\cdot z=0$, we can identify $z\cdot
p=\sqrt{z\circ z\ p\cdot p}$, and similarly for $\overline{z}$. To pull out
the factors corresponding to a given harmonic, one can use the completeness
relation of the Gegenbauer polynomial. To extract the $m$-th partial wave we
need to perform the integral

$\displaystyle\int_{-1}^{+1}(1-t^{2})^{\alpha-\frac{1}{2}}C^{\alpha}_{m}(t)P_{\ell}^{(h-2,\nu-\ell)}\left(t\right)\,\mathrm{d}t\,,$
(26)

which is quite complicated. The tactic we used is to rewrite the Gegenbauer
polynomial as a Jacobi Polynomial, and using eq. 7.391.9 of Gradshtein:2015aa
. The result is an expansion

$\displaystyle\widehat{G}_{\Delta,\ell}(p)=(p\cdot
p)^{\Delta-h}N_{\Delta,\ell}\sum_{m=0}^{\ell}\kappa_{\ell,m}(\Delta)\,\pi^{\ell}_{m}(z_{1},z_{2},\hat{p})\,.$
(27)

Rather miraculously, the final result takes a simple and palatable form. It
was first computed in Dobrev:1977qv , and is given by

$\displaystyle N_{\Delta,\ell}$
$\displaystyle=\frac{C_{\Delta,\ell}(-1)^{l}\pi^{h}}{l!}\frac{\Gamma(h-\Delta)}{\Gamma(\Delta-1)(\Delta+\ell-1)}\,,$
$\displaystyle\kappa_{\ell,m}(\Delta)$
$\displaystyle=\frac{\left(\overline{\Delta}+m-1\right)_{\ell-m}}{\left(\Delta+m-1\right)_{\ell-m}}\,.$
(28)

The overall constant $N_{\Delta,\ell}$ must be fixed by picking a
normalisation of the kernel. There is no single choice for this normalisation,
instead there are various ones suitable for different applications, and often
authors have used a panoply of them. As previously mentioned, since the
coefficients coming from the Fourier transform might be singular for some
values of the scaling dimension, it seems disingenuous to try to tune
$C_{\Delta,\ell}$ to remove all of this prefactor, as it encodes genuine
information and would hide what happens. Instead we take the much simpler
perspective that the only requirement is that the overall factor must be
positive (if need be, an overall minus sign can always be adjusted), making us
pick a simple normalisation for the 2pt. function

$\displaystyle C_{\Delta,\ell}=\frac{1}{(-1)^{\ell}\pi^{h}\ell!}\Rightarrow
N_{\Delta,\ell}=\frac{\Gamma(h-\Delta)}{\Gamma(\Delta-1)(\Delta+\ell-1)}\,.$
(29)

At regular points, $N_{\Delta,\ell}$ is not infinite nor zero, and can fix the
overall sign provided all the Pochhamer symbols have the same sign. This will
define the complementary series representations. The other points are called
exceptional, and generate the associated series.

Consider now the situation where $N_{\Delta,\ell}$ is regular. One must
require that all the partial wave coefficients share the same sign, meaning

$\displaystyle\frac{\kappa_{m+1}}{\kappa_{m}}=\frac{\Delta+m-1}{\overline{\Delta}+m-1}\geq
0\,.$ (30)

The strongest constraint comes from $m=0$, which imposes $\Delta>1$ and
$d-1-\Delta>0$. Normalisability is in principle still an issue, which we have
to investigate.

For $\ell=0$ there is no such constraint, as there is a single term whose sign
can be tuned by rescaling the intertwiner. The only remaining criteria is
normalisability of the inner product. The argument is particularly clear in
the “compact” picture where one can put the origin and infinity on the same
footing Sun:2021thf . In the non-compact picture we used up to now, one must
check that the asymptotic behaviour of the wavefunction is normalisable.
Instead, one can work with wavefunctions defined on $\Omega\in S_{d}$, where
the scalar intertwiner takes the form

$\displaystyle
G_{\Delta}(\Omega_{1},\Omega_{2})=\frac{2^{\Delta}C_{\Delta}}{(1-\Omega_{1}\cdot\Omega_{2})^{\Delta}}\,,$
(31)

Which can be obtained by projecting down an embedding space expression
Weinberg:2010ws ; Costa:2011wa  or performing an inverse stereographic
projection on the position space inner product. Clearly, normalisability of
the scalar inner-product for given $\Delta$ implies that of the spinning
kernel as well. To know whether this defines a positive function over
$S_{d}\equiv K$, we decompose this bilinear function over the spherical
harmonics121212This harmonics decomposition is different from the previous
one. Here, we effectively replaced the partial-wave decomposition over the
subgroup $\widetilde{N}$ of translation by a decomposition over the larger
subgroup $K=SO(d)$.

$\displaystyle G_{\Delta}$
$\displaystyle=\frac{C_{\Delta}}{\Gamma(\Delta)}\int_{0}^{\infty}\frac{\mathrm{d}s}{s}\,(2s)^{\Delta}e^{-s+s\Omega_{1}\cdot\Omega_{2}}$
(32)
$\displaystyle=C_{\Delta}\Gamma(h-\Delta)\frac{2^{d}\Gamma\left(\frac{d-1}{2}\right)}{4\sqrt{\pi}}\sum_{l}\frac{(d-1+2l)(\Delta)_{l}}{\Gamma(l+\overline{\Delta})}\,C_{l}^{\frac{d-1}{2}}(\Omega_{1}\cdot\Omega_{2})\,.$

Normalisability translates into the positivity of each of these coefficients,
which arises provided both $\Delta$ and $\overline{\Delta}$ are positive, i.e.
$0\leq\Delta\leq d$.

This analysis is clearly valid almost everywhere in $\Delta$, and defines the
range and properties of the complementary series $\mathcal{C}_{\Delta,\ell}$.
The caveats are the points where the harmonic decomposition is singular, which
defines exceptional points, admitting other representations.

###  2.4 Understanding Exceptional Points : Scalars

All representations can be found as subrepresentations of the elementary ones.
From our previous analysis, we have found that the representations are
irreducible for generic $\Delta$, and become unitary in some restricted range.
The caveats of our analysis are the exceptional points, where the
representation becomes reducible. Such a distinguished space arises when, for
special values of $\Delta$, the intertwiner develops a non-trivial kernel, and
the Fourier transform or partial wave expansion becomes singular. This is most
easily diagnosed by considering the automorphism generated by performing the
shadow transform forward and back, which is caracterised by a constant

$\displaystyle G_{\Delta,l}\circ G_{d-\Delta,l}=\mathcal{N}_{\Delta,l}\,,$
(33)

which is inversely-proportional to the Plancherel measure of $SO(1,d+1)$
Knapp:1971aa ; Karateev:2018oml ; Karateev:2018uk . It is simply given by
$N_{\Delta,l}N_{d-\Delta,l}$. It is, almost everywhere, a regular function of
$\Delta$. It admits poles and zeros at specific integer points, where the
isomorphism breaks down into an exact sequence.131313In what follows, we only
care about the zeros/irregularities which occur outside of the complementary
series. There are spurious poles for $\Delta=h+n$, $n\in\mathbb{N}$, due to
the kernel $(x-y)^{-2(h+n)}$ not being conformally invariant Bzowski:2013sza ;
Bzowski:2015pba ; Bzowski:2018fql . For points in the complementary series, we
can always regulate those by taking limits from the interior of the series,
but this is not even needed, nor an issue. At those points, we can use the
ultra-local kernel $(-\Box_{x})^{n}\delta^{d}(x-y)$ instead. For $n\leq h$,
this kernel is still the inverse of $G_{h-n}$, and the isomorphism is
preserved. There is no interesting invariant subspace generated by these
points. Outside of the complementary range, we must be more careful, as there
could be cancellation of zero and poles. For odd $d$, the zero and poles miss
each other by half-steps, and we also find there are no interesting invariant
subspaces. For even $d$ and $n>h$ however, the points where $G_{h+n}$ becomes
ultra-local coincides with the points where $G_{h-n}$ is polynomial and
singular, which might cause issue. An explicit computation shows that they
still annihilate each-other, and our discussion can still follow through.
Effectively, we can perform the analysis in dimensional regularisation keeping
$d$ as an external parameter. To understand how to deal with this situation
and its interpretation, it is useful to consider the case of the scalar
operator as an illustrative example of the general structure. For $l=0$, the
intertwiner is given by

$\displaystyle\widehat{G}_{\Delta}(p)$
$\displaystyle=\frac{\Gamma(h-\Delta)}{\Gamma(\Delta)}\absolutevalue{p}^{2\Delta-d},$
$\displaystyle\mathcal{N}_{\Delta}$
$\displaystyle=\frac{\Gamma(h-\Delta)\Gamma(\Delta-h)}{\Gamma(d-\Delta)\Gamma(\Delta)}\,.$
(34)

Figure 2: A view of the $\Delta\in\mathbb{R}$ line for scalars. At the centre,
there is the unitary complementary series. Outside of its limits in red, we
encounter the integer spaced exceptional points.

The Plancherel measure has a pole for $\Delta=-n\in-\mathbb{N}$. This is
linked to the original Fourier transform. Indeed, the 2pt. function $K_{-n}$
reduces to a polynomial in position space, and in momentum space it becomes a
derivative of Dirac delta function

$\displaystyle G_{-n}\propto(x-y)^{2n}\to\Box_{p}^{n}\delta^{d}(p)\,,$ (35)

the image of which is the set
$\imaginary(G_{-n})\equiv\mathcal{E}_{n}\subset\mathcal{F}_{-n}$ of
polynomials of order up to $n$. This forms an invariant subspace of
$\mathcal{F}_{-n}$. Since this is a finite dimensional representation and the
de Sitter group is not compact, it cannot be unitary except for the trivial
case $n=0$ corresponding to the identity. Although very important in some
applications with regards to weight-shifting operators Costa:2011dw ;
Karateev:2018uk ; Costa:2018mcg ; Baumann:2019oyu , we will not consider these
further. Likewise, the kernel
$\text{Ker}(G_{-n})\equiv\mathcal{V}_{n}\subset\mathcal{F}_{d+n}$ is also an
invariant subspace. These two spaces are candidate representations, which are
orthogonal under the natural pairing $(\cdot,\cdot)$, i.e.
$\mathcal{E}_{n}\perp\mathcal{V}_{n}$.

One might wonder whether their complements give other interesting subspaces.
The space $\nicefrac{{\mathcal{F}_{d+n}}}{{\mathcal{V}_{n}}}$ is finite
dimensional, and made up of functions which are polynomials of degree up to
$n$ in $p$, i.e. this is a finite dimensional space of differential operators.
The map from these differential operators to $\mathcal{E}_{n}$ is surjective,
linear, and has a trivial kernel ; this makes it a bijective map, making the
two spaces isomorphic. This logic follows through for the other quotient space
: $\mathcal{V}_{n}$ can be understood either as the subspace of states in
$\mathcal{F}_{d+n}$ for which the naive inner-product is ill-defined, or as
the equivalence class of states in $\mathcal{F}_{-n}$ defined up to
polynomials.

This situation can be efficiently encoded in an exact sequence of linear map,
which forms the degenerate form of the shadow transform

Figure 3: For exceptional points, the shadow isomorphism breaks into a series
of exact sequences. Two irreducible spaces survives, and a single one is
unitary.

Having settled on the study of $\mathcal{V}_{n}$, this leaves us with a
puzzle. Our candidate representation sits in $\text{Ker}(G_{-n})$, and so we
must find a different way to furnish it with an (unitary) inner product. The
procedure to do so is systematic, and relies on the quotient space
realisation. It answers our two questions

  1. 1.

What is the kernel which defines an invariant inner-product on this
irreducible subspace ? This is done through the limiting kernel procedure. It
manifest irreducibility, but makes unitarity hard to assess.

  2. 2.

Is this kernel unitary ? This is done by relating the inner-product just
defined to the one induced by the (isomorphic) quotient-space picture.

We start by finding a suitable kernel. The key is to notice that we have more
freedom in choosing the integrand than previously. Under a conformal
transformation, the intertwiner is now allowed to transform in-homogeneously;
up to a piece proportional to $G_{-n}$. This limiting kernel
$\widetilde{G}_{\Delta}$ can be obtained by picking out the (non-trivial)
subleading term in the limit to the exceptional point,

$\displaystyle\lim_{\Delta\to-n}\left.\frac{G_{\Delta}}{(\Delta+n)}\right\rvert_{\mathcal{V}_{n}}=\lim_{\Delta\to-n}\partial_{\Delta}G_{\Delta}\big{\rvert}_{\mathcal{V}_{n}}=\widetilde{G}_{-n}\,.$
(36)

This operator transforms inhomogeneously under conformal transformation, but
it is conformally invariant on the subspace $\mathcal{V}_{n}$ Dobrev:1977qv .
We use it to define an inner product

$\displaystyle\bra{\psi_{1}}\ket{\psi_{2}}$
$\displaystyle=\left(\psi_{1},\widetilde{G}_{-n}\psi_{2}\right)\,,\,\psi_{1},\psi_{2}\in\mathcal{V}_{n}\,,$
(37) $\displaystyle\widetilde{G}_{-n}(x;y)$
$\displaystyle\propto(x-y)^{2n}\log(\mu^{2}(x-y)^{2})\,.$

It is most interesting to note at this point that the natural inner-product is
logarithmic. Naively, this seems to signal a breaking of the de Sitter group,
as it introduces a scale into the system. This is the source of great a many
confusions relating to the discrete/exceptional series. It is then very
important to remember that the module we consider is made up of properly
smeared states, for which the scale $\mu$ is immaterial. Such a 2pt. function
was found while trying to impose a regularisation of the zero mode of a shift-
symmetric scalar in de Sitter Anninos:2023lin , and is the generally expected
result from a Gupta-Bleuler approach to the quantisation of such a gauge-
symmetric scalar Bros:2010wa ; Epstein:2014jaa . This scale is immaterial and
does not break in any way the isometry, as is most clear in the quotient space
picture which we will now explain.

It is naturally quite complicated to directly analyse the unitary properties
of this inner product ; but luckily, we do not have to do it like previously.
$\mathcal{V}_{n}$ is isomorphic to the quotient
$\nicefrac{{\mathcal{F}_{-n}}}{{\mathcal{E}_{n}}}$. Between these spaces,
$G_{d+n}$ is bijective, and can be inverted. Since
$\mathcal{E}_{n}\perp\mathcal{V}_{n}$, any ambiguity in this inverse map
leaves the inner-product invariant and can be ignored. Given any choice of
inverse map

$\displaystyle\varphi:\mathcal{V}_{n}\to\mathcal{F}_{-n}\,,\,\phi\circ
G_{d+n}\big{\rvert}_{\mathcal{V}_{n}}=1\,,$ (38)

it induces an inner-product on $\mathcal{V}_{n}$ from that of
$\mathcal{F}_{-n}$

$\displaystyle\psi_{1},\psi_{2}\in\mathcal{V}_{n}\,,\;\bra{\psi_{1}}\ket{\psi_{2}}$
$\displaystyle=\bra{\varphi(\psi_{1})}\ket{\varphi(\psi_{2})}_{\mathcal{F}_{-n}}$
(39) $\displaystyle=\big{(}\psi_{1},\varphi\circ
G_{d+n}\circ\varphi(\psi_{2})\big{)}=\big{(}\psi_{1},\varphi(\psi_{2})\big{)}\,.$

The question of unitarity is then transferred onto the quotient space, where
we can instead analyse the simpler kernel we have already considered earlier.
The limiting kernel procedure produces precisely a construction of an inverse
map $\varphi$. This follows directly by taking the limit of the whole shadow
automorphism

$\displaystyle\lim_{\Delta\to-n}\frac{G_{d-\Delta}\circ
G_{\Delta}}{\Delta+n}\bigg{\rvert}_{\mathcal{V}_{n}}=G_{d+n}\circ\widetilde{G}_{-n}\big{\rvert}_{\mathcal{V}_{n}}=\lim_{\Delta\to-n}\frac{\mathcal{N}_{\Delta}}{(\Delta+n)}=\widetilde{\mathcal{N}}_{-n}\,,$
(40)

where the other pieces drop out by virtue of being restricted to the kernel of
$G_{-n}$.

With this result in hand, we can properly address unitarity, by looking at
$\nicefrac{{\mathcal{F}_{-n}}}{{\mathcal{E}_{n}}}$. This is non-trivial, since
as we have seen, although for we can make the (scalar) 2-pt. function
seemingly positive for any $\Delta\in\mathbb{R}$ by adjusting the overall
coefficient, it is generally not normalisable for $\Delta\geq d$. The
representation $\mathcal{V}_{n}$ will only be unitary if the quotient excises
precisely the non-normalisable modes.

To summarise, the space $\mathcal{V}_{n}\in\mathcal{F}_{d+n}$ is unitary if
$G_{d+n}$ is unitary once quotiented by its kernel, and $\mathcal{E}_{n}$
would be if the analogous statement hold for $G_{-n}$. The inner product would
then be given by the limiting kernel, whose unitarity is induced from the
quotient space picture. To assess these, it is useful to consider again the
compact-picture form of the kernel, which we had expanded over spherical
harmonics

$\displaystyle G_{-n}(\Omega_{1},\Omega_{2})$
$\displaystyle\propto\sum_{l=0}^{n}(-1)^{l}\frac{(d-1+2l)(n-l+1)_{l}}{\Gamma(l+d+n)}C^{\frac{d-1}{2}}_{l}(\Omega_{1}\cdot\Omega_{2})\,,$
(41) $\displaystyle G_{d+n}(\Omega_{1},\Omega_{2})$
$\displaystyle\propto\sum_{l=n+1}^{\infty}\frac{(d-1+2l)(n-l+1)_{l}}{\Gamma(l+d+n)}C^{\frac{d-1}{2}}_{l}(\Omega_{1}\cdot\Omega_{2})\,.$

We find that $G_{-n}$ only contains harmonics with $l\leq n$ and alternating
signs, while the higher ones decouple. The finite-dimensional space of
surviving harmonics, $\mathcal{E}_{n}$, is non-unitary as it should be. The
space $\mathcal{V}_{n}$ is identified with functions on the sphere with
harmonics $l>n$. The expression for $G_{d+n}$ shows the converse, that the
first $n$ modes decouple, and that the remaining modes appear with positive
coefficients, showing it defines a unitary normalisable inner-product.

Through the relatively intricate construction outlined, we have constructed
the scalar exceptional series representations $\mathcal{V}_{n}$,
$n\in\mathbb{N}$. These representations are sometimes called “exceptional type
I” representations Sun:2021thf , to distinguish them from their spinning
cousins. We have shown how this module has a logarithmic inner product, which
does not induce any issue since the states which are actually part of the
representations are those restricted to the physicality conditon ; effectively
smearing them. The physicality condition acts like the Gupta-Bleuler
quantisation procedure, where one considers a bigger hilbert-space containing
unphysical states as well, and one restrict to a physical slice. The seemingly
growing modes, associated to polynomial states, are pure gauge, as a
consequence of the group-theoretic construction.

###  2.5 Spinning Exceptional Series

Armed with the understanding of the scalar case, we can turn to the more
intricate question of generic $\ell$. The spectral function will have poles
for $\Delta=d-1+k$, $k\in\mathbb{N}$ again, but the situation is more
involved. Indeed, we must differentiate the case $0\leq t<\ell$, for which
$G_{1-t,\ell}$ is not polynomial but still missing projectors, to those with
$k=\ell+n$, $0\leq n$ where, as in the scalar case, $G_{1-\ell-n,\ell}$ is a
pure polynomial. This is illustrated in the following figure

Figure 4: A view of the $\Delta\in\mathbb{R}$ line for tensors. Beyond the
complementary series, there are two regions: those lying at integer point
point below and above $\ell$.

Let us concentrate first on the points at $d-1+\ell+n$, and their cousins at
$1-\ell-n$. The scalar discussion is mimicked here, in that we can define the
space
$\mathcal{E}_{\ell,n}=\imaginary(G_{1-(\ell+n),\ell})\subset\mathcal{F}_{1-(\ell+n),\ell}$,
which is finite dimensional and non-unitary. Likewise, we can define the
kernel of this map, $\mathcal{V}_{\ell,n}$, and the two spaces are orthogonal.
It is useful to look at the structure of these spaces somewhat more. A generic
element of $\mathcal{E}_{\ell,n}$ is a polynomial annihilated by some finite
order differential operator, conversely any element of $\mathcal{V}_{\ell,n}$
must have at least a certain number of derivatives associated to it.
Exploiting these ideas, these spaces can be written in the form

$\displaystyle\mathcal{E}_{\ell,n}$ $\displaystyle=\big{\\{}\text{Polynomials
}p(x)\in\mathcal{F}_{1-(\ell+n),\ell}\mid(z\cdot\partial_{x})^{n}p=0\big{\\}}\,,$
(42) $\displaystyle\mathcal{V}_{\ell,n}$
$\displaystyle=\big{\\{}f(x)\in\mathcal{F}_{d-1+(\ell+n),\ell}\mid\exists
g\in\mathcal{F}_{d-1+\ell,\ell+n}\,,\;(\mathcal{D}_{z}\cdot\partial_{x})^{n}g=f\big{\\}}\,.$

In the first definition, the use of the polarisation $z$ is simply to allow
the most general tensor differential equation that annihilates the polynomial.
In the second, we use the Todorov operator141414The Todorov operator is an
interior derivative to the set $\mathbb{K}^{d}$ of null complex vectors. It is
used to free indices which have been contracted with polarisation vectors
while ensuring tracelessness of the resulting tensor. See Bargmann:1977gy ;
Dobrev:1977qv ; Costa:2011wa . which releases indices of tensors in the index-
free notation, and allowed the elements of $\mathcal{V}_{\ell,n}$ to be the
most generic tensorial-function which will be in the kernel. The functional
space in which the prefunction $g$ lives can be read up from the scaling and
rotational properties of the field and the number of derivatives. In this
perspective, we see it is interesting to introduce the two (dual) weight-
shifting operators Dobrev:1977qv

$\displaystyle d$ $\displaystyle=z\cdot\partial_{x},\quad$
$\displaystyle\quad\overline{d}$
$\displaystyle=\mathcal{D}_{z}\cdot\partial_{x}\,.$ (43)

These differential operators are intertwiners, operators that relates together
different representations. In our case, we see they connect the space with
integers shifts below and above $\ell$.

With this in mind, we can consider the doublet of spaces
$\mathcal{F}_{1-t,\ell}$ and $\mathcal{F}_{d-1+t,\ell}$, $0\leq t<\ell$. The
map $G_{1-t,\ell}$ is singular as the projectors with $0\leq m\leq t$
disappear. This allows us to define the invariant subspaces
$\mathcal{U}_{\ell,t}=\text{Ker}(G_{1-t,\ell})$ and
$\widetilde{\mathcal{V}}_{\ell,t}=\imaginary(G_{1-t,\ell})$. The space
$\mathcal{U}_{\ell,t}$ is analogous to $\mathcal{E}_{\ell,n}$, in that it is
annihilated by some finite order differential operator, which ensures it
propagates no helicity above $t$, but it is infinite dimensional since its
element are not polynomials; it is potentially unitary.

Figure 5: The quartet diagram of Dobrev:1977qv , where each directed sequence
is exact. It follows there are four potentially interesting irreducible
subspaces. A single one is (generically) unitary.

These new spaces are not totally unrelated to the previous ones however, since
weight-shifting operators connects them together. These relations are encoded
in a diagram of exact sequences given in fig. 5. For convenience, we have
summarised the main rules regarding the interpretation of this type of
diagrams in appendix B. Although it looks quite daunting, these simply express
some redundancy in the spaces we defined, showing relations in the different
invariant subspaces. Most notably, these imply $\mathcal{V}_{\ell,n}$ is
isomorphic to $\widetilde{\mathcal{V}}_{\ell+n,\ell}$, making the latter
redundant.151515The reader might wonder what is the goal of this analysis,
instead of directly analysing the unitarity of the map $G_{1-t,\ell}$ which
induces the inner product on $\widetilde{\mathcal{V}}_{\ell,t}$. The issue is
that the harmonic decomposition of $G_{1-t,\ell}$ is quite unwieldy : its
first $t+1$ components diverge with positive residue, while the remaining ones
are positive. Once we quotient out the modes in $\mathcal{U}_{\ell,t}$, we are
restricted to the diverging modes. Circumventing this difficulty is certainly
welcome. This can be seen through the rewriting of the quartet given in fig.
6.

Figure 6: The images of $d^{n}$ and $\overline{d}^{n}$ are
$\widetilde{\mathcal{V}}_{\ell+n,\ell}$ and $\mathcal{V}_{\ell,n}$, so they
are surjective maps. Taking the quotient, we can make them injective as well.
It follows that $\mathcal{V}_{\ell,n}$ and
$\widetilde{\mathcal{V}}_{\ell+n,\ell}$ are isomorphic.

This brings us down to two candidates representations, $\mathcal{V}_{\ell,n}$
which corresponds to the outer points, and $\mathcal{U}_{\ell,t}$ which
corresponds to the inner points. For each of these spaces, we can repeat the
limiting kernel construction, finding a logarithmic correlator. Unitarity is
assessed by recognising that the limiting kernel is an inverse of the map from
the quotients. The question then reduces to whether the quotient spaces have
positive and normalisable inner products.

Consider first the space $\mathcal{V}_{\ell,n}$. Its norm is induced from
$G_{d-1+\ell+n,\ell}$, quotiented by its finite dimensional kernel. Through
the harmonic decomposition we previously computed, we find

$\displaystyle
G_{d-1+\ell+n,\ell}\propto\sum_{m=0}^{\ell}(-1)^{\ell-m}\frac{(n+1)_{\ell-m}}{(d-1+\ell+n)_{\ell-m}}\pi^{\ell}_{m}\,.$
(44)

All the modes up to $\ell$ survive, but with alternating
coefficients.161616This might seem counterintuitive, since we previously
claimed that this map should have a non-trivial kernel. The point is that the
kernel is located not in the intrinsic spin, but in the orbital spin. To make
it manifest, one would have to repeat our discussion but in the compact-
picture, doing the harmonic analysis over $S_{d}$, as we did for the scalar.
This is however not particularly enlightening, since the explicit position
space form of the reciprocal maps makes the presence of the kernel completely
transparent already. This map is not positive, and so the representation
$\mathcal{V}_{\ell,n}$ is not unitary, except for $\ell=0$ where it reduces to
$\mathcal{V}_{n}$.

For the space $\mathcal{U}_{\ell,t}$, we have to look at $G_{d-1+t,\ell}$
instead. Only the partial waves with helicities $t<m\leq l$ survive, and they
all appear with positive coefficient

$\displaystyle G_{d-1+t,\ell}$
$\displaystyle\propto\sum_{m=t+1}^{\ell}\frac{(\ell-1-t)!}{(m-1-t)!(d-1+\ell+t)_{\ell-m}}\pi^{\ell}_{m}\,.$
(45)

The inner product is then positive, but it remains to check whether it is
finite over our representation. Proofs can be found in Dobrev:1977qv ;
Sun:2021thf . As a quick argument, one can note that the asymptotic behaviour
of the spinning wave functions is not worse than the scalar ones, which were
normalisable at these exceptional points as seen from the compact picture.

The representation $\mathcal{U}_{\ell,t}$, $t=0,1,\ldots\ell-1$ is a
depth-$r$, $r=(\ell-t)$ partially-conserved tensor representation, since it
contains a tensor field which is annihilated by taking $r$ divergences. It is
sometimes called “Exceptional type II” representation, or spinning exceptional
series.

##  3 Story of Fermi : Rarita-Schwinger And Higher

We have seen how to construct bosonic representations by considering the
positivity of a partial wave decomposition of the intertwining kernel. In this
section, we generalise this to fields which transform as tensor-spinors. This
requires us to redevelop some less common machinery, related to Clifford
Analysis.

Once this setup is done, we are in a position to make our first result,
showing that there are no complementary or exceptional series representation
for tensor-spinors. Beautifully, this boils down to a simple fact, that
$(\gamma\cdot p)^{2}=p\cdot p>0$, and so eigenvalue comes in pairs. Since the
harmonic decomposition used is only valid for $d>3$, lower-dimensional cases
require a more specific formalism. We relegate the case $d=2,3$ to the next
section. There, we find that there exist unitary exceptional spinor
representations for $d=3$ only.

###  3.1 Statement of The Problem

The goal is now to extend our range of representations by picking states to
transform under the subgroup $M$ under representation $\rho$ of the $Spin(d)$
group. The most straightforward generalisation of the symmetric traceless
tensor is the symmetric-tensor spinor with $\ell$ indices, the bulk
counterpart of which are generalised Rarita-Schwinger fields. We will call
such objects higher-spinors or spinor-tensors.

In more details, we considers the wavefunctions
$\psi_{\Delta}^{\mu_{1}\ldots\mu_{\ell}}(x)$ transforming as primaries with
scaling dimension $\Delta$ which are spinor-valued. One can again define an
index-free formalism. For irreducibility, we impose

$\displaystyle\begin{aligned}
\delta_{\mu_{i}\mu_{j}}\psi_{\Delta}^{\mu_{1}\ldots\mu_{\ell}}&=0\\\
\gamma_{\mu_{i}}\psi_{\Delta}^{\mu_{1}\ldots\mu_{\ell}}&=0\end{aligned}$
$\displaystyle\begin{aligned}
\psi_{\Delta}^{(\mu_{1}\ldots\mu_{\ell})}&=\psi_{\Delta}^{\mu_{1}\ldots\mu_{\ell}}\\\
\psi_{\Delta}(x,z)&=z_{\mu_{1}}\ldots
z_{\mu_{\ell}}\psi_{\Delta}^{\mu_{1}\ldots\mu_{\ell}}\end{aligned}$ (46)

The task of constructing an inner product reduces again to that of finding a
conformally covariant 2pt. functions for fields of this type,

$\displaystyle\bra{\psi_{1}}\ket{\psi_{2}}=\int\mathrm{d}^{d}x\,\mathrm{d}^{d}y\,\overline{\psi_{1}}(x)\cdot\mathcal{G}_{\overline{\Delta},\ell}(x,y)\cdot\psi_{2}(y)\,,$
(47)

where $\mathcal{G}_{\overline{\Delta}}$ is now Clifford valued. Since we are
working with euclidean spinors, the Dirac conjugate is simply given by taking
the hermetian conjugate.

As is always the case, there exists a principal series of UIR with
$\Delta=\frac{d}{2}+i\nu$, as the $L^{2}$ inner product is unitary and
irreducible.171717Strictly speaking, the intersection with the real axis is
reducible Hirai:1962aa ; Dobrev:1977qv , which does not happen for bosons.
This is simply due to the quotient of the null descendant induced by the Dirac
equation, as we comment on later. The bulk interpretation of the scaling
dimension is slightly different for fermions, as the quantisation of massive
fermions yields $\Delta=\frac{d}{2}+im$, with $m$ the renormalised, Lagrangian
mass of the field, which must be real for a real lagrangian
Pethybridge:2021rwf ; Schaub:2023scu ; Letsios:2023qzq . This in itself is not
enough to discard more exotic representations, since they could ultimately be
realised in non-Lagrangian scenarios.

To investigate the potential complementary or exceptional series, we need to
pick $\Delta\in\mathbb{R}$ and look at the non-trivial 2pt. function of
spinor-tensors. These can be built using standard CFT techniques
Iliesiu:2015qra ; Iliesiu:2017nrv ; Isono:2017grm ,

$\displaystyle\mathcal{G}_{\Delta,\ell}(x,y)=C_{\Delta,\ell+\frac{1}{2}}\frac{\left(z_{1}\cdot
I_{(x-y)}\cdot z_{2}\right)^{\ell-1}}{(x-y)^{2\Delta+1}}\big{(}$
$\displaystyle\gamma\cdot(x-y)\left(z_{1}\cdot I_{(x-y)}\cdot z_{2}\right)$
(48) $\displaystyle+\frac{\ell}{d-3+2\ell}\gamma\cdot
z_{1}\gamma\cdot(x-y)\gamma\cdot z_{2}\big{)}\,.$

As before $I_{(x-y)}^{\mu\nu}$ is the inversion tensor appearing in the 2pt.
function of vector fields. The first term is mandated from conformal symmetry,
while the second one is simply a subtraction to ensure $\gamma$-tracelessness.

This kernel is already much more intricate than its bosonic counterpart.
Before delving into the Fourier transform and subsequent partial wave
decomposition, let us gain some intuition for what the fermionic nature of the
field implies. Consider $\ell=0$, i.e. the Dirac spinor. In momentum space,
the kernel reduces to

$\displaystyle\widehat{\mathcal{G}}_{\Delta,0}(p)$
$\displaystyle=p^{2\Delta-d}C_{\Delta,\frac{1}{2}}\frac{4^{h-\Delta}\pi^{h}\Gamma(h-\Delta-\frac{1}{2})}{2\Gamma(\Delta+\frac{1}{2})}\gamma\cdot\hat{p}$
(49)
$\displaystyle=p^{2\Delta-d}C_{\Delta,\frac{1}{2}}\frac{4^{h-\Delta}\pi^{h}\Gamma(h-\Delta-\frac{1}{2})}{2\Gamma(\Delta+\frac{1}{2})}\left(P_{+}-P_{-}\right)\,.$

This expression does not form a positive definite inner product. The issue
with complementary series spinor is glaring : the matrix $\gamma\cdot\hat{p}$
possesses two eigenvalues, $+1$ and $-1$, hence once decomposed over
projectors $\gamma\cdot\hat{p}\,P_{\eta}=\eta P_{\eta}$, it always gives rise
to an indefinite inner product on the representation.

The only way out is to kill the projection, by imposing $\gamma\cdot p\
\psi=0$. This translates into a covariant equation of motion for the
wavefunctions Weinberg:1995mt , restricting us to a short-multiplet, which
enforces it to be a free massless Dirac field. This field lives at the bottom
of the principal series, for $\nu=m=0$. Clearly, this fields possesses null
descendant, which must simply be quotiented out quite trivially.

In what follows, we will show that this is precisely what happens for higher-
spinors as well. For $d>3$, we will find the only possible way out is to
restrict to wave functions satisfying $\gamma\cdot p\ \psi=0$. However these
are precisely the free fields satisfying the Dirac equation and for which
$\Delta=\frac{d}{2}$, that sits on the tail of the principal series.

To make this statement concrete, we must proceed with the construction of the
harmonic decomposition. As we did previously, we need to compute the momentum
space kernel. It follows from the bosonic kernel by taking one further
derivative and shifting the weight. Luckily, one does not actually need the
explicit expression, simply the schematic expression,

$\displaystyle\widehat{\mathcal{G}}_{\Delta,\ell}\approx(p\cdot
p)^{\Delta-h}\left(\gamma\cdot\hat{p}\ldots+\gamma\cdot\hat{p}\,\gamma_{\mu\nu}z^{\mu}\overline{z}^{\nu}\ldots+\gamma\cdot
z\ldots+\gamma\cdot\overline{z}\ldots\right)$ (50)

With $z$ and $\overline{z}$ the polarisations projected in the orthogonal
plane to $\hat{p}$. Each term can be written in term of some differential
operator acting on Jacobi polynomials, but their precise form is in the end,
not needed.

The crucial matter is clearly, how to decompose an object such as eq. (50)
into projectors; into irreducible pieces with respect to the stabiliser group
$Spin(d-1)$. The conceptual issue is that one must define the spinorial
equivalent of the harmonics, which is not as well known as the bosonic analog.
This is the goal of the next section.

###  3.2 Intermezzo : Clifford Harmonics And $Spin(d)$

We have seen how the harmonic decomposition story goes for bosonic
representations in much details. The logic has been setup to do the same for
fermions. We identify representations by their projectors. The technicality
arises in that there is a family of four type of projectors to consider to
have a full partition of unity on the space of spinor-tensors.

We are interested in performing a harmonic decomposition of an object which
transforms as a symmetric tensor taking value not in the field of real or
complex numbers as previously, but in the Spinor representation. We will
consider vector indices in $SO(d-1)$, and the associated projector $\Pi$ must
take value in the Clifford algebra $\mathcal{C}\ell_{d-1}$. We first look at
the irreducible pieces. The central issue one has to face is that such
irreducible representations are of two types, the $\left[m+\frac{1}{2}\right]$
irreps, which we call traceless, and $\left[m-\frac{1}{2}\right]$ pieces,
which is conceptually similar to a zonal component already, and that we call
traceful. The first one is $\gamma$-traceless, while the second one is a pure-
gamma trace, and is fixed by requiring that the sum of the two give a
partition of the tensors of spin $m$. All of this is simply the projector
reformulation of the well known factors from the study of the tensor product
of spinor and vector representations of $SO(d-1)$. We thus define the
projector

$\displaystyle\Pi^{m}_{\pm}{}_{\mu_{1}\ldots\mu_{m}}{}^{\nu_{1}\ldots\nu_{m}}$
$\displaystyle=\Pi^{m}_{\pm}{}_{(\mu_{1}\ldots\mu_{m})}{}^{(\nu_{1}\ldots\nu_{m})}\,,$
(51)
$\displaystyle\delta^{\mu_{i}\mu_{j}}\Pi^{m}_{\pm}{}_{\mu_{1}\ldots\mu_{m}}{}^{\nu_{1}\ldots\nu_{m}}$
$\displaystyle=0\,.$

Consider then the generic $\left[m+\frac{1}{2}\right]$ “traceless” case, with
$\Pi^{m}_{+}$, which satisfies

$\displaystyle\gamma^{\mu_{i}}\Pi^{m}_{+}{}_{\mu_{1}\ldots\mu_{m}}{}^{\nu_{1}\ldots\nu_{m}}$
$\displaystyle=0\,.$ (52)

Meanwhile, the $-$ “traceful” projector will not be annihilated by the
contraction, and it will come together with $\Pi^{m}_{+}$ to completely span
the space of symmetric-traceless-spinor. This translate into the requirement
that the traceless and traceful parts sum into the tensorial
projectors,181818This type of equation is known in the Clifford analysis
literature as a Fischer decomposition Clif:Van-Lancker:2001aa ; Clif1 . It
encodes that harmonics for spinors form a refinement of spherical harmonics.

$\displaystyle\pi^{m}{}_{\mu_{1}\ldots\mu_{m}}{}^{\nu_{1}\ldots\nu_{m}}$
$\displaystyle=\Pi^{m}_{+}{}_{\mu_{1}\ldots\mu_{m}}{}^{\nu_{1}\ldots\nu_{m}}+\Pi^{m}_{-}{}_{\mu_{1}\ldots\mu_{m}}{}^{\nu_{1}\ldots\nu_{m}}\,.$
(53)

As we did in the bosonic case, these projectors can be encoded as a (matrix
valued)-polynomial by contracting its indices with polarisations. The
projectors satisfy the properties

$\displaystyle\Pi^{m}_{\pm}(z,\overline{z})$ $\displaystyle=z^{\mu_{1}}\ldots
z^{\mu_{m}}\Pi^{m}_{\pm}{}_{\mu_{1}\ldots\mu_{m}}{}^{\nu_{1}\ldots\nu_{m}}\overline{z}_{\nu_{1}}\ldots\overline{z}_{\nu_{m}}\,,$
(54) $\displaystyle\Pi^{m}_{\pm}(\lambda z,\rho\overline{z})$
$\displaystyle=(\lambda\rho)^{m}\Pi^{m}_{\pm}(z,\overline{z})\,,$
$\displaystyle\partialderivative{z}\circ\partialderivative{z}\Pi^{m}_{\pm}(z,\overline{z})$
$\displaystyle=0\,,$
$\displaystyle\gamma\circ\partialderivative{z}\Pi^{m}_{+}(z,\overline{z})$
$\displaystyle=0\,.$

The first three conditions are unchanged from before. The fourth condition is
a stronger requirement than the third one, and implies it.191919In the
Clifford analysis literature, see for example Clif1 ; Clif2 ;
Clif:Delanghe:1992aa ; Clif:Van-Lancker:2009aa ; Clif:Van-Lancker:2015aa ,
this null-Dirac condition is called (left) monogenicity. Clifford analysis is
concerned with the study of monogenic functions. The term monogenic stems from
the equation being a single derivative constraint. The theory of monogenics
generalises the study of harmonics, while offering parallels to the theory of
holomorphic functions. We will refer to such function as Clifford harmonics.
They tell us that a given representation of the spinor-tensor, specified
through its projector, is equivalent to a homogeneous Clifford polynomial.
Such objects we call Clifford spherical harmonics. This construction forms the
$Spin(d)$ analogue of our construction of the $SO(d)$ representations.

From basic properties of Clifford algebra, we know the projector $\Pi^{m}$ can
be expanded on a basis,

$\displaystyle\Pi_{m}(z,\overline{z})$
$\displaystyle=\Pi^{0}_{m}+\gamma^{\mu}\Pi^{1}_{m,\mu}+\gamma^{\mu\nu}\Pi^{2}_{m,\mu\nu}\,,$
(55)

where each of the $\Pi^{k}_{m}$ are referred to as subprojectors, and are
valued in the identity element of the Clifford algebra. The subprojectors must
be totally antisymmetric homogeneous polynomial of degree $m$ in the variables
$z$ and $\overline{z}$. This implies directly that the expansion in
subprojectors truncate at second order in the Clifford algebra.

#### Remark on the dimension dependence:

We will pick $\gamma_{\mu}\in\mathcal{C}\ell_{d}$ but still take $\mu$ to live
in $d-1$ dimensions, since we ultimately want to work with zonal harmonics. As
is often the case regarding spinors, care must be given to the number of
dimension one works in. If the dimension is odd or even, the chiral matrix can
be incorporated as follows

  * •

If $d=2k$, $\gamma_{\star}$ could be accommodated by multiplying by
$\frac{1\pm\gamma_{\star}}{2}$.

  * •

If $d=2k+1$, we can identify $\gamma\cdot p$ with the $\gamma_{\star}$ of
$\mathcal{C}\ell_{d-1}$.

For low enough dimension, the algebra degenerates and one cannot separate the
zonal decomposition and the decomposition onto definite $\gamma\cdot p$
eigenvectors

  * •

For $d=3$, the Pauli matrices satisfy
$\gamma^{\mu\nu}=\epsilon^{\mu\nu}\gamma_{3}$, and we can without loss of
generality pick $\gamma\cdot p=\gamma_{3}$.

  * •

For $d=2$, there is the exceptional simplification whereby
$\gamma^{\mu\nu}=0$, and the algebra fully degenerates to level one.

Back to the projectors. Using the scaling and Lorentz properties of $\Pi$, we
can write ansätze for the subprojectors,

$\displaystyle\Pi^{0}_{m}$
$\displaystyle=(z^{2}\overline{z}^{2})^{\frac{m}{2}}f_{0}(t)\,,$ (56)
$\displaystyle\Pi^{1}_{m,\mu}$
$\displaystyle=(z^{2}\overline{z}^{2})^{\frac{m}{2}}\left(\frac{z_{\mu}}{\sqrt{z^{2}}}f(t)+\frac{\overline{z}_{\mu}}{\sqrt{\overline{z}^{2}}}g(t)\right)\,,$
$\displaystyle\Pi^{2}_{m,\mu\nu}$
$\displaystyle=(z_{\mu}\overline{z}_{\nu}-z_{\nu}\overline{z}_{\mu})(z^{2}\overline{z}^{2})^{\frac{m-1}{2}}f_{2}(t)\,.$

Let us investigate the traceless projector. This gives the differential
equation

$\displaystyle\not{\partial}_{z}\Pi_{m}=0$
$\displaystyle=\partial^{\mu}\Pi^{1}_{\mu}+\gamma^{\mu}\left(\partial_{\mu}\Pi^{0}+2\partial^{\nu}\Pi_{\nu\mu}^{2}\right)+\gamma^{\mu\nu}\partial_{\mu}\Pi^{1}_{\nu}\,,$
(57)

which translates into a set of coupled differential equations for the
functions $f_{0},f_{2},f$ and $g$

$\displaystyle 0$
$\displaystyle=(d+m-2)f-\left(t^{2}-1\right)g^{\prime}+mtg\,,$ $\displaystyle
0$ $\displaystyle=f^{\prime}+tg^{\prime}-mg\,,$ (58) $\displaystyle 0$
$\displaystyle=2\left(t^{2}-1\right)f_{2}^{\prime}+2(m-1)tf_{2}-tf_{0}^{\prime}+mf_{0}\,,$
$\displaystyle 0$ $\displaystyle=2(d+m-3)f_{2}+f_{0}^{\prime}\,.$

The last equation can be inputted back into the third one to give a second
order differential system for $f_{0}$, which turns out to be the $m$-th
Gegenbauer Polynomial defining equation again,

$\displaystyle f_{0}(t)$
$\displaystyle=\frac{2\alpha+m}{2(\alpha+m)}c_{(m)}C^{\alpha}_{m}(t)\,,$
$\displaystyle f_{2}(t)$ $\displaystyle=-\frac{\alpha
c_{(m)}}{4\alpha+m}C^{\alpha+1}_{m-1}(t)\,.$ (59)

This however, does not fix the $f$ and $g$ functions, and this is actually a
feature of the problem at hand. We postpone this issue slightly; let us
require that the projector be traceless from the right as well, which gives
the same set of equation but with the role of $f$ and $g$ interchanged. One
can check that the resulting equations are not compatible with each others,
setting $\Pi^{1}=0$.

Taking a step back, these results make sense when considering the
normalisation of the projectors. If $z^{2}=0=\overline{z}^{2}$, we must reduce
to an expression which looks like $(z\circ\overline{z})^{m}$ minus the
$\gamma$-traces, which can be computed using the Todorov operator. Going
through this exercise, we find the normalisation condition

$\displaystyle(z\circ\overline{z})^{m-1}\left(z\circ\overline{z}-\frac{m}{d-3+2m}\gamma\circ
z\gamma\circ\overline{z}\right)$ (60) $\displaystyle=$
$\displaystyle\frac{d-3+m}{d-3+2m}(z\circ\overline{z})^{m-1}\left(z\circ\overline{z}-m\gamma_{\mu\nu}z^{\mu}\overline{z}^{\nu}\right)\,.$

Clearly, the projector for the $m+\frac{1}{2}$ spin component, will have no
solitary $\gamma_{\mu}$ term. The previous solution for $f_{0}$ satisfies
precisely this normalisation.

Now, armed with insights, we can look back at the previous construction and
understand the general structure faster. Indeed

$\displaystyle\Pi^{m}_{++}$
$\displaystyle=\left(1-\frac{1}{2(\alpha+m)}z\circ\gamma\gamma\circ\partial\right)\pi_{m}$
(61)
$\displaystyle=\pi_{m}\left(1-\frac{1}{2(\alpha+m)}\overset{\leftarrow}{\overline{\partial}}\circ\gamma\gamma\circ\overline{z}\right)\,,$

is $\gamma$-traceless on both sides by construction, and satisfies all the
properties of a Clifford harmonic. Moreover, this expression makes it easy to
show that

$\displaystyle\Pi^{m}_{++}(z,\partial)\Pi^{m}_{++}(z,\overline{z})$
$\displaystyle=\left(1-\frac{1}{2(\alpha+m)}z\circ\gamma\gamma\circ\partial\right)^{2}\pi_{m}(z,\partial)\pi_{m}(z,\overline{z})$
(62) $\displaystyle=\Pi^{m}_{++}(z,\overline{z})\,,$

making it a good projector. An explicit computation shows it to be identical
to the one given previously. In this form however, the generalisation to find
the traceful projector, and the zonal Clifford harmonics is trivial,

$\displaystyle\Pi^{\ell,m}_{++}$
$\displaystyle=\left(1-\frac{1}{2(\alpha+m)}z\circ\gamma\gamma\circ\partial\right)\pi^{\ell}_{m}\,,$
(63) $\displaystyle\Pi^{\ell,m-1}_{--}$
$\displaystyle=\left(\frac{z\circ\gamma\gamma\circ\partial}{2(\alpha+m)}\right)\pi^{\ell}_{m}\,.$

These projectors enjoys the following expected properties, that follow from
those of the gamma matrices and zonal harmonics

$\displaystyle\Pi^{\ell,m}_{\alpha\alpha}(z,\partial)\Pi^{\ell,m^{\prime}}_{\beta\beta}(z,\overline{z})$
$\displaystyle=\delta_{m,m^{\prime}}\delta_{\alpha\beta}\Pi^{\ell,m}_{\alpha\alpha}(z,\overline{z})\,,$
(64)
$\displaystyle\sum_{m=0}^{\ell}\Pi^{\ell,m}_{++}(z,\overline{z})+\Pi^{\ell,m}_{--}(z,\overline{z})$
$\displaystyle=\left(z\cdot z\
\overline{z}\cdot\overline{z}\right)^{\ell/2}\frac{\ell!}{2^{\ell}(\alpha+\frac{1}{2})_{\ell}}C_{\ell}^{\alpha+\frac{1}{2}}\left(\frac{z\cdot\overline{z}}{\sqrt{z\cdot
z\ \overline{z}\cdot\overline{z}}}\right)\,.$

For later convenience, we defined the traceful projector with a shift in the
indices, so that $\Pi^{\ell,\ell}_{--}=0$, and $\Pi^{\ell,-1}_{--}=0$ of
course. In this convention, it always picks the component with spin
$\left[m+1/2\right]$, and there is a single component with spin
$\left[\ell+\frac{1}{2}\right]$. The definitions are chosen so that both
$\Pi^{\ell,m}_{\pm\pm}$ projects out a component which has spin
$\left[m+\frac{1}{2}\right]$ out of the reduction of
$[\ell]\otimes\left[\frac{1}{2}\right]$.

There is yet another expression for $\Pi_{++}^{l,m}$ which is useful in
derivations. Since the Dirac operator squares to the Laplacian, we can
sandwich the zonal harmonics with appropriate factors to retrieve the
projector just given. Through an explicit computation one finds

$\displaystyle\Pi^{l,m}_{++}$
$\displaystyle=\frac{(2\alpha+l+m+1)(2\alpha+l+m+2)\gamma^{\mu}\gamma^{\nu}}{4(l+1)(2\alpha+2l+1)(\alpha+m+1)^{2}}\partialderivative{z^{\mu}}\partialderivative{\overline{z}^{\nu}}\pi_{l+1,m+1}(z,\overline{z})\,$
(65)

The projectors just obtained are certainly an allowed solution to the defining
equations, but it does not fully solve our problem. In principle, the $f$ and
$g$ functions can be non-zero, for a projector which is traceless on one side,
traceful in the other. This arises from the group-theoretical fact that in the
harmonic decomposition of $\left[\frac{1}{2}\otimes j\right]^{spin(d-1)}$
there are multiple factors of the same representation
$\left[m+\frac{1}{2}\right]$ and $\left[(m+1)-\frac{1}{2}\right]$ which arise
from the $++$ or $--$ sector. Since there are two projectors picking out the
same group representations, there is an ambiguity in their action. This
implies the existence of a map that transforms one into the other. This is
provided by the two interpolating projectors $\Pi^{l,m}_{+-}$ and
$\Pi^{l,m}_{-+}$, which involve the non-trivial $f$ and $g$ factors, and a
single $\gamma$-matrix.

Directly solving the equations for $f$ and $g$ is hard to interpret, as the
interpolating projectors are only relevant in the zonal expansion. It is more
efficient to consider building them as differential operators acting on the
zonal spherical harmonics. A satisfying solution is found to be given by

$\displaystyle\Pi^{l,m}_{+-}$
$\displaystyle=\sqrt{\frac{2\alpha+l+m+1}{l-m}}\frac{z\cdot p}{2(\alpha+m+1)}\
\gamma\circ\partial\pi_{l,m+1}\,,$ (66) $\displaystyle\Pi^{l,m}_{-+}$
$\displaystyle=\pi_{l,m+1}\gamma\circ\overset{\leftarrow}{\overline{\partial}}\
\frac{\overline{z}\cdot p}{2(\alpha+m+1)}\sqrt{\frac{2\alpha+l+m+1}{l-m}}\,.$

Note that by our previous definitions, $\Pi_{\pm,\mp}^{l,l}=0$, as there is
only a single way to pick out a spin $\left[l+\frac{1}{2}\right]$ component
out of the reduction of $\left[l\right]\otimes\left[\frac{1}{2}\right]$. One
can derive straightforwardly the full set of identities encoding the algebra
of these projectors

$\displaystyle\Pi^{l,m}_{\alpha\beta}(z,\partial)\Pi^{l,m^{\prime}}_{\sigma\lambda}(z,\overline{z})$
$\displaystyle=\delta_{m,m^{\prime}}\delta_{\beta\sigma}\Pi^{l,m}_{\alpha\lambda}(z,\overline{z})\,,$
(67)

and we note that these operators can further be dressed by multiplying by
appropriate factors of the projector $P_{\pm}=\frac{1\pm\cdot p}{2}$ over
definite eigenvalue of $\gamma\cdot p$.

###  3.3 No Complementary Nor Exceptional Fermions For $d>3$

From the form of the different projectors and given the schematic form of the
kernel in eq. (50), one can see how the harmonic decomposition proceeds,

$\displaystyle\overline{\psi}\cdot\widehat{\mathcal{G}}_{\Delta,\ell}\cdot\psi$
$\displaystyle\propto\sum_{m=0}^{\ell}\overline{\psi}\left(\gamma\cdot{p}\left(a_{\ell,m}\Pi_{++}^{\ell,m}+d_{\ell,m}\Pi_{--}^{\ell,m}\right)+b_{\ell,m}\Pi_{+-}^{\ell,m}+c_{\ell,m}\Pi_{-+}^{\ell,m}\right)\psi$
(68)
$\displaystyle=\sum_{m=0}^{\ell}\sum_{\eta=\pm}\begin{pmatrix}\overline{\psi}\Pi^{\ell,m}_{+-}P_{\eta}&\overline{\psi}\Pi^{\ell,m}_{--}P_{\eta}\end{pmatrix}\begin{pmatrix}-a_{\ell,m}\eta&b_{\ell,m}\\\
c_{\ell,m}&d_{\ell,m}\eta\end{pmatrix}\begin{pmatrix}P_{\eta}\Pi^{\ell,m}_{-+}\psi\\\
P_{\eta}\Pi^{\ell,m}_{--}\psi\end{pmatrix}$
$\displaystyle=\sum_{m=0}^{\ell}\sum_{\eta=\pm}\overline{\Psi}_{\ell,m,\eta}M_{\ell,m,\eta}\Psi_{\ell,m,\eta}\,.$

This is the analog of the bosonic harmonic decomposition in eq. (27).
Positive-definiteness is now a matrix constraint, which requires that all the
$M_{\ell,m,\eta}$ matrices have eigenvalues all of the same sign. This is a
stringent requirement, which is much harder to satisfy than the equivalent
bosonic statement. In fact, this is really an impossible requirement,
independently of the exact form of each of the coefficients
$a_{\ell,m},b_{\ell,m},c_{\ell,m},d_{\ell,m}$.

Consider the $m=\ell$ contribution,

$\displaystyle\sum_{\eta}\overline{\Psi}_{\ell,\ell,\eta}M_{\ell,\ell,\eta}\Psi_{\ell,\ell,\eta}$
$\displaystyle=\sum_{\eta=\pm}\eta\left(a_{\ell,\ell}\overline{\psi}\Pi^{\ell}_{++}P_{\eta}\psi\right)\,.$
(69)

These two contributions from the leading $l+1/2$ representation degenerate
into single block, which always comes in a pair with opposite signs. This
means the inner product is always indefinite. There is no constraint with can
be put on the field which would make the inner product sign definite, hence
there is no salvaging the complementary series. This argument repeats
identically for all helicities. Since each $2\times 2$ matrices appears in
pairs with identical determinant but opposite trace, it must be that the
eigenvalue flip sign with $\eta$. This traces back simply to the sign
indefiniteness of $\gamma\cdot p$, as in the scalar case.

We have shown in general that there are no complementary series spinors. Could
there be exceptional ones ? If one could separate out the modes with opposite
norms, yes. This is not possible in general dimension however, precisely
because of the form of the decomposition : since the eigenvalues of
$M_{\ell,m,\eta}$ for given $\ell,m$ and are independent of $\eta$ apart from
an overall sign, singular-points cannot select out modes with definite signs.
To be able to single them out would imply an invariant equation of the form
$\gamma\cdot\partial\psi=(\ldots)\psi$, which is not conformally invariant
except for the vanishing case. This reduces back to the “massless” spinors,
with $\Delta=\frac{d}{2}$.

An alternative understanding of this impossible splitting of modes with
opposite helicities can be gathered by thinking of special conformal
transformations. Consider the Dirac spinor, under special conformal
transformation it behaves like a two-level system (the positive and negative
helicities), and the rotational part $\sim\gamma_{\mu\nu}$ mixes them
together. This features persist for higher-spinor, where the opposite modes,
which are simply related to the two helicities of the Dirac spinor tensor-
product with the tensor modes, mix together under special conformal
transformation.

In conclusion, we have shown that there are no unitary complementary or
exceptional spinor-tensor representations for $d>3$. The caveats of our
discussion is that our harmonic decomposition degenerates for low-dimensions.
Indeed, the Clifford algebra contracts down as we reduce the dimension, and
the structure $\gamma_{\mu\nu}$ in the harmonic projectors start to interact
with $\gamma\cdot p$, mixing together $l,m$ and $\eta$. Hence, it is only for
$d=1,2,3$ that one can hope to find exceptional spinorial representations,
beyond the one encoded by fields satisfying the Dirac equation. These lower-
dimensional discussions are the topic of the next section.

##  4 Peculiarities of Lower Dimensions

We have found from our general analysis that in $d>3$ spinors must reside in
the principal series. The goal of this section is to perform the analogue of
the discussion of sec. 2 to the lower-dimensional cases $d=3,2$ not captured
by sec. 3. This analysis is quite general, since it can covers both fermionic
and bosonic representations simultaneously.

###  4.1 Kernel in $d=3$

The highest dimension where our spinorial harmonic decomposition breaks down
is $d=3$. The projectors over definite $\gamma\cdot p$ eigenvalue is no longer
independent from the projectors $\Pi_{\pm\pm}$. To properly assess this case,
we return to a formalism which is more dimension-specific.

A general rotational representation $\rho$ of $Spin(3)$ is given by the spin-j
representation $[j]$, with $j\in\frac{1}{2}\mathbb{N}$. This representation
corresponds to states or fields
$\mathcal{O}_{\alpha_{1}\ldots\alpha_{j}}=\mathcal{O}_{(\alpha_{1}\ldots\alpha_{2j})}$,
sitting in the irreducible component of the tensor product of $2j$ spinors
Iliesiu:2015qra ; Iliesiu:2017nrv . The fundamental spinor representation is
isomorphic to $\mathbb{C}^{2}$, and the gamma matrices can be taken to be the
Pauli matrices

$\displaystyle\sigma_{1}$ $\displaystyle=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\,,$ $\displaystyle\sigma_{2}$
$\displaystyle=\begin{pmatrix}0&i\\\ -i&0\end{pmatrix}\,,$
$\displaystyle\sigma_{3}$ $\displaystyle=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\,.$ (70)

We define index-free fields by contracting them with commuting polarisation
spinors $\overline{s}^{\alpha}$, and $s_{\alpha}$. The Kernel must then reduce
to the unique conformally invariant 2pt. function of these fields

$\displaystyle G_{\Delta,j}(x,y)$
$\displaystyle=\expectationvalue{\mathcal{O}_{j}(x,\overline{s}_{1})\overline{\mathcal{O}}_{j}(y,s_{2})}=C_{\Delta,j}\frac{\left(i\,\overline{s}_{1}\sigma\cdot(x-y)s_{2}\right)^{2j}}{(x-y)^{2\Delta+2j}}\,,$
(71)

Clearly, the polarisation of the correlator is encoded in the vector
$z_{\mu}=\overline{s}_{1}\sigma_{\mu}s_{2}$. Note that this vector is
generically complex but not null. We can now look at the harmonic
decomposition, the first step of which is to perform the Fourier transform

$\displaystyle\widehat{G}_{\Delta,j}(p)=\frac{4^{h-\Delta-j}C_{\Delta,j}\pi^{h}\Gamma(h-\Delta-j)}{\Gamma(\Delta+j)}(-z\cdot\partial_{p})^{2j}(p\cdot
p)^{\Delta+j-h}\,.$ (72)

The rightmost element can be related to an analytically continued Gegenbauer
polynomial

$\displaystyle(-z\cdot\partial_{p})^{2j}(p\cdot p)^{-\alpha}$
$\displaystyle=\derivative[j]{\epsilon}\left(p\cdot p-2\epsilon z\cdot
p+\epsilon^{2}z\cdot z\right)^{-\alpha}$ (73) $\displaystyle=(2j)!(p\cdot
p)^{-\alpha}\left(\frac{z\cdot z}{p\cdot
p}\right)^{j}C_{2j}^{\alpha}\left(t=\frac{z\cdot p}{\sqrt{z\cdot zp\cdot
p}}\right)\,.$

the variable $t$ encodes the angle between the exchanged momenta and the
spinorial polarisations,

$\displaystyle z\cdot p=\sqrt{p\cdot p\ z\cdot z}\cos(\theta)=\sqrt{p\cdot p\
z\cdot z}\ t\,.$ (74)

Since its index is negative, this Gegenbauer should properly be defined as a
special hypergeometric function. With $\alpha=h-\Delta-j$, and
$h=\frac{3}{2}$. For simplicity, we now set $\Delta=h+\nu$, giving

$\displaystyle
C_{2j}^{\alpha}(t)=\frac{(-2\nu-2j)_{2j}}{(2j)!}{}_{2}F_{1}\left(-2j,-2\nu;\frac{1}{2}-\nu-j;\frac{1-t}{2}\right)\,.$
(75)

###  4.2 Harmonic Decomposition And $d=3$ Complementary Series

To perform the harmonic decomposition, we will take a slightly different root
as before, and use think of it more in term of an Hilbert space rather than a
decomposition over special functions. Our states transform in (rotational)
irreducible representations $[j]$ of $Spin(3)$, which is made of the span of
the states $\ket{j,m}$, $2j\in\mathbb{N}$, $m=-j,-j+1,\ldots,j$, following
Kravchuk:2016qvl ; Erramilli:2019njx . These states are realised as the
symmetrised tensor product of the fundamental, spin-$\frac{1}{2}$ irreducible
representation. The stabiliser group of a given vector $p^{\mu}$ in $Spin(3)$
is given by a $Spin(2)$ subgroup, which irreducible representation $[m]$ is
(complex) one-dimensional and transforms under rotation by angle $\phi$ as
$\ket{m}\to e^{im\phi}\ket{m}$. The branching rule from $Spin(3)\to Spin(2)$
is very simply

$\displaystyle\text{Res}^{Spin(3)}_{Spin(2)}[j]=\sum_{m=-j}^{j}[m]\,.$ (76)

By picking the basis of the fundamental representation to be the $\gamma\cdot
p$ eigenvectors basis, this the branching rules become trivial, as each the
states $\ket{j,m}$ directly encode the branching. Said otherwise, we can pick
our basis choice so that each basis element transforms irreducibly under the
stabiliser group. This means, to perform the harmonic decomposition, we simply
have to look at the decomposition of the kernel over the finite dimensional
basis.202020One is free to chose some other decomposition procedure, for
example Fourier series. This specific decomposition is simply the most
physical one and makes the action of conformal symmetry the clearest.

We can be more explicit now. We will label the basis states of the spinor
representation $\ket{\frac{1}{2},\pm\frac{1}{2}}=\ket{\uparrow/\downarrow}$,
which we take to be eigenstates of $\sigma\cdot p$. The polarisation spinor
$s_{\alpha}$ then specifies a definite orientation of the spinors

$\displaystyle
s=\xi_{\uparrow}\ket{\uparrow}+\xi_{\downarrow}\ket{\downarrow}\,.$ (77)

Products of spinor polarisation define states living in the tensor product.
The link to the irreducible spin-$j$ representations state is obtained through

$\displaystyle\ket{\uparrow}^{j+m}\ket{\downarrow}^{j-m}=i^{j+m}\binom{2j}{j+m}^{-\frac{1}{2}}\ket{j,m}\,.$
(78)

Naturally, in this basis we have an equivalence between a certain collection
of polarisation component and a projector

$\displaystyle\overline{s}_{1}As_{2}$
$\displaystyle=\binom{2j}{j+m}\overline{\xi}_{1,\downarrow}^{j-m}\overline{\xi}_{1,\uparrow}^{j+m}\overbrace{\bra{j,m}A\ket{j,m}}^{A_{j,m}}\xi_{2,\downarrow}^{j-m}\xi_{2,\uparrow}^{j+m}\,,$
(79) $\displaystyle A$ $\displaystyle=A_{j,m}\ket{j,m}\bra{j,m}\,.$

Which means that we can clear out our expressions, by “freeing” the indices,
replacing the components of the polarisation spinor by projectors.
Effectively, we take our previous expressions, evaluate them for the specific
decomposition (77), and replace

$\displaystyle\overline{\xi_{1,a}}\xi_{2,b}\to\ket{a}\bra{b}\,,$ (80)

and we will understand expressions with multiplications of these kets to act
with the tensor product, i.e. $(\ket{a}\bra{b})^{2}=\ket{a}^{2}\bra{b}^{2}$.
More prosaically, the kets $\ket{\uparrow/\downarrow}$ are to be viewed as
shorthand for specific functions of the polarisation spinors with definite
transformation properties.

Our goal is now to expand the kernel in term of projectors over the
orthonormal basis vectors $\ket{j,m}$. An explicit computation gives

$\displaystyle\sqrt{z\cdot z}$
$\displaystyle=\ket{\uparrow}\bra{\uparrow}+\ket{\downarrow}\bra{\downarrow}\,,$
$\displaystyle z\cdot p$
$\displaystyle=\ket{\uparrow}\bra{\uparrow}-\ket{\downarrow}\bra{\downarrow}\,.$
(81)

The first result follows from a Fierz identity. We then use an hypergeometric
identity to rewrite

$\displaystyle\widehat{G}_{\Delta,j}(p)=$ $\displaystyle
C_{\Delta,j}\frac{\pi^{h}(-1)^{2j}4^{j-\nu}\Gamma(j-\nu)}{\Gamma(\Delta+j)}(p\cdot
p)^{\nu}(\ket{\downarrow}\bra{\downarrow})^{2j}$ (82)
$\displaystyle\times{}_{2}F_{1}\left(-2j,\frac{1+2\nu-2j}{2};1+2\nu-2j;1+\frac{\ket{\uparrow}\bra{\uparrow}}{\ket{\downarrow}\bra{\downarrow}}\right)$
$\displaystyle=$ $\displaystyle(p\cdot
p)^{\nu}N_{\Delta,j}\sum_{m=-j}^{j}\kappa_{j,m}(\Delta)\ket{j,m}\bra{j,m}\,.$

The harmonic coefficients can be extracted out of the $(j+m)$-th term in the
series expansion of the function on the second line

$\displaystyle\frac{1}{(j+m)!}\binom{2j}{j+m}^{-1}\derivative[j+m]{x}{}{}_{2}F_{1}\left(-2j,\frac{1+2\nu-2j}{2};1+2\nu-2j;1+x\right)\,,$
(83)

Which can be simplified as

$\displaystyle\kappa_{j,m}(\Delta)$
$\displaystyle=(-1)^{j-m}\frac{\Gamma\left(\Delta+m-1\right)\Gamma\left(\Delta-m-1\right)}{\Gamma\left(\Delta+j-1\right)\Gamma\left(\Delta-j-1\right)}\,,$
(84) $\displaystyle N_{\Delta,j}$
$\displaystyle=C_{\Delta,j}\frac{4\pi(\Delta-1)\Gamma(2-2\Delta)}{\Delta+j-1}\sin(\pi(\Delta+j))\,.$

These match the result found in Karateev:2018oml ; Erramilli:2019njx  using
recursion relation, up to some global phase different due to the signature.
These coefficients satisfy the relations

$\displaystyle\kappa_{j,m}(\Delta)\kappa_{j,m}(d-\Delta)=1\,,$ (85)

Which imply the shadow transform indeed generates an automorphism, away from
exceptional points.

With these tools set, we can discuss whether the kernel $G_{\Delta,j}$ defines
a unitary inner product.

#### Normalisability :

The general discussion of the scalar case in sec. 2 still applies, as it was
dimension agnostic. The harmonic decomposition is positive provided
$0<\Delta<d$. Beyond this, some modes can become non-normalisable and must be
checked case by case, as in the scalar-exceptional series.

#### Bosonic Representations :

We set $j\in\mathbb{N}^{\star}$. The harmonic coefficients must not switch
signs for different values of m, which can be checked by requiring

$\displaystyle
0\leq\frac{\kappa_{j,m+1}}{\kappa_{j,m}}=\frac{\Delta+m-1}{\overline{\Delta}+m-1}\,.$
(86)

The strongest bound comes from $m=0$, and gives $1<\Delta<d-1$, as previously.
The kernel is irregular and admits unitary irreducible exceptional series
representations for $\Delta=-n$, $n\in\mathbb{N}$ for $j=0$, and $\Delta=-t$,
$0\leq t<j$. Again, this reduces to the general discussion of 2.

#### Fermionic Representation :

We have $j\in\frac{1}{2}+\mathbb{N}^{\star}$ and $m$ is an half-integer. The
previous argument can be reproduced, and the strongest bound comes
$m=-\frac{1}{2}$ which imposes $\Delta=\overline{\Delta}=\frac{3}{2}$, the tip
of the principal series. In fact, opposite $m$ have contributions with
identical norms but opposite signs

$\displaystyle\frac{\kappa_{l,m}}{\kappa_{l,-m}}=(-1)^{2m}\,,$ (87)

meaning there is no room for complementary series fermions. This is really
another avatar of the higher-dimensional pairing of eigenvalues. Given this
situation, it is extremely surprising that one can construct unitary
extraordinary fermionic representation. The procedure to do so, which we
investigate in the next section, hinges on a decoupling of the modes with
opposite $m$ at the exceptional points.

###  4.3 Harmonic Gap And $d=3$ Exceptional Series

The harmonic decomposition we have computed has the interesting feature that
for specific values of the scaling dimension, $\Delta=d-1+j-r=2+j-r$ with
$r=1,\ldots\lfloor j\rfloor$, the middle modes with $m\leq j-r$ decouple,
creating an harmonic gap. The opposite happens for the conjugate dimensions
$\overline{\Delta}=1-j+r$. With this gap, the modes with positive and negative
$m$ become isolated, which allows us to split up the inner product

$\displaystyle G_{2+j-r,j}\to G^{+}_{2+j-r,j}\oplus G^{-}_{2+j-r,j}\,.$ (88)

That this is a conformally invariant separation, is counter-intuitive : we
have written the most general 2-point function and obtained a sum over both
positive and negative helicities. In fact, one can check that under a special
conformal transformation, the modes with different values of $m$ mix together
into their nearest neighbors. For the kernel written in the form (82),
conformal invariance is equivalent to imposing invariance under $K_{\mu}$,
giving a recursion relation Karateev:2018oml

$\displaystyle(\Delta+m-2)\kappa_{j,m}$
$\displaystyle=(\Delta-m-1)\kappa_{j,m-1}\,,$ (89)

that is solved by our previous solution.212121In higher dimension we have seen
that the up and down helicities mix together under special conformal
transformation. This is the same thing happening here, the difference is that
in higher dimension the harmonic decomposition could factor out the projection
along $p\cdot\gamma$, and it acted as an overall factor. In $d=3$ meanwhile,
this interacts with the rest of the helicities label, since $K_{\mu}$ mixes
one of the $2j$ spinor substates with its opposite helicities, and overall
this relates states which differ by $m+1$,$m$ and $m-1$. The specific
coefficients are then not even needed to understand that the harmonic gap
decouples the two sectors. It is really the structure of the harmonic
decomposition which changed with respect to higher dimensions. What changes
now, is that for the exceptional points, the harmonic gap opens up, isolating
the two sets of modes. Hence, for this specific choice of dimensions, one can
write a kernel improved by contact and improvement terms, whose harmonic
decomposition only contains only one set of modes.

For concreteness, consider the fully-conserved (bosonic or fermionic) field,
i.e. $\Delta=j+1$, $r=1$. The modes $m=\pm j$ are totally isolated and do not
mix under conformal transformations. One can write out a conformally covariant
kernel which picks them out,

$\displaystyle\widehat{G}_{j+1,j}^{\pm}(p)$ $\displaystyle=(p\cdot
p)^{j-\frac{1}{2}}N_{j+1,j}\bra{j,\pm j}\ket{j,\pm j}\,,$ (90) $\displaystyle
G_{j+1,j}^{\pm}(x,y)$
$\displaystyle=\frac{(i\,\overline{s}_{1}\sigma\cdot(x-y)s_{2})^{2j}}{2\left(x-y\right)^{4j+2}}\pm\frac{\pi^{2}(\overline{s}_{1}s_{2})(i\,\overline{s}_{1}\sigma\cdot\partial_{x}s_{2})^{2j-1}}{(2i)^{2j}(2j-1)!}\delta^{3}(x-y)\,.$

Expressions for generic $r$ can be found case by case by an inverse Fourier
transform. They take the form of a sum of terms sprinkling in factors of
$\overline{s}_{1}s_{2}$ and delta-functions. These improvement terms appear
with an overall sign which picks out the positive or negative helicities. This
feature, unique to $d=3$, has implications for both bosonic and fermionic
representations.

#### Bosonic representations :

The analysis carried out in sec. 2 still applies, and the representation
$\mathcal{U}_{\ell,t}$ is unitary and can be identified with the one extracted
out of the previous discussion under the identification $t=\ell-r$. This
representation is however not irreducible anymore, since it has a conformally
invariant subspace made up of the positive and negative helicities. These two
subspaces are conjugate under reflections

$\displaystyle\mathcal{U}_{j,t}\underset{d=3}{\longrightarrow}\mathcal{U}_{j,t}^{+}\oplus\mathcal{U}_{j,t}^{-}\,.$
(91)

The two “chiral” representations, $\mathcal{U}_{\ell,t}^{\pm}$, are the
unitary irreducible representations relevant for 3-dimensional partially
conserved tensors or depth $j-t$. These representations happen to be discrete
series, meaning their group element are square integrable Dobrev:1977qv . In a
parity invariant theory, one will obtain both of these representations in
pairs and the relevant inner-product will possess no improvement terms.

#### Fermionic representation :

As we have seen, the fermions are plagued by opposite mode having opposite
signs. For the exceptional points at $\Delta=d-1+j+k$, $k\in\mathbb{N}$, there
is no harmonic gap and this problem persist. Even worse, one finds that modes
on both sides alternate in signs. The only hope is for the previously
mentioned representations with $\Delta=d-1+j-r$. There, because of the gap, we
can consistently flip the sign of the kernel for the positive and negative
helicity modes so that both give rise to positive inner product :
$G^{+}_{2+j-r,j}$ and $-G^{-}_{2+j-r,j}$ are both positive-definite, and both
give rise a different representation $\mathcal{U}_{j,t}^{\pm}$, which are a
conjugate under reflections. Their sum is a positive, conformally invariant
inner-product on the fermionic representation, in which only the improvement
terms survive . It is furthermore clear that such a representation breaks
parity explicitly. This was evident already in the treatment of
Letsios:2023qzq ; Letsios:2023awz ; Letsios:2024nmf  which had found a
positive inner-product over the basis of mode functions which involved
$\gamma_{\star}$.222222From a $d=3$ perspective, it is not very surprising
that (exceptional) fermions break parity. For example, the mass term for a
free $d=3$ fermion breaks parity as well. These representations are the
exceptional partially-conserved higher-spinors with depth $\lfloor
j-r\rfloor$.

###  4.4 The Complex Plane And $d=2$

We now investigate the case of $d=2$. In what follows, $h$ will be a generic
complex weight and not $d/2=1$. Famously, the spin-group is isomorphic to the
Möbius group $Spin(3,1)\simeq SL(2,\mathbb{C})$ Bargmann:1946me ;
Gelfand:1947aa ; Duc:1967aa ; Bonifacio:2023ban  . Owing to the decomposition
$SL(2,\mathbb{C})\sim SL(2,\mathbb{R})\times SL(2,\mathbb{R})$, we work with
states and wavefunctions in complex coordinates, with no holomorphicity
assumptions. These are totally equivalent to complex states and wavefunctions
of $2$-real variables under the identification

$\displaystyle(x,y)\to z=x+iy\,.$ (92)

Induced representations are those which transform with weight
$(h,\overline{h})$ under the $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$
subgroup, which is manifest in the $(z,\overline{z})$ coordinates
DiFrancesco:1997nk ,

$\displaystyle L_{0}\ket{h,\overline{h};0}$
$\displaystyle=h\ket{h,\overline{h};0}\,,$
$\displaystyle\overline{L}_{0}\ket{h;\overline{h},0}$
$\displaystyle=\overline{h}\ket{h,\overline{h};0}\,,$ (93) $\displaystyle
L_{-}\ket{h,\overline{h}}$ $\displaystyle=0\,,$
$\displaystyle\overline{L}_{-}\ket{h,\overline{h};0}$ $\displaystyle=0\,,$

and the states at generic coordinate $(z,\overline{z})$ is constructed through

$\displaystyle\ket{h,\overline{h};z}=e^{zL_{+}+\overline{z}\overline{L}_{+}}\ket{h,\overline{h};0}\,.$
(94)

Having specified the transformation laws, we can define the smeared states
Bonifacio:2023ban . We pick as a measure over the complex number the usual
Lebesgue measure over $\mathbb{R}^{2}$

$\displaystyle\int_{\mathbb{C}}\mathrm{d}^{2}z=\frac{i}{2}\int_{\mathbb{C}}\mathrm{d}z\wedge\mathrm{d}\overline{z}=\int_{\mathbb{R}^{2}}\mathrm{d}\real(z)\,\mathrm{d}\imaginary(z)\,.$
(95)

This measure is invariant under $z\leftrightarrow\overline{z}$, as is clear
from writing it in the real parametrisation. This complex-space realisation
can be slightly cumbersome at times, because one must be careful about how
complex-conjugation act on the function versus the coordinates. For us,
complex conjugation will always act on the function, leaving the coordinates
unchanged. Everything is now setup to define our smeared states

$\displaystyle\ket{\psi}=\int_{\mathbb{C}}\mathrm{d}^{2}z\
\psi(z)\ket{z;h,\overline{h}}\,.$ (96)

By going back to the real coordinates, we have the identification
$h+\overline{h}=\Delta$ and $h-\overline{h}=m$. Single-valuedness implies
$2m\in\mathbb{N}$. Such a state or wavefunction transforms in the (one-
dimensional) representation $[m]$ of the $Spin(2)\equiv U(1)$ subgroup.
Finally, in the complex variables, the asymptotic condition on the
wavefunctions becomes

$\displaystyle\lim_{\absolutevalue{z}\to\infty}\psi_{h,\overline{h}}(z,\overline{z})\approx
z^{h-1}\overline{z}^{\overline{h}-1}\,.$ (97)

These requirements define the elementary representation
$\mathcal{F}_{2-\Delta,m}$ appropriate to $Spin(3,1)$. We equip this space
with an inner product

$\displaystyle\bra{f_{1}}\ket{f_{2}}=\int_{\mathbb{C}^{2}}\mathrm{d}^{2}x\
\mathrm{d}^{2}y\ \overline{f_{1}(x)}f_{2}(y)G_{h,\overline{h}}(x,y)\,,$ (98)

that is sesquilinear and hermetian. The kernel can be identified as the 2pt.
function of quasi-primaries fields in an Euclidean 2D CFT

$\displaystyle G_{h,\overline{h}}(x,y)$
$\displaystyle=\frac{i^{2m}C_{h,\overline{h}}}{(x-y)^{2h}(\overline{x}-\overline{y})^{2\overline{h}}}\,,$
(99)

Which under simultaneous complex conjugation and exchange $x\leftrightarrow y$
is left invariant. This ensures that
$\bra{f_{1}}\ket{f_{2}}^{\dagger}=\bra{f_{2}}\ket{f_{1}}$.

To check for unitarity, we must decompose this kernel over a distinguished
basis. We can again rewrite the inner product in Fourier space, using the
complex-version of the Fourier transform

$\displaystyle\widehat{f}(p)$
$\displaystyle=\int_{\mathbb{C}}\mathrm{d}^{2}z\,e^{\overline{z}p-z\overline{p}}\,f(z)\,,$
$\displaystyle f(z)$
$\displaystyle=\int_{\mathbb{C}}\frac{\mathrm{d}^{2}p}{\pi^{2}}\,e^{\overline{p}z-p\overline{z}}\,\widehat{f}(p)\,,$
(100) $\displaystyle\delta^{2}(z)$
$\displaystyle=\int_{\mathbb{C}}\frac{\mathrm{d}^{2}p}{\pi^{2}}\,e^{\overline{p}z-p\overline{z}}\,,$
$\displaystyle\left(\widehat{f}(p)\right)^{\dagger}$
$\displaystyle=\int_{\mathbb{C}}\mathrm{d}^{2}z\,e^{\overline{z}p-z\overline{p}}\,f^{\dagger}(z)=\widehat{f}^{\dagger}(-p)\,.$

These types of relations and transform are sometimes called Wigner-Weyl
transform, and are used in Quantum optics in the study of coherent states, see
for example Gerry_Knight_2004 . The inner product becomes

$\displaystyle\bra{f_{1}}\ket{f_{2}}=\int_{\mathbb{C}}\frac{\mathrm{d}^{2}p}{\pi^{2}}\,\left(\widehat{f_{1}}(p)\right)^{\dagger}\,\widehat{f_{2}}(p)\,\widehat{G}_{1-\overline{h},1-h}(p)\,.$
(101)

Having set up these conventions, we go forward with the Fourier transform,

$\displaystyle\widehat{G}(p)$
$\displaystyle=i^{2m}C_{h,\overline{h}}\int_{0}^{\infty}\frac{\mathrm{d}r}{r^{2\Delta-1}}\int_{0}^{2\pi}\mathrm{d}\varphi\
e^{2i\absolutevalue{p}rcos(\varphi-\arg(p))-2im\varphi}\,.$ (102)

The exponential can be rewritten as an infinite sum of Bessel functions,

$\displaystyle\hat{G}_{h,\overline{h}}(p)$
$\displaystyle=i^{2m}C_{h,\overline{h}}\sum_{n=-\infty}^{\infty}\left(-ie^{-i\arg(p)}\right)^{n}\int_{0}^{\infty}\mathrm{d}r\,\frac{J_{n}(2\absolutevalue{p}r)}{r^{2\Delta-1}}\int_{0}^{2\pi}\frac{\mathrm{d}\phi}{(2\pi)^{2}}\,e^{i(n-2m)\phi}$
(103) $\displaystyle=p^{2\overline{h}-1}\ \overline{p}^{2h-1}\
\frac{C_{h,\overline{h}}}{2\pi}\frac{\Gamma\left(1+m-\Delta\right)}{\Gamma\left(m+\Delta\right)}\,.$

This kernel as quite a simple form, since we obtain two overall
$p^{2\Delta-d}$ with $\Delta\sim h,\overline{h}$ and $d\sim 1$, as we would
expect for a 1-dimensional Fourier transform of an $SL(2,\mathbb{R})$ 2pt.
function. This kernel is normalisable for $0<\Delta<2$, as it is expected from
the scalar analysis. To investigate the positivity, let us pick
$\bra{f}\ket{f}$, which is real, and expand the $\widehat{f}(p)$ in Fourier
modes over the argument of $p$

$\displaystyle\bra{f}\ket{f}$
$\displaystyle\propto\sum_{n,k}\int_{0}^{\infty}\mathrm{d}r\,r^{2\overline{\Delta}-1}\int_{-\pi}^{\pi}\mathrm{d}\phi\,e^{i(n-k-2m)\phi}\,f_{n}(r)f_{k}^{\dagger}(r)$
(104)
$\displaystyle=\sum_{n}\int_{0}^{\infty}\mathrm{d}r\,r^{2\overline{\Delta}-1}\real\left(f_{n}(r)f_{n+2m}^{\dagger}(r)\right)\,.$

This final sum is not over positive numbers, except if $m=0$. This implies
that it is impossible to generate non-trivial spinning exceptional
representations, since those would always have negative norm sectors. This can
be understood differently, by noting that the subspace $\mathcal{U}_{\ell,t}$
is zero-dimensional, hence we are only left with non-unitary invariant
subspaces at exceptional points.

We have established that the unitary irreducible representations are the
principal series $\mathcal{P}_{1+i\nu,m}$, and the scalar complementary series
$\mathcal{C}_{\Delta}$, for $\Delta\in[0,2]$. We can consistently restrict
ourselves to this half of the segment since the other half is equivalent under
the shadow transform.

It is very reasonable at this point to wonder about the exceptional
representations, which were so important in other dimensions. They turn out
not to be new representations in $d=2$. First, consider representations with
non-zero spin $m$, they clearly suffer from the same non-positivity as the
complementary spinning representation, and so are ruled out. Consider now the
scalar exceptional series, as constructed in sec. 2.4. The previous logic does
go through unchanged, and one can build these representations, but they are
redundant with the spinning principal series.232323This is actually a feature
which also appears in higher dimensions. One can replace an exceptional
representation by a principal series one with more involved spin structure.
Exceptional representation are defined as equivalence classes, i.e. they have
a gauge symmetry. From a field perspective, the logic is to build an invariant
field-strength that picks out a given equivalence class. See comments in
Dobrev:1977qv . The reasoning is summarised in fig. 7, which we now spell out.

Figure 7: In $2D$, the scalar exceptional representation is isomorphic to the
tip of the spinning principal series.

Consider the inner product for $\mathcal{F}_{-m,0}$, $m\in\mathbb{N}$, which
gives non-zero norm to the states isomorphic to $\mathcal{V}_{m}$. It is
generated by the kernel

$\displaystyle G_{1+\frac{m}{2},1+\frac{m}{2}}$
$\displaystyle\propto\frac{1}{(x-y)^{1+m}(\overline{x}-\overline{y})^{1+m}}\leadsto(-\partial_{x}\overline{\partial}_{x})^{1+m}\delta^{2}(x-y)$
(105)

which is effectively ultra-local because of Cauchy’s theorem.242424This can be
seen as an instance of the general non-conformality previously discussed, for
which the proper conformally invariant 2pt. function with scaling
$\Delta=\frac{d}{2}+k$ is the ultra-local one like here.  The inner product of
$f_{1},f_{2}\in\mathcal{V}_{m}\in\mathcal{F}_{-m,0}$ is then

$\displaystyle\bra{f_{1}}\ket{f_{2}}=\int\mathrm{d}^{2}z\,\left(\partial^{m+1}f_{1}(z)\right)^{\dagger}\partial^{m+1}f_{2}(z)=\left(\partial^{m+1}f_{1},\partial^{m+1}f_{2}\right)\,,$
(106)

where $\partial$ and $\overline{\partial}$ play the role of $d$ and
$\overline{d}$ in higher dimension as weight-shifting operators taking us from
one representation to another. Since they increase the weight $h$,
respectively $\overline{h}$, by $1$, we find that
$\partial^{m+1}f\in\mathcal{P}_{1,1+m}$. Since all of $\mathcal{V}_{m}$ is
outside of the kernel of $G_{1+\frac{m}{2},1+\frac{m}{2}}$ by construction,
this concludes showing the equivalence between the representations.

One can re-understand this construction in light of the usual CFT formalism
for the $2D$ free boson. The free boson $\phi$ has $\Delta=0$, which
correspond to $m=0$. Its correlation functions are logarithmic, as predicted
by the fact that it sits in an exceptional representation and is invariant
under constant shifts $\phi\to\phi+c$. One can however construct a primary
operator $\partial\phi$ which is shift-invariant, and which sits in the
principal series representation with $\Delta=1$. The output of this discussion
is that for euclidean conformal field theories, one has more unitary
representations of this type to consider, which are generalisations of this
construction with more derivatives.

##  5 Discussion

We have shown how to construct a few families of generic unitary irreducible
representations of the de Sitter group, the symmetric traceless tensor and
tensor-sinors. This extends the current literature, giving a novel account of
these representations, which are most relevant to the study of fermionic
operators in expanding space. The two important outputs of this work are the
following. First, our discussion of exceptional representations stresses the
great departure from the usual rules of conformal field theories as
encountered in Lorentzian signature. Exceptional representations have
logarithmic correlators, which far from breaking de Sitter isometries, are
mandated by them. Second, we have shown and explained the structure of
fermionic representations. We explained why spinor fields only exist in the
principal series, except in $d=3$, which allows remarkably for the existence
of partially-massless unitary representations because. This boils down to the
fact that the spinor representation possesses a pair of opposite helicities
for the general case, and to a degeneracy of the Clifford algebra in $d=3$ for
the exceptional one.

The question of the field theoretic realisation of these unitary
representations remains an open problem, specifically given the naive complex-
valued Lagrangian description of massless higher spinors Pilch:1984aw ;
Letsios:2023qzq ; Bonifacio:2023prb . This is part of a broader problem
regarding exceptional representations in de Sitter. The question for $d=1$
discrete series scalars has only recently been resolved Anninos:2023lin .
There is moreover evidence for an analogue of Weinberg-Witten theorem
Weinberg:1980kq  for partially-massless tensors Sleight:2021iix , hinting at a
rich structure to be explored in the future.

As already acknowledged in the introduction, there are plenty of future
directions to be pursued. Notably, given the construction of fermionic
representations given, it would be extremely useful to use it to extend our
knowledge of multi-particle states in de Sitter following Dobrev:1976vr ;
Penedones:2023uqc . For example, it would be very interesting to see whether
the tensor-product of principal series spinor overlaps with bosonic
exceptional representations, and whether the tensor-product of bosons and
fermions can overlap with the exceptional spinors in $dS_{4}$. These
considerations are essential to the full development of the bootstrap approach
to cosmological correlators Hogervorst:2021uvp . A related issue is the study
of the characters, Hirai:1965aa ; Harish-Chandra:1966aa  and their link with
quasinormal modes Du:2004jt ; Lopez-Ortega:2006aal ; Lopez-Ortega:2012xvr ;
Jafferis:2013qia ; Sun:2020sgn ; Law:2023ohq . This would help gain more
understanding of fermionic contributions to the path integral in de Sitter in
generic dimensions, Anninos:2020hfj ; Law:2020cpj ; Muhlmann:2021aa ;
Law:2023ohq , which hopefully would then build further into the investigation
of the properties of the cosmological horizon and the dynamic in the static-
patch Anninos:2019aa ; Albrychiewicz:2020ruh ; Anninos2021 ; Harris2021 ;
Mirbabayi:2022gnl .

###### Acknowledgements.

The author would like to thank V. Letsios for first bringing the partially-
massless spinor to his attention and useful discussion regarding his work, D.
Anninos, T. Anous,, A. Rios-Fukelman and B. Pethybridge for discussions
regarding the discrete series representations, and P. Kravchuk for his
insights about the conformal group and field theory. Thanks are given to C.
Herzog, D. Anninos for their comments on the drafts and constant
encouragements. The author is grateful to the organisers of the workshop
“Holography in & Beyond the AdS Paradigm” at Instituto de Fisica de La Plata
which overlapped with the elaboration of this work, and to M. Downing, G.
Watts, for useful discussions about $2D$ CFT. This work was supported by the
U.K. Science & Technology Facilities Council Grant ST/P000258/1.

##  Appendix A Discrete Versus Exceptional And Parity of $d$

The unitary representation split between representations which are regular
(principal and complementary series), on which the shadow transform defines an
isomorphism, and the exceptional series. The exceptional representations have
many interesting features : their inner product is logarithmic, naively
breaking the isometries, their Hilbert space has a quotient structure, they
are non-trivially unitary, etc. The analysis of the unitary representations we
performed is fully general, and include as a sub-case the relevant discrete
series representations.

The discrete series form a subset of the exceptional representations, for
which the group-element are square integrable over the group manifold Harish-
Chandra:1965aa ; Harish-Chandra:1966aa ; Dobrev:1977qv . More concretely, for
$g\in G=SO(1,d+1)$, and $\psi$ transforming in some representation
$\mathcal{R}$, define $g\circ\psi=R(g)[\psi]$. $R(g)$ is a functional which is
the realisation of $g$ on the representation $\mathcal{R}$. Such a
representation is said to be discrete if

$\displaystyle\int_{G}\absolutevalue{\expectationvalue{\psi_{1},R(g)\psi_{2}}}^{2}\mathrm{d}g$
$\displaystyle=\frac{\absolutevalue{\expectationvalue{\psi_{1},\psi_{1}}\expectationvalue{\psi_{2},\psi_{2}}}^{2}}{d_{R}}\,,\quad
d_{R}>0\,.$ (107)

Where $\mathrm{d}g$ is the Haar measure over the group $G$. Peeling back the
complexity, picking an orthonormal basis of the representation we see this is
simply the statement that the group-average norm of any elements of the
representation matrices are equal, proportional to the identity, and finite.
$d_{R}$ is called the formal dimension of the representation. For most
representations of non-compact Lie groups, this integral diverge, and there is
no such non-zero $d_{R}$.

This normalisability, is highly dimensionally dependant, and totally unrelated
to the normalisability requirement for the element of the unitary
representations. This gives rise to a peculiar picture which can offer a
different lighting on some of the results of this work. We would like to
stress however, that the fact that a given exceptional representation is
discrete does not seem to bring special properties outside of concerns
regarding group-averaging.

To understand the pattern of the discrete series, one must (re)-consider the
size of the group-factors. The general statement is due to Harish-Chandra
Harish-Chandra:1965aa , and is that discrete series representations exist when
the maximal compact subgroup has the same rank as the group. Our
representations are constructed in the the non-compact picture from the
factors $(A,M)$, with $M=SO(d)$. let us call $H_{G}$ the Cartan subalgebra of
a group $G$. Our induced representations are then constructed of the full
Cartan $A\oplus H_{M}$, which is non-compact and has dimension
$1+\text{dim}(H_{M})$. The non-compactness of the Cartan translates into the
generic non-normalisability of the group elements over the group manifold.

Consider now the compact picture, where one uses the Iwasawa decomposition
$KAN$, with $K=SO(d+1)$. If $d=2k$, there are the Lie-algebra identifications
$M\equiv D_{k}$ and $K\equiv C_{k}$, and both have the same rank, and the non-
compact Cartan subalgebra has dimension $k+1$. However, if $d=2k+1$, then
$M\equiv C_{K}$ and $K\equiv D_{k+1}$. Then, the Cartan has dimension $k+1$,
and it can be picked to be the compact $H_{K}$. This gives rise to integrable
representations, which are the discrete series.

$\displaystyle SO(2k)$
$\displaystyle=D_{k}\quad\includegraphics[valign={c},width=86.72267pt]{diagram/dynkin1}\,,$
(108) $\displaystyle SO(2k+1)$
$\displaystyle=B_{k}\quad\includegraphics[valign={c},width=108.405pt]{diagram/dynkin2}\,.$

These group theoretical considerations explain the pattern, though it is far
removed from the more physical intertwining kernel perspective, which is what
actually builds up the representation. From the point of view of the Hilbert
space construction, nothing drastic happens from one dimension to the next.
Take for example the spinning exceptional series $\mathcal{U}_{\ell,t}$. For
$d=3$, these happen to break in two pieces, which are discrete series. This is
also the case for the scalar exceptional series $\mathcal{V}_{n}$ for $d=1$.
None of the interesting physical properties of these representations is due to
the discreteness per se, since these features survive, unchanged, in arbitrary
dimensions.

Part of the features of fermionic representations can be understood through
this lense. The partially massless tensor-spinors, the would-be fermionic
equivalents of $\mathcal{U}_{\ell,t}$, are non-unitary in all dimensions,
except when they fall in the discrete series and this only happens in $d=3$.
In $d=1$, a similar process occurs for the fermionic equivalents of
$\mathcal{V}_{n}$ Kitaev:2017hnr ; Sun:2021thf . The fact that these
representations are not unitary for other dimension is however much better
understood as a feature of inner-product on the representation, rather than
some rather global, topological statement regarding the representation of
group elements.

##  Appendix B Reminder About Exact Sequences

In this work, we sometimes encounter exact sequences of a particular type,
where all spaces are vector spaces and all the maps are linear. These are all
split exact sequences. For convenience, we here collect some of the basic
properties we use throughout.

An exact sequence is a diagram where spaces are connected by series of map
with definite properties, as summarised in fig. 8.

Figure 8: Typical exact sequence, with $\imaginary(f_{1})=\text{Ker}(f_{2})$.
In this work, all maps will be linear, between vector spaces, making these
split sequences.

In complicated diagrams, the rules are that going two steps gives zero. If the
endpoints are empty sets, then the maps have further properties, illustrated
in fig. 9

(a) $f_{1}$ is injective.

(b) $f_{2}$ is surjective.

(c) $f_{3}$ is a bijective map which defines an isomorphism $A\simeq B$.

Figure 9: Examples of properties encoded in exact sequences. We usually omit
the leftmost $0$ for injective maps. In particular, for $A\subset B$, we only
write the inclusion map $\iota$ from $A$ to $B$.

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