Superspin projection operators and off-shell higher-spin supermultiplets on
Minkowski and anti-de Sitter superspace

Daniel Hutchings
Supervisor: Prof. Sergei M. Kuzenko
Co-supervisor: A/Prof. Evgeny I. Buchbinder

This thesis is presented for the degree of Doctor of Philosophy
The University of Western Australia
Department of Physics
April 2023

## Abstract

  * This thesis is dedicated to the construction of (super)spin projection operators on maximally symmetric (super)space backgrounds in three and four dimensions. On such a background, the irreducible representations of the associated isometry (super)algebra may be realised on the space of tensor (super)fields satisfying certain differential constraints. The (super)spin projectors isolate the component of an unconstrained (super)field which furnishes the irreducible representation with maximal (super)spin. The explicit form of these (super)projectors are found and an array of novel applications are investigated.

We begin the first half of this thesis by reviewing and elaborating upon the
tale of spin projection operators in three-(3$d$) and four-(4$d$) dimensional
Minkowski space and the supersymmetric generalisation of the latter. Special
emphasis is placed on the interplay between (super)projectors and
(super)conformal higher-spin theory as this is a recurring theme of the
thesis. We derive the superprojectors in 3$d$ ${\cal N}$-extended Minkowski
superspace, for $1\leq{\cal N}\leq 6$, which are utilised to construct
linearised off-shell actions for conformal higher-spin supergravity in terms
of unconstrained prepotentials. These theories are then deformed to generate
models for massive ${\cal N}$-extended higher-spin supermultiplets.

In the second half of this thesis we study extensions of the (super)spin
projectors, and their applications detailed above, to anti-de Sitter (AdS)
(super)space. In particular, the superspin projection operators in 4$d$ ${\cal
N}=1$ AdS superspace are derived. It is demonstrated that the poles of such
superprojectors are naturally associated with partially massless
supermultiplets. This allows us to provide a systematic discussion of how to
realise all massive and (partially-)massless irreducible representations of
the 4$d$ ${\cal N}=1$ AdS superalgebra in terms of on-shell superfields.
Similar results are derived in the case of 3$d$ AdS (super)space. We also
compute the superprojectors in 4$d$ ${\cal N}=2$ AdS superspace.

Another major component of this thesis consists of a detailed study of
massless higher-spin gauge models with ${\cal N}=2$ AdS supersymmetry in three
dimensions. Here, there exist two types of ${\cal N}=2$ AdS supersymmetry
which are referred to as $(1,1)$ and $(2,0)$ supersymmetry. To better
understand the differences between $(1,1)$ and $(2,0)$ higher-spin
supermultiplets, we complete the reduction of all known $(1,1)$ supersymmetric
massless higher-spin models to ${\cal N}=1$ AdS superspace. We find that every
known higher-spin theory with $(1,1)$ AdS supersymmetry decomposes into a sum
of two off-shell $(1,0)$ supermultiplets which belong to three series of
inequivalent higher-spin gauge models. A similar analysis is performed for all
massive higher-spin gauge models with $(1,1)$ AdS supersymmetry.

## Authorship Declaration

This thesis is based on four published papers [1, 2, 3, 4]. Their details are
as follows:

  1. 1.

E. I. Buchbinder, D. Hutchings, J. Hutomo and S. M. Kuzenko,
Linearised actions for $\mathcal{N}$-extended (higher-spin) superconformal
gravity,
JHEP 1908, 077 (2019) [arXiv:1905.12476 [hep-th]].
Location in thesis: Chapter 3.

  2. 2.

D. Hutchings, J. Hutomo and S. M. Kuzenko,
Higher-spin gauge models with $(1,1)$ supersymmetry in AdS3: Reduction to
$(1,0)$ superspace,
Phys. Rev. D 103, no.12, 126023 (2021) [arXiv:2011.14294 [hep-th]].
Location in thesis: Chapters 2, 5 and 6.

  3. 3.

E. I. Buchbinder, D. Hutchings, S. M. Kuzenko and M. Ponds,
AdS superprojectors,
JHEP 2104, 074 (2021) [arXiv:2101.05524 [hep-th]]. Location in thesis: Chapter
4.

  4. 4.

D. Hutchings, S. M. Kuzenko and M. Ponds,
AdS (super)projectors in three dimensions and partial masslessness,
JHEP 2110, 090 (2021), [arXiv:2107.12201 [hep-th]].
Location in thesis: Chapters 2 and 5.

Permission has been granted to use the above work.

Evgeny Buchbinder

Jessica Hutomo

Sergei Kuzenko

Michael Ponds

## Acknowledgements

The completion of this thesis has been one of the most enjoyable, yet
challenging, journeys that I have ever embarked on. There are many people who
have aided my academic pursuits and I would like to take a moment to thank
them all for their support.

##### Field theory and quantum gravity group at UWA

First and foremost, I would like to thank my supervisor Prof. Sergei Kuzenko
for his continuous support and guidance throughout my candidature. Your
passion and commitment to the pursuit of advancing theoretical physics has
been inspiring and instrumental in shaping my academic journey. Thank you for
giving me the opportunity to explore the world of theoretical physics. I am
also grateful to my co-supervisor Assoc Prof. Evgeny Buchbinder for the
enjoyable collaborations over the whiteboard and his guidance on this thesis.

Notably, I would like to express my deepest gratitude to my dear friend,
collaborator and mentor Dr. Michael Ponds. Your unwavering support, guidance
and patience were invaluable in completing this thesis. I appreciate the time
that you took to not only discuss the grandeurs of physics but also life with
me. A special thanks to my collaborator and friend Dr. Jessica Hutomo. Thank
you for your mentorship and collaboration in the early stages of my
candidature.

To my close friend and office companion Emmanouil Raptakis, I am grateful that
we were able to share this academic journey together, which began in our first
semester of university. I will fondly reminisce our office antics and
discussions on physics and life. I would also like to thank Benjamin Stone for
his friendship and support throughout my candidature. Finally, I would like to
thank all current and past members of the field theory and quantum gravity
group at The University of Western Australia. In particular Dr. Darren Grasso,
James La Fontaine, Nowar Koning, Joshua Pinelli, Jake Stirling and Kai Turner.

##### Family and friends

Firstly, thank you to my wonderful parents, Vicki and Richard. I am forever
indebted to you both for everything you have provided me in life. The
sacrifices you have made to afford me every opportunity will never be
forgotten. A special thanks to my brother Alex for always providing me with a
well welcomed escape from the stresses of this thesis.

The completion of this thesis would not have possible without the endless
support and encouragement from my beautiful partner Wahida. Thank you for
always being by my side and supporting me through the challenging periods of
my candidature. I am also thankful to my close friends, Conway, Lance, Luke,
Mcleod, Samuel and Tyron for always supporting my academic pursuits and
providing me with refreshing comedic relief.

It would be remiss of me to not thank my awesome housemates, Eliza, Michael
and Sam for their constant encouragement and understanding. I am also grateful
to Tahira and Sam for always providing me with good company and delicious food
throughout my candidature. Finally, I would like to express my deepest
gratitude to anyone whose support I have not mentioned.

##### Financial support

This research was supported by a Jean Rogerson Postgraduate Scholarship and an
Australian Government Research Training Program (RTP) Scholarship at The
University of Western Australia.

##### Examiners

I would like to thank Prof. Ruben Manvelyan and Prof. Rikard von Unge for
examining my thesis and for their insightful and stimulating comments which
have motivated future research directions.

###### Contents

  1. 1 Introduction
  2. 2 Background material
    1. 2.1 Four-dimensional Minkowski space
      1. 2.1.1 Facets of four-dimensional Minkowski space
      2. 2.1.2 Irreducible representations of the Poincaré algebra
      3. 2.1.3 Irreducible field representations
      4. 2.1.4 Spin projection operators
      5. 2.1.5 Conformal higher-spin theory
    2. 2.2 Four-dimensional $\mathcal{N}=1$ Minkowski superspace
      1. 2.2.1 Irreducible representations of the Poincaré superalgebra
      2. 2.2.2 Irreducible superfield representations
      3. 2.2.3 Superprojectors
      4. 2.2.4 Superconformal higher-spin theory
    3. 2.3 Three-dimensional Minkowski space
      1. 2.3.1 Irreducible representations of the Poincaré algebra
      2. 2.3.2 Irreducible field representations
      3. 2.3.3 Spin projection operators
      4. 2.3.4 Conformal higher-spin theory
      5. 2.3.5 Massless higher-spin theories
      6. 2.3.6 New topologically massive theories
  3. Appendix
    1. 2.A Four-dimensional notation and identities
    2. 2.B Three-dimensional notation and identities
  4. 3 Three-dimensional ${\cal N}$-extended Minkowski superspace
    1. 3.1 Irreducible representations of the super-Poincaré algebra
    2. 3.2 Irreducible superfield representations
      1. 3.2.1 Massive superfields
      2. 3.2.2 Massless superfields
    3. 3.3 Superspin projection operators
      1. 3.3.1 Superspin projection operators in $\mathbb{M}^{3|2}$
      2. 3.3.2 Superspin projection operators in $\mathbb{M}^{3|4}$
      3. 3.3.3 Superspin projection operators in $\mathbb{M}^{3|6}$
      4. 3.3.4 Superspin projection operators in $\mathbb{M}^{3|8}$
      5. 3.3.5 Superspin projection operators in $\mathbb{M}^{3|10}$
      6. 3.3.6 Superspin projection operators in $\mathbb{M}^{3|12}$
    4. 3.4 Superconformal higher-spin theory
      1. 3.4.1 Linearised higher-spin super-Cotton tensor in $\mathbb{M}^{3|2}$
      2. 3.4.2 Linearised higher-spin super-Cotton tensor in $\mathbb{M}^{3|4}$
      3. 3.4.3 Linearised higher-spin super-Cotton tensor in $\mathbb{M}^{3|6}$
      4. 3.4.4 Linearised higher-spin super-Cotton tensor in $\mathbb{M}^{3|8}$
      5. 3.4.5 Linearised higher-spin super-Cotton tensor in $\mathbb{M}^{3|10}$
      6. 3.4.6 Linearised higher-spin super-Cotton tensor in $\mathbb{M}^{3|12}$
    5. 3.5 Massless higher-spin theories in $\mathbb{M}^{3|2}$
      1. 3.5.1 Massless integer superspin action in $\mathbb{M}^{3|2}$
      2. 3.5.2 Massless half-integer superspin action in $\mathbb{M}^{3|2}$
    6. 3.6 New topologically massive theories
    7. 3.7 Summary of results
  5. Appendix
    1. 3.A Superconformal primary multiplets
  6. 4 Four-dimensional anti-de Sitter (super)space
    1. 4.1 Four-dimensional anti-de Sitter space
      1. 4.1.1 Irreducible representations of the anti-de Sitter algebra
      2. 4.1.2 Irreducible field representations
      3. 4.1.3 Spin projection operators
      4. 4.1.4 Conformal higher-spin theory
    2. 4.2 Four-dimensional ${\cal N}=1$ anti-de Sitter superspace
      1. 4.2.1 Irreducible representations of ${\cal N}=1$ AdS4 superalgebra
      2. 4.2.2 ${\cal N}=1$ AdS superspace
      3. 4.2.3 Irreducible superfield representations
      4. 4.2.4 Component analysis of on-shell supermultiplets
      5. 4.2.5 Superspin projection operators
      6. 4.2.6 Superconformal higher-spin theory
      7. 4.2.7 Off-shell models in AdS4|4
    3. 4.3 Four-dimensional ${\cal N}=2$ anti-de Sitter superspace
      1. 4.3.1 Facets of four-dimensional ${\cal N}=2$ AdS superspace
      2. 4.3.2 Superconformal higher-spin theory
      3. 4.3.3 Superspin projection operators
      4. 4.3.4 On-shell superfields
    4. 4.4 Summary of results
  7. Appendix
    1. 4.A AdS4|4 superspace toolkit
    2. 4.B Partially massless gauge symmetry
  8. 5 Three-dimensional anti-de Sitter (super)space
    1. 5.1 Three-dimensional anti-de Sitter space
      1. 5.1.1 Irreducible representations of the AdS3 algebra
      2. 5.1.2 Irreducible field representations
      3. 5.1.3 Spin projection operators
      4. 5.1.4 Linearised higher-spin Cotton tensors
      5. 5.1.5 (Fang-)Fronsdal-type actions
      6. 5.1.6 New topologically massive actions
    2. 5.2 Three-dimensional ${\cal N}=1$ anti-de Sitter superspace
      1. 5.2.1 Irreducible representations of the AdS3 superalgebra
      2. 5.2.2 Geometry of AdS3|2
      3. 5.2.3 Irreducible superfield representations
      4. 5.2.4 Superspin projection operators
      5. 5.2.5 Linearised higher-spin super-Cotton tensors
      6. 5.2.6 Massless higher-spin theories
      7. 5.2.7 Component structure of ${\cal N}=1$ supersymmetric higher-spin actions in AdS3
      8. 5.2.8 Topologically massive theories
    3. 5.3 Summary of results
  9. Appendix
    1. 5.A Generating function formalism
  10. 6 Higher-spin gauge models with $(1,1)$ supersymmetry in AdS3
    1. 6.1  (1,1) $\rightarrow$ (1,0) AdS superspace reduction
      1. 6.1.1 (1,1) AdS superspace: Complex basis
      2. 6.1.2 (1,1) AdS superspace: Real basis
      3. 6.1.3 Reduction from $(1,1)\to(1,0)$ AdS superspace
      4. 6.1.4 The (1,1) AdS supersymmetric actions in AdS3|2
    2. 6.2 Massless half-integer superspin: Transverse formulation
      1. 6.2.1 Transverse formulation
      2. 6.2.2 Reduction of gauge prepotentials to $\text{AdS}^{3|2}$
      3. 6.2.3 Massless higher-spin ${\cal N}=1$ supermultiplets
      4. 6.2.4 Second supersymmetry transformations
    3. 6.3 Massless half-integer superspin: Longitudinal formulation
      1. 6.3.1 Longitudinal formulation
      2. 6.3.2 Reduction of gauge prepotentials to AdS3|2
      3. 6.3.3 Massless higher-spin ${\cal N}=1$ supermultiplets
      4. 6.3.4 Second supersymmetry transformations
    4. 6.4 Massless integer superspin: Longitudinal formulation
      1. 6.4.1 Longitudinal formulation
      2. 6.4.2 Reduction of gauge prepotentials to AdS3|2
      3. 6.4.3 Massless higher-spin ${\cal N}=1$ supermultiplets
      4. 6.4.4 Second supersymmetry transformations
    5. 6.5 Massless integer superspin: Transverse formulation
      1. 6.5.1 Transverse formulation
      2. 6.5.2 Massless higher-spin ${\cal N}=1$ supermultiplets
      3. 6.5.3 Second supersymmetry transformations
    6. 6.6 Topologically massive theories
    7. 6.7 Discussion and summary of results
  11. Appendix
    1. 6.A $(1,1)$ AdS superspace toolkit
  12. 7 Conclusion and outlook

##  Chapter 1 Introduction

Einstein’s theory of special relativity revolutionised our theoretical
understanding of the structure of the Universe, unifying the previously
independent entities of space and time into the framework of spacetime. This
profound result was founded on the two fundamental ideas that in any inertial
system: (i) the laws of physics have the same form; and (ii) the speed of
light is constant. These principles are equivalent to the mathematical
statement that the Poincaré group is a symmetry group of any closed system in
Minkowski space. The successful use of symmetry in the formulation of special
relativity ushered in a paradigm shift in theoretical physics, which saw this
Poincaré, or relativistic, symmetry become the new guiding principle in the
construction of theories attempting to describe the physical processes of
nature.

Symmetry is ubiquitous in modern physics as all physical systems should be
compatible with special relativity, i.e. possess Poincaré symmetry. This
allows for the universal classification of elementary particles as unitary
irreducible representations of the Poincaré group. The fundamental
correspondence between elementary particles propagating in four-dimensional
Minkowski space $\mathbb{M}^{4}$ and unitary irreducible representations
(UIRs) of the Poincaré algebra $\mathfrak{iso}(3,1)$ was established by Wigner
in his seminal work [6] in 1939. Here, Wigner completed the classification of
the UIRs of $\mathfrak{iso}(3,1)$ and showed that they are characterised by
two parameters, mass ${m}$ and spin $s$. This groundbreaking identification
resides at the heart of all modern relativistic field theories describing
elementary particles.

Our best theoretical descriptions of the physical phenomenon that we observe
in Nature are the Standard Model (SM) of particle physics and Einstein’s
General relativity (GR). These are examples of particle theories that realise
UIRs of the Poincaré algebra at the linearised level. The Standard Model is a
quantum field theory which describes the fundamental interactions of Nature
dominant at the subatomic scale (the electromagnetic, the strong and the weak
forces). The particle content encapsulated by the SM includes
spin-$\frac{1}{2}$ fermions (matter), spin-$1$ gauge bosons which mediate
interactions and the Higgs field (scalar boson). On the other hand, GR is a
classical gauge theory of the Poincaré group which concerns the physics
governing Nature at large scales. Einstein’s gravity is conjectured to be
mediated by a massless spin-$2$ particle known as the graviton. Both of these
theories have proven to be very successful, predicting many physical processes
which have been experimentally verified to great accuracy. It appears that the
current theories which best describe the physical processes observed in Nature
are realised in terms of particles carrying only spin $s\leq 2$. However, from
a mathematical viewpoint, there is nothing preventing the existence of
particles with spin $s>2$ in Wigner’s framework. A field (or particle) is said
to be of higher-spin if it possesses spin $s>2$.

The study of higher-spin fields has attracted much attention recently due to
its inextricable connection to various topics in fundamental physics.111There
have been extensive contributions to the development of higher-spin theory.
For more information on these works, and also the current state of affairs, we
recommend the review articles [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]
and references therein. Its origin in four-dimensional Minkowski space can be
attributed to the pioneering contributions of Majorana [19], Dirac [20], and
Fierz & Pauli [21, 22] in the 1930’s, which culminated in the development of
the relativistic field equations for massive fields of arbitrary spin [21,
22]. The field-theoretic formulation of massive higher-spin fields [21, 22]
appeared before the seminal group-theoretic work of Wigner [6]. The field- and
group-theoretic pictures were eventually unified in the later work of Bargmann
and Wigner [23] in 1948. Here, it was shown that the wave equation that a
field must satisfy in order to realise a massive UIR of $\mathfrak{iso}(3,1)$
is equivalent to that appearing in the Fierz-Pauli equations [21, 22]. In this
thesis, we will refer to the appropriate conditions which ensures that a
tensor field furnishes a UIR of the isometry algebra as on-shell
conditions.222Some people pay homage by referring to these conditions as the
Bargmann-Wigner equations, while others refer to them as the Fierz-Pauli
equations. They are also known as irreducible field representations.

The on-shell conditions are an essential ingredient in the formulation of any
relativistic field theory in particle physics. Naturally, they should result
from the Euler-Lagrange equations (the equations of motion) derived from an
appropriate Lagrangian. The quest of seeking Lagrangian formulations which
generate the Bargmann-Wigner equations for arbitrary spin was initiated by
Fierz and Pauli in $\mathbb{M}^{4}$, who constructed Lagrangian theories for
fields of spins $s=\frac{3}{2},2$ [21, 22].333Fierz and Pauli [21, 22] did not
comment on the gauge invariance of the massless spin $s=\frac{3}{2},2$
actions. The gauge symmetry for the massless spin $s=\frac{3}{2}$ action was
first discussed by Rarita and Schwinger in [24]. This program, which is aptly
known as the Fierz-Pauli program, was completed for massive fields of
arbitrary spin by Singh and Hagen [25, 26] 35 years later in 1974. Building on
this work, free field theories describing massless particles of arbitrary spin
were later derived in four-dimensional Minkowski space [27, 28] and anti-de
Sitter (AdS4) space[29, 30] by Fang and Fronsdal. Lagrangian descriptions of
free field theories are essential as they are the starting point for:
constructing interacting (non-linear) extensions; and analysing the quantum
properties of these theories.

In this thesis, we will be interested in studying a certain type of non-local
differential operator, the so-called spin projection operator, which extracts
the physical part of an unconstrained field, i.e. the component satisfying the
on-shell conditions with fixed maximal spin in $\mathbb{M}^{4}$. Our main
objective is to construct novel generalisations of these operators in
maximally symmetric backgrounds in three and four dimensions, and their
supersymmetric analogues. Before describing the nature of these spin
projection operators, it is necessary to first elaborate on the constraint
space which these operators project onto.

#####  1.0.0.1 Irreducible (super)field representations

The program of determining the on-shell conditions for (spin-)tensor fields of
arbitrary rank in $\mathbb{M}^{4}$ was initiated by Bargmann and Wigner
[23].444This program is aptly referred to as the Bargmann-Wigner program in
the higher-spin community. Let us denote by $G(m,s)$ the massive UIR of
$\mathfrak{iso}(3,1)$ carrying mass $m>0$, integer spin $s\geq 0$ and positive
definite energy. A totally symmetric real rank-$s$ tensor field
$\phi_{a_{1}\ldots a_{s}}(x)=\phi_{(a_{1}\ldots a_{s})}(x)\equiv\phi_{a(s)}$
in $\mathbb{M}^{4}$ is said to furnish $G(m,s)$ if it satisfies the on-shell
conditions [23]

$\displaystyle\eta^{bc}\phi_{bca(s-2)}$ $\displaystyle=$ $\displaystyle 0~{},$
(1.1a) $\displaystyle\partial^{b}\phi_{ba(s-1)}$ $\displaystyle=$
$\displaystyle 0~{},$ (1.1b) $\displaystyle(\Box-{m}^{2})\phi_{a(s)}$
$\displaystyle=$ $\displaystyle 0~{}.$ (1.1c)

In other words, a field $\phi_{a(s)}$ that is traceless (1.1a), transverse
(divergenceless) (1.1b) and satisfies the Klein-Gordon equation (1.1c)
describes a bosonic particle carrying mass ${m}$ and spin $s$. Note that in
the case $s=0$, a massive field only satisfies (1.1c), while for the case
$s=1$ , it obeys (1.1b) and (1.1c).

For integer $s>0$, a symmetric rank-$s$ Majorana tensor-spinor field
$\mbox{\boldmath$\psi$}_{a_{1}\ldots
a_{s}}(x)=\mbox{\boldmath$\psi$}_{(a_{1}\ldots
a_{s})}(x)\equiv\mbox{\boldmath$\psi$}_{a(s)}$555We have omitted the spinor
index of the Majorana spinor field $\mbox{\boldmath$\psi$}_{a(s)}$. realises
$G({m},s+\frac{1}{2})$ if it obeys the on-shell conditions [23]

$\displaystyle\partial^{b}\mbox{\boldmath$\psi$}_{ba(s-1)}$ $\displaystyle=$
$\displaystyle 0~{},$ (1.2a)
$\displaystyle\gamma^{b}\mbox{\boldmath$\psi$}_{ba(s-1)}$ $\displaystyle=$
$\displaystyle 0~{},$ (1.2b) $\displaystyle({\rm
i}\gamma^{b}\partial_{b}+{m})\mbox{\boldmath$\psi$}_{a(s)}$ $\displaystyle=$
$\displaystyle 0~{}.$ (1.2c)

Thus, a Majorana spinor-tensor field $\mbox{\boldmath$\psi$}_{a(s)}$ which is
transverse (1.2a), gamma traceless (1.2b) and obeys the Dirac equation (1.2c)
describes a fermionic particle carrying mass ${m}$ and spin-$(s+\frac{1}{2})$.
A massive spin-$\frac{1}{2}$ field only satisfies the Dirac equation (1.2c).

In the massless case $m=0$, the system of equations (1.1) and (1.2) admit a
gauge symmetry, which accounts for the additional non-physical degrees of
freedom. The on-shell conditions for massless gauge fields of arbitrary spin
in $\mathbb{M}^{4}$ were first studied in detail by Fronsdal and Fang in [27,
28].666Bargmann and Wigner [23] found that a field which furnishes the
massless UIR of $\mathfrak{iso}(3,1)$ with discrete spin must satisfy the
massless Klein-Gordon equation. They did not detail the supplementary
conditions, nor the gauge symmetry, associated with these fields.

The on-shell conditions for a superfield with arbitrary superspin in four-
dimensional ${\cal N}=1$ Minkowski superspace $\mathbb{M}^{4|4}$ were first
derived by Sokatchev [31] in 1975 (see also [32, 33]). Here, it was shown that
chiral superfields realise irreducible representations of the super Poincaré
algebra. In the same year, the on-shell conditions which select out the
maximal superspin component of a higher-rank superfield were given by Howe,
Stelle and Townsend [34] in the context of irreducible supercurrents. The on-
shell conditions of [32, 33, 34] will be treated in more detail in section
2.2.

#####  1.0.0.2 (Super)spin projection operators

A totally symmetric traceful field in $\mathbb{M}^{4}$ (which is otherwise
unconstrained) satisfying the Klein-Gordon equation (1.1c) realises the
reducible representation of $\mathfrak{iso}(3,1)$

$G(m,s)\oplus G(m,s-1)\oplus
2G(m,s-2)\oplus\cdots=\bigoplus^{s}_{j=0}\big{(}\lfloor\frac{j}{2}\rfloor+1\big{)}G(m,s-j)~{},$
(1.3)

which contains a collection of UIRs with varying multiplicities. The purpose
of the spin projection operator is to select the maximal spin UIR $G(m,s)$
from this decomposition.

The spin projection operators, or transverse and traceless (TT) projectors,
were first derived in $\mathbb{M}^{4}$ by Fronsdal in the late 1950’s [35]
(see also [36]).777Isaev and Podoinitsyn [37, 38] refer to the spin projection
operators, which appeared in the literature for the first time in the work of
Behrends and Fronsdal [36], as Behrends-Fronsdal spin projection operators. We
will refrain from using this terminology as the authors of [36] credit the
derivation of the spin projectors to Fronsdal, citing the work [35] which was
published in the following year. For any integer $s\geq 2$, let us denote
${V}^{\text{T}}_{(s)}$ the space of totally symmetric traceful rank-$s$ tensor
fields ${\phi}^{\text{T}}_{a(s)}$, and ${V}_{(s)}$ the space of symmetric
traceless rank-$s$ tensor fields $\phi_{a(s)}$. Fronsdal [35] constructed the
spin projection operator $\Pi^{\perp}_{(s)}$ which is defined by its action on
${V}^{\text{T}}_{(s)}$ via the rule

$\displaystyle\Pi^{\perp}_{(s)}:{V}^{\text{T}}_{(s)}$
$\displaystyle\longrightarrow$ $\displaystyle V_{(s)}~{},$ (1.4)
$\displaystyle{\phi}^{\text{T}}_{a(s)}$ $\displaystyle\longmapsto$
$\displaystyle\Pi^{\perp}_{a(s)}({\phi}^{\text{T}})~{},\qquad\eta^{bc}{\phi}^{\text{T}}_{bca(s-2)}\neq
0~{},$

where the image
$\Pi^{\perp}_{a(s)}({\phi}^{\text{T}})\equiv\Pi^{\perp}_{(s)}{\phi}^{\text{T}}_{a(s)}$
satisfies the properties:

  1. 1.

Idempotent: the operator $\Pi^{\perp}_{(s)}$ is a projector,

$\Pi^{\perp}_{(s)}\Pi^{\perp}_{(s)}{\phi}^{\text{T}}_{a(s)}=\Pi^{\perp}_{(s)}{\phi}^{\text{T}}_{a(s)}~{}.$
(1.5a)

  2. 2.

Symmetric: the operator $\Pi^{\perp}_{(s)}$ maps ${\phi}^{\text{T}}_{a(s)}$ to
a totally symmetric field,

$\Pi^{\perp}_{a_{1}\cdots
a_{s}}({\phi}^{\text{T}})=\Pi^{\perp}_{a(s)}({\phi}^{\text{T}})~{}.$ (1.5b)

  3. 3.

Traceless: the operator $\Pi^{\perp}_{(s)}$ maps ${\phi}^{\text{T}}_{a(s)}$ to
a traceless field,

$\eta^{bc}\Pi^{\perp}_{bca(s-2)}({\phi}^{\text{T}})=0~{}.$ (1.5c)

  4. 4.

Transverse: the operator $\Pi^{\perp}_{(s)}$ maps ${\phi}^{\text{T}}_{a(s)}$
to a transverse field,

$\partial^{b}\Pi^{\perp}_{ba(s-1)}({\phi}^{\text{T}})=0~{}.$ (1.5d)

The constraints (1.5b), (1.5c) and (1.5d) eliminate all lower spin modes
contained in the projected field $\phi^{\text{T}}_{a(s)}$, thus ensuring that
the highest-spin component of the bosonic field is selected. In other words,
the spin projection operator $\Pi^{\perp}_{(s)}$ maps an unconstrained tensor
field $\phi^{\text{T}}_{a(s)}$ to a pure spin-$s$ field. Moreover, if the
field $\phi^{\text{T}}_{a(s)}$ is on the mass-shell (1.1c), it follows from
(1.1) that the operator $\Pi^{\perp}_{(s)}$ singles out the component of
$\phi^{\text{T}}_{a(s)}$ that furnishes the UIR $G({m},s)$ with maximal spin
$s$.

The fermionic counterpart is computed in a similar manner. Let us denote
${\mbox{\boldmath$V$}}^{\text{T}}_{(s+\frac{1}{2})}$ the space of four-
component (gamma) traceful spin-tensor fields
${\mbox{\boldmath$\psi$}}^{\text{T}}_{a(s)}$, and
$\mbox{\boldmath${V}$}_{(s+\frac{1}{2})}$ the space of totally symmetric
(gamma) traceless rank-$s$ tensor fields $\mbox{\boldmath${\psi}$}_{a(s)}$.
For integer $s\geq 2$, the spin projection operator
$\Pi^{\perp}_{(s+\frac{1}{2})}$ acts on
${\mbox{\boldmath$V$}}^{\text{T}}_{(s+\frac{1}{2})}$ according to the rule
[35]

$\displaystyle\Pi^{\perp}_{(s+\frac{1}{2})}:{\mbox{\boldmath$V$}}^{\text{T}}_{(s+\frac{1}{2})}$
$\displaystyle\longrightarrow$
$\displaystyle{\mbox{\boldmath$V$}}_{(s+\frac{1}{2})}~{},$ (1.6a)
$\displaystyle{\mbox{\boldmath$\psi$}^{\text{T}}}_{a(s)}$
$\displaystyle\longmapsto$
$\displaystyle\Pi^{\perp}_{a(s)}({\mbox{\boldmath$\psi$}}^{\text{T}})~{},\qquad\gamma^{b}{\mbox{\boldmath$\psi$}}^{\text{T}}_{ba(s-1)}\neq
0~{}.$ By definition, the image
$\Pi^{\perp}_{a(s)}(\mbox{\boldmath$\psi$}^{\text{T}})\equiv\Pi^{\perp}_{(s+\frac{1}{2})}\mbox{\boldmath$\psi$}^{\text{T}}_{a(s)}$
satisfies the properties (1.5), but the traceless property (1.5c) is replaced
by:

  1. 1.

Gamma traceless: the operator $\Pi^{\perp}_{(s+\frac{1}{2})}$ maps
$\mbox{\boldmath$\psi$}^{\text{T}}_{a(s)}$ to a gamma traceless field

$\gamma^{b}\Pi^{\perp}_{ba(s-1)}(\mbox{\boldmath$\psi$}^{\text{T}})=0~{}.$
(1.6b)

Analogous to the bosonic case, the spin projector
$\Pi^{\perp}_{(s+\frac{1}{2})}$ maps an arbitrary spinor-tensor field
${\mbox{\boldmath$\psi$}}^{\text{T}}_{a(s)}$ to a pure spin-$(s+\frac{1}{2})$
state. Let us consider a traceful spin-tensor field which satisfies the Dirac
equation (1.2c). It follows from the defining properties of
$\Pi^{\perp}_{(s+\frac{1}{2})}$ that the spin projection operator
$\Pi^{\perp}_{(s+\frac{1}{2})}$ selects the physical component from
${\mbox{\boldmath$\psi$}}^{\text{T}}_{a(s)}$ which furnishes
$G({m},s+\frac{1}{2})$ with maximal spin $(s+\frac{1}{2})$.

It was shown by Aurilia and Umezawa [39, 40] that the bosonic (1.5) and
fermionic (1.6) spin projection operators can be expressed in terms of the
Casimir operators of $\mathfrak{iso}(3,1)$. In two-component spinor notation,
the spin projection operators were generalised to spinor fields of arbitrary
rank (including mixed symmetry fields) by Siegel and Gates [33] (see also
[41]). The TT projectors were first generalised to $d$-dimensional Minkowski
space for $d>2$ by Segal [42] (see also [43, 44, 45, 37, 38]) in the bosonic
case, and later by Isaev and Podoinitsyn [38] for fermionic fields.

Many applications for the TT projectors have been found within the landscape
of high-energy physics. They are known to determine the structural form of
propagators for massive fields of arbitrary spin [46] (see also [38]).888The
propagators for CHS fields were formulated in terms of the TT projectors in
$\mathbb{M}^{4}$ in [47]. The spin projection operators were also essential in
Fronsdal’s Lagrangian formulation for a massive spin-$\frac{3}{2}$ particle
[35], which was later extended to spins $s\leq 4$ by Chang [48]. Specifically,
given a traceful bosonic field $\phi^{\text{T}}_{a(s)}$, Fronsdal [35]
introduced the equation

$m^{2}\phi^{\text{T}}_{a(s)}=\Box\Pi^{\perp}_{(s)}\phi^{\text{T}}_{a(s)}~{}.$
(1.7)

Making use of the projector properties (1.5), it can be easily shown that
equation (1.7) encodes the massive on-shell conditions (1.1). An analogous
equation was also found for half-integer spin fields in [35] .

The spin projectors are also important tools which can be used to decompose
any symmetric tensor field of arbitrary rank into irreducible components. For
example, the irreducible decomposition of a rank-$2$ field was crucial in the
canonical formulation of general relativity, see e.g. [49, 50, 51, 52, 53].
Specifically, the projectors $\Pi^{\perp}_{(s)}$ with $s\leq 2$ can be used to
decompose a rank-$2$ symmetric tensor field $g^{\text{T}}_{ab}$ in the
following manner

$g^{\text{T}}_{ab}=g^{\text{TT}}_{ab}+\partial_{(a}g^{\perp}_{b)}+\partial_{a}\partial_{b}g+\eta_{ab}\tilde{g}~{},$
(1.8)

where $g$ and $\tilde{g}$ are scalar, $g^{\perp}_{a}$ is transverse and
$g^{\text{TT}}_{ab}$ is traceless and transverse. The defining properties of
these irreducible components are inherited immediately from
$\Pi^{\perp}_{(s)}$, thus illustrating their importance in the decomposition
scheme. Furthermore, these decompositions prove invaluable in the framework of
path-integral quantisation (see e.g. [55, 54]).

The superspin projection operators (or superprojectors) in four-dimensional
${\cal N}=1$ Minkowski superspace were introduced by Salam and Strathdee [56]
in the case of a scalar superfield. Shortly after, these operators were
extended to certain constrained superfields of arbitrary rank by Sokatchev
[31] and were formulated in terms of Casimir operators. A few years later
Rittenberg and Sokatchev [57] made use of a similar method to construct the
superprojectors in ${\cal N}$-extended Minkowski superspace
$\mathbb{M}^{4|4{\cal N}}$ (see also [32]). An alternative powerful
construction of the superprojectors in $\mathbb{M}^{4|4{\cal N}}$ was given in
[33, 41], which requires no knowledge of the Casimir operators of the ${\cal
N}$-extended Minkowski superalgebra. In the early days of supersymmetry, the
superspin projection operators of [57, 33] were used in the construction of
superfield equations of motion [58, 59] and gauge-invariant actions [60, 61]
in $\mathbb{M}^{4|4}$. One of the main goals of this thesis is to compute the
(super)spin projection operators in three-dimensional ${\cal N}$-extended
superspace.

#####  1.0.0.3 (Super)conformal higher-spin theory

One of the alluring applications of the spin projection operators is that they
can be used as a tool to construct linearised conformal higher-spin theories.
Conformal higher-spin (CHS) theories were first studied in the pioneering work
of Fradkin and Tseytlin [62] as generalisations of Maxwell’s electrodynamics
$(s=1)$, the conformal gravitino model $(s=\frac{3}{2})$ and conformal gravity
$(s=2)$ in $\mathbb{M}^{4}$. By construction, the linearised CHS actions of
[62] describe pure spin fields of arbitrary spin off the mass-shell (see also
[63, 64, 42]). For $s\geq 1$, the free bosonic CHS model of [62] is described
in terms of a totally symmetric real traceful rank-$s$ tensor field
${h}^{\text{T}}_{a(s)}$ which is defined modulo gauge transformations of the
form999The gauge freedom of $\lambda_{a(s-2)}$ can be used to make
${h}^{\text{T}}_{a(s)}$ traceless i.e. ${h}^{\text{T}}_{a(s)}=h_{a(s)}$, where
$\eta^{bc}h_{bca(s-2)}=0$. In this thesis, we will only work with totally
symmetric and traceless fields.

$\delta_{\xi,\lambda}{h}^{\text{T}}_{a(s)}=\partial_{(a_{1}}\xi_{a_{2}...a_{s})}+\eta_{(a_{1}a_{2}}\lambda_{a_{3}...a_{s})}~{},\qquad\eta^{bc}\xi_{bca(s-3)}=0~{}.$
(1.9)

The gauge parameters $\xi_{a(s-1)}$ and $\lambda_{a(s-2)}$ are symmetric while
$\xi_{a(s-1)}$ is also traceless. The differential and algebraic gauge
transformations in (1.9) can be identified as higher-spin generalisations of
linearised diffeomorphisms and Weyl transformations of conformal gravity
respectively.

The spin-$s$ linearised conformal model of [62] is given by

$S^{(s)}_{\text{CHS}}[{h}]\propto\int{\rm
d}^{4}x~{}{h}^{\text{T}a(s)}\Box^{s}\Pi_{(s)}^{\perp}{h}^{\text{T}}_{a(s)}~{}.$
(1.10)

It must be noted that the spin projection operators $\Pi_{(s)}^{\perp}$ were
not given for spins $s>2$ in [62]. The properties (1.5) of the projector
ensure that the action (1.10) is invariant under the gauge transformations
(1.9). The fermionic spin projection operator $\Pi^{\perp}_{(s+\frac{1}{2})}$
(1.6) also appears explicitly in the fermionic CHS action [62].

It is apparent from the structural form of the action (1.10) that there is an
intimate connection between CHS theories and spin projection
operators.101010The free CHS actions of [62] were purely schematic for spins
$s>2$, as only the explicit form of the spin projection operators were given
for the cases $s\leq 2$. This correspondence follows naturally from the
definition of the spin projectors of [35], which we recall are the unique
differential operators which select the maximal pure spin component from an
unconstrained (spin-)tensor. Thus, the presence of the spin projectors in the
CHS actions ensure that the theory describes a single pure spin state off the
mass-shell.111111The 3$d$ CHS actions of [65, 66] were recently constructed in
terms of spin projectors in [67].

Many efforts have been invested into the development of CHS theories in the
past decade due to their desirable properties. For example, in contrast to the
massless case where no local non-linear Lagrangian description exists,121212At
the level of the equations of motion, a fully interacting theory of massless
fields was derived in (A)dS space by Vasiliev [68, 69]. Segal [42] (see also
[64, 70, 71, 45]) constructed the Lagrangian for a complete interacting
bosonic CHS theory in even $d-$dimensional Minkowski space, for $d\geq 4$. The
existence of a consistent non-linear action functional for CHS fields also
allows for the study of its quantum properties [64, 70, 77, 72, 78, 73, 74,
47, 75, 76]. In addition, CHS theories play a natural role in the context of
the AdS/CFT correspondence [84, 82, 64, 42, 79, 80, 81, 83].

Supersymmetric generalisations of CHS fields were first discussed by Howe,
Stelle and Townsend in the context of supercurrent multiplets via the Noether
procedure in 1981 [34]. The SCHS superfields provided here were only for half-
integer superspin. The first on-shell131313A supermultiplet is said to be off-
shell if the algebra of supersymmetry transformations closes off the equations
of motion. Otherwise, the supermultiplet is called on-shell. Lagrangian
description of the half-integer supermultiplets of [34] was given by Fradkin
and Tseytlin [62] in 1985. Off-shell superconformal higher-spin multiplets and
their corresponding actions were constructed over 30 years later in 2017 by
Kuzenko, Manveylan and Theisen [85] in four-dimensional Minkowski and anti-de
Sitter superspace. The off-shell ${\cal N}=1$ SCHS actions of [85] were
generalised two years later to conformally-flat backgrounds by Kuzenko and
Ponds [86]. The ${\cal N}$-extended superconformal free higher-spin models
were recently formulated in conformally-flat backgrounds by Kuzenko and
Raptakis in [87]. An interacting theory of SCHS fields was recently proposed
in [88] (see also [89]).

All of the above SCHS theories were constructed independently of superspin
projection operators. Recasting these theories in terms of superprojectors is
beneficial for a variety of reasons. Firstly, it elucidates the defining
feature of SCHS theories that they describe maximal superspin states off the
mass-shell. In addition, it also makes properties of the action such as gauge
invariance manifest. The interplay of (S)CHS theories and (super)spin
projection operators will be a prominent theme of this thesis.

#####  1.0.0.4 (Super)spin projection operators in (A)dS

The (super)spin projection operators have proven to be very useful in
Minkowski (super)space due to their many interesting applications. For this
reason, it is natural to consider generalisations of these operators to
(super)space backgrounds other than Minkowski (super)space. The next logical
step is to consider the formulation of the (super)spin projection operators in
curved (super)space backgrounds, with the simplest starting point being
(anti-)de Sitter space. Upon initiating this program, one immediately
encounters problems due to the presence of non-vanishing curvature, which
makes constructing the (super)spin projectors technically challenging. Despite
this, recent progress was made in this pursuit by Kuzenko and Ponds in 2020
[90], where the spin projection operators were computed in four-dimensional
(anti-)de Sitter space (A)dS4.

One of the important outcomes of [90] was a new understanding of the so-called
partially massless fields in AdS4. First discovered by Deser and Nepomechie
[91] in 1983, partially massless fields are an exotic species of on-shell
fields in AdS4 which have no direct analogue in Minkowski space. As the name
suggests, the defining properties of these fields are intermediate of both
massive and massless fields, in the sense that they possess a higher-depth
gauge symmetry. Partially massless fields have been at the centre of extensive
studies for the past 35 years [95, 91, 96, 97, 98, 99, 100, 101, 94, 92, 93,
102, 103, 104]. It was shown in [90] that the poles of the spin projection
operators correspond to partially massless fields. Thus, the spin projection
operators provided a novel program to complete the dictionary of on-shell
fields in AdS4.

Another goal of this thesis is to study extensions of the spin projection
operators to (anti-)de Sitter space, and their corresponding supersymmetric
generalisations. In particular, we will compute the projection operators in
three-dimensional ${\cal N}=0$ and ${\cal N}=1$ AdS (super)space and four-
dimensional ${\cal N}$-extended AdS superspace for ${\cal N}=1$ and ${\cal
N}=2$. We will also initiate the construction of the superspin projectors in
$3d$ ${\cal N}=(1,1)$ AdS superspace.

#####  1.0.0.5 Free off-shell supersymmetric massless higher-spin theories

The study of Lagrangian descriptions of free higher-spin massless fields,
particularly in three dimensions, will also be a point of interest in this
thesis. There are two different approaches in which supersymmetric field
theories can be formulated: the component (on-shell) formalism; and the
superfield (off-shell) formalism.141414For a pedagogical treatment of the
superfield formalism, see e.g. [105, 41]. The former deals with a multiplet of
bosonic and fermionic fields on spacetime, known as component fields, which
are related via supersymmetry transformations. In the component approach,
supersymmetry is not manifest, and the supersymmetry algebra only closes on
the equations of motion (EoM).151515 In order for it to close off the EoM, one
needs to introduce auxiliary (non-dynamical) fields. The off-shell formalism
deals with superfields which live on a superspace. The superfield is comprised
of a multiplet of component fields, both physical and non-physical
(auxiliary), which ensures that the superalgebra closes off the EoM. The main
benefit of working in superspace is that supersymmetry is always manifest. In
this thesis, we will always work in the superfield formulation, however, we
will also make contact with the component formulation via a component
reduction.

Four-dimensional supersymmetric field theories realising massless
superspin-$\hat{s}$ multiplets, with
$\hat{s}=\frac{1}{2},1,\frac{3}{2},\cdots$, describe two massless fields which
each carry spin $\hat{s}$ and $\hat{s}+\frac{1}{2}$, respectively. These
actions with on-shell Poincaré supersymmetry were first computed in the
component formalism by Curtight [106] and Vasiliev [107] in the metric- and
frame- like formulations respectively. A superfield realisation of the
massless half-integer supermultiplet in $\mathbb{M}^{4|4}$ was first derived
by Kuzenko, Postnikov and Sibiryakov [108] in 1993. In the same year, the off-
shell integer superspin massless multiplets were found by Kuzenko and
Sibiryakov [109]. The formulations of [108, 109] were extended to ${\cal N}=1$
anti-de Sitter superspace in [110]. The Lagrangian formulation of free ${\cal
N}=2$ higher-spin supermultiplets in terms of ${\cal N}=1$ superfields was
achieved in [111, 112]. Recently in 2021, Buchbinder, Ivanov and Zaigraev
[113] computed an off-shell formulation for massless multiplets of integer
superspin in ${\cal N}=2$ harmonic superspace [114, 115, 116].

The story of off-shell higher-spin supersymmetric massless field theories in
three dimensions is considerably different to its four-dimensional cousin. It
is well known that these higher-superspin theories are purely kinematical, as
a consequence of the fact that massless fields with spins $s\geq\frac{3}{2}$
do not to propagate any physical degrees of freedom. Despite this, the study
of massless theories is attractive since they can be deformed into the so-
called topologically massive higher-spin actions [117] following the
philosophy of the seminal works [118, 119, 120, 121]. Off-shell massless
higher spin supermultiplets were first derived in $3d$ ${\cal N}=2$ Minkowski
superspace by Kuzenko and Ogburn [122] in 2016. The ${\cal N}=1$ higher-spin
supermultiplets were later computed in the same year by Kuzenko and Tsulaia
[123] by applying a ${\cal N}=2\rightarrow{\cal N}=1$ superspace reduction to
the massless supermultiplets of [122].

Three-dimensional supersymmetric field theories with $(p,q)$ AdS
supersymmetry, where the integers $p\geq q\geq 0$ are such that ${\cal
N}=p+q$, can naturally be realised on the so-called $(p,q)$ AdS superspaces
[124]. In the case of ${\cal N}=2$ AdS supersymmetry, off-shell massless
superspin-$\hat{s}$ theories can be formulated on either of the inequivalent
$(2,0)$ or $(1,1)$ AdS superspaces respectively. All known (half-)integer off-
shell massless supermultiplets in $(1,1)$ AdS superspace were computed by
Hutomo, Kuzenko and Ogburn in [125]. Soon after, the massless multiplets of
half-integer superspin were found by Hutomo and Kuzenko in $(2,0)$ AdS
superspace [126].161616The Lagrangian description of the massless integer
superspin multiplet in $(2,0)$ AdS superspace is still unknown. Direct
comparison of the massless $(1,1)$ [125] and $(2,0)$ [126] theories describing
the same superspin content is difficult as they are formulated in different
superspaces. In order to make contact between these supermultiplets, the
theories need to be reduced to ${\cal N}=1$ AdS superspace. The massless
multiplets with half-integer superspin on $(2,0)$ AdS superspace [126] have
already been reduced to ${\cal N}=1$ AdS superspace by Hutomo and Kuzenko in
[127] in 2019.

In this thesis, we reduce all known massless supermultiplets on $(1,1)$ AdS
superspace to ${\cal N}=1$ AdS superspace. This will allow for the differences
of three-dimensional ${\cal N}=2$ massless theories living in $(1,1)$ and
$(2,0)$ AdS superspace to be elucidated. To compliment this, we also reduce
all known massless supermultiplets in ${\cal N}=1$ AdS superspace to AdS
space. Brief discussions concerning the massless on-shell conditions these
theories realise will also be provided.

#####  1.0.0.6 Thesis structure

This thesis consists of six chapters, excluding the introduction chapter 1.
The first chapter 2 reviews essential background material, with emphasis
placed on the notation and conventions which will be employed in this thesis.
The original content of this thesis can be found in the research chapters 3-6.
These chapters are based on the publications [1, 2, 3, 4] and the current work
in progress [5]. A detailed description regarding the location of each of the
above works can found in the authorship declaration section and at the
beginning of each chapter. A summary of the general results of this thesis and
discussion concerning possible future research directions is given in the
final chapter 7. Each of the chapters 2-6 are accompanied by various technical
appendices.

In general, the structure of the chapters (and of particular sections) of this
thesis are organised following a certain prescription, but each are clearly
distinguished by the dimension and (super)space background being studied.
These chapters (or sections) begin with a review of pertinent aspects
concerning the representation and field theory of the (super)space backgrounds
under consideration. In particular, the on-shell (super)fields which furnish
irreducible representations of the corresponding isometry (super)algebra are
presented. This provides all the ingredients necessary to introduce the
(super)spin projection operators. With the (super)spin projectors at our
disposal, we are able to study numerous applications. In particular, we
establish their connection with theories describing the propagation of (S)CHS
fields. The chapters (or sections) which are strictly based in three-
dimensions also include a discussion concerning free theories that propagate
massless and massive particles.

Let us specify the particular dimensions and backgrounds covered in each
chapter (or section) of this thesis. In the mathematical background chapter 2,
we will review and expand upon known results in four-dimensional Minkowski
space, its ${\cal N}=1$ supersymmetric extension and three-dimensional
Minkowski space. These are studied in sections 2.1, 2.2 and 2.3 respectively.
The first research chapter 3 is strictly based in three-dimensional ${\cal
N}$-extended Minkowski superspace. Chapter 4 is set in four-dimensional anti-
de Sitter backgrounds. The first section concerns AdS space 4.1, with the
subsequent sections 4.2 and 4.3 investigating ${\cal N}=1$ and ${\cal N}=2$
AdS superspace respectively. The first section of chapter 5 focuses on three-
dimensional AdS space, while section 5.2 deals with its ${\cal N}=1$
extension. The final research chapter 6 is concerned with three-dimensional
$(1,1)$ AdS superspace.

We have chosen to present the original results from the papers [1, 2, 3, 4],
which are essential to this thesis, as if it was the first time that they have
appeared. In accordance with this, we will not provide citations for these
results in the main body of this thesis. However, a summary of the main
original results obtained, with relevant citations, is given in the last
section of each chapter. As stated in the authorship declaration, some results
from [3] are not strictly part of this thesis (e.g. the partially massless
supermultiplets), however, they are included for completeness. These
particular results will be credited accordingly.

##  Chapter 2 Background material

In this chapter we review essential background material pertaining to
classical field theory and representation theory which is necessary for the
study of (super)spin projection and their applications in three- and four-
dimensional Minkowski (super)space. Particular emphasis is placed on the
notations and conventions which will be employed throughout this thesis. Most
of the content presented is well-established within the literature, thus
important ideas are simply stated. The presentation of this chapter is not
intended to be mathematically rigorous, nor complete, but written in such a
way as to ensure that important themes are self-contained.

###  2.1 Four-dimensional Minkowski space

In this section we will review facets of classical field theory in four-
dimensional Minkowski space. We begin by introducing the unitary irreducible
representations of the four-dimensional Poincaré algebra $\mathfrak{iso}(3,1)$
and their corresponding field realisations on $\mathbb{M}^{4}$. Once
established, we study the spin projection operators [28, 35, 33, 41] and their
corresponding applications. In particular, we will explore the intrinsic
relationship between these operators and conformal higher-spin theory [62].
The notation and conventions employed in this section coincide with those used
in [105].

####  2.1.1 Facets of four-dimensional Minkowski space

Minkowski space in $d$-dimensions, which we denote by $\mathbb{M}^{d}$, is
simply the manifold $\mathbb{R}^{d}$ endowed with the metric
$\eta_{ab}=\text{diag}(-1,1,\cdots,1)$. The spacetime $\mathbb{M}^{d}$ is
parametrised by the local Cartesian coordinates
$x^{a}=(x^{0},x^{1},\cdots,x^{d-1})$, where $x^{0}=t$ is the time coordinate
and $\vec{x}=(x^{1},x^{2},\cdots,x^{d-1})$ are the space coordinates.111From
the onset and throughout, we will make use of natural units $c=\hbar=1$. In
this thesis, we will use indices from the beginning of the Latin alphabet to
denote Lorentz indices.

The isometry group of $\mathbb{M}^{d}$ is the Poincaré group
$\mathsf{IO}(d-1,1)$, which is the group of coordinate transformations
$(\Lambda,a)$

$x^{\prime a}=\Lambda^{a}{}_{b}x^{b}+a^{a}~{},$ (2.1)

that preserve the spacetime interval ${\rm d}s^{2}=\eta_{ab}{\rm d}x^{a}{\rm
d}x^{b}$ on $\mathbb{M}^{d}$. Here, $a^{a}\in\mathbb{R}^{d}$ is a constant
$d$-vector and $\Lambda$ is an element of the Lorentz group
$\mathsf{O}(d-1,1)$ which satisfies the defining master equation

$\Lambda^{\text{T}}\eta\Lambda=\eta~{},$ (2.2)

where $\Lambda^{\text{T}}$ is the matrix transpose of $\Lambda$.

We are primarily interested in the subgroup of $\mathsf{IO}(d-1,1)$, known as
the proper orthochronous Poincaré group $\mathsf{ISO}_{0}(d-1,1)$. The group
$\mathsf{ISO}_{0}(d-1,1)$ describes the coordinate transformations
$(\Lambda,a)$, where instead, the matrix $\Lambda$ is an element of the proper
orthochronous Lorentz group $\mathsf{SO}_{0}(d-1,1)$.222We will commonly refer
to the proper orthochronous Lorentz group as the Lorentz group. In addition to
obeying (2.2), every group element $\Lambda\in\mathsf{SO}_{0}(d-1,1)$ also
satisfies the properties

$\text{det}(\Lambda)=1~{},\qquad\Lambda^{0}{}_{0}\geq 1~{}.$ (2.3)

The conditions (2.3) ensure that the spatial orientation and the direction of
time are preserved, respectively.

The Lorentz algebra $\mathfrak{so}(d-1,1)$ is spanned by the antisymmetric
generators $M_{ab}=-M_{ba}$ which satisfy the commutation relation

$[M_{ab},M_{ce}]=\eta_{ac}M_{be}-\eta_{bc}M_{ae}+\eta_{be}M_{ac}-\eta_{ae}M_{bc}~{}.$
(2.4)

The quadratic Casimir operator of $\mathfrak{so}(d-1,1)$ is $M^{ab}M_{ab}$. In
four dimensions, there exists an additional quadratic Casimir operator of
$\mathfrak{so}(3,1)$, which takes the form $\varepsilon_{abce}M^{ab}M^{ce}$.

The Poincaré algebra $\mathfrak{iso}(d-1,1)$ is spanned by the generators of
spacetime translations $P_{a}$ and Lorentz transformations $J_{ab}$. These
generators satisfy the commutation relations

$\displaystyle[P_{a},P_{b}]$ $\displaystyle=$ $\displaystyle 0~{},$ (2.5a)
$\displaystyle\ [J_{ab},P_{c}]$ $\displaystyle=$ $\displaystyle{\rm
i}\big{(}\eta_{ac}P_{b}-\eta_{bc}P_{a}\big{)}~{},$ (2.5b) $\displaystyle\
[J_{ab},J_{ce}]$ $\displaystyle=$ $\displaystyle{\rm
i}\big{(}\eta_{ac}J_{be}-\eta_{ae}J_{bc}+\eta_{be}J_{ac}-\eta_{bc}J_{ae}\big{)}~{}.$
(2.5c)

These hold in an arbitrary representation of the Poincaré algebra.333In any
unitary representation of $\mathfrak{iso}(d-1,1)$, the energy-momentum $P_{a}$
and Lorentz $J_{ab}$ generators are Hermitian.

The quadratic Casimir operator $C_{1}$ of $\mathfrak{iso}(d-1,1)$ takes the
following form

$C_{1}=-P^{a}P_{a}~{},\qquad[C_{1},P_{a}]=[C_{1},J_{ab}]=0~{}.$ (2.6)

There exist higher-order Casimir operators of $\mathfrak{iso}(d-1,1)$,
however, they are dependent on the dimension $d$ of the spacetime under
consideration (see e.g. [128]). For the remainder of this section, we will
restrict our attention to $4d$. In $4d$, there exists a Casimir operator
$C_{2}$ which is quartic in the generators of $\mathfrak{iso}(3,1)$,

$C_{2}=\mathbb{W}^{a}\mathbb{W}_{a}~{},\qquad[C_{2},P_{a}]=[C_{2},J_{ab}]=0~{},$
(2.7)

where $\mathbb{W}_{a}$ is the Pauli-Lubanski pseudovector

$\mathbb{W}_{a}=\frac{1}{2}\varepsilon_{abce}J^{bc}P^{e}~{}.$ (2.8)

Here, $\varepsilon_{abce}$ is the totally antisymmetric Levi-Civita
tensor.444We choose the normalisation of the Levi-Civita tensor to be
$\varepsilon_{01\dots d-1}=-\varepsilon^{01\dots d-1}=-1$. Using the algebra
(2.5), it can be shown that the Pauli-Lubanski pseudovector (2.8) satisfies
the properties

$\displaystyle\mathbb{W}^{a}P_{a}$ $\displaystyle=$ $\displaystyle 0~{},$
(2.9a) $\displaystyle\ [\mathbb{W}_{a},P_{b}]$ $\displaystyle=$ $\displaystyle
0~{},$ (2.9b) $\displaystyle\ [J_{ab},\mathbb{W}_{c}]$ $\displaystyle=$
$\displaystyle{\rm
i}\big{(}\eta_{ac}\mathbb{W}_{b}-\eta_{bc}\mathbb{W}_{a}\big{)}~{},$ (2.9c)
$\displaystyle\ [\mathbb{W}_{a},\mathbb{W}_{b}]$ $\displaystyle=$
$\displaystyle{\rm i}\varepsilon_{abce}\mathbb{W}^{c}P^{e}~{}.$ (2.9d)

The properties (2.9) are useful in showing that the quadratic Casimir operator
$C_{2}$ commutes with the generators of $\mathfrak{iso}(3,1)$.

####  2.1.2 Irreducible representations of the Poincaré algebra

In this section we review salient aspects concerning the classification of the
massive and massless helicity UIRs of $\mathfrak{iso}(3,1)$ which carry
positive energy $p^{0}\equiv E>0$. The classification of these UIRs was first
completed by Wigner [6] (see also [129]) via the method of induced
representations.555The classification of the UIRs of the Poincaré algebra has
been generalised to $d$ dimensions, for $d>2$. See the works [130, 131, 132]
for an extensive treatment.

The massive UIRs of $\mathfrak{iso}(3,1)$ are labelled by the quantum numbers,
mass $m$ and spin $s$ [6]. It was shown by Bargmann and Wigner [23] that these
quantum numbers are determined by the eigenvalues of the Casimir operators
$C_{1}$ and $C_{2}$

$C_{1}=P^{a}P_{a}=-\mbox{\boldmath$m$}^{2}\mathds{1}~{},\qquad
C_{2}=\mathbb{W}^{a}\mathbb{W}_{a}=s(s+1)\mbox{\boldmath$m$}^{2}\mathds{1}~{},$
(2.10)

where the mass $\mbox{\boldmath$m$}>0$ is strictly positive and the spin $s$
takes positive (half-)integer values $s=0,\frac{1}{2},1,\frac{3}{2},\cdots$ in
different representations. The massive irreducible representations of
$\mathfrak{iso}(3,1)$ are characterised by the conditions (2.10). We will
denote by $G(\mbox{\boldmath$m$},s)$ the massive UIR of $\mathfrak{iso}(3,1)$
which carries mass $m$ and spin $s$.666The notation $G(\mbox{\boldmath$m$},s)$
for a UIR of $\mathfrak{iso}(3,1)$ is unconventional. We choose it to help
distinguish between the UIRs of different isometry (super)algebras, which will
appear in later sections.  For a given $s$, the UIR $G(\mbox{\boldmath$m$},s)$
describes $(2s+1)$ physical degrees of freedom.

It was also shown in [23] that the massless helicity UIRs cannot be
characterised by the eigenvalues of the Casimir operators of
$\mathfrak{iso}(3,1)$, as they are both vanishing777There exists another class
of massless UIRs, known as continuous-spin representations (see [133] for a
modern review). These UIRs are characterised by $C_{2}=\mu^{2}$, where $\mu$
is a real parameter with unit mass dimension. These UIRs are not covered
within the scope of this thesis.

$C_{1}=P^{a}P_{a}=0~{},\qquad C_{2}=\mathbb{W}^{a}\mathbb{W}_{a}=0~{}.$ (2.11)

The massless UIRs888In the hope that no confusion arises, we will refer to
massless helicity representations as massless representations from this point
onwards. are instead characterised by Wigner’s equation,

$\mathbb{W}_{a}=\lambda P_{a}~{}.$ (2.12)

In every massless UIR, the Pauli-Lubanski pseudovector is proportional to the
energy-momentum generator $P_{a}$ (2.12) where the quantum number $\lambda$ is
known as helicity. The helicity can take (half-)integer values
$\lambda=0,\pm\frac{1}{2},\pm 1,\pm\frac{3}{2},\cdots$ in different massless
UIRs. It is common to say that a massless UIR has spin $|\lambda|$. Thus for
$s>0$, a massless representation carries a single physical degree of freedom
corresponding to the helicity state it describes.

####  2.1.3 Irreducible field representations

Let us denote by $V_{(m,n)}$ the space of totally symmetric complex spinor
fields $\phi_{\alpha(m){\dot{\alpha}}(n)}(x)$ of Lorentz type
$(\frac{m}{2},\frac{n}{2})$ on $\mathbb{M}^{4}$.999 It is convenient to work
in two-component spinor notation since fermionic and bosonic fields can both
be treated in the same framework. See appendix 2.A for a review on the two-
component spinor notation.  For integers $m,n\geq 1$, a spinor field
$\phi_{\alpha(m){\dot{\alpha}}(n)}(x)$ on $V_{(m,n)}$ is said to be an on-
shell field if it satisfies the conditions

$\displaystyle\partial^{\beta{\dot{\beta}}}\phi_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}(x)$
$\displaystyle=$ $\displaystyle 0~{},$ (2.13a)
$\displaystyle(\Box-\mbox{\boldmath$m$}^{2})\phi_{\alpha(m){\dot{\alpha}}(n)}(x)$
$\displaystyle=$ $\displaystyle 0~{},$ (2.13b)

where mass $\mbox{\boldmath$m$}\geq 0$ is non-negative. The d’Alembertian
operator
$\Box=-\frac{1}{2}\partial^{\beta{\dot{\beta}}}\partial_{\beta{\dot{\beta}}}$
is defined in terms of the Lorentz covariant derivatives
$\partial_{\beta{\dot{\beta}}}$ of $\mathbb{M}^{4}$. In the case of a scalar
field and spinor fields of Lorentz type $(\frac{m}{2},0)$ and
$(0,\frac{n}{2})$, an on-shell field is defined to only satisfy the Klein-
Gordon equation (2.13b). Note that from this point onwards, we will suppress
the spacetime dependence of tensor fields
$\phi_{\alpha(m){\dot{\alpha}}(n)}(x)\equiv\phi_{\alpha(m){\dot{\alpha}}(n)}$.
In the following sections, we will demonstrate how on-shell fields realise
massive and massless UIRs of $\mathfrak{iso}(3,1)$, following the discussion
of [105].

#####  2.1.3.1 Massive field representations

An on-shell field (2.13) carrying strictly positive mass
$\mbox{\boldmath$m$}>0$ furnishes the massive UIR $G(\mbox{\boldmath$m$},s)$
of $\mathfrak{iso}(3,1)$. We say that such an on-shell field is massive,
carrying spin $s=\frac{1}{2}(m+n)$ and mass $m$. We wish to demonstrate that a
massive field (2.13) does indeed realise the massive UIR
$G(\mbox{\boldmath$m$},s)$. To do this, it suffices to show that the
eigenvalues of the Casimir operators $C_{1}$ and $C_{2}$ of
$\mathfrak{iso}(3,1)$ take the fixed values (2.10).

In the field representation, the Poincaré generators take the following form

$\displaystyle P_{a}$ $\displaystyle=$ $\displaystyle-{\rm i}\partial_{a}~{},$
(2.14a) $\displaystyle J_{ab}$ $\displaystyle=$ $\displaystyle{\rm
i}\big{(}x_{b}\partial_{a}-x_{a}\partial_{b}\big{)}-{\rm i}M_{ab}~{}.$ (2.14b)

One can check that the generators (2.14) do indeed satisfy the Poincaré
algebra (2.5). Note that in this thesis, we use the notation $M_{ab}$ to
denote the Lorentz generators which, when acting on any tensor field, return
the spin contribution only (and not the orbital part $L_{ab}={\rm
i}(x_{b}\partial_{a}-x_{a}\partial_{b})$ in (2.14b)). The action of the
Lorentz generators $M_{ab}$ on a tensor field is given in appendix 2.A.

Using the generators (2.14), it can be shown that the Pauli-Lubanski
pseudovector (2.8) takes the following form in the field representation

$\mathbb{W}_{\alpha{\dot{\alpha}}}=-{\rm
i}\partial^{\beta}{}_{\dot{\alpha}}M_{\beta\alpha}+{\rm
i}\partial_{\alpha}{}^{\dot{\beta}}\bar{M}_{{\dot{\beta}}{\dot{\alpha}}}~{}.$
(2.15)

It follows from (2.15) that the action of the Casimir operator $C_{2}$ (2.7)
on $V_{(m,n)}$ is

$\mathbb{W}^{2}\phi_{\alpha(m){\dot{\alpha}}(n)}=s(s+1)\Box\phi_{\alpha(m){\dot{\alpha}}(n)}+mn\partial_{\alpha{\dot{\alpha}}}\partial^{\beta{\dot{\beta}}}\phi_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}~{},$
(2.16)

where we have made use of the identities

$\partial_{\alpha}{}^{\dot{\beta}}\partial^{\beta}{}_{\dot{\beta}}=\delta_{\alpha}{}^{\beta}\Box~{},\qquad\partial^{\beta}{}_{{\dot{\alpha}}}\partial_{\beta}{}^{\dot{\beta}}=\delta_{\dot{\alpha}}{}^{\dot{\beta}}\Box~{}.$
(2.17)

One immediately sees that the action of the Casimir operator $C_{2}$ (2.16) on
a massive on-shell field (2.13) reduces to

$\mathbb{W}^{2}\phi_{\alpha(m){\dot{\alpha}}(n)}=s(s+1)\mbox{\boldmath$m$}^{2}\phi_{\alpha(m){\dot{\alpha}}(n)}~{},$
(2.18)

where $s=\frac{1}{2}(m+n)$. It follows from (2.13b) and (2.18) that the
eigenvalues of the Casimir operators $C_{1}$ and $C_{2}$ on the space of
massive fields are equivalent to those which characterise the massive UIR
$G(\mbox{\boldmath$m$},s)$ (cf. (2.10)), thus demonstrating that massive on-
shell fields do indeed realise the UIR $G(\mbox{\boldmath$m$},s)$. Note that
in (2.16) we have adopted the condensed notation (2.352).

Next we wish to show that a massive field describes $(2s+1)$ physical degrees
of freedom, as required by a field which furnishes the massive UIR
$G(\mbox{\boldmath$m$},s)$ with fixed spin $s$. Counting the physical degrees
of freedom, a totally symmetric spinor field
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ has $(m+1)(n+1)$ independent components.
The component dependence which arises from the transverse constraint (2.13a)
is equivalent to the number of independent components described by the tensor
field $\phi_{\alpha(m-1){\dot{\alpha}}(n-1)}$, which is $mn$. Thus, there are
$(m+1)(n+1)-mn=2s+1$ independent components remaining, as required.

Let us denote by $V_{(m,n)}^{[\boldsymbol{m}]}$ the space of massive on-shell
fields. It turns out that the UIR $G(\mbox{\boldmath$m$},s)$ can be
equivalently realised on spaces other than $V_{(m,n)}^{[\boldsymbol{m}]}$. In
particular, it can be realised on the space of massive fields
$V^{[\boldsymbol{m}]}_{(m+t,n-t)}$ for $1\leq t\leq n$, or
$V^{[\boldsymbol{m}]}_{(m-t,n+t)}$ for $1\leq t\leq m$. To observe this, let
us introduce the operator $\Delta_{\alpha{\dot{\alpha}}}$ [41]

$\Delta_{\alpha{\dot{\alpha}}}=\frac{1}{\sqrt{\Box}}\partial_{\alpha{\dot{\alpha}}}=\frac{1}{m}\partial_{\alpha{\dot{\alpha}}}~{},$
(2.19)

which is invertible

$\Delta_{\alpha}{}^{\dot{\beta}}\Delta^{\beta}{}_{\dot{\beta}}=\delta_{\alpha}{}^{\beta}~{},\quad\Delta^{\beta}{}_{\dot{\alpha}}\Delta_{\beta}{}^{\dot{\beta}}=\delta_{\dot{\alpha}}{}^{\dot{\beta}}~{}.$
(2.20)

Since we are strictly working with massive fields (2.13), the operators (2.19)
will always be well defined. The operator $\Delta_{\alpha{\dot{\alpha}}}$ is
important as it can be used to map the space $V^{[\boldsymbol{m}]}_{(m,n)}$ to
either $V^{[m]}_{(m+t,n-t)}$

$\displaystyle\Delta^{t}:V^{[\boldsymbol{m}]}_{(m,n)}$
$\displaystyle\longrightarrow$ $\displaystyle
V^{[\boldsymbol{m}]}_{(m+t,n-t)}~{},\qquad 1\leq t\leq n~{},$ (2.21a)
$\displaystyle\phi_{\alpha(m){\dot{\alpha}}(n)}$ $\displaystyle\longmapsto$
$\displaystyle\phi_{\alpha(m+t){\dot{\alpha}}(n-t)}=\Delta_{\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\Delta_{\alpha_{t}}{}^{{\dot{\beta}}_{t}}\phi_{\alpha(m){\dot{\alpha}}(n-t){\dot{\beta}}(t)}~{},$
(2.21b)

or $V^{[\boldsymbol{m}]}_{(m-t,n+t)}$

$\displaystyle\widehat{\Delta}^{t}:V^{[\boldsymbol{m}]}_{(m,n)}$
$\displaystyle\longrightarrow$ $\displaystyle
V^{[\boldsymbol{m}]}_{(m-t,n+t)}~{},\qquad 1\leq t\leq m~{},$ (2.22a)
$\displaystyle\phi_{\alpha(m){\dot{\alpha}}(n)}$ $\displaystyle\longmapsto$
$\displaystyle\phi_{\alpha(m-t){\dot{\alpha}}(n+t)}=\Delta^{\beta_{1}}{}_{{\dot{\alpha}}_{1}}\cdots\Delta^{\beta_{t}}{}_{{\dot{\alpha}}_{t}}\phi_{\alpha(m-t)\beta(t){\dot{\alpha}}(n)}~{}.$
(2.22b)

The mappings (2.21) and (2.22) are invertible since
$\Delta_{\alpha{\dot{\alpha}}}$ is invertible (2.20). It is an easy exercise
to show that if the field $\phi_{\alpha(m){\dot{\alpha}}(n)}$ furnishes the
UIR $G(\mbox{\boldmath$m$},s)$, then so do the fields
$\phi_{\alpha(m+t){\dot{\alpha}}(n-t)}$ and
$\phi_{\alpha(m-t){\dot{\alpha}}(n+t)}$. Thus, the spaces of massive fields
$V^{[\boldsymbol{m}]}_{(2s,0)},V^{[\boldsymbol{m}]}_{(2s-1,1)},\cdots,V^{[\boldsymbol{m}]}_{(1,2s-1)},V^{[\boldsymbol{m}]}_{(0,2s)}$
can all furnish the massive UIR $G(\mbox{\boldmath$m$},s)$.

###### Real massive field representations

We have only treated massive UIRs of $\mathfrak{iso}(3,1)$ on the space of
complex fields in our discussions so far. A complex massive field
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ on $V^{[\boldsymbol{m}]}_{(m,n)}$
possesses a complex conjugate $\bar{\phi}_{\alpha(n){\dot{\alpha}}(m)}$, which
is obtained from $\phi_{\alpha(m){\dot{\alpha}}(n)}$ via the operation of
complex conjugation

$\displaystyle{}^{*}:V_{(m,n)}$ $\displaystyle\longrightarrow$ $\displaystyle
V_{(n,m)}~{},$ (2.23a) $\displaystyle\phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle\longmapsto$
$\displaystyle\big{(}\phi_{\alpha(m){\dot{\alpha}}(n)}\big{)}^{*}=\bar{\phi}_{\alpha(n){\dot{\alpha}}(m)}~{}.$
(2.23b)

In general, a massive field (2.13) and its complex conjugate are independent.
We will be interested in studying real massive fields, which are a class of
complex massive fields $\phi_{\alpha(m){\dot{\alpha}}(n)}$ supplemented with
some reality condition which ensures that $\phi_{\alpha(m){\dot{\alpha}}(n)}$
is related to its complex conjugate $\bar{\phi}_{\alpha(n){\dot{\alpha}}(m)}$.
Note that this review of real massive fields follows the work of Siegel and
Gates in [33] (see also [41]).

We already have the tools at our disposal to find such suitable reality
conditions on $V^{[\boldsymbol{m}]}_{(m,n)}$. Let us make use of the operator
$\Delta_{\alpha{\dot{\alpha}}}$ (2.19) to construct the mapping

$\displaystyle\Delta:V^{[\boldsymbol{m}]}_{(m,n)}\longrightarrow
V^{[\boldsymbol{m}]}_{(n,m)}~{},$ (2.24a)
$\displaystyle\phi_{\alpha(m){\dot{\alpha}}(n)}\longmapsto\big{(}\Delta\phi\big{)}_{\alpha(n){\dot{\alpha}}(m)}=e^{{\rm
i}\lambda}\Delta_{\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\Delta_{\alpha_{n}}{}^{{\dot{\beta}}_{n}}\Delta^{\beta_{1}}{}_{{\dot{\alpha}}_{1}}\cdots\Delta^{\beta_{m}}{}_{{\dot{\alpha}}_{m}}\phi_{\beta(m){\dot{\beta}}(n)}~{},$
(2.24b)

for some $\lambda\in\mathbb{R}$. Note that the domain and co-domain of both ∗
(2.23) and $\Delta$ (2.24) are identical, thus, it is natural to consider the
subspace of $V^{[\boldsymbol{m}]}_{(m,n)}$ on which ∗ and $\Delta$ coincide

$\bar{\phi}_{\alpha(n){\dot{\alpha}}(m)}=e^{{\rm
i}\lambda}\Delta_{\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\Delta_{\alpha_{n}}{}^{{\dot{\beta}}_{n}}\Delta^{\beta_{1}}{}_{{\dot{\alpha}}_{1}}\cdots\Delta^{\beta_{m}}{}_{{\dot{\alpha}}_{m}}\phi_{\beta(m){\dot{\beta}}(n)}~{}.$
(2.25)

We will refer to (2.25) as a reality condition, as it ensures that
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ and its complex conjugate
$\bar{\phi}_{\alpha(n){\dot{\alpha}}(m)}$ are dependent.

In accordance with the discussion above, a field
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ is said to be a real massive field if it
satisfies the properties

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial^{\beta{\dot{\beta}}}\phi_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}~{},$
(2.26a) $\displaystyle 0$ $\displaystyle=$
$\displaystyle(\Box-\mbox{\boldmath$m$}^{2})\phi_{\alpha(m){\dot{\alpha}}(n)}~{},$
(2.26b) $\displaystyle\bar{\phi}_{\alpha(n){\dot{\alpha}}(m)}$
$\displaystyle=$ $\displaystyle
e^{i\lambda}\Delta_{\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\Delta_{\alpha_{n}}{}^{{\dot{\beta}}_{n}}\Delta^{\beta_{1}}{}_{{\dot{\alpha}}_{1}}\cdots\Delta^{\beta_{m}}{}_{{\dot{\alpha}}_{m}}\phi_{\beta(m){\dot{\beta}}(n)}~{}.$
(2.26c)

For different values of $\lambda$, the corresponding linear space of fields
satisfying (2.26) are all isomorphic. Let us look at the reality conditions
(2.26c) for the archetypal examples $m=n$ and $m=n+1$.

In the case $m=n$, the reality condition (2.26c) coincides with the expected
result

$\phi_{\alpha(n){\dot{\alpha}}(n)}=\bar{\phi}_{\alpha(n){\dot{\alpha}}(n)}~{},$
(2.27)

where we have made use of the properties (2.26a), (2.26b) and fixed
$\lambda=0$. Without loss of generality, the reality condition (2.25) in the
case $m=n+1$ coincides with the Dirac-type equation

$\bar{\phi}_{\alpha(n){\dot{\alpha}}(n){\dot{\beta}}}=e^{i\lambda}\Delta^{\beta}{}_{\dot{\beta}}\phi_{\beta\alpha(n){\dot{\alpha}}(n)}\quad\Longrightarrow\quad\mbox{\boldmath$m$}\bar{\phi}_{\alpha(n){\dot{\alpha}}(n){\dot{\beta}}}=e^{i\lambda}\partial^{\beta}{}_{\dot{\beta}}\phi_{\beta\alpha(n){\dot{\alpha}}(n)}~{}.$
(2.28)

For the case $\lambda=\frac{\pi}{2}$, this simply reduces to the Dirac
equation.

Let us show that a complex massive field $\phi_{\alpha(m){\dot{\alpha}}(n)}$
decomposes into two real massive fields when a reality condition is imposed.
To do this, it is necessary to introduce the conjugation operator $K$ which is
defined by its action on $V^{[\boldsymbol{m}]}_{(m,n)}$ [33, 41]

$\displaystyle
K\phi_{\alpha(m){\dot{\alpha}}(n)}=e^{i\lambda}\Delta_{\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\Delta_{\alpha_{m}}{}^{{\dot{\beta}}_{m}}\Delta^{\beta_{1}}{}_{{\dot{\alpha}}_{1}}\cdots\Delta^{\beta_{n}}{}_{{\dot{\alpha}}_{n}}\bar{\phi}_{\beta(n){\dot{\beta}}(m)}~{}.$
(2.29)

It is a simple exercise to show that the conjugation operator is involutive,
$K^{2}\phi_{\alpha(m){\dot{\alpha}}(n)}=\phi_{\alpha(m){\dot{\alpha}}(n)}$.101010Since
the operator $K$ is involutive, it follows that the operators
$\frac{1}{2}(1\pm K)$ are orthogonal projectors. With this, let us bisect a
complex massive field in the following manner

$\phi_{\alpha(m){\dot{\alpha}}(n)}=\phi^{(+)}_{\alpha(m){\dot{\alpha}}(n)}+\phi^{(-)}_{\alpha(m){\dot{\alpha}}(n)}~{},$
(2.30)

where we have introduced the fields

$\phi^{(\pm)}_{\alpha(m){\dot{\alpha}}(n)}=\frac{1}{2}(1\pm
K)\phi_{\alpha(m){\dot{\alpha}}(n)}~{}.$ (2.31)

Using the on-shell properties (2.13) and the fact that $K$ is involutive, it
can be shown that the fields $\phi^{(\pm)}_{\alpha(m){\dot{\alpha}}(n)}$
satisfy the conditions

$\displaystyle\partial^{\beta{\dot{\beta}}}\phi^{(\pm)}_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}$
$\displaystyle=$ $\displaystyle 0~{},$ (2.32a)
$\displaystyle\big{(}\Box-\mbox{\boldmath$m$}^{2}\big{)}\phi^{(\pm)}_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=$ $\displaystyle 0~{},$ (2.32b) $\displaystyle\big{(}1\mp
K\big{)}\phi^{(\pm)}_{\alpha(m){\dot{\alpha}}(n)}$ $\displaystyle=$
$\displaystyle 0~{}.$ (2.32c)

The last equation (2.32c) is nothing more than the reality condition (2.26c)
written in terms of the conjugation operator. In accordance with (2.26), the
fields $\phi^{(\pm)}_{\alpha(m){\dot{\alpha}}(n)}$ can be identified as real
massive fields.

#####  2.1.3.2 Massless field representations

The formulation of massless fields can be approached in two ways: in terms of
gauge fields $\phi_{\alpha(m){\dot{\alpha}}(n)}$ or in terms of gauge-
invariant field strengths $C_{\alpha(p){\dot{\alpha}}(q)}(\phi)$. We begin by
reviewing the construction of massless gauge fields.

For integers $m,n\geq 1$, let us consider a field
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ on $V_{(m,n)}$ which satisfies the on-
shell conditions

$\displaystyle\partial^{\beta{\dot{\beta}}}\phi_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}$
$\displaystyle=$ $\displaystyle 0~{},$ (2.33a)
$\displaystyle\Box\phi_{\alpha(m){\dot{\alpha}}(n)}$ $\displaystyle=$
$\displaystyle 0~{}.$ (2.33b) The equations (2.33) are nothing more than the
on-shell conditions (2.13) in the massless limit $m\rightarrow 0$. It can be
shown that these conditions are compatible with the gauge symmetry
$\delta_{\zeta}\phi_{\alpha(m){\dot{\alpha}}(n)}=\partial_{\alpha{\dot{\alpha}}}\zeta_{\alpha(m-1){\dot{\alpha}}(n-1)}~{},$
(2.33c) given that the complex gauge parameter
$\zeta_{\alpha(m-1){\dot{\alpha}}(n-1)}$ is itself on-shell
$\displaystyle\Box\zeta_{\alpha(m-1){\dot{\alpha}}(n-1)}$ $\displaystyle=$
$\displaystyle 0~{},$ (2.33d)
$\displaystyle\partial^{\beta{\dot{\beta}}}\zeta_{\beta\alpha(m-2){\dot{\beta}}{\dot{\alpha}}(n-2)}$
$\displaystyle=$ $\displaystyle 0~{}.$ (2.33e)

We say that a field satisfying the on-shell conditions (2.33) is a massless
(gauge) field. For the case $m=n=0$, $m=0,n=1$ and $m=1,n=0$, the
corresponding massless fields only satisfy the massless Klein-Gordon equation
(2.33b). It follows from (2.33a) and (2.33b) that the eigenvalues of the
Casimir operators vanish on the space of massless fields (2.33), which is
consistent for a tensor field which realises a massless UIR (cf. (2.11)).

However, studying the helicity content of the massless field (2.33) reveals

$\displaystyle\mathbb{W}_{\beta{\dot{\beta}}}\phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=\frac{1}{2}(m-n)P_{\beta{\dot{\beta}}}\phi_{\alpha(m){\dot{\alpha}}(n)}-m\varepsilon_{\beta\alpha}P^{\gamma}{}_{\dot{\beta}}\phi_{\alpha(m-1)\gamma{\dot{\alpha}}(n)}+n\varepsilon_{{\dot{\beta}}{\dot{\alpha}}}P_{\beta}{}^{\dot{\gamma}}\phi_{\alpha(m){\dot{\alpha}}(n-1){\dot{\gamma}}}~{}.$
(2.34)

Recall that in a massless UIR of $\mathfrak{iso}(3,1)$, the helicity $\lambda$
is determined from the Wigner equation,
$\mathbb{W}_{\beta{\dot{\beta}}}\phi_{\alpha(m){\dot{\alpha}}(n)}=\lambda
P_{\beta{\dot{\beta}}}\phi_{\alpha(m){\dot{\alpha}}(n)}$ (cf. (2.12)), with
$\lambda$ taking (half-)integer values. It is apparent from (2.1.3.2) that
this condition does not hold in the case of massless gauge fields (2.33). In
other words, massless gauge fields do not furnish massless UIRs of
$\mathfrak{iso}(3,1)$. This is consistent with the fact that massless gauge
fields (2.33) are not truly irreducible due to the presence of the gauge
symmetry (2.33c). It follows that the solution space of massless fields
(2.33a) and (2.33b) possesses non-trivial invariant subspaces which are mapped
to one another by the gauge transformations (2.33c). To make the solution
space irreducible, we need to factor out these redundant gauge degrees of
freedom by completely fixing the gauge symmetry (2.33c).

Going about this, it will be useful to switch to momentum space, i.e.
$\phi_{\alpha(m){\dot{\alpha}}(n)}(x)\rightarrow\phi_{\alpha(m){\dot{\alpha}}(n)}(p)$.
In momentum space, the constraints (2.33a) and (2.33e) assume the following
form

$p^{\beta{\dot{\beta}}}\phi_{\beta\alpha(m-1)\beta{\dot{\alpha}}(n-1)}=0~{},\qquad
p^{\beta{\dot{\beta}}}\zeta_{\beta\alpha(m-2){\dot{\beta}}{\dot{\alpha}}(n-2)}=0~{}.$
(2.35)

Since the conditions (2.44) are invariant under Poincaré transformations, we
are free to transition to the standard momentum frame $p^{a}=(E,0,0,E)$, where
energy $E$ is strictly positive. Converting to two-component spinor notation
yields

$p^{a}=(E,0,0,E)\quad\Longrightarrow\quad
p^{\alpha{\dot{\alpha}}}=2E\begin{pmatrix}~{}0~{}&0~{}\\\
~{}0~{}&1~{}\end{pmatrix}~{},\quad
p_{\alpha{\dot{\alpha}}}=2E\begin{pmatrix}~{}1~{}&0~{}\\\
~{}0~{}&0~{}\end{pmatrix}~{}.$ (2.36)

The equations (2.35) assume the following form in the rest frame

$\phi_{2\alpha(m-1)\dot{2}{\dot{\alpha}}(n-1)}(p)=0~{},\qquad\zeta_{2\alpha(m-2)\dot{2}{\dot{\alpha}}(n-2)}(p)=0~{}.$
(2.37)

Additionally, the gauge transformation (2.33c) takes the form,

$\delta\phi_{1\alpha(m-1)\dot{1}{\dot{\alpha}}(n-1)}(p)\propto
p_{1\dot{1}}\zeta_{\alpha(m-1){\dot{\alpha}}(n-1)}(p)~{},$ (2.38)

in the frame (2.36), which allows us to impose the gauge

$\phi_{1\alpha(m-1)\dot{1}{\dot{\alpha}}(n-1)}(p)=0~{}.$ (2.39)

It follows from (2.37) and (2.39) that the only non-vanishing components of
$\phi_{\alpha(m){\dot{\alpha}}(n)}(p)$ are $\phi_{1\dots
1\dot{2}\dots\dot{2}}(p)$ and $\phi_{2\dots 2\dot{1}\dots\dot{1}}(p)$. Note
that we have completely fixed the gauge freedom.

Elucidating the helicity content of the two non-vanishing components of
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ (2.1.3.2), we find

$\displaystyle\mathbb{W}_{\beta{\dot{\beta}}}\phi_{1\dots
1\dot{2}\dots\dot{2}}(p)$ $\displaystyle=$
$\displaystyle\phantom{-}\frac{1}{2}(m+n)\phi_{1\dots
1\dot{2}\dots\dot{2}}(p)~{},$ (2.40a)
$\displaystyle\mathbb{W}_{\beta{\dot{\beta}}}\phi_{2\dots
2\dot{1}\dots\dot{1}}(p)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(m+n)\phi_{2\dots 2\dot{1}\dots\dot{1}}(p)~{}.$
(2.40b)

Upon completely fixing the gauge symmetry, it follows from (2.33b) and (2.40)
that the massless field (2.33) furnishes a massless UIR of
$\mathfrak{iso}(3,1)$ (cf. (2.12)). In particular, the massless field
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ describes two physical components,
$\phi_{1\dots 1\dot{2}\dots\dot{2}}$ and $\phi_{2\dots 2\dot{1}\dots\dot{1}}$,
which describe the two helicity modes $\frac{1}{2}(m+n)$ and
$-\frac{1}{2}(m+n)$, respectively.

It is worth taking a moment to comment on the subtleties concerning the
irreducibility of the massless field representations (2.33). In accordance
with (2.12), massless irreducible representations of the connected component
of the Poincaré group $\textsf{ISO}_{0}(3,1)$ are characterised by a single
helicity value $\lambda=0,\pm\frac{1}{2},\pm 1,\pm\frac{3}{2},\cdots$. Thus
the massless fields (2.33) realise a reducible representation of
$\textsf{ISO}_{0}(3,1)$, since they describe the two helicity modes
$\lambda=\frac{1}{2}(m+n)$ and $\lambda=-\frac{1}{2}(m+n)$ which differ by a
sign factor. However, the massless fields (2.33) realise an irreducible
representation with respect to the full Poincaré group. $\textsf{ISO}(3,1)$.
The reason for this is that parity transformations change the sign of the
helicity $\lambda$. Thus under a parity transformation, the massless
representations characterised by the helicity $\frac{1}{2}(m+n)$ will map to
the representation with helicity $-\frac{1}{2}(m+n)$, and vice versa. Hence, a
massless field which furnishes a UIR of $\textsf{ISO}(3,1)$ describes two
helicity modes with identical spin $\lambda$, but different sign. The field
equations (2.33) are the desired on-shell conditions for a parity invariant
theory that describing the propagation of a massless spin-$s$ field.

As seen, it is sometimes cumbersome to study massless dynamics in terms of the
massless gauge fields, due to difficulties concerning irreducibility. However,
these difficulties are immediately circumvented if massless dynamics are
studied within the framework of gauge-invariant field strengths. Let us
consider a field strength $C_{\alpha(p){\dot{\alpha}}(q)}(\phi)$, which is
invariant under the gauge transformations (2.33c)

$\delta_{\zeta}C_{\alpha(p){\dot{\alpha}}(q)}(\phi)=0~{}.$ (2.41)

For integers $p,q\geq 1$, the field strength
$C_{\alpha(p){\dot{\alpha}}(q)}(\phi)$ furnishes a massless UIR of
$\mathfrak{iso}(3,1)$ if it obeys the constraints

$\displaystyle\partial^{\beta}{}_{\dot{\beta}}C_{\beta\alpha(p-1){\dot{\alpha}}(q)}(\phi)=0~{},$
(2.42a)
$\displaystyle\partial_{\beta}{}^{{\dot{\beta}}}C_{\alpha(p){\dot{\beta}}{\dot{\alpha}}(q-1)}(\phi)=0~{}.$
(2.42b)

Conditions (2.42) imply that the the eigenvalues of the Casimir operators
$C_{1}$ and $C_{2}$ vanish on the space of field strengths
$C_{\alpha(p){\dot{\alpha}}(q)}(\phi)$. Note that for the cases $p>0,q=0$ and
$p=0,q>0$, the conditions (2.42a) and (2.42b) describe the massless UIRs,
respectively.

The helicity content of a massless field is easily attainable when working in
terms of field strengths. Acting on $C_{\alpha(p){\dot{\alpha}}(q)}(\phi)$
with the Pauli-Lubanski pseudovector (2.15) yields

$\mathbb{W}_{\beta{\dot{\beta}}}C_{\alpha(p){\dot{\alpha}}(q)}(\phi)=\frac{1}{2}\big{(}p-q\big{)}P_{\beta{\dot{\beta}}}C_{\alpha(p){\dot{\alpha}}(q)}(\phi)~{},$
(2.43)

thus demonstrating that $C_{\alpha(p){\dot{\alpha}}(q)}(\phi)$ carries
helicity $\lambda=\frac{1}{2}(p-q)$ (cf. (2.12)).

Next, we wish to show that $C_{\alpha(p){\dot{\alpha}}(q)}(\phi)$ describes
the correct physical degrees of freedom for a massless field. Transitioning to
momentum space, the constraints (2.42) take the following form

$\displaystyle
p^{\beta}{}_{\dot{\beta}}C_{\beta\alpha(p-1){\dot{\alpha}}(q)}(\phi)=0~{},$
(2.44a) $\displaystyle
p_{\beta}{}^{\dot{\beta}}C_{\alpha(p){\dot{\alpha}}(q-1){\dot{\beta}}}(\phi)=0~{}.$
(2.44b)

Again, we are free to move into the standard momentum frame (2.36), where the
conditions (2.44) demonstrate that the only non-vanishing component of
$C_{\alpha(p){\dot{\alpha}}(q)}(\phi)$ is $C_{{1\ldots
1}{{\dot{1}\ldots\dot{1}}}}(\phi)$. In accordance with the classification of
massless UIRs (cf.(2.12)), it follows from (2.43) that the field strength
$C_{\alpha(p){\dot{\alpha}}(q)}(\phi)$ describes a massless field with
helicity $\lambda=\frac{1}{2}(p-q)$.

In the case $p>q$, $C_{\alpha(p){\dot{\alpha}}(q)}(\phi)$ can always be
expressed in terms of the gauge-invariant field strength
$C_{\alpha(p-q)}(\phi)$ which only carries undotted spinor indices,

$C_{\alpha(p){\dot{\alpha}}(q)}(\phi)=(\partial_{\alpha{\dot{\alpha}}})^{q}C_{\alpha(p-q)}(\phi)~{},$
(2.45)

given that $C_{\alpha(p-q)}(\phi)$ satisfies the on-shell condition

$\partial^{\beta}{}_{\dot{\beta}}C_{\beta\alpha(p-q-1)}(\phi)=0~{}.$ (2.46)

Similarly for the case $q>p$, the field $C_{\alpha(p){\dot{\alpha}}(q)}(\phi)$
can always be written in terms of the gauge-invariant field strength
$C_{{\dot{\alpha}}(q-p)}(\phi)$ which only has dotted spinor indices

$C_{\alpha(p){\dot{\alpha}}(q)}(\phi)=(\partial_{\alpha{\dot{\alpha}}})^{p}C_{{\dot{\alpha}}(q-p)}(\phi)~{},$
(2.47)

given that $C_{{\dot{\alpha}}(q-p)}(\phi)$ satisfies the on-shell condition

$\partial_{\beta}{}^{\dot{\beta}}C_{{\dot{\beta}}{\dot{\alpha}}(q-p-1)}(\phi)=0~{}.$
(2.48)

In other words, we need only to work with constrained field strengths of a
single index type to describe massless dynamics.

It proves instructive to perform an on-shell analysis of a massless theory in
order to demonstrate how their dynamics can be studied in the two frameworks
detailed above. As an example, we will study the action for a massless
spin-$s$ field, which was formulated by Fronsdal [27] in 1978. In two-
component notation, Fronsdal’s massless action is described in terms of two
real bosonic fields $\phi_{\alpha(s){\dot{\alpha}}(s)}$ and
$\varphi_{\alpha(s-2){\dot{\alpha}}(s-2)}$. Note that the analysis presented
in this section follows that given by Buchbinder and Kuzenko in [105].

The Fronsdal action [27] assumes the following form in two-component spinor
notation

$\displaystyle S^{(s)}_{\text{F}}[\phi,\varphi]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{4}x~{}\Big{\\{}\phi^{\alpha(s){\dot{\alpha}}(s)}\Box\phi_{\alpha(s){\dot{\alpha}}(s)}-\frac{s}{2}\partial_{\beta{\dot{\beta}}}\phi^{\beta\alpha(s-1){\dot{\beta}}{\dot{\alpha}}(s-1)}\partial^{\gamma{\dot{\gamma}}}\phi_{\gamma\alpha(s-1){\dot{\gamma}}{\dot{\alpha}}(s-1)}$
(2.49)
$\displaystyle-s(s-1)\varphi^{\alpha(s-2){\dot{\alpha}}(s-2)}\partial^{\beta{\dot{\beta}}}\partial^{\gamma{\dot{\gamma}}}\phi_{\beta\gamma\alpha(s-2){\dot{\beta}}{\dot{\gamma}}{\dot{\alpha}}(s-2)}-s(2s-1)\varphi^{\alpha(s-2){\dot{\alpha}}(s-2)}\Box\varphi_{\alpha(s-2){\dot{\alpha}}(s-2)}$
$\displaystyle-\frac{1}{2}s(s-2)^{2}\partial_{\beta{\dot{\beta}}}\varphi^{\beta\alpha(s-3){\dot{\beta}}{\dot{\alpha}}(s-3)}\partial^{\gamma{\dot{\gamma}}}\varphi_{\gamma\alpha(s-3){\dot{\gamma}}{\dot{\alpha}}(s-3)}\Big{\\}}~{}.$

The action (2.49) is invariant under the gauge transformations

$\displaystyle\delta_{\zeta}\phi_{\alpha(s){\dot{\alpha}}(s)}$
$\displaystyle=$
$\displaystyle\partial_{\alpha{\dot{\alpha}}}\zeta_{\alpha(s-1){\dot{\alpha}}(s-1)}~{},$
(2.50a) $\displaystyle\delta_{\zeta}\varphi_{\alpha(s-2){\dot{\alpha}}(s-2)}$
$\displaystyle=$
$\displaystyle\frac{1}{s^{2}}(s-1)\partial^{\beta{\dot{\beta}}}\zeta_{\beta\alpha(s-2){\dot{\beta}}{\dot{\alpha}}(s-2)}~{},$
(2.50b)

where the gauge parameter $\zeta_{\alpha(s-1){\dot{\alpha}}(s-1)}$ is real
unconstrained. The equations of motion corresponding to the action (2.49) read

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\Box\phi_{\alpha(s){\dot{\alpha}}(s)}+s\partial_{\alpha{\dot{\alpha}}}\partial^{\beta{\dot{\beta}}}\phi_{\beta\alpha(s-1){\dot{\beta}}{\dot{\alpha}}(s-1)}-s(s-1)\partial_{\alpha{\dot{\alpha}}}\partial_{\alpha{\dot{\alpha}}}\varphi_{\alpha(s-2){\dot{\alpha}}(s-2)}~{},$
(2.51a) $\displaystyle 0$ $\displaystyle=$
$\displaystyle(s-1)\partial^{\beta{\dot{\beta}}}\partial^{\gamma{\dot{\gamma}}}\phi_{\beta\gamma\alpha(2s-2){\dot{\beta}}{\dot{\gamma}}{\dot{\alpha}}(s-2)}+2(2s-1)\Box\varphi_{\alpha(s-2){\dot{\alpha}}(s-2)}$
(2.51b)
$\displaystyle-(s-2)^{2}\partial_{\alpha{\dot{\alpha}}}\partial^{\beta{\dot{\beta}}}\varphi_{\beta\alpha(s-3){\dot{\beta}}{\dot{\alpha}}(s-3)}~{}.$

We can immediately make use of the gauge freedom (2.50b) to impose the gauge

$\varphi_{\alpha(s-2){\dot{\alpha}}(s-2)}=0~{}.$ (2.52)

The gauge parameter of the residual symmetry is then constrained by

$\partial^{\beta{\dot{\beta}}}\zeta_{\beta\alpha(s-2){\dot{\beta}}{\dot{\alpha}}(s-2)}=0~{}.$
(2.53)

In the gauge (2.52), the equations of motion (2.51) reduce to

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\Box\phi_{\alpha(s){\dot{\alpha}}(s)}+s\partial_{\alpha{\dot{\alpha}}}\partial^{\beta{\dot{\beta}}}\phi_{\beta\alpha(s-1){\dot{\beta}}{\dot{\alpha}}(s-1)}~{},$
(2.54a) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial^{\beta{\dot{\beta}}}\partial^{\gamma{\dot{\gamma}}}\phi_{\beta\gamma\alpha(2s-2){\dot{\beta}}{\dot{\gamma}}{\dot{\alpha}}(s-2)}~{}.$
(2.54b)

Computing the variation of
$\partial^{\beta{\dot{\beta}}}\varphi_{\beta\alpha(s-1){\dot{\beta}}{\dot{\alpha}}(s-1)}$
yields

$\delta_{\zeta}\big{(}\partial^{\beta{\dot{\beta}}}\varphi_{\beta\alpha(s-1){\dot{\beta}}{\dot{\alpha}}(s-1)}\big{)}=-\frac{2}{s}\Box\zeta_{\alpha(s-1){\dot{\alpha}}(s-1)}~{}.$
(2.55)

Taking into account (2.54b), it follows from (2.55) that
$\partial^{\beta{\dot{\beta}}}\varphi_{\beta\alpha(s-1){\dot{\beta}}{\dot{\alpha}}(s-1)}$
can be gauged away

$\partial^{\beta{\dot{\beta}}}\varphi_{\beta\alpha(s-1){\dot{\beta}}{\dot{\alpha}}(s-1)}=0~{}.$
(2.56)

The gauge parameter of the residual gauge freedom is then constrained by

$\Box\zeta_{\alpha(s-1){\dot{\alpha}}(s-1)}=0~{}.$ (2.57)

In the gauge (2.56), the equation of motion (2.54a) reduces to

$0=\Box\phi_{\alpha(s){\dot{\alpha}}(s)}~{}.$ (2.58)

It follows from the on-shell analysis of the Fronsdal action (2.49) that the
only surviving field $\phi_{\alpha(s){\dot{\alpha}}(s)}$ satisfies the
massless on-shell conditions (2.33) for the case $m=n=s$, as a consequence of
the conditions (2.50a), (2.53), (2.56), (2.58) and (2.57). Thus from the
perspective of massless gauge fields, the model (2.49) describes two massless
spin-$s$ modes, with helicities $s$ and $-s$ respectively.

Moreover, it can be shown that Fronsdal’s theory (2.49) possesses two gauge-
invariant field strengths associated with the field
$\phi_{\alpha(s){\dot{\alpha}}(s)}$

$\displaystyle C_{\alpha(2s)}(\phi)$ $\displaystyle=$
$\displaystyle\partial_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\partial_{\alpha_{s}}{}^{{\dot{\beta}}_{s}}\phi_{\alpha_{s+1}\ldots\alpha_{2s}){\dot{\beta}}(s)}~{},$
(2.59a) $\displaystyle\bar{C}_{{\dot{\alpha}}(2s)}(\phi)$ $\displaystyle=$
$\displaystyle\partial_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\cdots\partial_{{\dot{\alpha}}_{s}}{}^{\beta_{s}}\phi_{\beta(s){\dot{\alpha}}_{s+1}\ldots{\dot{\alpha}}_{2s})}~{}.$
(2.59b)

By the on-shell analysis given above, it can be seen that the field strengths
(2.59) are non-vanishing, and satisfy the differential constraints

$\partial^{\beta{\dot{\gamma}}}C_{\beta\alpha(2s-1)}(\phi)=0~{},\quad\partial^{\gamma{\dot{\beta}}}\bar{C}_{{\dot{\beta}}{\dot{\alpha}}(2s-1)}(\phi)=0~{}.$
(2.60)

The constraints (2.60) coincide with the conditions for a massless field
within the framework of field strengths (2.42). In particular, the field
strengths $C_{\alpha(2s)}(\phi)$ and $\bar{C}_{{\dot{\alpha}}(2s)}(\phi)$ can
be identified as massless fields carrying helicity $s$ and $-s$ respectively.
Thus in both frameworks, we consistently find that Fronsdal’s action describes
two massless spin-$s$ modes with helicities $s$ and $-s$.

####  2.1.4 Spin projection operators

In this section we review the spin projection operators on $\mathbb{M}^{4}$.
Special attention is given to the study of the properties of these operators,
and their corresponding applications. For integers $m,n\geq 1$, the spin
projection operator $\Pi^{\perp}_{(m,n)}$ is defined by its action on
$V_{(m,n)}$ via the rule

$\displaystyle\Pi^{\perp}_{(m,n)}:{V}_{(m,n)}$ $\displaystyle\longrightarrow$
$\displaystyle{V}_{(m,n)}~{},$ (2.61a)
$\displaystyle\phi_{\alpha(m){\dot{\alpha}}(n)}$ $\displaystyle\longmapsto$
$\displaystyle\Pi^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}=:\Pi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}(\phi)~{}.$
(2.61b)

For fixed $m$ and $n$, the operator $\Pi^{\perp}_{(m,n)}$ satisfies the
defining properties:

  1. 1.

Idempotence: $\Pi^{\perp}_{(m,n)}$ is a projector in the sense that it squares
to itself

$\Pi^{\perp}_{(m,n)}\Pi^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}=\Pi^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(2.62a)

  2. 2.

Transversality: $\Pi^{\perp}_{(m,n)}$ maps any field
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ to a transverse field

$\partial^{\beta{\dot{\beta}}}\Pi^{\perp}_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}(\phi)=0~{}.$
(2.62b)

It is easy to see that projector $\Pi_{(m,n)}^{\perp}$ maps an unconstrained
field $\phi_{\alpha(m){\dot{\alpha}}(n)}\in V_{(m,n)}$ on the mass-shell
(2.13b) to a massive field (2.13),

$\displaystyle\partial^{\beta{\dot{\beta}}}\Pi^{\perp}_{(m,n)}\phi_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}=0~{},$
(2.63a)
$\displaystyle(\Box-\mbox{\boldmath$m$}^{2})\Pi^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$
(2.63b)

Hence, the projector $\Pi_{(m,n)}^{\perp}$ selects the physical component
which furnishes the massive UIR $G(\mbox{\boldmath$m$},\frac{1}{2}(m+n))$ from
an unconstrained field $\phi_{\alpha(m){\dot{\alpha}}(n)}$ satisfying the
Klein-Gordon equation.

The spin projection operators (2.62) take the following explicit form on
$V_{(m,n)}$ [41]

$\displaystyle\Pi^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=$
$\displaystyle\frac{1}{\Box^{n}}\partial_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\cdots\partial_{{\dot{\alpha}}_{n})}{}^{\beta_{n}}\partial_{(\beta_{1}}{}^{{\dot{\beta}}_{1}}\cdots\partial_{\beta_{n}}{}^{{\dot{\beta}}_{n}}\phi_{\alpha_{1}\ldots\alpha_{m}){\dot{\beta}}(n)}~{},$
(2.64a)
$\displaystyle\widehat{\Pi}^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=$
$\displaystyle\frac{1}{\Box^{m}}\partial_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\partial_{\alpha_{m})}{}^{{\dot{\beta}}_{m}}\partial_{({\dot{\beta}}_{1}}{}^{\beta_{1}}\cdots\partial_{{\dot{\beta}}_{m}}{}^{\beta_{m}}\phi_{\beta(m){\dot{\alpha}}_{1}\ldots{\dot{\alpha}}_{n})}~{}.$
(2.64b)

It is immediately apparent that for $\phi_{\alpha(n)}$ off the mass-shell, the
projected fields $\Pi^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}$ and
$\widehat{\Pi}^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}$ are non-local
due to the presence of the inverse d’Alembertian operators. Locality can be
restored if the field being projected satisfies the massive Klein-Gordon
equation (2.13b). In addition, it appears that the spin projection operators
diverge if the field being mapped satisfies the massless Klein-Gordon equation
(2.33b).

Thus we are able to draw some interesting conclusions from the structural form
of the spin projection operators. Specifically, the relativistic wave
equations which make the projectors local are characteristic of a massive
field. On the other hand, the wave equations for which the projectors diverge
are attributed to massless gauge fields. Accompanying the fields satisfying
these wave equations with the defining constraint of the spin projection
operator on $V_{(m,n)}$, which is the transverse condition, allows one to
obtain the complete dictionary of on-shell fields in $\mathbb{M}^{4}$. This
observation will be very useful in providing some intuition concerning the
poles of the projection operators in AdS backgrounds, as we will see in the
later sections of this thesis.

It can be shown after some work that the spin projection operators (2.64a) and
(2.64b) are indeed equivalent on the space $V_{(m,n)}$

$\Pi^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}=\widehat{\Pi}^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(2.65)

Due to this equivalence, we will only consider $\Pi^{\perp}_{(m,n)}$ on
$V_{(m,n)}$ unless stated otherwise.

Additionally, the projector $\Pi^{\perp}_{(m,n)}$ acts like the identity
operator (is surjective) on the space of transverse fields
$\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}$

$\partial^{\beta{\dot{\beta}}}\phi^{\perp}_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}=0\qquad\Longrightarrow\qquad{\Pi}^{\perp}_{(m,n)}\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}=\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(2.66)

The spin projection operators (2.64) for fields of arbitrary rank, including
mixed symmetry $m\neq n$, were first derived by Siegel and Gates [33] (see
also [41]). For the cases $m=n$ and $m=n+1$, the spin projection operators are
equivalent to those derived in the pioneering work of Fronsdal [35]. In vector
notation, the explicit form of these operators is very complicated and
consequently proves very difficult work with. However, in the two-component
spinor formalism, the form of the spin projection operators take the
remarkably simple and elegant form (2.64) which is more tractable to use. This
is due to the fact that in two component spinor notation, we work with
irreducible spinor fields which are automatically traceless and
symmetric.111111We can start with a traceless tensor field in vector notation.
For this to hold in general, it is necessary derive an operator which maps an
arbitrary traceful tensor field to a traceless field.

It is possible to derive an alternative set of spin projection operators which
can be written solely in terms of the Casimir operators of
$\mathfrak{iso}(3,1)$. To this end, let us introduce the operator
$\tilde{\Pi}^{\perp}_{(m,n)}$ which acts on $V_{(m,n)}$ in the following
manner

$\displaystyle\tilde{\Pi}^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=$
$\displaystyle\frac{m!}{(m+n)!n!}\frac{1}{\Box^{n}}\prod_{j=0}^{n-1}\big{(}\mathbb{W}^{2}-(s-j)(s-j-1)\Box\big{)}\phi_{\alpha(m){\dot{\alpha}}(n)}~{},\hskip
28.45274pt$ (2.67a) $\displaystyle=$
$\displaystyle\frac{n!}{(m+n)!m!}\frac{1}{\Box^{m}}\prod_{j=0}^{m-1}\big{(}\mathbb{W}^{2}-(s-j)(s-j-1)\Box\big{)}\phi_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(2.67b)

It can be shown that $\tilde{\Pi}^{\perp}_{(m,n)}$ satisfies the defining
properties of spin projection operator (2.62). The formulation of projection
operators in terms of Casimir operators (2.67) in $\mathbb{M}^{4}$ was first
studied by Aurilia and Umezawa in [39, 40] for the cases of spin-$s$ and
spin-$(s+\frac{1}{2})$ fields. For $s\geq 1$, the projection operators given
in [39, 40] take the explicit form121212Note that the Casimir operator $W$
given in [39, 40] is equivalent to $W^{2}$ (2.7). Additionally, we make the
redefinition $p^{2}\mapsto-p^{2}$ to ensure consistency between the
conventions of [39, 40] and those employed in this thesis.

$\displaystyle\tilde{\Pi}^{\perp}_{(s)}$ $\displaystyle=$
$\displaystyle\prod_{j=0}^{s-1}\frac{1}{\big{(}s(s+1)-j(j+1)\big{)}\Box}\big{(}\mathbb{W}^{2}-j(j+1)\Box\big{)}~{},$
(2.68a) $\displaystyle\tilde{\Pi}^{\perp}_{(s+\frac{1}{2})}$ $\displaystyle=$
$\displaystyle\prod_{j=0}^{s-1}\frac{1}{\big{(}(s+\frac{1}{2})(s+\tfrac{3}{2})-(j+\frac{1}{2})(j+\frac{3}{2})\big{)}\Box}\big{(}\mathbb{W}^{2}-\frac{1}{4}(2j+1)(2j+3)\Box\big{)}~{}.\hskip
28.45274pt$ (2.68b)

Upon reversing the order of the products, the expressions (2.68a) and (2.68b)
can be shown to coincide with the spin projection operators (2.67a) for the
cases $m=n=s$ and $m=n+1=s+1$, respectively. The operator
$\tilde{\Pi}^{\perp}_{(m,n)}$ also acts like the identity operator on the
space of transverse fields $\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}$

$\tilde{\Pi}^{\perp}_{(m,n)}\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}=\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(2.69)

Although it is not immediately obvious, it turns out that the spin projection
operators $\Pi^{\perp}_{(m,n)}$ and $\tilde{\Pi}^{\perp}_{(m,n)}$ are
equivalent on $V_{(m,n)}$. Proving this equivalence follows quite simply from
the properties of the spin projectors. Specifically, let us consider the
action of $\tilde{\Pi}^{\perp}_{(m,n)}$, followed by the action of
$\Pi^{\perp}_{(m,n)}$ on an arbitrary field
$\phi_{\alpha(m){\dot{\alpha}}(n)}$. Since
$\tilde{\Pi}^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}$ is transverse,
and $\Pi^{\perp}_{(m,n)}$ acts like the identity operator on this space, we
find

$\Pi^{\perp}_{(m,n)}\tilde{\Pi}^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}=\tilde{\Pi}^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(2.70a) Next, we perform the same procedure but in the opposite order,
$\tilde{\Pi}^{\perp}_{(m,n)}\Pi^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}={\Pi}^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(2.70b)

Since $\tilde{\Pi}^{\perp}_{(m,n)}$ is composed strictly from Casimir
operators, and hence commutes with $\Pi^{\perp}_{(m,n)}$, it follows from
(2.70a) and (2.70b) that the two projectors are equivalent on $V_{(m,n)}$

$\Pi^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}=\tilde{\Pi}^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(2.71)

This argument can be used to prove that the spin projection operators
$\Pi^{\perp}_{(m,n)}$ are unique on $\mathbb{M}^{4}$.

The form of the projector which one chooses to work with is dependent on the
problem under consideration. It is natural to work with the projector in the
form (2.64a) when proving that the operator satisfies the defining properties
of idempotence and transversality. One of the immediate benefits of writing
the projectors in the form (2.67) is that they automatically factorise into a
product of second-order differential operators. Additionally, the operator in
the form $\tilde{\Pi}^{\perp}_{(m,n)}$ allows one to easily show that the spin
projection operators act like the unit operator on the space of transverse
fields.

The projectors $\tilde{\Pi}^{\perp}_{(m,n)}$ display some interesting
properties if they are no longer restricted to the space $V_{(m,n)}$. The
first is that the spin projection operators $\Pi^{\perp}_{(m,n)}$ (2.64a) and
$\tilde{\Pi}^{\perp}_{(m,n)}$ (2.67) are no longer equivalent on $V_{(j,k)}$,
where $1\leq k\leq m-1$ and $1\leq j\leq n-1$. As a matter of fact, the
operators $\Pi^{\perp}_{(m,n)}$ (2.64a) and $\tilde{\Pi}^{\perp}_{(m,n)}$
(2.67) are no longer idempotent and transverse on $V_{(k,j)}$. It can be shown
that $\tilde{\Pi}^{\perp}_{(m,n)}$ (2.67) annihilates all rank-$(m-1,n-1)$
fields

$\tilde{\Pi}^{\perp}_{(m,n)}\phi_{\alpha(m-1){\dot{\alpha}}(n-1)}=0~{}.$
(2.72)

We wish to study the properties of the operator $\tilde{\Pi}_{(m,n)}$, for the
archetypal cases of $m=n=s$ and $m=n+1=s$, on certain spaces $V_{(k,j)}$.

In the case $m=n=s$, the operator $\tilde{\Pi}_{(s,s)}^{\perp}$ takes the
explicit form

$\tilde{\Pi}_{(s,s)}^{\perp}=\frac{1}{\Box^{s}(2s)!}\prod_{j=0}^{s-1}\big{(}\mathbb{W}^{2}-j(j+1)\Box\big{)}~{}.$
(2.73)

It can be shown on $V_{(k,k)}$, where $1\leq k\leq s-1$, the operator
$\tilde{\Pi}_{(s,s)}^{\perp}$ annihilates any lower rank field
$\phi_{\alpha(k){\dot{\alpha}}(k)}$

$\tilde{\Pi}_{(s,s)}^{\perp}\phi_{\alpha(k){\dot{\alpha}}(k)}=0~{}.$ (2.74)

In the case $m=n+1=s$, the operator $\tilde{\Pi}^{\perp}_{(s,s-1)}$ assumes
the form

$\displaystyle\tilde{\Pi}^{\perp}_{(s,s-1)}$ $\displaystyle=$
$\displaystyle\frac{s}{(2s-1)!}\frac{1}{\Box^{s-1}}\prod_{j=0}^{s-2}\Big{(}\mathbb{W}^{2}-\frac{1}{4}(2j+1)(2j+3)\Box\Big{)}$
(2.75a) $\displaystyle=$
$\displaystyle\frac{1}{s(2s-1)!}\frac{1}{\Box^{s}}\prod_{j=0}^{s-1}\Big{(}\mathbb{W}^{2}-\frac{1}{4}(2j-1)(2j+1)\Box\Big{)}~{}.$
(2.75b)

It can be shown that on $V_{(k,k-1)}$, where $1\leq k\leq s-1$, the operator
$\tilde{\Pi}^{\perp}_{(s,s-1)}$ annihilates all lower rank fields
$\phi_{\alpha(k){\dot{\alpha}}(k-1)}$

$\tilde{\Pi}^{\perp}_{(s,s-1)}\phi_{\alpha(k){\dot{\alpha}}(k-1)}=0~{}.$
(2.76)

The properties (2.74) and (2.76) have interesting implications on fields in
(vector)four-component spinor notation. For example, let us consider a
rank-$s$ traceful tensor fields $\phi^{\text{T}}_{a(s)}$. Converting to two-
component spinor notation yields

$\phi^{\text{T}}_{\alpha_{1}{\dot{\alpha}}_{1},\alpha_{2}{\dot{\alpha}}_{2},\ldots,\alpha_{s}{\dot{\alpha}}_{s}}=\phi_{\alpha(s){\dot{\alpha}}(s)}+\text{trace
contributions}~{},$ (2.77)

where $\phi_{\alpha(s){\dot{\alpha}}(s)}$ is the traceless component of
$\phi^{\text{T}}_{a(s)}$, while the other terms consist of lower rank bosonic
fields associated with the trace. Due to property (2.74), one can conclude
that the operator $\tilde{\Pi}_{(s,s)}^{\perp}$ selects out the TT component
of $\phi^{\text{T}}_{a(s)}$

$\partial^{b}\phi^{\text{TT}}_{ba(s-1)}=0~{},\qquad\eta^{bc}\phi^{\text{TT}}_{bca(s-2)}=0~{},\qquad\tilde{\Pi}_{(s,s)}^{\perp}\phi^{\text{T}}_{a(s)}:=\phi^{\text{TT}}_{a(s)}~{}.$
(2.78)

Therefore, the spin-$s$ projection operator is a TT projector when acting on a
symmetric traceful rank-$s$ field $\phi^{\text{TT}}_{a(s)}$.

#####  2.1.4.1 Longitudinal projectors and lower-spin extractors

Let us introduce the orthogonal complement of $\Pi^{\perp}_{(m,n)}$ on
$V_{(m,n)}$

$\Pi_{(m,n)}^{\parallel}=\mathds{1}-\Pi_{(m,n)}^{\perp}~{},$ (2.79)

which automatically satisfies the following properties by construction

$\Pi_{(m,n)}^{\parallel}\Pi_{(m,n)}^{\parallel}=\Pi_{(m,n)}^{\parallel}~{},$
(2.80a)
$\qquad\Pi_{(m,n)}^{\parallel}\Pi_{(m,n)}^{\perp}=\Pi_{(m,n)}^{\perp}\Pi_{(m,n)}^{\parallel}=0~{}.$
(2.80b)

Explicitly evaluating $\Pi_{(m,n)}^{\parallel}$ on $V_{(m,n)}$, it can be
shown that $\Pi_{(m,n)}^{\parallel}$ projects onto the longitudinal part of
$\phi_{\alpha(m){\dot{\alpha}}(n)}$

$\Pi_{(m,n)}^{\parallel}\phi_{\alpha(m){\dot{\alpha}}(n)}=\partial_{\alpha{\dot{\alpha}}}\phi_{\alpha(m-1){\dot{\alpha}}(n-1)}~{},$
(2.81)

where $\phi_{\alpha(m-1){\dot{\alpha}}(n-1)}$ is unconstrained. The operator
$\Pi_{(m,n)}^{\parallel}$ is called the longitudinal projector, as a result of
(2.80a) and (2.81).

Let
$\phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}=\partial_{\alpha{\dot{\alpha}}}\phi_{\alpha(m-1){\dot{\alpha}}(n-1)}$
be some longitudinal field. Since the spin projector
$\tilde{\Pi}_{(m,n)}^{\perp}$ annihilates any rank-$(m-1,n-1)$ field (2.72),
it follows that it also annihilates any rank-$(m,n)$ longitudinal field

$\tilde{\Pi}_{(m,n)}^{\perp}\phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$
(2.82)

An immediate consequence of the identity (2.82) is that the longitudinal
projector $\Pi_{(m,n)}^{\parallel}$ acts as the identity operator on the space
of rank-$(m,n)$ longitudinal fields

$\displaystyle\tilde{\Pi}_{(m,n)}^{\perp}\phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=$ $\displaystyle~{}0\quad$ $\displaystyle\Longrightarrow$
$\displaystyle\quad\Pi_{(m,n)}^{\parallel}\phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=\phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}~{}.$ (2.83)

These properties will prove useful in the subsequent section.

We can make use of the fact that the projectors $\Pi^{\perp}_{(m,n)}$ and
$\Pi_{(m,n)}^{\parallel}$ resolve the identity (2.79) to decompose any tensor
field $\phi_{\alpha(m){\dot{\alpha}}(n)}$ in the following manner

$\phi_{\alpha(m){\dot{\alpha}}(n)}=\big{(}\Pi^{\perp}_{(m,n)}+\Pi^{\parallel}_{(m,n)}\big{)}\phi_{\alpha(m){\dot{\alpha}}(n)}=\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}+\partial_{\alpha{\dot{\alpha}}}\phi_{\alpha(m-1){\dot{\alpha}}(n-1)}~{},$
(2.84)

where $\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}$ is transverse and
$\phi_{\alpha(m-1){\dot{\alpha}}(n-1)}$ is unconstrained and thus reducible.
It follows from properties the (2.69) and (2.80) that the transverse projector
$\tilde{\Pi}^{\perp}_{(m,n)}$ selects the component
$\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}$ from the decomposition (2.84). In
other words, the spin projection operators extract the pure spin state with
maximal spin $s$ from an arbitrary field off the mass-shell.

We can continue this prescription iteratively in order to obtain the general
decomposition of any field $\phi_{\alpha(m){\dot{\alpha}}(n)}$ into
irreducible components

$\phi_{\alpha(m){\dot{\alpha}}(n)}=\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}+\sum_{j=1}^{n-1}(\partial_{\alpha{\dot{\alpha}}})^{j}\phi^{\perp}_{\alpha(m-j){\dot{\alpha}}(n-j)}+(\partial_{\alpha{\dot{\alpha}}})^{n}\phi_{\alpha(m-n)}~{},$
(2.85)

where we have assumed, without loss of generality, that $m\geq n$. The
collection of fields
$\big{\\{}\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)},\phi^{\perp}_{\alpha(m-1){\dot{\alpha}}(n-1)},\cdots,\phi^{\perp}_{\alpha(m-n+1){\dot{\alpha}}}\big{\\}}$
appearing in the decomposition are transverse, while $\phi_{\alpha(m-n)}$ does
not satisfy any differential constraints.

We can introduce operators which extract the bosonic
$\phi^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k)}$ and fermionic
$\phi^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}$ fields from the
decomposition (2.85). In particular, the spin-$(s-k)$ and
spin-$(s-k-\frac{1}{2})$ components can be extracted via

$\displaystyle\phi_{\alpha(s){\dot{\alpha}}(s)}$ $\displaystyle\mapsto$
$\displaystyle\phi^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k)}=\big{(}S^{\perp}_{(s-k)}\phi\big{)}_{\alpha(s-k){\dot{\alpha}}(s-k)}\equiv
S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k)}(\phi)~{},$ (2.86a)
$\displaystyle\phi_{\alpha(s){\dot{\alpha}}(s-1)}$ $\displaystyle\mapsto$
$\displaystyle\phi^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}=\big{(}S^{\perp}_{(s-k-\frac{1}{2})}\phi\big{)}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}\equiv
S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}(\phi)~{},\hskip 28.45274pt$
(2.86b)

where we have defined the operators

$\displaystyle S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k)}(\phi)$
$\displaystyle=$ $\displaystyle
a_{k}\Box^{-k}\tilde{\Pi}^{\perp}_{(s-k,s-k)}\big{(}\partial^{\beta{\dot{\beta}}}\big{)}^{k}\phi_{\beta(k)\alpha(s-k){\dot{\beta}}(k){\dot{\alpha}}(s-k)}~{},$
(2.87a) $\displaystyle S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}(\phi)$
$\displaystyle=$ $\displaystyle
b_{k}\Box^{-k}\tilde{\Pi}^{\perp}_{(s-k,s-k-1)}\big{(}\partial^{\beta{\dot{\beta}}}\big{)}^{k}\phi_{\beta(k)\alpha(s-k){\dot{\beta}}(k){\dot{\alpha}}(s-k-1)}~{},$
(2.87b) with the constants $a_{k}$ and $b_{k}$ taking the form
$a_{k}=(-1)^{k}\binom{s}{k}^{2}\binom{2s-k+1}{k}^{-1}~{},\quad
b_{k}=(-1)^{k}\binom{s}{k}\binom{s-1}{k}\binom{2s-k}{k}^{-1}~{}.$ (2.87c)

It is clear that $S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k)}(\phi)$ and
$S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}(\phi)$ are transverse

$\partial^{\beta{\dot{\beta}}}S^{\perp}_{\beta\alpha(s-k-1){\dot{\beta}}{\dot{\alpha}}(s-k-1)}(\phi)=0~{},\qquad\partial^{\beta{\dot{\beta}}}S^{\perp}_{\beta\alpha(s-k-1){\dot{\beta}}{\dot{\alpha}}(s-k-2)}(\phi)=0~{},$
(2.88)

due to the presence of the spin projection operators. The operators
$S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k)}(\phi)$ and
$S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}(\phi)$ are called the
spin-$(s-k)$ and spin-$(s-k-\frac{1}{2})$ transverse extractors respectively.
Note that the extractors (2.87) are not idempotent because they are
dimensionful.

####  2.1.5 Conformal higher-spin theory

In this section we illustrate an important application of the spin projection
operators in the context of conformal higher-spin theory in $\mathbb{M}^{4}$.
In the spirit of the pioneering work of Fradkin and Tseytlin [62], we show
that the actions for CHS fields with mixed symmetry can be recast in terms of
the spin projectors (2.64). Before doing this, it is necessary to review the
construction of CHS theory in two-component spinor notation [85].

For fixed integers $m,n\geq 1$, a CHS gauge field in $\mathbb{M}^{4}$ is
described by a complex spinor field $h_{\alpha(m){\dot{\alpha}}(n)}$ of
Lorentz type $(\frac{m}{2},\frac{n}{2})$ which is defined modulo gauge
transformations of the form [85]

$\delta_{\xi}h_{\alpha(m){\dot{\alpha}}(n)}=\partial_{\alpha{\dot{\alpha}}}\xi_{\alpha(m-1){\dot{\alpha}}(n-1)}~{}.$
(2.89)

Here, the gauge parameter $\xi_{\alpha(m-1){\dot{\alpha}}(n-1)}$ is complex
unconstrained. The bosonic ($m=n$) and fermionic ($m=n+1$ and $n=m+1$) CHS
fields were first given in [62]. They were later extended to the cases of
bosonic and fermionic fields with mixed symmetry in the works [134] and [85],
respectively.

Associated with the CHS field $h_{\alpha(m){\dot{\alpha}}(n)}$ and its complex
conjugate $\bar{h}_{\alpha(n){\dot{\alpha}}(m)}$ are the linearised higher-
spin Weyl tensors [85]

$\displaystyle{{W}}_{\alpha(m+n)}(h)$ $\displaystyle=$
$\displaystyle\partial_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\partial_{\alpha_{n}}{}^{{\dot{\beta}}_{n}}h_{\alpha_{n+1}\ldots\alpha_{m+n}){\dot{\beta}}(n)}~{},$
(2.90a) $\displaystyle{{W}}_{\alpha(m+n)}(\bar{h})$ $\displaystyle=$
$\displaystyle\partial_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\dots\partial_{\alpha_{m}}{}^{{\dot{\beta}}_{m}}\bar{h}_{\alpha_{m+1}\dots\alpha_{m+n}){\dot{\beta}}(m)}~{}.$
(2.90b)

The complex conjugates of the field strengths (2.90),
$\overline{W}_{{\dot{\alpha}}(m+n)}(h)=\big{(}W_{\alpha(m+n)}(\bar{h})\big{)}^{*}$
and
$\overline{W}_{{\dot{\alpha}}(m+n)}(\bar{h})=\big{(}W_{\alpha(m+n)}(h)\big{)}^{*}$,
can be computed via complex conjugation. The higher-spin Weyl tensors (2.90)
are invariant under the gauge transformations (2.89)

$\delta_{\xi}{{W}}_{\alpha(m+n)}(h)=0~{},\qquad\delta_{\xi}{{W}}_{\alpha(m+n)}(\bar{h})=0~{}.$
(2.91)

Due to the properties (2.91), it follows that the linearised conformal higher-
spin action [85]

$S_{\text{CHS}}^{(m,n)}[h,\bar{h}]={\rm i}^{m+n}\int{\rm
d}^{4}x~{}{{W}}^{\alpha(m+n)}(h){{W}}_{\alpha(m+n)}(\bar{h})~{}+~{}\text{c.c.}~{},$
(2.92)

is invariant under the gauge transformations (2.89). Note that by
$+\text{c.c.}$ we mean to add the complex conjugate. The presence of this
complex conjugate contribution ensures that the actions being studied are
real. The CHS actions were first expressed in terms of the linearised higher-
spin Weyl tensors in [135] by Fradkin and Linetsky, for the cases $m=n$ and
$m=n+1$.

As the name suggests, the CHS action (2.92) is also invariant under conformal
transformations. In this thesis, we do not elaborate on the conformal
properties of CHS theories, as in general, these are well studied. Instead, we
will direct the interested reader to the appropriate resources which detail
the conformal properties of the CHS actions under consideration. For example,
for more information concerning the conformal properties of the CHS theories
(2.92) in $\mathbb{M}^{4}$, see [85, 86]. We are primarily interested in
studying CHS theories due to their connection to spin projection operators.

Integrating by parts, the CHS action (2.92) can be written in the alternative
form

$\displaystyle S_{\text{CHS}}^{(m,n)}[h,\bar{h}]$ $\displaystyle=$
$\displaystyle{\rm i}^{m+n}\int{\rm
d}^{4}x~{}\bar{h}^{\alpha(n){\dot{\alpha}}(m)}{{B}}_{\alpha(n){\dot{\alpha}}(m)}(h)+{\mathrm{c.c.}}~{}$
(2.93a) $\displaystyle=$ $\displaystyle{\rm i}^{m+n}\int{\rm
d}^{4}x~{}\bar{h}^{\alpha(n){\dot{\alpha}}(m)}\widehat{B}_{\alpha(n){\dot{\alpha}}(m)}({h})+{\mathrm{c.c.}}~{},$
(2.93b)

in which the linearised higher-spin Bach tensors appear

$\displaystyle{{B}}_{\alpha(n){\dot{\alpha}}(m)}(h)$ $\displaystyle=$
$\displaystyle\partial_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\cdots\partial_{{\dot{\alpha}}_{m})}{}^{\beta_{m}}{{W}}_{\alpha(n)\beta(m)}(h)~{},$
(2.94a) $\displaystyle{\widehat{B}}_{\alpha(n){\dot{\alpha}}(m)}({h})$
$\displaystyle=$
$\displaystyle\partial_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\partial_{\alpha_{n})}{}^{{\dot{\beta}}_{n}}{\overline{W}}_{{\dot{\alpha}}(m){\dot{\beta}}(n)}({h})~{}.$
(2.94b)

The higher-spin Bach tensors (2.94) are gauge-invariant

$\delta_{\xi}{{B}}_{\alpha(n){\dot{\alpha}}(m)}(h)=0~{},\qquad\delta_{\xi}{\widehat{B}}_{\alpha(n){\dot{\alpha}}(m)}({h})=0~{},$
(2.95)

and transverse

$\partial^{\beta{\dot{\beta}}}{{B}}_{\beta\alpha(n-1){\dot{\beta}}{\dot{\alpha}}(m-1)}(h)=0~{},\qquad\partial^{\beta{\dot{\beta}}}{\widehat{B}}_{\beta\alpha(n-1){\dot{\beta}}{\dot{\alpha}}(m-1)}({h})=0~{}.$
(2.96)

After some work, the Bach tensors (2.94a) and (2.94b) can be shown to be
equivalent

${{B}}_{\alpha(n){\dot{\alpha}}(m)}(h)={\widehat{B}}_{\alpha(n){\dot{\alpha}}(m)}({h})~{}.$
(2.97)

For the remainder of this section, we will only work with
${{B}}_{\alpha(n){\dot{\alpha}}(m)}(h)$.

Due to the transverse nature (2.96) of
${{B}}_{\alpha(n){\dot{\alpha}}(m)}(h)$, it is natural to recast the Bach
tensors in terms of the unique transverse projection operators (2.64) as
follows

$\displaystyle{B}_{\alpha(n){\dot{\alpha}}(m)}(h)$ $\displaystyle=$
$\displaystyle\Box^{n}\partial_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\cdots\partial_{{\dot{\alpha}}_{m-n}}{}^{\beta_{m-n}}\Pi_{(m,n)}^{\perp}h_{\beta(m-n)\alpha(n){\dot{\alpha}}_{m-n+1}\ldots{\dot{\alpha}}_{m})}~{},~{}~{}m\geq
n~{},$ (2.98a) $\displaystyle{B}_{\alpha(n){\dot{\alpha}}(m)}(h)$
$\displaystyle=$
$\displaystyle\Box^{m}\partial_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\partial_{\alpha_{n-m}}{}^{{\dot{\beta}}_{n-m}}\Pi_{(m,n)}^{\perp}h_{\alpha_{n-m+1}\ldots\alpha_{n}){\dot{\alpha}}(m){\dot{\beta}}(n-m)}~{},~{}~{}n\geq
m~{}.\hskip 28.45274pt$ (2.98b)

One of the immediate benefits of expressing the Bach tensors (2.98) in terms
of spin projection operators is that the defining properties of divergenceless
and gauge-invariance are made manifest.

Since the Bach tensor (2.94) is a descendent of the Weyl tensors (2.90), it
follows that they can also be realised in terms of the spin projection
operators (2.64)

$\displaystyle{W}_{\alpha(m+n)}(h)$ $\displaystyle=$
$\displaystyle\partial_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\partial_{\alpha_{n}}{}^{{\dot{\beta}}_{n}}\Pi^{\perp}_{(m,n)}h_{\alpha_{n+1}\ldots\alpha_{m+n}){\dot{\beta}}(n)}~{},$
(2.99a) $\displaystyle{W}_{\alpha(m+n)}(\bar{h})$ $\displaystyle=$
$\displaystyle\partial_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\partial_{\alpha_{m}}{}^{{\dot{\beta}}_{m}}\Pi^{\perp}_{(n,m)}\bar{h}_{\alpha_{m+1}\ldots\alpha_{m+n}){\dot{\beta}}(m)}~{}.$
(2.99b)

It follows that the gauge-invariant CHS action (2.92) can be written in the
form

$\displaystyle S^{(m,n)}_{\text{CHS}}[h,\bar{h}]$ $\displaystyle=$
$\displaystyle{\rm i}^{m+n}\int{\rm
d}^{4}x~{}\bar{h}^{\alpha(n){\dot{\alpha}}(m)}\Box^{n}\big{(}\partial_{\dot{\alpha}}{}^{\beta}\big{)}^{m-n}\Pi_{(m,n)}^{\perp}h_{\beta(m-n)\alpha(n){\dot{\alpha}}(n)}+\text{c.c.}~{},$
(2.100a) $\displaystyle S^{(m,n)}_{\text{CHS}}[h,\bar{h}]$ $\displaystyle=$
$\displaystyle{\rm i}^{m+n}\int{\rm
d}^{4}x~{}\bar{h}^{\alpha(n){\dot{\alpha}}(m)}\Box^{m}\big{(}\partial_{\alpha}{}^{\dot{\beta}}\big{)}^{n-m}\Pi_{(m,n)}^{\perp}h_{\alpha(m){\dot{\beta}}(n-m){\dot{\alpha}}(m)}+\text{c.c.}~{},\hskip
28.45274pt$ (2.100b)

where we have assumed $m>n$ in (2.100a) and $n>m$ in (2.100b). There are many
benefits to recasting the CHS actions (2.100) in terms of the spin projection
operators. Besides manifest gauge invariance, which follows from (2.82), the
presence of the projectors makes the defining feature of CHS theories, that
they describe pure spin $s=\frac{1}{2}(m+n)$ states off the mass-shell,
manifest. Moreover, they are useful in showing that the CHS actions factorise
into products of second order differential equations.

Let us demonstrate this property for the bosonic ($m=n=s$) CHS action. We
begin by imposing the reality condition
$h_{\alpha(s){\dot{\alpha}}(s)}=\bar{h}_{\alpha(s){\dot{\alpha}}(s)}$. Under
reality, the bosonic action (2.100) becomes

$S^{(s,s)}_{\text{CHS}}[h]=(-1)^{s}\int{\rm
d}^{4}x~{}h^{\alpha(s){\dot{\alpha}}(s)}\Box^{s}\Pi^{\perp}_{(s,s)}h_{\alpha(s){\dot{\alpha}}(s)}~{},$
(2.101)

which is equivalent to the conformal spin-$s$ action (1.10) given in [62].
Next, let us decompose the CHS field $h_{\alpha(s){\dot{\alpha}}(s)}$ into
transverse and longitudinal parts using (2.85). Since the spin projection
operator $\Pi^{\perp}_{(s,s)}$ acts as the unit operator on the space of
transverse fields (2.69), and annihilates all bosonic fields of lower rank
(2.74), we find that the CHS action can be written in terms of the transverse
component of the CHS field

$S^{(s,s)}_{\text{CHS}}[h^{\perp}]=(-1)^{s}\int{\rm
d}^{4}x~{}h^{\perp\alpha(s){\dot{\alpha}}(s)}\Box^{s}h^{\perp}_{\alpha(s){\dot{\alpha}}(s)}~{}.$
(2.102)

Thus we see that the bosonic CHS action (2.102) can be factorised into second-
order differential operators. This procedure utilised above is equivalent to
fixing the gauge symmetry (2.89) by imposing the transverse gauge condition
$h_{a(s)}\equiv h^{\perp}_{a(s)}$, since the action (2.102) no longer has
gauge symmetry.

In the fermionic case $m=n+1=s+1$, the CHS action (2.100) takes the form

$S^{(s+1,s)}_{\text{CHS}}[h,\bar{h}]={\rm i}(-1)^{s}\int{\rm
d}^{4}x~{}\bar{h}^{\alpha(s){\dot{\alpha}}(s+1)}\Box^{s}\Pi^{\perp}_{(s+\frac{1}{2})}\partial_{({\dot{\alpha}}_{1}}{}^{\beta}h_{\beta\alpha(s){\dot{\alpha}}_{2}\ldots{\dot{\alpha}}_{s+1})}~{},$
(2.103)

where we have defined $\Pi^{\perp}_{(s+1,s)}:=\Pi^{\perp}_{(s+\frac{1}{2})}$.
The action (2.103) is equivalent to the conformal spin-$(s+\frac{1}{2})$
action given in [62]. Following the argument presented in the bosonic case, it
follows that the CHS action (2.103) cannot be factorised entirely in terms of
second-order operators

$S^{(s,s+1)}_{\text{CHS}}[h^{\perp},\bar{h}^{\perp}]={\rm i}(-1)^{s}\int{\rm
d}^{4}x~{}\bar{h}^{\perp\alpha(s){\dot{\alpha}}(s+1)}\Box^{s}\partial_{({\dot{\alpha}}_{1}}{}^{\beta}h^{\perp}_{\beta\alpha(s){\dot{\alpha}}_{2}\ldots{\dot{\alpha}}_{s+1})}~{}.$
(2.104)

However, it can be shown that the equation of motion of (2.104) can be
factorised purely in terms of second-order differential operators

$0=\Box^{s+1}h^{\perp}_{\alpha(s+1){\dot{\alpha}}(s)}~{}.$ (2.105)

Analogously, it can be shown that the equations of motion for the CHS action
(2.100) for mixed CHS fields can also be factorised in terms of second-order
differential operators.

###  2.2 Four-dimensional $\mathcal{N}=1$ Minkowski superspace

In this section we extend the discussion and results of section 2.1 to ${\cal
N}=1$ Minkowski superspace. We start by reviewing the unitary irreducible
representations of the Poincaré superalgebra [136, 137, 138] and their
corresponding realisations on the space of tensor superfields [31, 32, 33,
34]. This will provide the necessary background material to introduce
supersymmetric generalisations of the spin projection operators in
$\mathbb{M}^{4|4}$ [31, 33]. A variety of applications for these operators are
explored, including their use in the formulation of superconformal higher-spin
theory [85]. The presentation and material covered in this section is inspired
by the work [105], to which we direct the reader for a pedagogical treatment.

####  2.2.1 Irreducible representations of the Poincaré superalgebra

The simplest supersymmetric extension of the Poincaré algebra (2.5),131313It
was shown by Haag, Lopuszanski and Sohnius [139] that this is the most general
extension of $\mathfrak{iso}(3,1)$ by fermionic generators. known as the
${\cal N}=1$ Poincaré superalgebra $S\mathscr{P}$,141414The ${\cal N}=1$
Poincaré superalgebra is simply referred to as the supersymmetry algebra.  was
introduced in the pioneering work of Golfand and Likhtman [140] in 1971. The
${\cal N}=1$ Poincaré superalgebra is spanned by the Poincaré generators
$P_{a}$ and $J_{ab}$ and the supersymmetry generators $Q_{\alpha}$ and
$\bar{Q}_{\dot{\alpha}}$.151515The $4d$ ${\cal N}$-extended Poincaré
superalgebra was introduced by Akulov and Volkov [141], in which
$\mathfrak{iso}(3,1)$ is extended by including the supercharges
$Q^{I}_{\alpha}$ and $\bar{Q}^{I}_{\dot{\alpha}}$, where $I=1,2,\ldots,{\cal
N}$.  The generators of the supersymmetry algebra satisfy the
(anti-)commutation relations

$\displaystyle[P_{a},P_{b}]$ $\displaystyle=0~{},\qquad\hskip
56.9055pt[J_{ab},P_{c}]={\rm i}\eta_{ac}P_{b}-{\rm i}\eta_{bc}P_{a}~{},$
(2.106a) $\displaystyle\ [P_{a},Q_{\alpha}]$ $\displaystyle=0~{},\qquad\hskip
55.48277pt[P_{a},\bar{Q}_{{\dot{\alpha}}}]=0~{},$ (2.106b) $\displaystyle\
[J_{ab},Q_{\alpha}]$ $\displaystyle={\rm
i}(\sigma_{ab})_{\alpha}{}^{\beta}Q_{\beta}~{},\qquad\
[J_{ab},\bar{Q}^{{\dot{\alpha}}}]={\rm
i}({\tilde{\sigma}}_{ab})^{{\dot{\alpha}}}{}_{{\dot{\beta}}}\bar{Q}^{\dot{\beta}}~{},$
(2.106c) $\displaystyle\ [J_{ab},J_{ce}]$ $\displaystyle={\rm
i}\eta_{ac}J_{be}-{\rm i}\eta_{ae}J_{bc}+{\rm i}\eta_{be}J_{ac}-{\rm
i}\eta_{bc}J_{ae}~{},$ (2.106d) $\displaystyle\\{Q_{\alpha},Q_{\beta}\\}$
$\displaystyle=0~{},\qquad\hskip
48.36958pt\\{\bar{Q}_{{\dot{\alpha}}},\bar{Q}_{{\dot{\beta}}}\\}=0~{},$
(2.106e) $\displaystyle\\{Q_{\alpha},\bar{Q}_{{\dot{\alpha}}}\\}$
$\displaystyle=2(\sigma^{a})_{\alpha{\dot{\alpha}}}P_{a}~{}.$ (2.106f)

The relations (2.106) define an arbitrary representation of $S\mathscr{P}$. In
a UIR of $S\mathscr{P}$, the Poincaré generators are Hermitian, while the
supersymmetry generators are Hermitian conjugates to each other,
$\bar{Q}_{\dot{\alpha}}=(Q_{\alpha})^{+}$. It is clear from the commutation
relations (2.106a) and (2.106d) that the Poincaré algebra
$\mathfrak{iso}(3,1)$ is indeed a subalgebra of $S\mathscr{P}$.

Since the momentum generator $P_{a}$ commutes with the supersymmetry
generators $Q_{\alpha}$ and $\bar{Q}_{\dot{\alpha}}$ (2.106b), it follows that
the quadratic Casimir operator $\mathbb{C}_{1}$ of $S\mathscr{P}$ has the same
form as the quadratic Casimir operator $C_{1}$ of $\mathfrak{iso}(3,1)$ (2.6)

$\displaystyle\mathbb{C}_{1}=-P^{a}P_{a}~{},$ (2.107a)
$\displaystyle[\mathbb{C}_{1},P_{a}]=[\mathbb{C}_{1},J_{ab}]=[\mathbb{C}_{1},Q_{\alpha}]=[\mathbb{C}_{1},\bar{Q}_{\dot{\alpha}}]=0~{}.$

However, the form of the Casimir operator $C_{2}$ of $\mathfrak{iso}(3,1)$
(2.7) does not directly yield a Casimir operator of $S\mathscr{P}$. This is
due to the fact that $C_{2}$ contains the generator $J_{ab}$, which does not
commute with the fermionic generators $Q_{\alpha}$ and
$\bar{Q}_{\dot{\alpha}}$ (cf. (2.106c)). The Casimir operator $\mathbb{C}_{2}$
of $S\mathscr{P}$, which can be considered the supersymmetric generalisation
of $C_{2}$, takes the following form

$\displaystyle\mathbb{C}_{2}=Z_{a}P^{a}Z_{b}P^{b}-Z_{a}Z^{a}P_{b}P^{b}~{},\qquad$
(2.107b)
$\displaystyle[\mathbb{C}_{2},P_{a}]=[\mathbb{C}_{2},J_{ab}]=[\mathbb{C}_{2},Q_{\alpha}]=[\mathbb{C}_{2},\bar{Q}_{\dot{\alpha}}]=0~{}.$

Here, $Z_{a}$ is the supersymmetric generalisation of the Pauli-Lubanski
pseudovector (2.8)

$Z_{a}=\mathbb{W}_{a}-\frac{1}{8}({\tilde{\sigma}}_{a})^{{\dot{\beta}}\beta}[Q_{\beta},\bar{Q}_{\dot{\beta}}]~{},$
(2.108)

which can be shown to satisfy the properties

$[Z_{a},P_{b}]=0~{},\qquad[Z_{a},Q_{\alpha}]=\frac{1}{2}Q_{\alpha}P_{a}~{},\qquad[Z_{a},Z_{b}]={\rm
i}\varepsilon_{abce}Z^{c}P^{e}~{}.$ (2.109)

The commutation relations (2.109) are useful in proving that the operator
$\mathbb{C}_{2}$ commutes with the generators of $S\mathscr{P}$. The Casimir
operator $\mathbb{C}_{2}$ is often called the superspin operator.

Let us review the classification of the massive and massless UIRs of
$S\mathscr{P}$. These particular representations are supersymmetric
generalisations of the UIRs of $\mathfrak{iso}(3,1)$ which were presented in
section 2.1.2. An important feature of the UIRs of $S\mathscr{P}$ is that they
all possess positive energy by construction, in comparison to their non-
supersymmetric counterparts of section 2.1.2, for which the negative energy
representations were discarded by hand.

#####  2.2.1.1 Massive UIRs of $S\mathscr{P}$

The massive UIRs of $S\mathscr{P}$ are classified by the quantum numbers mass
$m$ and superspin $s$. It can be shown that the eigenvalues of the Casimir
operators (2.107) label the massive UIRs of $S\mathscr{P}$ as follows [136,
137, 138]

$\mathbb{C}_{1}=-\mbox{\boldmath$m$}^{2}\mathds{1}~{},\qquad\mathbb{C}_{2}=\mbox{\boldmath$m$}^{4}s(s+1)\mathds{1}~{},$
(2.110)

where the mass $m$ is strictly positive $\mbox{\boldmath$m$}>0$ and the
superspin $s$ takes positive (half-)integer values
$s=0,\frac{1}{2},1,\frac{3}{2},\cdots$. We denote by
$\mathbb{G}(\mbox{\boldmath$m$},s)$ a massive UIR of $S\mathscr{P}$ carrying
mass $m$ and superspin $s$. Since the Poincaré algebra is a subalgebra of the
supersymmetry algebra, it can be shown that the UIRs of $S\mathscr{P}$
describe a reducible representation of $\mathfrak{iso}(3,1)$. Specifically for
$s>0$, the UIR $\mathbb{G}(\mbox{\boldmath$m$},s)$ of $S\mathscr{P}$ can be
decomposed into the direct sum of four UIRs of $\mathfrak{iso}(3,1)$,161616In
the case $s=0$, the UIR $G(\mbox{\boldmath$m$},s-\frac{1}{2})$ is not present
in the decomposition (2.111). with each carrying mass $m$ and the respective
spins $s-\frac{1}{2},s,s,s+\frac{1}{2}$

$\mathbb{G}(\mbox{\boldmath$m$},s)=G(\mbox{\boldmath$m$},s-\frac{1}{2})\oplus
G(\mbox{\boldmath$m$},s)\oplus G(\mbox{\boldmath$m$},s)\oplus
G(\mbox{\boldmath$m$},s+\frac{1}{2})~{}.$ (2.111)

The massive UIRs of $\mathfrak{iso}(3,1)$ encoded in
$\mathbb{G}(\mbox{\boldmath$m$},s)$ all possess the same mass since the
Casimir operator $C_{1}$ of $\mathfrak{iso}(3,1)$, which classifies the mass