algebraic aspects of higher spin symmetries

The Pennsylvania State UniversityThe Graduate School

Eberly College of Science

ALGEBRAIC ASPECTS OF HIGHER SPIN SYMMETRIES

A Dissertation inPhysics

byKaran Govil

© 2015 Karan Govil

Submitted in Partial Fulfillmentof the Requirementsfor the Degree of

Doctor of Philosophy

August 2015

The dissertation of Karan Govil was reviewed and approved∗ by the following:

Murat GünaydinProfessor of PhysicsDissertation Advisor, Chair of Committee

Martin BojowaldProfessor of Physics

Richard RobinettDirector of Graduate StudiesDepartment of Physics

Ping XuProfessor of Mathematics

∗Signatures are on file in the Graduate School.

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Abstract

Massless conformal scalar fields in four-dimensional and six-dimensional Minkowskispace-time correspond to minimal unitary representations of SO(4, 2) ∼ SU(2, 2) andSO(6, 2) ∼ SO∗(8), respectively. The Fradkin-Vasiliev type higher spin algebras aredefined as the symmetry algebras of massless Klein-Gordon equation for a scalar fieldwhich implies that the universal enveloping algebras of the minimal representation arejust the conformal higher spin algebras in four and six dimensions, or the AdS5 and AdS7

higher spin algebras, respectively.In this thesis, using the quasiconformal methods developed by Günaydin, Koepsell,

and Nicolai, we formulate the minimal unitary representation for SU(2, 2) and SO∗(8) interms of non-linear twistorial oscillators that transform non-linearly under the respectiveLorentz groups, SL(2,C) in four dimensions, and SL(2,H) in six dimensions. Using thisformulation, we will define the conformal higher spin algebras hs(4, 2) and hs(6, 2). Wewill also define a one-parameter family of continuous deformations, hs(4, 2; ζ), and discretedeformations hs(6, 2; t) of these higher spin algebras. Here ζ is the continuous helicity ofmassless conformal fields in four dimensions and t is the spin of an SU(2) subgroup of thelittle group of massless particles, SO(4), in six dimensions. Our results imply the existenceof a family of (supersymmetric) HS theories in AdS5 and AdS7 which are dual to free(super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric)CFTs in four and six dimensions, respectively.

Using the quasiconformal methods, we also construct the minimal unitary representa-tions (and its deformations) for the exceptional supergroup D(2, 1;λ) which is the mostgeneral N = 4 superconformal group in one dimension. We shall also review Lorentzcovariant twistorial oscillator construction for SO(3, 2) ∼ Sp(4,R), SU(2, 2), and SO∗(8).

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Table of Contents

List of Tables vii

Acknowledgments ix

Chapter 1Introduction 11.1 On singletons, doubletons, AdS/CFT , and minimal representations . . . . 41.2 Vasiliev’s higher spin theories and their underlying algebras . . . . . . . . 71.3 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Chapter 2Minimal Unitary Representation of D(2, 1;λ) 112.1 D(2, 1;λ) basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Minimal unitary representation for D(2, 1;λ): A review . . . . . . . . . . . 122.3 SU(2) Deformations of the minimal unitary supermultiplet of D(2, 1;λ) . 142.4 SU(2) deformed minimal unitary representations as positive energy unitary

supermultiplets of D(2, 1;λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.1 Compact 3-grading . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.2 Unitary supermultiplets of D(2, 1;λ) . . . . . . . . . . . . . . . . . . 23

2.5 SU(2) Deformations of the minimal unitary representation of D(2, 1;λ)using both bosons and fermions and OSp(2n∗|2m) superalgebras . . . . . 38

2.6 SU(2) deformed minimal unitary supermultiplets of D(2, 1;α) and N = 4superconformal mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.6.1 N = 4 Superconformal Quantum Mechanical Models . . . . . . . . 402.6.2 Mapping between the harmonic superspace generators of N = 4

superconformal mechanics and generators of deformed minimalunitary representations of D(2, 1;α) . . . . . . . . . . . . . . . . . . 42

2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Chapter 3Conformal Higher Spin Algebras in d = 4: hs(4, 2; ζ) 473.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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3.2 3d conformal algebra SO(3, 2) ∼ Sp(4,R) and its minimal unitary realization 483.2.1 Twistorial oscillator construction of SO(3, 2) . . . . . . . . . . . . . 493.2.2 SO(3, 2) algebra in conformal three-grading and 3d covariant twistors 50

3.3 Conformal and superconformal algebras in four dimensions . . . . . . . . 523.3.1 Covariant twistorial oscillator construction of the doubletons of

SO(4, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.2 Quasiconformal approach to the minimal unitary representation of

SO(4, 2) and its deformations . . . . . . . . . . . . . . . . . . . . . 563.3.3 Deformations of the minimal unitary representation of SU(2, 2) . . 583.3.4 Minimal unitary supermultiplet of SU(2, 2|4), its deformations and

deformed twistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Higher spin (super-)algebras, Joseph ideals and their deformations . . . . 63

3.4.1 Joseph ideal for SO(3, 2) singletons . . . . . . . . . . . . . . . . . . 653.4.2 Joseph ideal of SO(4, 2) . . . . . . . . . . . . . . . . . . . . . . . . . 663.4.3 Deformations of the minrep of SO(4, 2) and their associated ideals 703.4.4 Higher spin algebras and superalgebras and their deformations . . 72

Chapter 4AdS7/CFT6 Higher spin algebras in d = 6: hs(6, 2; t) 774.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Realizations of the 6d conformal algebra SO(6, 2) ∼ SO∗(8) and its super-

symmetric extension OSp(8∗|4) . . . . . . . . . . . . . . . . . . . . . . . . . 794.2.1 Covariant twistorial oscillator construction of the massless repre-

sentations (doubletons) of 6d conformal group SO(6,2) . . . . . . . 794.2.2 Quasiconformal approach to minimal unitary representation of

SO(6, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2.3 Deformations of the minimal unitary representation of SO(6, 2) . 854.2.4 Minimal unitary supermultiplet of OSp(8∗|4) and its deformations 87

4.3 AdS7/CFT6 higher spin (super-)algebras, Joseph ideals and their defor-mations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.3.1 Joseph ideal for minimal unitary representation of SO(6, 2) . . . . 924.3.2 Deformations of the minimal unitary representation of SO(6,2) and

the Joseph ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.3.3 Comparison with the covariant twistorial oscillator realization . 954.3.4 AdS7/CFT6 Higher spin algebras and superalgebras and their

deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Chapter 5Discussion 995.1 AdS5/CFT4 higher spin holography . . . . . . . . . . . . . . . . . . . . . . 995.2 AdS7/CFT6 higher spin holography . . . . . . . . . . . . . . . . . . . . . . 101

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Appendix A102

A.1 Spinor conventions for SO(2, 1) . . . . . . . . . . . . . . . . . . . . . . . . . 102A.2 Minimal unitary supermultiplet of OSp(N |4,R) . . . . . . . . . . . . . . . 102A.3 The quasiconformal realization of the minimal unitary representation of

SO(4, 2) in compact three-grading . . . . . . . . . . . . . . . . . . . . . . . 104A.4 Commutation relations of deformed twistorial oscillators for SO(4, 2) . . . 105

Appendix B106

B.1 Clifford algebra conventions for doubleton realization . . . . . . . . . . . . 106B.2 Clifford algebra conventions for deformed twistors . . . . . . . . . . . . . . 107B.3 The quasiconformal realization of the minimal unitary representation of

SO(6, 2) in compact 3-grading . . . . . . . . . . . . . . . . . . . . . . . . . . 107B.4 Commutation relations of deformed twistorial oscillators for SO(6, 2) . . . 111

Bibliography 113

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List of Tables

2.1 Decomposition of SU(2) deformed minimal unitary lowest energy su-permultiplets of D(2, 1;λ) with respect to SU(2)T × SU(2)A × SU(1, 1)K .The conformal wavefunctions transforming in the (t, a) representation ofSU(2)T × SU(2)A with conformal energy ω are denoted as Ψω

(t,a). The firstcolumn shows the super tableaux of the lowest energy SU(2|1) supermulti-plet, the second column gives the eigenvalue of the U(1) generator H. Theallowed range of λ in this case is λ > 1/(4s+ 2). . . . . . . . . . . . . . . . 30


(t,a). The firstcolumn shows the super tableaux of the lowest energy SU(2|1) supermulti-plet, the second column gives the eigenvalue of the U(1) generator H. Theallowed range of λ in this case is λ < −1/(4s+ 2). . . . . . . . . . . . . . . 32


(t,a). The firstcolumn shows the super tableaux of the lowest energy SU(2|1) supermulti-plet, the second column gives the eigenvalue of the U(1) generator H. Theallowed range of λ in this case is λ < −3/(4s+ 2). . . . . . . . . . . . . . . 35


(t,a). The firstcolumn shows the super tableaux of the lowest energy SU(2|1) supermulti-plet, the second column gives the eigenvalue of the U(1) generator H. Theallowed range of λ in this case is λ > 3/(4s+ 2). . . . . . . . . . . . . . . . 37

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2.5 Below we give the correspondence between the quantum mechanical oper-ators of N = 4 superconformal mechanics and the operators of minimalunitary supermultiplet of D(2, 1;λ) deformed by a pair of bosons. TheSU(2) indices on the left column are raised and lowered by the Levi-Civitatensor εij (with ε12 = ε21 = 1). . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6 Below we give the mapping between the symmetry generators of N = 4superconformal mechanics in harmonic superspace and the minimal unitaryrepresentation of D(2, 1;λ) deformed by a pair of bosons. The first columnlists the Symmetry generators of N = 4 superconformal quantum mechanicsin harmonic superspace (N = 4 SCQM in HSS) and the second column liststhe generators for the minimal unitary realization of D(2, 1;λ) deformedby a pair of bosons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.7 Below we give the mapping between the quantum numbers of the spectrumof N = 4 superconformal mechanics of [115] and the minimal unitarysupermultiplets of D(2, 1;λ) deformed by a pair of bosons. . . . . . . . . . 46

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Acknowledgments

I am deeply indebted to my advisor, Murat Günaydın, for introducing me to incredibleideas of symmetry and group theory, for patiently teaching me so many different things,and for his unwavering support and encouragement. It was always a special pleasure todiscuss physics and mathematics and collaborate with Professor Günaydın. He has beena constant inspiration for the kind of physicist I always wanted to become.

I am also grateful to my thesis committee members Prof. Martin Bojowald, Prof.Richard Robinett and Prof. Ping Xu for serving on my committee on a short notice andtheir flexibility regarding the date of the thesis defense.

I benefited greatly from many stimulating conversations with Mikhail Vasiliev, EvgenySkvortsov and Massimo Taronna during our collaborations at AEI Potsdam and PennState. I would also like to thank Hermann Nicolai for inviting and hosting me at AEI.

I would like to express my gratitude to my friends, Suddho Brahma and UmutBüyükçam, for providing an intellectually refreshing environment and for their constantsupport and encouragement. My special thanks go to John Hopkins for being flexiblewith my teaching commitments, and to Carol Deering for her readiness to assist andknowledge of intricacies of formal requirements which have been a great help to me.

And last, but not the least, I would like to thank my parents and Megan for theireverlasting patience, love, and support throughout my time at the graduate school.

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Dedicated To My Parents and Megan

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Chapter 1 |Introduction

“The miracle of the appropriateness of the language of mathematics for theformulation of the laws of physics is a wonderful gift which we neither understandnor deserve.”– Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural

Sciences [1]

The role of symmetry in the formulations of laws of nature and invariance principlesis ubiquitous in theoretical physics, and group theory is the language of symmetry. Eventhough symmetry and conservation laws have been around in physics dating back toNewton, Kepler and even Greeks who were fascinated by the symmetries of regularobjects and believed that these might be reflected in the principles of nature itself, it wasEinstein, in particular, who regarded symmetry principles as fundamental and insistedthat the various conservation laws and transformation laws be consequences of symmetry(Noether’s theorem [2] explains the connection between symmetry and conservation laws).A great triumph for the symmetry principle was the equivalence principle which governsthe dynamics of gravity and of space-time itself, and is a statement of invariance of thelaws of nature under the local changes of the space-time coordinates.

The development of quantum mechanics in the early 20th century rendered theneed for a more mathematically rigorous treatment of symmetry principles using grouptheoretical ideas. Wigner laid the foundations for the application of group theory inquantum mechanics in a series of seminal papers during 1926-1935. A study of theinfinite dimensional unitary irreducible representations of Lorentz group was started byWigner 1 [3] and Dirac [4] which served as the inspiration for the works of Bargmann [5],

1According to George Mackey’s annotations to the first volume of Wigner’s Collected Works, it wasvon Neumann who pointed out to Wigner that if the Hamiltonian of a quantum system is unchanged bysome group of symmetries G, then the eigenspaces of H span representations for G, which was indeedstated by Wigner in the introduction of the second paper in Volume I.

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Gel’fand-Naimark [6] and Harish-Chandra [7] who performed a mathematically rigorousand systematic study of the unitary representations of SL(2,R) and SL(2,C). Theseworks had shown the fruitfulness of investigations of unitary representations of groupsrelevant to physics and they were followed by intense works on representation theory ofsemisimple Lie groups and their applications in physics by a number of physicists andmathematicians alike.

Quantum field theory, the relativistic version of quantum mechanics, describes thedynamics and interactions of fundamental particles and forces. The Standard Model ofelementary particles, discovered by Glashow, Weinberg and Salam [8–10], is a quantumfield theory based on the local gauge group SU(3)×SU(2)×U(1) and describes all the knownelementary particles and their interactions except gravity. It is one of the most preciselytested theory of physics and the Higgs boson, one of the most important pieces of StandardModel, was recently discovered at the Large Hadron Collider at CERN [11,12]. The twomost important characteristic energy scales for the standard model are ΛQCD ∼ 200 MeV,which is the energy scale at which QCD (Quantum Chromodynamics, a gauge theoryof strong interactions) becomes strongly coupled, and, mW,Z,H ∼ 100 GeV, the scale atwhich electromagnetic and weak interaction becomes unified.

With the Standard Model, we have been able to formulate quantum theories ofelectromagnetism, weak, and strong interaction; however, serious difficulties arise whenone tries to reconcile gravity or general relativity with quantum mechanics. One of theproblems with quantum gravity is that it is non-renormalizable in d ≥ 4, which meansthat the predictive power of the theory at high energy scales breaks down. Generalrelativity describes gravity classically at large length scales and standard model is only aneffective theory which does not take into account quantum gravity effects at high energyscales. The characteristic energy scale at which gravity becomes strong is the Planckscale MPl ∼ 1019 GeV, which is not accessible at the high energy particle acceleratorswhich are currently operating at fifteen orders of magnitude below Planck scale. As aresult, the research in quantum gravity is mainly driven by thought experiments andsymmetry principles.

The search for unified field theories can be traced back to Weyl [13] where he triedto use gauge invariance to relate gravitation and electricity. Since then we have comea long way and String Theory has emerged as one of most promising candidates for aunified field theory, especially as a theory of quantum gravity. One of the most importantaspects of String Theory that has kept physicists busy for over a decade is the anti-deSitter-conformal field theory, i.e. AdS/CFT correspondence, which is also known asgauge-gravity duality. It was first proposed by Maldacena in reference [14] where herealized that the type IIB superstring theory on AdS5 × S5 background is equivalent

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to N = 4 super Yang-Mills theory on the four dimensional boundary. In the large Nlimit, the AdS/CFT duality is a statement about quantum gravity and strings in anasymptotically AdS background involving black holes being dual to a conformal fieldtheory of the boundary of the AdS, which is a non gravitational quantum field theory.Even though we know of many examples of the duality in various dimensions and manychecks have been performed, there are still many open questions with the most importantbeing a mathematical proof of the conjecture.

The duality between AdS5 × S5 superstring and N = 4 super Yang-Mills is one of themost abundantly studied AdS/CFT dualities but it is far from being the simplest. Themodel of a duality between large N conformal field theories in d dimensions containingN−component vector field and theories of infinite number of higher spin massless gaugefields in AdSd+1 was put forward in reference [15]. This turns out to be one of the simplestexamples of AdS/CFT duality with the dual conformal field theory being a free theorybecause of the presence of infinitely many conserved charges. This infinite symmetry iscalled the higher spin symmetry and forms the basis of the study of higher spin gaugetheories in curved backgrounds initiated by Fradkin and Vasiliev in the 1980s [16, 17].There has been an enormous amount of work on higher spin theories by Vasiliev andvarious collaborators (we refer to [18] and [19] for detailed reviews on higher spin theories).This work has been mainly targeted at higher spin theories in AdS4 where the existenceof Lorentz covariant spinorial variables makes it convenient to formulate the higher spinalgebra under which the infinite massless higher spin gauge fields transform irreducibly.However, from the AdS/CFT and M/Superstring Theory point of view, AdS5 and AdS7

are also extremely important. The AdSd higher spin algebras for d > 4 are complicatedand the progress in these theories has been slow due to the lack of a general formulationof irreducible higher spin algebras that are realized unitarily. It is the aim of this thesisto construct AdS5,7 higher spin algebras using the quasiconformal approach (developedfirst for the exceptional group E8 in references [20,21] and generalized and extended tosuperalgebras in references [22–24]) and elucidate novel features such as one parameterfamily of deformations of these algebras. The important ingredients for the constructionof these higher spin algebras are group theoretic origins of AdS/CFT correspondence,minimal unitary representations, and higher spin theories of Vasiliev, and we shall spendrest of this chapter on reviewing these concepts.

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1.1 On singletons, doubletons, AdS/CFT , and minimal repre-sentations

In 1963 [25], Dirac discovered remarkable representations of the four dimensional anti-deSitter (AdS4) group SO(3, 2) which are known to have a singular Poincaré limit [26,27].These representation are also known as Di and Rac or singleton representations. It wasshown in references [26–30] that the singleton tensor products Di⊗Di, Rac⊗ Rac andDi⊗Rac decompose into an infinite set of massless unitary irreducible representationswhich have a smooth Poincaré limit in d = 4 and go over to the massless representationsof the Poincaré group labeled by helicity. The singletons are the only two conformalmassless representations in three dimensions where SO(3, 2) acts as conformal group. Itshould be noted that the tensor products of two singletons, which are massless in fourdimensional (AdS4) sense are in fact massive when considered as representations of thethree dimensional conformal group. The analogues of singletons for AdS5 and AdS7 groupsSO(4, 2) and SO(6, 2) respectively are called doubleton 2 representations and were firstobtained in reference [32] for SO(4, 2) ∼ SU(2, 2) and in reference [33] for SO(6, 2) ∼ SO∗(8).A similar story holds for the AdS4 supergroup OSp(N |4) where the tensor product ofthe singleton supermultiplet decomposes into an infinite set of massless supermultipletswhich have a Poincaré limit in four dimensions [28,34–36]. The doubleton supermultipletsin AdS5 (SU(2, 2|N)) and AdS7 (OSp(8∗|2N)) share the same features, i.e. their tensorproducts decompose into infinitely many massless supermultiplets [32,33,36–39].

The issue of masslessness in AdS is a subtle one because the Poincaré mass operatoris not a Casimir invariant of the AdS group. Thus the following definition of a masslessrepresentation or a supermultiplet of an AdS supergroup was proposed in reference [40]:

A representation (or a supermultiplet) of an AdS group (or supergroup) ismassless if it occurs in the decomposition of the tensor product of two singletonor two doubleton representations (or supermultiplets).

The group theoretical origin of AdSd+1/CFTd correspondence lies in the fact theconformal group, SO(d, 2), in d Minkowski space-time dimensions is same as the AdSdisometry group. The hints of duality were already known back in 1984 [32] when the

2Oscillator methods for constructing unitary representations of non-compact groups and supergroupsusing bosonic and fermionic oscillators was formulated in references [29,31]. Given a semisimple Lie groupG admitting lowest weight representation with maximal compact subgroup of the form H × U(1) (forcompact subgroup H), the oscillator method describes unitary irreducible representations by realizing thegenerators of G as creation and annihilation operators on some Fock space, and have them transformin the (anti-)fundamental representation of H. It turns out that for symplectic groups Sp(2n,R), theminimum number of sets of oscillators needed is one and the result is singleton representations whereasfor SU(n,m) and SO∗(2n) type groups, one needs a minimum of two sets of bosonic oscillators and suchrepresentations were termed as doubletons.

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Kaluza-Klein spectrum of type IIB supergravity was first obtained by repeated tensoringof CPT self-conjugate doubleton supermultiplet of SU(2, 2|4) (the N = 4 superconformalgroup in four dimensions) with itself repeatedly and restricting to the CPT self-conjugateshort supermultiplets. Moreover, it was pointed out in reference [32] that the CPTself-conjugate doubleton supermultiplet of SU(2, 2|4) does not have a Poincaré limit in fivedimensions and its field theory lives on the boundary of AdS5, on which SU(2, 2) ∼ SO(4, 2)acts as the conformal group and that the unique candidate for this theory is the conformallyinvariant N = 4 super Yang-Mills theory in four dimensions. Similarly the Kaluza-Kleinspectra of the compactifications of eleven dimensional supergravity over AdS4 × S7 andAdS7 × S4 were obtained by tensoring the singleton supermultiplet of OSp(8|4,R) [35]and scalar doubleton supermultiplet of OSp(8∗|4,R) [33], respectively. It was also pointedout in reference [33, 35] that the field theories of the singleton and scalar doubletonsupermultiplets live on the boundaries of AdS4 and AdS7, respectively, as conformal fieldtheories. Thus, within the framework of Kaluza-Klein supergravities, these are someof the earliest works on AdS/CFT correspondence. The modern era of AdS/CFT wasstarted by the extension of these ideas to Superstring and M-theory arena [14,41,42].

From a purely mathematical point of view, these singleton and scalar doubletonrepresentations are very special and are known as minimal unitary representations. Theminimal unitary representation (minrep) of a non-compact Lie group is defined as aunitary representation over a Hilbert space of functions with smallest or minimal numberof variables possible. The idea of the minrep was introduced by Joseph in reference [43]in the context of determining the least number of degrees of freedom for which a quantummechanical system admits a given semisimple Lie algebra as a spectrum generatingsymmetry. The corresponding representations were termed as minimal representations.Such representations are also known in the literature as singular representations 3 [44].The minreps have a low Gelfand-Kirillov dimension 4 which suggests that there areperhaps some “missing” states in the representation similar to massless gauge bosonssuch as photon, graviton etc, which lack the longitudinal states. Thus minreps play animportant role in the description of massless states in gauge invariant quantum fieldtheories.

The minrep of a Lie algebra exponentiates to a unitary representation of the corre-sponding non-compact group over a Hilbert space of functions depending on the smallest(minimal) number of variables possible. Joseph presented the minimal realizations of the

3The unitary dual of a semisimple Lie group contains the tempered representations which enter thePlancherel decomposition of L2(G) and their complement, “singular representations” [44].

4Let (π), V be an irreducible representation of a Lie algebra g on a vector space V , v be any vector inV , and Un(g) be the subspace of universal enveloping algebra consisting at most n elements of g. TheGelfand-Kirillov dimension [45] then measures the size of (π, V ) in terms of growth rate of dim(Un(g) · v)as n→∞

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complex forms of classical Lie algebras and of G2 in a Cartan-Weil basis. The existenceof the minimal unitary representation of E8(8) using Langland’s classification was firstshown by Vogan [46]. The minimal unitary representations of simply laced groups werestudied by Kazhdan and Savin [47], and Brylinski and Kostant [48, 49]. The minimalrepresentations of quaternionic real forms of exceptional Lie groups were later studied byGross and Wallach [50]. For a review and more complete list of references on the subjectin the mathematics literature prior to 2000, we refer to the lectures of Jian-Shu Li [51].Pioline, Kazhdan and Waldron [52] reformulated the minimal unitary representations ofsimply laced groups given in reference [47] and gave explicit realizations of the simpleroot (Chevalley) generators in terms of pseudo-differential operators for the simply lacedexceptional groups as well as the spherical vectors necessary for the construction ofmodular forms.

The construction of the minrep for the exceptional group E8(8) using the quasiconformalmethods was first obtained in reference [20]5. Quasiconformal realizations exist for differentreal forms of all non-compact groups as well as for their complex forms [21, 53]. Theterm “quasiconformal” refers to the fact that the geometric action of the generators inthis realization leaves invariant a light-cone defined by a quartic distance function in ageneralized space time. Remarkably, upon quantization of this geometric realization, weget the minrep of the corresponding group. This was first shown explicitly for the splitexceptional group E8(8) with the maximal compact subgroup SO(16) [20]. Consequently,the minrep of the three dimensional U-duality group E8(−24) of the exceptional supergravity[54] was obtained in reference [55].

A unified formulation of the minreps of non-compact groups using the quantization ofquasiconformal action was presented in reference [22]. This was also extended to minrepsof non-compact supergroups G whose even subgroups are of the form H×SL(2,R) with Hcompact6 . These supergroups include G(3) with even subgroup G2 × SL(2,R), F (4) witheven subgroup Spin(7)× SL(2,R), D (2, 1;σ) with even subgroup SU(2)× SU(2)× SU(1, 1)and OSp (N |2,R). Inspired by the AdS/CFT applications, the minrep for supergroupsof the form SU(n,m|p + q) and OSp(2N∗|2M) was obtained in reference [23, 24, 56] andapplied to conformal superalgebras in four and six dimensions.

An isomorphism between the minrep of SU(2, 2) and the scalar doubleton repre-sentation was established in reference [23]. It was further shown that the minrep inquasiconformal formulation admits a one parameter family (ζ) of deformations that can

5For the largest exceptional group E8(8), the quasiconformal action is the first and only knowngeometric realization of E8(8) and leaves invariant a generalized light-cone with respect to a quarticdistance function in 57 dimensions [21].

6We shall be mainly working at the level of Lie algebras and Lie superalgebras and be cavalier aboutusing the same symbol to denote a (super)group and its Lie (super)algebra.

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be identified with helicity, which can be continuous. The resulting unitary irreduciblerepresentation of SU(2, 2) corresponds to a 4d massless conformal field transforming in(

0 , ζ2)(for ζ > 0) or

(− ζ2 , 0

)(for ζ < 0) representation of the Lorentz subgroup, SL(2,C).

These deformed minimal representations for integer values of ζ turn out to be isomorphicto the doubletons of SU(2, 2) [32,37,38]. Similarly the minimal unitary supermultipletof SU(2, 2|N) turns out to be the CPT self-conjugate (scalar) doubleton supermultiplet,and for PSU(2, 2|4) it is simply the four dimensional N = 4 Yang-Mills supermultiplet.Once again it was shown that there exists a one-parameter family of deformations ofthe minimal unitary supermultiplet of SU(2, 2|N) and its deformations with integer ζ areisomorphic to the unitary doubleton supermultiplets studied in reference [32,37,38].

Analogous to the AdS5 case, the minrep of AdS7 or 6d conformal group SO(6, 2) =SO∗(8) and its deformations were studied in reference [24]. The minrep was againshown to be isomorphic to the scalar doubleton representation and the minrep admitsdeformations labelled by the eigenvalues of the Casimir of an SU(2)T subgroup of the littlegroup, SO(4), of massless particles in six dimensions. These deformed minreps labeledby spin t of SU(2)T are positive energy unitary irreducible representations of SO∗(8)that describe massless conformal fields in six dimensions. Quasiconformal constructionof the minimal unitary supermultiplet of OSp(8∗|2N) and its deformations were givenin reference [24, 56]. The minimal unitary supermultiplet of OSp(8∗|4) is the masslessconformal (2, 0) supermultiplet whose interacting theory is believed to be dual to M-theoryon AdS7 × S4 . It is isomorphic to the scalar doubleton supermultiplet of OSp(8∗|4) firstconstructed in reference [33].

The minimal unitary representation for symplectic groups Sp(2N,R) constructed usingthe covariant oscillator realization coincides with the one obtained from quantization ofits quasiconformal action [22]. This is due to the fact that the quartic invariant operatorthat enters the quasiconformal construction vanishes for symplectic groups and hencethe resulting generators involve only bilinears of oscillators. Therefore the minreps ofSp(4,R) are simply the scalar and spinor singletons that were called the remarkablerepresentations of anti-de Sitter group by Dirac [25] as discussed earlier.

1.2 Vasiliev’s higher spin theories and their underlying algebras

One of the most important motivations for studying higher spin gauge theories comes fromSuperstring Theory where the spectrum includes massive infinite higher spin excitationsof all spins [57]. Although these higher spin excitations are massive, one can contemplatethe existence of a symmetric phase where all these higher spin excitations are masslessand acquire mass by spontaneous symmetry breaking.

7

The Lagrangians for free massive [58,59] and massless [60, 61] higher spin fields havebeen known since the 1970s. However there are several no-go theorems (Weinberg [62],Aragone-Deser [63], Weinberg-Witten [64]) for massless fields due to inconsistencies whicharise when one turns on the interactions for the higher spin fields by replacing the ordinaryderivatives by covariant derivatives. We refer to reference [65] for an excellent review onno-go theorems for massless higher spin fields.

Non-trivial consistent cubic interactions in higher spin gauge theories were constructedin references [66–73], but these did not include gravitational interactions, which wasproblematic because of the universal role of gravity. The problems with coupling gravityto higher spin gauge fields were elucidated in references [63,74,75] and can be summarizedas follows: in order to couple higher spin fields with gravity, we need to covariantize thederivatives, i.e. ∂ → D = ∂ − Γ, which breaks the gauge invariance of higher spin fields.The reason is that when one tries to take the variation of the action under the higherspin gauge transformations, the derivatives need to be commuted and the commutator ofcovariant derivatives is proportional to the Riemann tensor and it is not clear how tocancel these terms by adding terms to the action or modifying the gauge transformations.

This problem was resolved in reference [76,77] where it was shown that one needs toconsider (anti-) de Sitter background to construct consistent higher spin gravitationalinteractions because the gauge invariant gravitational interaction term in AdS containsterms that are inversely proportional to the cosmological constant and consequentlydiverge in the flat space limit. We should stress the non-analyticity in the cosmologicalconstant is needed for preserving higher spin gauge symmetries, which are expectedto be broken in a realistic physical model/phase and thus the expectation value of thecosmological constant will also be modified in such a physical phase. Subsequently, theequations of motion for full non-linear theories of interacting higher spin gauge fields inAdSd were developed by Vasiliev in reference [78] (we refer to the reviews [79, 80] andreferences therein for details) in the so-called unfolded formulation that is a generalizationof Cartan’s geometric formulation of Einstein gravity. However, a complete Lagrangianformulation for interacting higher spin gauge fields is still missing.

One of the most important ingredients for the formulation of higher spin gaugetheories is the higher spin symmetry algebra, under which the higher spin fields transformirreducibly. The AdS4 higher spin algebras and superalgebras were constructed inreference [81,82] and the underlying principle was the fact that the tensor product of a pairof singletons decomposes into infinite AdS4 massless representations of SO(3, 2) ∼ Sp(4,R)(Flato-Fronsdal theorem [27, 83]) played an important role. The extensions of Flato-Fronsdal theorem for AdS5 and AdS7 were obtained in reference [32] and [33], respectively,as discussed in the previous section.

8

The bosonic AdSd higher spin algebra used in reference [78] is the infinite dimensionalsymmetry algebra of the massless Klein-Gordon equation, i.e. symmetry algebra of themassless scalar in d− 1 dimensions. One can also consider symmetry algebra (bosonic) ofa boundary spinor, which for AdS4, turns out to be isomorphic to the scalar symmetryalgebra. Based on these algebraic ideas, a higher spin/vector model hologrpahic dualitywas proposed in reference [15] (see also [84–92]). This holographic duality is regarded asone of the simplest non-trivial examples of AdS/CFT correspondence. It was shown inreference [93] that the spectrum of operators in the vector model (CFT) is not renormalizedat infinite N and thus the spectrum of fields in the bulk is simple. The most non-trivialpart of this duality thus lies in the exact agreement between the correlation functionson the boundary conformal field theory i.e. the vector model and those of Vasiliev’shigher spin gauge theory in the bulk. This agreement was checked explicitly by directcalculations in the seminal papers of Giombi and Yin [94,95] where they computed thethree-point functions of higher spin currents and showed that they matched with those offree and critical O(N) vector models. Since then, a considerable progress has been madein higher spin holography in the last few years including generalizing the conjecture toparity violating Vasiliev theories and Chern-Simons vector models [93,96], structure ofcorrelation functions [97–99], proof of CFT/higher spin version of the Coleman-Mandulatheorem [100,101], exact large N computations in Chern-Simons vector models [93,102,103],supersymmetric extension and symmetry breaking, and connection between Vasiliev’shigher spin gauge theory and string theory [96]. Some approaches towards deriving thehigher spin/vector model duality from first principles were investigated in reference [104]and [105–108] (see also [109] for relevant earlier work). Recently, a dS/CFT version ofthe duality was also proposed [110], and further studied in references [111,112]. Therehas also been exciting development in the AdS3/CFT2 version of higher spin holography(see [113] and references therein), as well as in higher dimensions [114].

As we discussed in the previous section, there are only two conformally masslessrepresentations for SO(3, 2), namely Di and Rac or the singletons. For AdS5,7, however,we have an infinite number of doubletons and thus one would expect to get a family ofhigher spin algebras. These families of infinite dimensional higher spin algebras in AdS5

and AdS7 and associated higher spin holographic dualities are the subject of this thesis.

1.3 Overview of the Thesis

We have explored various aspects of minimal representations in quasiconformal formal-ism and higher spin symmetries in preceding sections. This thesis explores algebraicaspects of higher spin symmetry by constructing higher spin algebras and their unitary

9

representations in AdS5 and AdS7.In Chapter two, we study the minimal representations of the exceptional supergroup

D(2, 1;λ) which happens to be the most general N = 4 superconformal group in onedimension. This will serve as an exercise in using quasiconformal methods to constructminimal representations and their deformations. We also establish a correspondencebetween the generators of the minimal representation and certain models of N = 4superconformal quantum mechanics in harmonic superspace [115].

Chapter three is devoted to the study of AdS5 higher spin algebras hs(4, 2; ζ) or the fourdimensional conformal higher spin algebras. We will review the oscillator constructionfor SO(3, 2) and SO(4, 2) and then formulate the quasiconformal realization obtainedin reference [23] in terms of non-linear twistors that transform non-linearly under theLorentz group SO(3, 1). We shall then proceed to define the higher spin algebras asenveloping algebras of the minimal representations and its deformations.

Similarly in Chapter four, we study AdS7 higher spin algebras hs(6, 2; t) or the sixdimensional conformal higher spin algebras. We will review the oscillator construction forSO(6, 2) and then formulate the quasiconformal realization obtained in references [24, 56]in terms of non-linear twistors that transform non-linearly under the Lorentz groupSO(5, 1). As in Chapter three, we define the higher spin algebras as enveloping algebrasof the minimal representations and its deformations.

Our results suggest the existence of a family of (supersymmetric) higher spin theoriesin AdS5 and AdS7 which are dual to free (super)conformal field theories or to interactingbut integrable (supersymmetric) conformal field theories in four and six dimensions,respectively. We conclude in Chapter five by discussing applications of our results tohigher spin holography and address some open problems.

10

Chapter 2 |Minimal Unitary Representationof D(2, 1;λ)

In this chapter we will study the minimal unitary supermultiplet for the exceptionalsupergroup D(2, 1;λ) and its SU(2) deformations, and its connection with certain modelsof superconformal quantum mechanics in harmonic superspace. This chapter will serveas a warm-up to using quasiconformal methods to obtain minreps and studying theirdeformations. These results were published in [116] which we follow closely in our reviewin this section.

The plan of this chapter is as follows. We start by discussing the basics of D(2, 1;λ) andin section 2.2, we review the minimal unitary realization of D(2, 1;σ) as constructed in [22]using quasiconformal techniques. In section 2.3 we construct the SU(2) deformations ofthe minrep of D(2, 1;λ) using bosonic oscillators in the non-compact 5-graded basis. InSection 2.4 we reformulate the results of section 2.3 in the compact 3-graded basis andshow that the deformations of the minrep of D(2, 1;λ) are positive “energy” ( unitarylowest weight ) representations of D(2, 1;λ). We then present the corresponding unitarysupermultiplets. Section 2.5 discusses deformations of the minrep using both bosons andfermions and how the deformed D(2, 1;λ) commutes with a non-compact super algebraOSp(2n∗|2m) with the even subgroup SO∗(2m)×USp(2n) constructed using “deformation”bosons and fermions. In section 2.6 we review some of the results of work on N = 4superconformal mechanics and show how its symmetry generators and spectrum mapinto the generators of D(2, 1;λ) deformed by a pair of bosonic oscillators and the resultingunitary supermultiplets. We conclude with a brief discussion of our results.

11

2.1 D(2, 1;λ) basics

A complete classification of Lie superalgebras was given by Kac in [117]. The algebrasD(2, 1;λ) are a one-parameter family of non-ismorphic superalgebras of dimension 17, andis isomorphic to OSp(4|2,R) for λ = −1/2. It is the most interesting of the exceptionalsuperalgebras in the sense that there is no Lie algebraic counterpart i.e. there does notexist a family of non-isomorphic simple Lie algebras with the same dimension, and it isalso the most general N = 4 superconformal algebra in one dimension. The irreduciblerepresentations of D(2, 1;λ) were first studied in [118]. In the present chapter, we willfocus on obtaining the positive energy or lowest weight representations, in particular theminrep, by using the quasiconformal methods.

The even or bosonic part of D(2, 1;λ) is SU(2)×SU(2)×SU(2), however among all thepossible non-compact real forms, only the real form with even part SU(2)×SU(2)×SU(1, 1)admits lowest energy unitary representation that are relevant for physical applications.We will only study this real form in this chapter and label the even subalgebras asSU(2)A × SU(2)T × SU(1, 1)K with odd generators transforming in the (1/2, 1/2, 12/)representation of the even part.

The representation of OSp(4|2,R) (i.e. D(2, 1;−1/2)) were studied using twistorialoscillators in [34] and the minrep of D(2, 1;λ) in terms of one bosonic and four fermioniccoordinates using quasiconformal methods was studied in [22]. We shall reformulate theminrep that was obtained in [22] and obtain a one-parameter family of deformations ofthe minrep. Note that in [22], the parameter λ was called σ and the two are related asfollows:

σ = 2λ+ 13 (2.1.1)

In the next section, we shall first briefly review the construction of [22] and thenreformulate it in a manner suitable to obtaining one-parameter deformations of theminrep.

2.2 Minimal unitary representation for D(2, 1;λ): A review

Let us start by reviewing the construction of D(2, 1;σ) minrep given in [22]. The Liesuper algebra of D(2, 1;σ) can be given a 5-graded decomposition of the form

D(2, 1;σ) = g(−2) ⊕ g(−1) ⊕ g(0) ⊕ g(+1) ⊕ g(+2) (2.2.1)

12

where the grade ±2 subspaces are one dimensional and the grade zero sub algebra is

g(0) = su(2)A ⊕ su(2)T ⊕ so(1, 1)∆ (2.2.2)

The grade ±2 generators together with ∆ in g(0) form the su(1, 1)K subalgebra. Theremaining 8 odd generators transforming in the (1/2, 1/2) representation of SU(2)A×SU(2)Tsubgroup belong to g(±1)

We can label the generators in various subspaces according to how they transformunder SU(2)A × SU(2)T × SO(1, 1)∆ subgroup as follows:

D (2, 1;σ) = (0,0)−2 ⊕ (1/2,1/2)−1 ⊕ (su (2)A ⊕ su (2)T ⊕∆)

⊕ (1/2,1/2)+1 ⊕ (0,0)+2 (2.2.3)

D (2, 1;σ) = E ⊕ Eα,α ⊕(Mα,β

(1) +M α,β(2) + ∆

)⊕ Fα,α ⊕ F (2.2.4)

The single bosonic coordinate and its canonical momentum (x, p) satisfy

[x , p] = i (2.2.5)

The four fermionic “coordinates” Xα,α satisfy the anti-commutation relations [22]:Xα,α , Xβ,β

= εαβεαβ (2.2.6)

where α, α, .. denote the spinor indices of SU(2)A and SU(2)T subgroups and εαβ and εαβ

are the Levi-Civita tensors in the respective spaces. The generators belonging to thenegative and zero grade subspaces are realized as bilinears

E = 12x

2 Eα,α = xXα,α ∆ = 12 (xp+ px) (2.2.7)

Mα,β(1) = 1

4εαβ(Xα,αXβ,β −Xβ,βXα,α

)(2.2.8)

M α,β(2) = 1

4εαβ(Xα,αXβ,β −Xβ,βXα,α

)(2.2.9)

13

They satisfy the (super)commutation relations[Mα,β

(1) ,Mλ,µ(1)

]= ελβMα,µ

(1) + εµαMβ,λ(1)[

M α,β(2) ,M

λ,µ(2)

]= ελβM α,µ

(2) + εµαM β,λ(2)[

Mα,β(1) ,M

λ,µ(2)

]= 0

Eα,α, Eβ,β

= 2 εαβεαβ E

(2.2.10)

The quadratic Casimirs of SU(2)A and SU(2)T are as follows:

I4 = εαβελµMαλ(1)M

βµ(1) J4 = εαβελµM

αλ(2)M

βµ(2) (2.2.11)

They differ by a c-number I4 + J4 = − 32 thus we can use either one of them to express

the generator F of g+2 subspace as

F = 12p

2 + σ

x2

(I4 + 3

4 + 98σ)

(2.2.12)

Using the closure of algebra and the property of graded spaces ([gm , gn] ⊂ gm+n), thegrade +1 generators are given by

Fαα = −i[Eαα, F

](2.2.13)

and one finds Fαα, F ββ

= 2εαβεαβF

[Fαα, F

]= 0 (2.2.14)

and Fαα, Eββ

= εαβεαβ∆− (1− 3σ) iεαβM αβ

(2) − (1 + 3σ) iεαβMαβ(1) (2.2.15)

As mentioned before, for σ = 0 the superalgebra D (2, 1, σ) is isomorphic to OSp(4|2,R)and for the values σ = ± 1

3 it reduces to

SU (1, 1|2)× SU (2)

2.3 SU(2) Deformations of the minimal unitary supermultipletof D(2, 1;λ)

Although the construction in the previous section is suitable to obtain the minrep,we need to reformulate it in order to deform the minrep. We shall first rewrite the(super)commutation relations of its generators in a split basis in which the U(1) generators

14

of the two SU(2) subgroups are diagonalized

SU(2)A =⇒ A+, A−, A0 (2.3.1)

SU(2)T =⇒ T+, T−, T0 (2.3.2)

and the fermionic “coordinates” Xα,α are written as fermionic annihilation and creationoperators α(α†) and β(β†) with definite values of U(1) charges

α, α† = 1 = β, β† (2.3.3)

α, β† = 0 = β, α† (2.3.4)

We can now choose a fermionic Fock vacuum such that

α|0〉F = 0 = β|0〉F (2.3.5)

The generators of SU(2)T are then given by the following bilinears of these fermionicoscillators:

T+ = α†β T− = β†α T0 = 12 (Nα −Nβ) (2.3.6)

where Nα = α†α and Nβ = β†β are the respective number operators. They satisfy thefollowing commutation relations:

[T+ , T−] = 2T0 [T0 , T±] = ±T± (2.3.7)

with the CasimirT 2 = T0

2 + 12 (T+T− + T−T+) (2.3.8)

Similarly, the generators of the subalgebra su(2)A are given by the following bilinears offermionic oscillators:

A+ = α†β†

A− = (A+)† = βα

A0 = 12 (Nα +Nβ − 1)

(2.3.9)

The quadratic Casimir of the subalgebra su(2)A is

C2 [su(2)A] = A2 = A02 + 1

2 (A+A− +A−A+) . (2.3.10)

15

As before, the quadratic Casimirs T 2 and A2 are not independent and are related to eachother as follows:

T 2 +A2 = 34 (2.3.11)

The Fock space of two pairs of fermionic oscillators, (α, α†) and (β, β†), is four dimensionaland the states transform in the (1/2, 1/2) representation of SU(2)A × SU(2)T , We shalllabel them by their eigenvalues under T0 and A0 as |mt;ma〉 as follows:

T0|mt;ma〉 = mt|mt;ma〉 (2.3.12)

A0|mt;ma〉 = ma|mt;ma〉 (2.3.13)

More explicitly we have

|0〉F =∣∣0,− 1

2⟩F

= |0, ↓〉Fα†β† |0〉F =

∣∣0, 12 ,⟩F

= |0, ↑〉Fα† |0〉F =

∣∣ 12 , 0⟩F

= | ↑, 0〉Fβ† |0〉F =

∣∣− 12 , 0⟩F

= | ↓, 0〉F

where ↑ denotes +1/2 and ↓ denotes −1/2 eigenvalue of the respective U(1) generator.The grade -1 generators can then be written as bilinears of the coordinate x with the

fermionic oscillators:

Q = xα

S = xβ

Q† = xα†

S† = xβ†(2.3.14)

They close into K− under anti-commutation:

Q , Q†

= 2K− (2.3.15)

The grade zero generator, ∆, that gives the 5-grading, is same as before:

∆ = 12 (xp+ px) (2.3.16)

The generator K− of grade -2 subspace and the 4 supersymmetry generators in the grade-1 subspace form a super Heisenberg algebra and they do not change in going over to thedeformed minreps.

One makes an ansatz for grade +2 generator K+ of the form

K+ = 12p

2 + 1x2

(c1T

2 + c2A2 + c3

)(2.3.17)

16

where the constants c1, ..c3 are to be determined. The grade +1 generators are defined bythe commutators :

Q = i [Q , K+] Q† =(Q)†

= i[Q† , K+

]S = i [S , K+] S† =

(S)†

= i[S† , K+

] (2.3.18)

Using the Jacobi identities and closure of algebra one determines the four unknownconstants in terms of λ and obtain

K+ = 12p

2 + 1x2

(2λT 2 + 2

3(λ− 1)A2 + 38 + 1

2λ(λ− 1))

K+ = 12p

2 + 1x2

(23(2λ+ 1)T 2 + λ2

2 + 1)

(2.3.19)

We shall now introduce bosonic oscillators am, bm and their hermitian conjugatesam = (am)†, bm = (bm)† (m,n, · · · = 1, 2) that satisfy the commutation relations:

[am , an] = [bm , bn] = δnm [am , an] = [am , bn] = [bm , bn] = 0 (2.3.20)

To obtain the deformations of the minrep, we use the bosonic oscillators to introduce anSU(2)S Lie algebra whose generators are as follows:

S+ = ambm S− = (S+)† = ambm S0 = 1

2 (Na −Nb) (2.3.21)

where Na = amam and Nb = bmbm are the respective number operators. They satisfy:

[S+ , S−] = 2S0 [S0 , S±] = ±S± (2.3.22)

The quadratic Casimir of su(2)S is

C2 [su(2)S ] = S2 = S02 + 1

2 (S+S− + S−S+)

= 12 (Na +Nb)

[12 (Na +Nb) + 1

]− 2a[mbn] a[mbn]

(2.3.23)

where square bracketing a[mbn] = 12 (ambn − anbm) represents antisymmetrization of weight

one. The bilinears a[mbn] and a[mbn] close into U(P ) generated by the bilinears

Umn = aman + bmbn (2.3.24)

under commutation and all together they form the Lie algebra of non-compact groupSO∗(2P ) with the maximal compact subgroup U(P ). The group SO∗(2P ) thus generated

17

commutes with SU(2)S as well as with D(2, 1;λ).In order to obtain the SU(2) deformations of the minrep of D(2, 1;λ), we extend the

SU(2)T subalgebra with the diagonal subgroup of SU(2)T and SU(2)S. We shall call thisas SU(2)T and its generators are given below:

su(2)T =⇒ su(2)S ⊕ su(2)T (2.3.25)

T+ = S+ + T+ = ambm + α†β

T− = S− + T− = bmam + β†α

T0 = S0 + T0 = 12 (Na −Nb +Nα −Nβ)

(2.3.26)

with the quadratic Casimir

C2 [su(2)T ] = T 2 = T02 + 1

2 (T+T− + T−T+) . (2.3.27)

The generator ∆ and the negative grade generators defined by it remain unchanged ingoing over to the deformed minreps.

However as the zero grade subalgebra has changed, we need to make a new ansatz forK+:

K+ = 12p

2 + 1x2

(c1T 2 + c2S

2 + c3A2 + c4

)(2.3.28)

where c1, ..c4 are some constant parameters. Once again, using the closure of the algebra,we determine these four unknown constants in terms of λ and obtain:

K+ = 12p

2 + 14x2

(8λT 2 + 8

3(λ− 1)A2 + 32 + 8λ(λ− 1)S2 + 2λ(λ− 1)

)(2.3.29)

We can now solve for the +1 grade generators by using the grading property as follows:

Q = i [Q , K+] Q† =(Q)†

= i[Q† , K+

]S = i [S , K+] S† =

(S)†

= i[S† , K+

] (2.3.30)

This gives the following result for the +1 grade generators:

Q = −pα+ 2ix

[λ

(T0 + 3

4

)α+ T−β

− λ− 1

3

(A0 −

34

)α− 2A−β†

]Q† = −pα† − 2i

x

[λ

(T0 −

34

)α† + T+β

†− λ− 1

3

(A0 + 3

4

)α† − 2A+β

]

18

S = −pβ − 2ix

[λ

(T0 −

34

)β − T+α

− λ− 1

3

(A0 + 3

4

)β −A−α†

]S† = −pβ† + 2i

x

[λ

(T0 + 3

4

)β† − T−α†

− λ− 1

3

(A0 −

34

)β† −A+α

](2.3.31)

The complete commutation relations of the algebra are given as follows:Q , Q†

= −∆− 2iλT0 + i(λ+ 1)A0

Q† , Q

= −∆ + 2iλT0 − i(λ+ 1)A0S , S†

= −∆ + 2iλT0 + i(λ+ 1)A0

S† , S

= −∆− 2iλT0 − i(λ+ 1)A0 (2.3.32)

Q , S

= +2i(λ+ 1)A−

Q , S†

= −2iλT− (2.3.33)

Q† , S†

= −2i(λ+ 1)A+

Q† , S

= +2iλT+ (2.3.34)

S , Q

= −2i(λ+ 1)A−

S , Q†

= −2iλT+ (2.3.35)

S† , Q†

= +2i(λ+ 1)A+

S† , Q

= +2iλT− (2.3.36)

Q , Q

=Q† , Q†

=S , S

=S† , S†

= 0 (2.3.37)

[Q , K−

]= iQ[

S , K−

]= i S

[Q† , K−

]= iQ†[

S† , K−

]= i S†

(2.3.38)

The quadratic Casimir of su(1, 1)K generated by K±2 and ∆ is

C2 [su(1, 1)K ] = K2 = 12(K+K− +K−K+)− 1

4∆2

= λT 2 + λ− 13 A2 + λ(λ− 1)S2 + λ(λ− 1)

4 (2.3.39)

There exists a one parameter family of quadratic Casimir elements C2(µ) that commute

19

with all the generators of D(2, 1;λ).

C2(µ) = µ

4K2 − λ

4 (µ− 8)T 2 − 112 (16 + 8λ+ µ(λ− 1)))A2 + i

4F(Q,S)

= λ

4 (8 + µ(λ− 1))(S2 + 1

4

)(2.3.40)

where

F(Q,S) =[Q , Q†

]+[Q† , Q

]+[S , S†

]+[S† , S

](2.3.41)

is the contribution from the odd generators.The quadratic Casimir C2 depends only on the Casimir S2 of SU(2)S and hence the

eigenvalues s(s+ 1) i.e. the spin s of SU(2)S can be used to label the deformed unitarysupermultiplets of D(2, 1;λ).

2.4 SU(2) deformed minimal unitary representations as positiveenergy unitary supermultiplets of D(2, 1;λ)

2.4.1 Compact 3-grading

As shown in previous section, the Lie superalgebra D(2, 1;λ) admits a 5-graded decompo-sition where the grading is given by the generator ∆ :

D(2, 1;λ) = g(−2) ⊕ g(−1) ⊕ [su(2)T ⊕ su(2)A ⊕ so(1, 1)∆]⊕ g(+1) ⊕ g(+2)

= K− ⊕[Q , Q† , S , S†

]⊕ [A±,0 , T±,0 , ∆]⊕

[Q , Q† , S , S†

]⊕K+

In addition to the 5-grading, D(2, 1;λ) also admits a 3-graded decomposition with respectto its compact subsuperalgebra osp(2|2)⊕ u(1) = su(2|1)⊕ u(1) , which we shall refer to ascompact 3-grading :

D(2, 1;λ) = C− ⊕ C0 ⊕ C+ (2.4.1)

D(2, 1;λ) = (A−, B−,Q−,S−)⊕(T±,0,J ,H,Q0,S0,Q

†0,S

†0

)⊕ (A+, B+,Q+,S+) (2.4.2)

The generators belonging to grade -1 subspace C− are as follows:

A− = βα (2.4.3)

B− = i

2 [∆ + i (K+ −K−)] (2.4.4)

20

= 14 (x+ ip)2 − 1

x2

(L2 + 3

16

)Q− = 1

2(Q− iQ) (2.4.5)

= 12(x+ ip)α+ 1

x

[(2λ+ 1

3

G0 + 3

4

+ λS0

)α+

2λ+ 1

3 G− + λS−

β

]= 1

2(x+ ip)α+ 1x

[(2λ+ 1)

(14 −

β†β

2

)α+ λ

((amam − bmbm)α

2 + bmamβ

)]S− = 1

2(S − iS) (2.4.6)

= 12(x+ ip)β − 1

x

[(2λ+ 1

3

G0 −

34

+ λS0

)β −

2λ+ 1

3 G+ + λS+

α

]= 1

2(x+ ip)β − 1x

[(2λ+ 1)

(α†α

2 − 14

)β

+λ(

(amam − bmbm)β2 − ambmα

)]where

L2 = λT 2 + 13(λ− 1)A2 + λ(λ− 1)S2 + 1

4λ(λ− 1) (2.4.7)

The grade +1 generators in C+ are obtained by Hermitian conjugation of grade −1generators:

A+ = α†β† (2.4.8)

B+ = − i2 [∆− i (K+ −K−)] (2.4.9)

= 14 (x− ip)2 − 1

x2

(L2 + 3

16

)Q+ = (Q−)† = 1

2(Q† + iQ†) (2.4.10)

= 12(x− ip)α† + 1

x

[(2λ+ 1

3

G0 −

34

+ λS0

)α† +

2λ+ 1

3 G+ + λS+

β†]

= 12(x− ip)α† + 1

x

[(2λ+ 1)

(14 −

β†β

2

)α† + λ

((amam − bmbm)α†

2 + ambmβ†)]

S+ = (S−)† = 12(S† + iS†) (2.4.11)

= 12(x− ip)β† − 1

x

[(2λ+ 1

3

G0 + 3

4

+ λS0

)β† −

2λ+ 1

3 G− + λS−

α†]

= 12(x− ip)β† − 1

x

[(2λ+ 1)

(α†α

2 − 14

)β† + λ

((amam − bmbm)β†

2 − bmamα†)]

The grade 0 fermionic generators in C0 are given by

Q0 = 12(Q+ iQ) (2.4.12)

21

= 12(x− ip)α− 1

x

[(2λ+ 1

3

G0 + 3

4

+ λS0

)α+

2λ+ 1

3 G− + λS−

β

]= 1

2(x− ip)α− 1x

[(2λ+ 1)

(14 −

β†β

2

)α+ λ

((amam − bmbm)α

2 + bmamβ

)]

S0 = 12(S + iS) (2.4.13)

= 12(x− ip)β + 1

x

[(2λ+ 1

3

G0 −

34

+ λS0

)β −

2λ+ 1

3 G+ + λS+

α

]= 1

2(x− ip)β + 1x

[(2λ+ 1)

(α†α

2 − 14

)β + λ

((amam − bmbm)β

2 − ambmα)]

Q†0 = 12(Q† − iQ†) (2.4.14)

= 12(x+ ip)α† − 1

x

[(2λ+ 1

3

G0 −

34

+ λS0

)α† +

2λ+ 1

3 G+ + λS+

β†]

= 12(x+ ip)α† − 1

x

[(2λ+ 1)

(14 −

β†β

2

)α† + λ

((amam − bmbm)α†

2 + ambmβ†)]

S†0 = 12(S† − iS†) (2.4.15)

= 12(x+ ip)β† + 1

x

[(2λ+ 1

3

G0 + 3

4

+ λS0

)β† −

2λ+ 1

3 G− + λS−

α†]

= 12(x+ ip)β† + 1

x

[(2λ+ 1)

(α†α

2 − 14

)β† + λ

((amam − bmbm)β†

2 − bmamα†)]

The grade zero odd generators together with the SU(2)T generators T+, T−, T0 and U(1)generator

J = (λ+ 1)A0 + 12 (K+ +K−) (2.4.16)

generate the sub-supergroup SU(2|1). They satisfy the anticommutation relationsQ0 , Q

†0

= −λT0 + J

Q0 , S

†0

= −λT−

S0 , S†0

= +λT0 + J

S0 , Q

†0

= −λT+

[T0 , Q0] = − 12Q0 [T0 , S0] = +1

2S0[T0 , Q

†0

]= + 1

2Q†0

[T0 , S

†0

]= − 1

2S†0

[J , Q0] = −λ2Q0 [J , S0] = −λ2S0[J , Q†0

]= +λ

2Q†0

[J , S†0

]= +λ

2S†0

[T+ , Q0] = −S0 [T− , S0] = −Q0[T+ , Q

†0

]= +S†0

[T− , S†0

]= +Q†0

(2.4.17)

22

The anticommutation relations of grade zero fermionic generators with grade ±1 generatorsin the compact 3-grading are as follows

Q0 , Q+ = 2B+ S0 , Q+ = 0Q†0 , Q+

= 0

S†0 , Q+

= 2(λ+ 1)A+

Q0 , S+ = 0 S0 , S+ = 2B+Q†0 , S+

= −2(λ+ 1)A+

S†0 , S+

= 0

Q0 , Q− = 0 S0 , Q− = 2(λ+ 1)A−Q†0 , Q−

= 2B−

S†0 , Q−

= 0

Q0 , S− = −2(λ+ 1)A− S0 , S− = 0Q†0 , S−

= 0

S†0 , S−

= 2B−

(2.4.18)

The generator H that determines the compact three grading is given by

H = 12 (K+ +K−) +A0

= B0 + 12 (Nα +Nβ − 1) (2.4.19)

where

B0 = 14(p2 + x2) + 1

x2

(L2 + 3

16

)(2.4.20)

= 14(p2 + x2) + 1

x2

(λT 2 + 1

3(λ− 1)A2 + λ(λ− 1)S2 + 14λ(λ− 1) + 3

16

)

2.4.2 Unitary supermultiplets of D(2, 1;λ)

The generators B− and B+ defined above close into B0 under commutation and generatethe distinguished su(1, 1)K subalgebra

[B− , B+] = 2B0 [B0 , B+] = +B+ [B0 , B−] = −B− . (2.4.21)

The generator B0 can be written as follows:

HConf = 2B0 = 12(x2 + p2)+ g2

x2 (2.4.22)

with g2 = (2L2 + 38 ). This can be interpreted as Hconf, the Hamiltonian for conformal

quantum mechanics [119] or of a singular oscillator [120]. The role of coupling constantfor the inverse square potential is played by g2.

Since we are looking at SU(1, 1)K which is a non-compact group, the unitary repre-sentations are infinite dimensional. We are interested in the physically relevant positive

23

energy representations which are bounded from below. These are also known as lowestweight representations and are uniquely determined by a state |ψα0 〉 with the lowesteigenvalue of B0 that is annihilated by B−:

B−|ψω0 〉 = 0 (2.4.23)

In the coordinate (x) representation its wave function is given by

ψω0 (x) = C0 xω e−x

2/2 (2.4.24)

where C0 is the normalization constant, ω is given by

ω = 12 +

(14 + 2g2

)1/2(2.4.25)

and g2 is the eigenvalue of (2L2 + 38 )

(2L2 + 38)|ψω0 〉 = g2|ψω0 〉 (2.4.26)

We shall denote the functions obtained by the repeated action of differential operatorsB+ on ψω0 (x) in the coordinate representation as ψωn (x) and the corresponding states inthe Hilbert space as |ψωn 〉 :

ψωn (x) = cn(B+)nψω0 (x) (2.4.27)

where the normalization constant is given as

cn = (−1)2n

√Γ(ω + 1/2)√

n! Γ(n+ ω + 1/2)(2.4.28)

The wave functions ψωn (x) can be written as

ψωn (x) =

√2(n!)

Γ(n+ ω + 1/2) L(ω−1/2)n (x2)xωe−x

2/2 (2.4.29)

where L(ω−1/2)n (x2) is the generalized Laguerre polynomial.

We shall use |Ω〉 to uniquely label the irreducible lowest weight (positive energy)representations of D(2, 1;λ). Being the lowest weight states, |Ω〉 are all annihilated bythe generators B−, A−,Q− and S− in C− and transform irreducibly under C0 subalgebraOSp(2/2)× U(1).

As explained earlier, the positive energy refers to the fact that the spectrum of H isbounded from below and continuing the same terminology, we shall refer to H as the

24

(total) Hamiltonian and its eigenvalues as total energy. Each state in the set |Ω〉 is alowest (conformal) energy state of a positive energy irrep of SU(1, 1)K , since they are allannihilated by B−. The conditions

B−|Ω〉 = 0

A−|Ω〉 = 0

Q−|Ω〉 = 0

S−|Ω〉 = 0 (2.4.30)

imply that the states |Ω〉 must be linear combinations of the tensor product states of theform

|F 〉 × |B〉 × |ψω0 〉 (2.4.31)

where the state |F 〉 in (2.4.31) is either the fermionic Fock vacuum

|0〉F = |mt = 0;ma = −1/2〉F = |0, ↓〉 (2.4.32)

or one of the following SU(2)T doublet of states:

α†|0〉F ≡ |mt = 1/2;ma = 0〉F = |↑, 0〉 , (2.4.33)

β†|0〉F ≡ |mt = −1/2;ma = 0〉F = |↓, 0〉 (2.4.34)

and the state |B〉 in (2.4.31) is any one of the states

am1 · · · amkbmk+1 · · · bm2s |0〉B (2.4.35)

where |0〉B is the bosonic Fock vacuum annihilated by the bosonic oscillators am andbm (m = 1, 2, · · · , P ). For fixed n the states of the form (2.4.35) transform in the spin s

representation of SU(2)S. They also form representations of SO∗(2P ) generated by thebilinears

Umn = aman + bmbn, Umn = (ambn − anbm), Umn = (ambn − anbm) (2.4.36)

and which commutes with D(2, 1;λ). Therefore as far as the SU(2) deformations of theminrep of D(2, 1;λ) are concerned we can restrict our analysis to P = 1. Then for P = 1we simply have

S+ = a†b

S− = b†a

25

andS0 = 1

2(a†a− b†b)

Then the states |B〉 belong to the set

|s,ms〉B ≡ (a†)k (b†)2s−k|0〉B (2.4.37)

where ms = k − s ( k = 0, 1, · · · , 2s ) and transform in spin s representation of SU(2)S:

S2|s,ms〉B = s(s+ 1)|s,ms〉B (2.4.38)

S0|s,ms〉B = ms|s,ms〉B (2.4.39)

The action of raising operator S+ = a†b and lowering operator S− = b†a on this state isthen given as

S+ |s,ms = k − s〉B =√

(k + 1)(2s− k) |s, k − s+ 1〉B (2.4.40)

S− |s,ms = k − s〉B =√k(2s− k + 1) |s, k − s− 1〉B (2.4.41)

The eigenvalues (1/4 + 2g2) of (4L2 + 1) on the above states determine the values of ωlabeling the eigenstates |ψω0 〉 of B0 annihilated by B− :

(4L2 + 1)|mt = 0;ma = −1/2〉F × |s,ms〉B = λ2(2s+ 1)2|0;−1/2〉F × |s,ms〉B (2.4.42)

(4L2 + 1)|mt = ±1/2;ma = 0〉F × |s,ms〉B = [λ(2s+ 1) + 1]2| ± 1/2; 0〉F × |s,ms〉B (2.4.43)

The SU(2) subalgebra of SU(1|2) is the diagonal subalgebra SU(2)T of SU(2)S and SU(2)T .Therefore we shall work in a basis where T 2 and T0 are diagonalized and denote thesimultaneous eigenstates of B0, T 2, T0 , A2 and A0 as1

|ω; t,mt; a,ma〉 (2.4.44)

where ω is the eigenvalue of B0.The set of states |Ω〉 must be linear combinations of the tensor product states of

the form |F 〉 × |B〉 × |ψωo 〉 where the state |F 〉 could be either |0〉F = |0, ↓〉F or one of the1We introduce this notation for the states because the tensor product states of the form |F 〉×|B〉×|ψω0 〉

are not always definite eigenstates of T 2and T0 but it is easier to understand the structure of supermultipletsin terms of these tensor product states. Thus we will use both notations for states.

26

SU(2)T doublet of states |↑, 0〉F or |↓, 0〉F . We will now study the unitary representationsfor these two cases.

2.4.2.1 |F 〉 = |0〉F

For states with t = 0, we can use equation (2.4.25) to write

ω = 12 ± λ(2s+ 1) (2.4.45)

where the sign of the square root is determined by the sign of λ and the range of λ isdetermined by the square integrability of the states and the positivity of g2. This leadsto the following restriction on λ

|λ| > 12(2s+ 1) (2.4.46)

Let us first consider the case s = 0 and with the positive square root taken in theabove equations. Then the lowest energy state |0, ↓〉F × |0〉B ×

∣∣∣ψλ+1/20

⟩, annihilated by

all the generators in C−1 , is a singlet of the grade zero super algebra SU(1|2) since

Q0 |0, ↓〉F × |0〉B ×∣∣∣ψλ+1/2

0

⟩= 0

S0 |0, ↓〉F × |0〉B ×∣∣∣ψλ+1/2

0

⟩= 0

Q†0 |0, ↓〉F × |0〉B ×∣∣∣ψλ+1/2

0

⟩= 0

S†0 |0, ↓〉F × |0〉B ×∣∣∣ψλ+1/2

0

⟩= 0 (2.4.47)

States generated by action of grade +1 generators C+ on this lowest weight state∣∣∣ψ(λ+1/2)

0

⟩B+ |0, ↓〉F × |0〉B ×

∣∣∣ψλ+1/20

⟩= |0, ↓〉F × |0〉B ×

∣∣∣ψλ+1/21

⟩A+ |0, ↓〉F × |0〉B ×

∣∣∣ψλ+1/20

⟩= |0, ↑〉F × |0〉B ×

∣∣∣ψλ+1/20

⟩Q+ |0, ↓〉F × |0〉B ×

∣∣∣ψλ+1/20

⟩= |↑, 0〉F × |0〉B ×

∣∣∣ψλ+3/20

⟩S+ |0, ↓〉F × |0〉B ×

∣∣∣ψλ+1/20

⟩= |↓, 0〉F × |0〉B ×

∣∣∣ψλ+3/20

⟩(2.4.48)

form a supermultiplet transforming in the representation with super tableau of

SU(2|1). The states Q+ |0, ↓〉F × |0〉B ×∣∣∣ψλ+1/2

0

⟩and S+ |0, ↓〉F × |0〉B ×

∣∣∣ψλ+1/20

⟩are both

lowest weight vectors of SU(1, 1)K transforming as a doublet of SU(2)T . The state

27

A+ |0, ↓〉F × |0〉B ×∣∣∣ψλ+1/2

0

⟩is a lowest weight vector of SU(1, 1)K and together with

|0, ↓〉F ×|0〉B×∣∣∣ψλ+1/2

0

⟩form a doublet of SU(2)A. The commutator of two susy generators

in C+ satisfies

[Q+,S+] |0, ↓〉F × |0〉B ×∣∣∣ψλ+1/2

0

⟩∝ α†β†B+ |0〉F × |0〉B ×

∣∣∣ψλ+1/20

⟩Hence one does not generate any new lowest weight vectors of SU(1, 1)K by further actionsof grade +1 supersymmetry generators.

The conformal group in d dimensions is SO(d, 2) and positive energy unitary or lowestweight representations correspond to conformal fields in d (Minkowski) dimensions withthe conformal dimension given by the eigenvalue of the SO(2) generator. In the case ofD(2, 1;λ), we are working in one dimension and the positive energy unitary representationsare instead identified with conformal wave functions. We shall denote the conformal wavefunction associated with a positive energy unitary representation of SO(2, 1) with lowestweight vector

∣∣∣ψ(ω)0

⟩as Ψω(x). The conformal wave functions transforming in the (t, a)

representation of SU(2)T × SU(2)A will then be denoted as

Ψω(t,a)(x)

Thus the unitary supermultiplet of D(2, 1;λ) with the lowest weight vector |0〉F × |0〉B ×∣∣∣ψλ+1/20

⟩decomposes as:

Ψ(λ+1)/2(0,1/2) ⊕Ψ(λ+2)/2

(1/2,0) (2.4.49)

This is simply the minimal unitary supermultiplet of D(2, 1;λ) and for λ = −1/2 coincideswith the singleton supermultiplet of OSp(4|2,R) = D(2, 1;−1/2).

Next we consider the representations for the case s 6= 0 with λ > 0. In this case thelowest energy states

|0, ↓〉F × |s,ms = (k − s)〉B × |ψω0 〉 , (2.4.50)

where ω = 1/2 + λ(2s + 1), and k = 0, . . . , 2s, are not annihilated by all supersymmetrygenerators of SU(2|1):

Q0 |0, ↓〉F × |s, k − s〉B × |ψω0 〉 = 0

S0 |0, ↓〉F × |s, k − s〉B × |ψω0 〉 = 0

Q†0 |0, ↓〉F × |s, k − s〉B × |ψω0 〉 = λ(2s− k) |↑, 0〉F × |s, k − s〉B ×

∣∣ψω−10

⟩− λ

√(k + 1)(2s− k) |↓, 0〉F × |s, k − s+ 1〉B ×

∣∣ψω−10

⟩S†0 |0, ↓〉F × |s, k − s〉B × |ψ

ω0 〉 = λk |↓, 0〉F × |s, k − s〉B ×

∣∣ψω−10

⟩28

−λ√k(2s− k + 1) |↑, 0〉F × |s, k − s− 1〉B ×

∣∣ψω−10

⟩[Q+ , S+] |0, ↓〉F × |s, k − s〉B × |ψ

ω0 〉 = 2s |0, ↑〉F × |s, k − s〉B ×

∣∣ψω−20

⟩(2.4.51)

Since the supersymmetry generators Q†0 and S†0 transform in the spin 1/2 represen-tation of SU(2)T acting on the states with spin t = s one would expect to obtain stateswith both t = s± 1/2. However setting k = 2s in the above formulas we find

Q†0 |0, ↓〉F × |s, s〉B × |ψω0 〉 = 0

S†0 |0, ↓〉F × |s, s〉B × |ψω0 〉 = 2λs |↓, 0〉F × |s, s〉B ×

∣∣ψω−10

⟩−λ√

2s |↑, 0〉F × |s, s− 1〉B ×∣∣ψω−1

0⟩

(2.4.52)

which implies that we only get states with t = s−1/2. Therefore the lowest energy supermul-tiplets of SU(2|1) that uniquely determine the deformed minimal unitary supermultipletsof D(2, 1;λ) transform in the representation with the super tableau · · · ︸︷︷︸

2swhich decomposes under the even subgroup SU(2)T × U(1)J as

· · · ︸︷︷︸2s

= ( · · ·︸︷︷︸2s

, 0)⊕ ( · · ·︸︷︷︸(2s−1)

,λ

2 ) (2.4.53)

By acting with grade +1 generators of the compact 3-grading on these states witht = s and t = s− 1/2 to obtain states with t = s± 1/2 and t = s :

B+ |0, ↓〉F × |s, k − s〉B × |ψω0 〉 = |0, ↓〉F × |s, k − s〉B × |ψ

ω1 〉

A+ |0, ↓〉F × |s, k − s〉B × |ψω0 〉 = |0, ↑〉F × |s, k − s〉B × |ψ

ω0 〉

Q+ |0, ↓〉F × |s, k − s〉B × |ψω0 〉 = |↑, 0〉F × |s, k − s〉B ×

∣∣ψω+10

⟩−λ(2s− k) |↑, 0〉F × |s, k − s〉B ×

∣∣ψω−10

⟩+λ√

(k + 1)(2s− k) |↓, 0〉F × |s, k − s+ 1〉B ×∣∣ψω−1

0⟩

S+ |0, ↓〉F × |s, k − s〉B × |ψω0 〉 = |↓, 0〉F × |s, k − s〉B ×

∣∣ψω+10

⟩−λ(2s− k) |↓, 0〉F × |s, k − s〉B ×

∣∣ψω−10

⟩+λ√k(2s− k + 1) |↑, 0〉F × |s, k − s− 1〉B ×

∣∣ψω−10

⟩(2.4.54)

The commutator of two supersymmetry generators does not generate any new lowest

29

weight vector of SU(1, 1)K :

[Q+ , S+] |0, ↓〉F × |s, k − s〉B × |ψω0 〉 ∝ |0, ↑〉F × |s, k − s〉B × |ψ

ω1 〉

= α†β†B+ |0, ↓〉F × |s, k − s〉B × |ψω0 〉

(2.4.55)

Thus the complete supermultiplet is simply

Ψp(s−1/2,0) ⊕Ψp+1/2

(s,1/2) ⊕Ψp+1(s+1/2,0) (2.4.56)

where p = λ(2s + 1)/2. We have summarized the deformed supermultiplets for lowestweight states with t = s and λ > 1/(4s+ 2) in Table 2.1.

Table 2.1. Decomposition of SU(2) deformed minimal unitary lowest energy supermultipletsof D(2, 1;λ) with respect to SU(2)T × SU(2)A × SU(1, 1)K . The conformal wavefunctionstransforming in the (t, a) representation of SU(2)T × SU(2)A with conformal energy ω aredenoted as Ψω

(t,a). The first column shows the super tableaux of the lowest energy SU(2|1)supermultiplet, the second column gives the eigenvalue of the U(1) generator H. The allowedrange of λ in this case is λ > 1/(4s+ 2).

SU(2|1)l.w.v H SU(1, 1)K × SU(2)T × SU(2)A

1 λ/2 Ψ(λ+1)/2(0,1/2) ⊕Ψ(λ+2)/2

(1/2,0)

λ Ψλ(0,0) ⊕Ψλ+1/2

(1/2,1/2) ⊕Ψλ+1(1,0)

3λ/2 Ψ3λ/2(1/2,0) ⊕Ψ(3λ+1)/2

(1,1/2) ⊕Ψ(3λ/2+1(3/2,0)

......

......

......

· · · ︸︷︷︸2s

(2s+ 1)λ/2 Ψp(s−1/2,0) ⊕Ψp+1/2

(s,1/2) ⊕Ψp+1(s+1/2,0)

p = (2s+ 1)λ/2

So far we have considered the representations for λ > 0 when the lowest weight state

30

has t = s. Now we take a look at the case when λ < 0 and ω is then given as

ω = 12 − λ(2s+ 1) (2.4.57)

The action of grade 0 supersymmetry generators on these states produces states witht = s + 1/2. This is different from the case with λ > 0 where we obtained states witht = s − 1/2 by the action of grade 0 supersymmetry generators. Thus the the lowestenergy supermultiplets of SU(2|1) that uniquely determine the deformed minimal unitarysupermultiplets of D(2, 1;λ) transform in the representation with the super tableau

· · · ︸︷︷︸2s+1

which decomposes under the even subgroup SU(2)T × U(1)J as

· · · ︸︷︷︸2s+1

= ( · · ·︸︷︷︸2s+1

, 0)⊕ ( · · ·︸︷︷︸2s

,λ

2 ) (2.4.58)

By acting with the grade +1 supersymmetry generators on these states, we complete theD(2, 1;λ) supermultiplet given as:

Ψp(s+1/2,0) ⊕Ψp+1/2

(s,1/2) ⊕Ψp+1(s−1/2,0) (2.4.59)

where p = |λ|(2s + 1)/2. We have summarized the deformed supermultiplets for lowestweight states with t = s and λ < −1/(4s+ 2) in Table 2.2.

2.4.2.2 |F〉 =(|↑, 0〉F|↓, 0〉F

)

If we choose the doublet of states |F 〉 = |↑, 0〉F and |F 〉 = |↓, 0〉F as part of the lowestenergy supermultiplet, the states |ω, t,mt, a,ma〉 satisfying the conditions given in (2.4.30)will have t = s± 1/2 and can be written as

|−λ(2s+ 1)/2, s+ 1/2,mt, 0, 0〉 = 1√2s+ 1

√s+ 1/2 +mt |↑, 0〉F × |s,mt − 1/2〉B × |ψ

ω0 〉

+√s+ 1/2−mt |↓, 0〉F × |s,mt + 1/2〉B × |ψ

ω0 〉(2.4.60)

where ω = −1/2 − λ(2s + 1) with λ < 0. The range of λ is determined by the squareintegrability of the states and the positivity of g2. This leads to the following restrictionon λ

λ < − 32(2s+ 1) (2.4.61)

31


(t,a). The first column shows the super tableaux of the lowest energy SU(2|1)supermultiplet, the second column gives the eigenvalue of the U(1) generator H. The allowedrange of λ in this case is λ < −1/(4s+ 2).


|λ|/2 Ψ|λ|/2(1/2,0) ⊕Ψ(|λ|+1)/2(0,1/2)

|λ| Ψ|λ|(1,0) ⊕Ψ|λ|+1/2(1/2,1/2) ⊕Ψ|λ|+1

(0,0)

3|λ|/2 Ψ3|λ|/2(3/2,0) ⊕Ψ(3|λ|+1)/2

(1,1/2) ⊕Ψ3|λ|/2+1(1/2,0)

......

......

......

· · · ︸︷︷︸2s+1

(2s+ 1)|λ|/2 Ψp(s+1/2,0) ⊕Ψp+1/2

(s,1/2) ⊕Ψp+1(s−1/2,0)

p = (2s+ 1)|λ|/2

On the other hand for t = s− 1/2, we have

|λ(2s+ 1)/2, s− 1/2,mt, 0, 0〉 = 1√2s+ 1

√s+ 1/2 +mt |↓, 0〉F × |s,mt + 1/2〉B × |ψ

ω0 〉

−√s+ 1/2−mt |↑, 0〉F × |s,mt − 1/2〉B × |ψ

ω0 〉(2.4.62)

where ω = −1/2 + λ(2s + 1) with λ > 0. The range of λ is determined by the squareintegrability of the states and the positivity of g2. This leads to the following restrictionon λ

λ >3

2(2s+ 1) (2.4.63)

Let us now study the simplest case when s = 0. The lowest energy states that areannihilated by grade -1 generators are |±1/2, 0〉F × |0〉B ×

∣∣∣ψ|λ|−1/20

⟩where λ < 0. The

32

action of grade 0 supersymmetry generators on these states gives:

Q0 |↑, 0〉F × |0〉B ×∣∣∣ψ|λ|−1/2

0

⟩= |0, ↓〉F × |0〉B ×

∣∣∣ψ|λ|+1/20

⟩S0 |↑, 0〉F × |0〉B ×

∣∣∣ψ|λ|−1/20

⟩= 0

Q†0 |↑, 0〉F × |0〉B ×∣∣∣ψ|λ|−1/2

0

⟩= 0

S†0 |↑, 0〉F × |0〉B ×∣∣∣ψ|λ|−1/2

0

⟩= 0 (2.4.64)

Q0 |↓, 0〉F × |0〉B ×∣∣∣ψ|λ|−1/2

0

⟩= 0

S0 |↓, 0〉F × |0〉B ×∣∣∣ψ|λ|−1/2

0

⟩= |0, ↓〉F × |0〉B ×

∣∣∣ψ|λ|+1/20

⟩Q†0 |↓, 0〉F × |0〉B ×

∣∣∣ψ|λ|−1/20

⟩= 0

S†0 |↓, 0〉F × |0〉B ×∣∣∣ψ|λ|−1/2

0

⟩= 0 (2.4.65)

Thus the action of grade 0 supersymmetry generators on states with t = 1/2 producestates with t = 0 , but not t = 1 as might be expected. Next we examine the action ofgrade +1 supersymmetry generators on these states.

Q+ |↑, 0〉F × |0〉B ×∣∣∣ψ|λ|−1/2

0

⟩= 0

S+ |↑, 0〉F × |0〉B ×∣∣∣ψ|λ|−1/2

0

⟩= |0, ↑〉F × |0〉B ×

∣∣∣ψ|λ|+1/20

⟩(2.4.66)

Q+ |↓, 0〉F × |0〉B ×∣∣∣ψ|λ|−1/2

0

⟩= |0, ↑〉F × |0〉B ×

∣∣∣ψ|λ|+1/20

⟩S+ |↓, 0〉F × |0〉B ×

∣∣∣ψ|λ|−1/20

⟩= 0 (2.4.67)

Thus the complete supermultiplet in this case is

Ψ|λ|/2(1/2,0) ⊕Ψ(|λ|+1)/2(0,1/2) , λ < 0 (2.4.68)

Let us now consider the action of grade 0 and grade+1 supersymmetry generators onstates with s 6= 0 given in (2.4.60) and (2.4.62). The action of grade 0 generators ont = s+ 1/2 states is given as

Q0 ||λ|(2s+ 1)/2, s+ 1/2,mt, 0, 0〉 =√s+ 1/2 +mt

2s+ 1 |0, ↓〉F × |s,mt − 1/2〉B ×∣∣ψω+1

0⟩

33

S0 ||λ|(2s+ 1)/2, s+ 1/2,mt, 0, 0〉 =√s+ 1/2−mt

2s+ 1 |0, ↓〉F × |s,mt + 1/2〉B ×∣∣ψω+1

0⟩

Q†0 ||λ|(2s+ 1)/2, s+ 1/2,mt, 0, 0〉 = −2λ√s+ 1/2−mt

2s+ 1 (s+ 1/2 +mt)×

|0, ↑〉F × |s,mt + 1/2〉B ×∣∣ψω−1

0⟩

S†0 ||λ|(2s+ 1)/2, s+ 1/2,mt, 0, 0〉 = −2λ√s+ 1/2 +mt

2s+ 1 (s+ 1/2−mt)×

|0, ↑〉F × |s,mt − 1/2〉B ×∣∣ψω−1

0⟩

(2.4.69)

Let us now evaluate the action of +1 grade supersymmetry generators on the stateswith t = s+ 1/2.

Q+ ||λ|(2s+ 1)/2, s+ 1/2,mt, 0, 0〉 =√s+ 1/2−mt

2s+ 1

|0, ↑〉F × |s,mt + 1/2〉B ×

∣∣ψω+10

⟩+2λ(s+ 1/2 +mt) |0, ↑〉F × |s,mt + 1/2〉B ×

∣∣ψω−10

⟩=

√s+ 1/2−mt

2s+ 1

|0, ↑〉F × |s,mt + 1/2〉B ×

∣∣ψω−11

⟩+(2mt − 1)λ |0, ↑〉F × |s,mt + 1/2〉B ×

∣∣ψω−10

⟩(2.4.70)

S+ ||λ|(2s+ 1)/2, s+ 1/2,mt, 0, 0〉 =√s+ 1/2 +mt

2s+ 1

|0, ↑〉F × |s,mt − 1/2〉B ×

∣∣ψω+10

⟩+2λ(s+ 1/2−mt) |0, ↑〉F × |s,mt − 1/2〉B ×

∣∣ψω−10

⟩ ]=

√s+ 1/2 +mt

2s+ 1

|0, ↑〉F × |s,mt − 1/2〉B ×

∣∣ψω−11

⟩−(2mt + 1)λ |0, ↑〉F × |s,mt − 1/2〉B ×

∣∣ψω−10

⟩(2.4.71)

[Q+ , S+] ||λ|(2s+ 1)/2, s+ 1/2,mt, 0, 0〉 = 0 (2.4.72)

Next we need to evaluate the action of +1 grade supersymmetry generators on the statesobtained in (2.4.69) which are of the form |0, ↓〉F × |s,mt ± 1/2〉B ×

∣∣ψω+10

⟩. From the

previous section we would expect states with t = s± 1/2 but the states with t = s+ 1/2obtained in this fashion are excitations so the only new states we obtain are the stateswith t = s− 1/2.

The lowest energy supermultiplet for t = s+ 1/2 corresponds to the following SU(2|1)

34

supertableau

· · · ︸︷︷︸2s+1

=(

· · ·︸︷︷︸2s+1

, 1)⊕(

· · ·︸︷︷︸2s

,)

(2.4.73)

and the resulting unitary supermultiplet of D(2, 1;λ) decomposes as

Ψp(s+1/2,0) ⊕Ψp+1/2

(s,1/2) ⊕Ψp+1s−1/2,0 (2.4.74)

where p = (2s+ 1)|λ|/2 with λ < 0. We have summarized the deformed supermultipletsfor lowest weight states with t = s + 1/2 in Table 2.3. Note that these occur only forλ < −3/(4s+ 2).


(t,a). The first column shows the super tableaux of the lowest energy SU(2|1)supermultiplet, the second column gives the eigenvalue of the U(1) generator H. The allowedrange of λ in this case is λ < −3/(4s+ 2).


|λ|/2 Ψ|λ|/2(1/2,0) ⊕Ψ(|λ|+1)/2(0,1/2)

|λ| Ψ|λ|(1,0) ⊕Ψ|λ|+1/2(1/2,1/2) ⊕Ψ|λ|+1

(0,0)

3|λ|/2 Ψ3|λ|/2(3/2,0) ⊕Ψ(3|λ|+1)/2

(1,1/2) ⊕Ψ3|λ|/2+1(1/2,0)

......

......

......

· · · ︸︷︷︸2s+1

(2s+ 1)|λ|/2 Ψp(s+1/2,0) ⊕Ψp+1/2

(s,1/2) ⊕Ψp+1(s−1/2,0)

p = (2s+ 1)|λ|/2

Next we look at the states with t = s− 1/2. The action of grade 0 generators on these

35

states is given below:

Q0 |λ(2s+ 1)/2, s− 1/2,mt, 0, 0〉 = −√s+ 1/2−mt

2s+ 1 |0, ↓〉F × |s,mt − 1/2〉B ×∣∣ψω+1

0⟩

S0 |λ(2s+ 1)/2, s− 1/2,mt, 0, 0〉 =√s+ 1/2 +mt

2s+ 1 |0, ↓〉F × |s,mt + 1/2〉B ×∣∣ψω+1

0⟩

Q†0 |λ(2s+ 1)/2, s− 1/2,mt, 0, 0〉 = 2λ√s+ 1/2 +mt

2s+ 1 (s+ 1/2−mt)×

|0, ↑〉F × |s,mt + 1/2〉B ×∣∣ψω−1

0⟩

S†0 |λ(2s+ 1)/2, s− 1/2,mt, 0, 0〉 = −2λ√s+ 1/2−mt

2s+ 1 (s+ 1/2 +mt)×

|0, ↑〉F × |s,mt − 1/2〉B ×∣∣ψω−1

0⟩

(2.4.75)

The action of +1 grade supersymmetry generators on the states with t = s− 1/2 isgiven as:

Q+ |λ(2s+ 1)/2, s− 1/2,mt, 0, 0〉 =√s+ 1/2 +mt

2s+ 1

|0, ↑〉F × |s,mt + 1/2〉B ×

∣∣ψω+10

⟩−2λ(s+ 1/2−mt) |0, ↑〉F × |s,mt + 1/2〉B ×

∣∣ψω−10

⟩=

√s+ 1/2 +mt

2s+ 1

|0, ↑〉F × |s,mt + 1/2〉B ×

∣∣ψω−11

⟩+(2mt − 1)λ |0, ↑〉F × |s,mt + 1/2〉B ×

∣∣ψω−10

⟩(2.4.76)

S+ |λ(2s+ 1)/2, s− 1/2,mt, 0, 0〉 = −√s+ 1/2−mt

2s+ 1

|0, ↑〉F × |s,mt − 1/2〉B ×

∣∣ψω+10

⟩−2λ(s+ 1/2 +mt) |0, ↑〉F × |s,mt − 1/2〉B ×

∣∣ψω−10

⟩ ]=

√s+ 1/2−mt

2s+ 1

|0, ↑〉F × |s,mt − 1/2〉B ×

∣∣ψω−11

⟩−(2mt + 1)λ |0, ↑〉F × |s,mt − 1/2〉B ×

∣∣ψω−10

⟩(2.4.77)

[Q+ , S+] |λ(2s+ 1)/2, s− 1/2,mt, 0, 0〉 = 0 (2.4.78)

Next we need to evaluate the action of +1 grade supersymmetry generators on the statesobtained in (2.4.75) which are of the form |0, ↓〉F × |s,mt ± 1/2〉B ×

∣∣ψω+10

⟩. From the

previous section we would expect states with t = s± 1/2 but the states with t = s− 1/2obtained in this fashion are excitations so the only new states we obtain are the stateswith t = s+ 1/2.

36

The lowest energy super multiplet for t = s− 1/2 corresponds to the following SU(2|1)supertableau

· · · ︸︷︷︸

2s

=(

· · ·

︸︷︷︸2s

, 1)⊕(

· · ·︸︷︷︸2s

,)

(2.4.79)

and leads to the supermultiplet

Ψp(s−1/2,0) ⊕Ψp+1/2

(s,1/2) ⊕Ψp+1(s+1/2,0) (2.4.80)

where p = (2s+ 1)λ/2 with λ > 0. We have summarized the deformed supermultiplets forlowest weight states with t = s− 1/2 in Table 2.4. Note that these occur only for s > 1/2and λ > 3/(4s+ 2).


(t,a). The first column shows the super tableaux of the lowest energy SU(2|1)supermultiplet, the second column gives the eigenvalue of the U(1) generator H. The allowedrange of λ in this case is λ > 3/(4s+ 2).


λ Ψλ

(0,0) ⊕Ψλ+1/2(1/2,1/2) ⊕Ψλ+1

(1,0)

3λ/2 Ψ3λ/2(1/2,0) ⊕Ψ(3λ+1)/2

(1,1/2) ⊕Ψ3λ+1(3/2,0)

2λ Ψ2λ(1,0) ⊕Ψ2λ+1/2

(3/2,1/2) ⊕Ψ2λ+1(2,0)

......

......

......

· · · ︸︷︷︸

2s

(2s+ 1)λ/2 Ψp(s−1/2,0) ⊕Ψp+1/2

(s,1/2) ⊕Ψλ+1(s+1/2,)

p = (2s+ 1)λ/2

We note the similarities of Table 2.1 with Table 2.4, and that of Table 2.2 with Table2.3. This shows that the supermultiplets obtained for lowest weight states with t = s

and t = s− 1/2 and λ > 0 are the same and the supermultiplets for lowest weight stateswith t = s and t = s+ 1/2 and λ < 0 are same. The difference between these two types ofsupermultiplets is that the SU(1, 1)K spin (labeled as p) gets interchanged between states

37

with t = s+ 1/2 and t = s− 1/2 as we change the sign of λ.

2.5 SU(2) Deformations of the minimal unitary representationof D(2, 1;λ) using both bosons and fermions and OSp(2n∗|2m)superalgebras

Above we obtained unitary supermultiplets of D(2, 1;λ) which are SU(2) deformations ofthe minimal unitary representation. This was achieved by introducing bosonic oscillatorsan and bn and extending the SU(2)T generators to the generators of the diagonal subgroupof SU(2)T and SU(2)S realized as bilinears of the bosonic oscillators

S+ = anbn (2.5.1)

S− = bnan (2.5.2)

S0 = 12(Na −Nb) (2.5.3)

As stated above, the non-compact group SO∗(2n) generated by the bilinears

Amn = ambn − anbm (2.5.4)

Amn = ambn − anbm (2.5.5)

Umn = aman + bnbm (2.5.6)

commutes with the generators of D(2, 1;λ). One can similarly obtain SU(2) deformationsof the minimal unitary supermultiplet of D(2, 1;λ) by introducing fermionic oscillators ρrand σs (r, s, .. = 1, ..n) satisfying

ρr , ρs = σr , σs = δsr ρr , ρs = ρr , σs = σr , σs = 0 (2.5.7)

and extend the generators of SU(2)T to the generators of the diagonal subgroup of SU(2)Tand SU(2)F generated by

F+ = ρrσr (2.5.8)

F− = σrρr (2.5.9)

F0 = 12(ρrρr − σrσr) (2.5.10)

In this case the compact USp(2n) generated by the fermion bilinears

Srs = ρrσs + ρsσr (2.5.11)

38

Srs = σrρs + σsρr (2.5.12)

Srs = ρrρs − σsσr (2.5.13)

commute with the generators of D(2, 1;λ). One can deform the minimal unitary repre-sentation of D(2, 1;λ) using fermions and bosons simultaneously. This is achieved byreplacing the SU(2)T generators by the diagonal generators of SU(2)T × SU(2)S × SU(2)F, which we shall denote as U+, U− and U0

U+ = T+ + S+ + F+ (2.5.14)

U− = T− + S− + F− (2.5.15)

U0 = T0 + S0 + F0 (2.5.16)

and substituting the quadratic Casimir of SU(2)T in the Ansatz for K− with the quadraticCasimir of SU(2)D. Remarkably, in this case the resulting generators of D(2, 1;λ) commutewith the generators of the non-compact superalgebra OSp(2n∗|2m) generated by thegenerators of SO∗(2n) and USp(2m) given above and the supersymmetry generators:

Πmr = amσr − bmρr, Πmr = (Πmr)† = amσr − bmρr (2.5.17)

Σrm = amρr + bmσ

r, Σmr = (Σrm)† = amρr + bmσr (2.5.18)

The (anti) commutation relations for the OSp(2n∗|2m) algebra are given below:

[Aij , A

kl

]= δkjA

il − δilAkj , [Srs , Stu] = δtsS

ru − δruSts[

Aij , Akl

]= δkjAil − δki Ajl, [Srs , Stu] = δtsSru + δtrSus[

Aij , Akl]

= δilAjk − δjlAik, [Srs , Stu] = −δsuSrt − δruSts

[Aij , A

kl]

= −δkjAli + δljAki − δliAkj + δki A

lj

[Srs , S

tu]

= −δtsSur − δtrSus − δusStr − δur Sts

39

Πmr , Πns

= δsrA

nm − δnmSsr ,

Σrm , Σns

= δrs A

nm + δnm S

rs

Πmr , Σsn = δsrAmn,

Πmr , Σns

= −δnm Srs[Amn , Πkr] = −δmk Πnr, [Amn , Σrk] = −δmk Σrn[Amn , Πkr] = −δmk Σnr + δnk Σmr , [Amn , Σrk] = −δnk Πm

r + δmk Πnr

[Amn , Πkr] = 0, [Amn , Σrk] = 0[Srs , Πmt] = −δrtΠms, [Srs , Σtm] = δtsΣrm[Srs , Πmt] = δstΣrm + δrtΣsm, [Srs , Σtm] = 0[Srs , Πmt] = 0, [Srs , Σtm] = −δtrΠms − δtsΠmr

2.6 SU(2) deformed minimal unitary supermultiplets of D(2, 1;α)and N = 4 superconformal mechanics

2.6.1 N = 4 Superconformal Quantum Mechanical Models

Conformal quantum mechanical models have been subjects of interest since the seventiesand one of the main reason is that they characterize an important integrable (completelysolvable) system discovered by Calogero in [121, 122]. A concrete study of conformalmechanics at both classical and quantum levels was first performed in [119]. The interestresurfaced in connection with AdS/CFT correspondence [14] after it was shown thatthe dynamics of a super-particle near the horizon of an extremal Reissner-Nordströmblack-hole, in the limit of large black-hole mass, is governed by a superconformal quantummechanical model in [123].

An N = 4 superconformal quantum mechanical model that is invariant under D(2, 1;α)symmetry algebra was studied by in [115, 124]. In this section we will review thatconstruction following [115]

The on shell component action was shown to take the form [115]

S = Sb + Sf , (2.6.1)

Sb =∫dt[xx+ i

2(zkz

k − ˙zkzk)− α2(zkzk)2

4x2 −A(zkz

k − c) ], (2.6.2)

Sf = −i∫dt(ψkψ

k − ˙ψkψk)

+ 2α∫dtψiψkz(izk)

x2 + 23 (1 + 2α)

∫dtψiψkψ(iψk)

x2 (2.6.3)

Here x, zi and ψj (i, j = 1, 2) are d = 1 bosonic and fermionic “fields”, respectively. Thefields zi form a complex doublet of the R-symmetry group SU(2). The last term in (2.6.2)represents the constraint

zkzk = c , (2.6.4)

40

and A is the Lagrange multiplier.Upon quantization of the action given in (2.6.1), the dynamical variables were pro-

moted to quantum mechanical operators with following commutators:

[X,P ] = i , [Zi, Zj ] = δij , Ψi, Ψj = − 12 δ

ij (i, j = 1, 2). (2.6.5)

As the action is invariant under the group D(2, 1;α), the corresponding symmetry gen-erators can be obtained by the Noether procedure. The results as given in [115] are:

Qi = PΨi + 2iαZ(iZk)Ψk

X+ i(1 + 2α) 〈ΨkΨkΨi〉

X, (2.6.6)

Qi = P Ψi − 2iαZ(iZk)Ψk

X+ i(1 + 2α) 〈Ψ

kΨkΨi〉X

, (2.6.7)

Si = −2XΨi + tQi, Si = −2XΨi + t Qi . (2.6.8)

H = 14 P

2 + α2 (ZkZk)2 + 2ZkZk

4X2 − 2αZ(iZk)Ψ(iΨk)

X2 (2.6.9)

− (1 + 2α) 〈ΨiΨi ΨkΨk〉2X2 + (1 + 2α)2

16X2 ,

K = X2 − t 12 X,P+ t2 H , (2.6.10)

D = − 14 X,P+ tH , (2.6.11)

Jik = i[Z(iZk) + 2Ψ(iΨk)

], (2.6.12)

I1′1′ = −iΨkΨk , I2′2′ = iΨkΨk , I1′2′ = − i2 [Ψk, Ψk] . (2.6.13)

where t is time variable and the symbol 〈...〉 denotes Weyl ordering:

〈ΨkΨkΨi〉 = ΨkΨkΨi + 12 Ψi , 〈ΨkΨkΨi〉 = ΨkΨkΨi + 1

2 Ψi

〈ΨiΨi ΨkΨk〉 = 12

ΨiΨi, ΨkΨk

− 1

4 = ΨiΨi ΨkΨk −ΨiΨi + 14 ,

and Qi = −(Qi)+, Si = −

(Si)+.

In the above set of generators Qi and Si are supertranslation and superconformalboost generators respectively. The generators H, K and D are the Hamiltonian, specialconformal transformations and dilatation generators respectively and they form an su(1, 1)algebra. The remaining generators Jik and Ii′k′ are the generators of two su(2) algebras.

41

2.6.2 Mapping between the harmonic superspace generators of N = 4 su-perconformal mechanics and generators of deformed minimal unitary repre-sentations of D(2, 1;α)

The symmetries of N = 2 supersymmetric sigma models with a quaternionic Kählerbackground that couple to supergravity in d = 4 were studied in harmonic superspacewere studied in [125] and a precise correspondence with the minrep of the isometrygroup was established. It was further suggested in [126] that the correspondence can beextended to the full quantum correspondence on both sides by reducing the N = 2 sigmamodelto one dimension and quantizing it to get supersymmetric quantum mechanics.The bosonic spectrum of this quantum mechanics should provide the minrep for thesymmetry group. The results presented in this chapter and the D(2, 1;α) superconformalquantum mechanics reviewed in previous section [115] provides an opportunity to testthis proposal.

The basic quantum mechanical operators in the minrep and their deformations wasgiven in section 2.3 are the coordinate x and its momentum p , fermionic oscillatorsα†, α, β† andβ and the bosonic oscillators a†, a, b† and b with the following commutationrelations: [

a , a†]

= 1 =[b , b†

],α , α†

= 1 =

β , β†

(2.6.14)

The generators of quantized N = 4 superconformal mechanics in harmonic superspace goover to the generators of minimal unitary realization of D(2, 1;λ) deformed by a pair ofbosonic oscillators if we make the simple substitutions listed in Table 2.5

As expected from the results of [125, 126] we find a one-to-one correspondencebetween the symmetry generators of D(2, 1;α) superconformal quantum mechanics andthe generators of the minimal unitary representations of D(2, 1;α) deformed by a pairof bosons, which we present in Table 2.6. Using this mapping it is easy to see that thequadratic Casimir obtained in equation (4.26) of [115] is the same as the one obtained byour construction given in (2.3.40) for µ = 4. The quantum spectra of N = 4 superconformalmechanics were also studied in [115] using the realization reviewed in the previous section.To relate the quantum spectra of these models to the minimal unitary realizations ofD(2, 1;λ) we tabulate the correspondence between the SU(1, 1), SU(2)R, SU(2)L quantumnumbers of [115] and the SU(1, 1)K , SU(2)T , SU(2)A spins of our construction in Table2.7. Using this table we see that the superfield contents of the quantum spectra of thesemodels as given in Table 2 of [115] are exactly the same as supermultiplets described inTables 2.1, 2.2, 2.3 and 2.4 above.

42

Table 2.5. Below we give the correspondence between the quantum mechanical operatorsof N = 4 superconformal mechanics and the operators of minimal unitary supermultiplet ofD(2, 1;λ) deformed by a pair of bosons. The SU(2) indices on the left column are raised andlowered by the Levi-Civita tensor εij (with ε12 = ε21 = 1).Operators of N = 4 Superconformal Operators of minimal unitary representationMechanics in Harmonic superspace of D(2, 1;λ) deformed by a pair of bosons

ψ1 − i√2α†

ψ2 − i√2α†

ψ1i√2α

ψ2i√2β

Z1 −ia†

Z2 −ib†

Z1 ia

Z2 ib

2.7 Discussion

In this chapter we reformulated the minrep of D(2, 1;λ) and constructed the SU(2)deformed minimal unitary supermultiplets of D(2, 1;λ) using quasiconformal methods.Similar deformations of the minrep were obtained for the 4d and 6d superconformalalgebras in [23, 24, 56]. We also established a correspondence between deformationsobtained by a pair of bosons and the quantum spectra of the N = 4 superconformalmechanical models studied in [115]. However we found a more general class of deformationsinvolving an arbitrary numbers of pairs of bosons and/or pairs of fermions which begs thequestion of what kind of superconformal models correspond to these general deformations.

As discussed in previous section, the results of [125,126] indicate these deformationsto describe the spectra of superconformal quantum mechanical models with quaternionicKähler sigma models that descend from 4d, N = 2 supersymmetric sigma models thatcouple to N = 2 supergravity. Another interesting problem is to understand the preciseconnection between the above results and the d = 2, N = 4 supersymmetric gauged WZWmodels studied in [127, 128] that extend the results of [129] on N = 2 supersymmetricgauged WZW models.

These gauged WZW models correspond to realizations over spaces of the form

GcH × SU(2) × SU(2)× U(1) (2.7.1)

43

where Gc/(H ×SU(2)) is a compact quaternionic symmetric space. On the other hand thequaternionic Kähler manifolds that couple to 4d, N = 2 supergravity are non-compact.

44

Table 2.6. Below we give the mapping between the symmetry generators of N = 4 supercon-formal mechanics in harmonic superspace and the minimal unitary representation of D(2, 1;λ)deformed by a pair of bosons. The first column lists the Symmetry generators of N = 4 super-conformal quantum mechanics in harmonic superspace (N = 4 SCQM in HSS) and the secondcolumn lists the generators for the minimal unitary realization of D(2, 1;λ) deformed by a pair ofbosons.

N = 4 SCQM in HSS Deformed Minreps of D(2, 1;λ)

iI1′1′ A+

−iI2′2′ A−

iI1′2′ A0

−iJ11 T+

iJ22 T−

iJ12 T0

− i√2Q

1 Q1

− i√2Q

2 S1

− i√2 Q1 Q1

− i√2 Q2 S1

− i√2S

1 Q1

− i√2S

2 S1

− i√2S1 Q1

− i√2S2 S1

2H K+

−2D ∆12K K−

45

Table 2.7. Below we give the mapping between the quantum numbers of the spectrum of N = 4superconformal mechanics of [115] and the minimal unitary supermultiplets of D(2, 1;λ) deformedby a pair of bosons.

Quantum spectrum of N = 4 SCQM Deformed Minreps of D(2, 1;λ)

r0 p

j t

i a

c 2s

46

Chapter 3 |Conformal Higher Spin Algebrasin d = 4: hs(4, 2; ζ)

3.1 Introduction

Higher spin algebras are defined as Lie algebra of global symmetries of a theory that hashigher spin particles in its spectrum. A more mathematically precise definition was givenby Eastwood in [130] where a higher spin algebra in AdSd+1 space was defined as theuniversal enveloping algbera U(so(d, 2)) of the conformal group SO(d, 2) in d Minkowskidimensions, quotiented by a two sided ideal J(so(d, 2)), known as the Joseph ideal [43],which is the annihilator of the minrep. In other words, higher spin algebra is theenveloping algebra of the minrep of SO(d, 2). Thus in order to construct the higher spinalgebra in any dimension, all we need to do is construct the minrep and its universalenveloping algebra is just the higher spin algebra.

Most of the work on higher spin theories that has been done in the literature is for AdS4

and AdS3 where the structure of higher spin algebras is understood fairly well. Howeverfrom the string theory and M-theory point of view, AdS5 and AdS7 higher spin theoriesare very important and the associated AdS/CFT dualities can provide new insightsinto higher spin theories in these dimensions. However, the lack of convenient spinorformulations in AdS5 and AdS7 spaces makes it harder to formulate higher spin algebrasin these dimensions. In this chapter we will focus on formulating the minrep of thefour dimensional conformal group SO(4, 2) ∼ SU(2, 2) as bilinears of deformed twistorialoscillators that transform nonlinearly under the Lorentz group SO(3, 1). The minrep (andits deformations) for SU(2, 2) was first obtained in [23] using the quasiconformal methdosand we shall reformulate their results and show that the highly nonlinear realization canbe conveniently rewritten as bilinears of deformed twistorial oscillators which themselvesare nonlinear and have nontrivial commutation relations. These results were published

47

in [131] which we follow closely in our review in this section.The plan of the chapter is as follows: In section 3.2 we review the covariant twistorial

oscillator (singleton) construction of the conformal group in three dimensions SO(3, 2) ∼Sp(4,R) and its superextension OSp(N |4,R). Then we review the covariant twistorialoscillator (doubleton) construction for the four dimensional conformal group SO(4, 2) ∼SU(2, 2) in section 3.3.1. In section 3.3.2, we present the minimal unitary representationof SU(2, 2) obtained by the quasiconformal approach [23] in terms of certain deformedtwistorial oscillators that transform nonlinearly under the Lorentz group. We thendefine a one-parameter family of these deformed twistors, which we call helicity deformedtwistorial oscillators and express the generators of a one parameter family of deformationsof the minrep given in [23] as bilinears of the helicity deformed twistors. They describemassless conformal fields of arbitrary helicity which can be continuous. In section 3.3.4,we use the deformed twistors to realize the superconformal algebra PSU(2, 2|4) and itsdeformations in the quasiconformal framework. In section 3.4, we review the Eastwood’sformula for the generator J of the annihilator of the minrep (Joseph ideal) and showby explicit calculations that it vanishes identically for the singletons of SO(3, 2) and theminrep of SU(2, 2) obtained by quasiconformal methods. We then present the generatorJ of the Joseph ideal in 4d covariant indices and use them to define the deformations Jζ

that are the annihilators of the deformations of the minrep. In section 3.4.4, we use thefact that annihilators vanish identically to identify the AdS5/Conf4 higher spin algebra(as defined by Eastwood [130]) and define its deformations as the enveloping algebras ofthe deformations of the minrep within the quasiconformal framework. In section 4.4 wediscuss the extension of these results to higher spin superalgebras.

3.2 3d conformal algebra SO(3, 2) ∼ Sp(4,R) and its minimalunitary realization

In this section we shall review the twistorial oscillator construction of the unitaryrepresentations of the conformal groups SO(3, 2) in d = 3 dimensions that correspond toconformally massless fields in d = 3 following [35, 132]. These representations turn out tobe the minimal unitary representations and are also called the singleton (scalar and spinorsingleton) representations of Dirac [25]. Since the quartic invariant vanishes identically forthe symplectic groups Sp(2N,R), the quasiconformal and covariant oscillator constructionsof Sp(2N,R) coincide [22] and thus we will only review the oscillator construction ofSp(4,R) following [35,132].

48

3.2.1 Twistorial oscillator construction of SO(3, 2)

The covering group of the three dimensional conformal group or the AdS4 isometry groupSO(3, 2) is isomorphic to the non-compact symplectic group Sp(4,R) with the maximalcompact subgroup U(2). Commutation relations of its generators can be written as

[MAB , MCD] = i(ηBCMAD − ηACMBD − ηBDMAC + ηADMBC) (3.2.1)

where ηAB = diag(−,+,+,+,−) and A,B = 0, 1, . . . , 4. The spinor representation of SO(3, 2)can be realized in terms of four-dimensional gamma matrices γµ that satisfy

γµ , γν = −2ηµν

where ηµν = diag(−,+,+) and µ, ν = 0, 1, . . . , 3 and γ5 = γ0γ1γ2γ3 as follows:

Σµν = − i4 [γµ , γν ] , Σµ4 = −12γµ (3.2.2)

We adopt the following conventions for gamma matrices in four dimensions:

γ0 =(12 00 −12

), γm =

(0 −σmσm 0

), γ5 = i

(12 00 12

)(3.2.3)

where σm (m = 1, 2, 3 are Pauli matrices. Consider now a pair of bosonic oscillators ai, a†i( i = 1, 2) that satisfy [

ai , a†j

]= δij . (3.2.4)

and define a twistorial (Majorana) spinor Ψ and its Dirac conjugate in terms of theseoscillators Ψ = Ψ†γ0

Ψ =

a1

−ia2

ia†2

−a†1

, Ψ =(a†1 ia

†2 ia2 a1

)(3.2.5)

Then the bilinears MAB = 2ΨΣABΨ satisfy the commutation relations (3.2.1) of SO(3, 2)Lie algebra.

The Fock space of these oscillators decompose into two ireducible unitary representa-tions of Sp(4,R) that are simply the two remarkable representations of Dirac [25] whichwere called Di and Rac in [133]. As representations of AdS4, these representations do nothave a Poincaré limit in 4d and their field theories live on the boundary of AdS4 whichcan be identified with the conformal compactification of three dimensional Minkowskispace [134]. It should be noted that Di and Rac (singletons) are conformal massless

49

representations in three dimensions but not of AdS4. The tensor products of two singletonsdecomposes into infinite irreducible representations which are massless in AdS4 sense butmassive as representations of three dimensional conformal group. By massless in AdS4 wemean that they become massless in the Poincaré limit since the P 2 is no longer a Casimirinvariant for the AdS4 group.

3.2.2 SO(3, 2) algebra in conformal three-grading and 3d covariant twistors

The conformal algebra in d dimensions can be given a three graded decomposition withrespect to the non-compact dilatation generator ∆ as follows:

so(d, 2) = Kµ ⊕ (Mµν ⊕∆)⊕ Pµ (3.2.6)

We shall call this conformal 3-grading. The commutation relations of the algebra in thisbasis are given as follows:

[Mµν , Mρτ ] = i (ηνρMµτ − ηµρMντ − ηντMµρ + ηµτMνρ)

[Pµ , Mνρ] = i (ηµν Pρ − ηµρ Pν)

[Kµ , Mνρ] = i (ηµν Kρ − ηµρKν)

[∆ , Mµν ] = [Pµ , Pν ] = [Kµ , Kν ] = 0

[∆ , Pµ] = +i Pµ [∆ , Kµ] = −iKµ

[Pµ , Kν ] = 2i (ηµν ∆ +Mµν)

(3.2.7)

where Mµν (µ, ν = 0, 1, ..., (d− 1)) are the Lorentz groups generators. Pµ and Kµ are thegenerators of translations and special conformal transformations.

In d = 3 dimensions the Greek indices µ, ν, . . . run over 0, 1, 2 and dilatation generatoris simply

D = −M34 (3.2.8)

and translations Pµ and special conformal transformations Kµ are given by:

Pµ = Mµ4 +Mµ3 (3.2.9)

Kµ = Mµ4 −Mµ3 (3.2.10)

In order to make connection with higher spin (super-)algebras it is best to write thealgebra in SO(2, 1) covariant spinorial oscillators. Let us now introduce linear combinations

50

of ai, a†i which we shall call 3d twistors 1:

κ1 = i

2

(a1 + a†2 + a†1 + a2

), µ1 = i

2

(a1 − a†2 − a

†1 + a2

)(3.2.11)

κ2 = 12

(a1 + a†2 − a

†1 − a2

), µ2 = 1

2

(a1 − a†2 + a†1 − a2

)(3.2.12)

They satisfy the following commutation relations:

[κα , µ

β]

= δβα (3.2.13)

Using these we can write (spinor conventions for SO(2, 1) are given in appendix A.1):

Pαβ = (σµPµ)αβ = −κακβ (3.2.14)

Kαβ = (σµKµ)αβ = −µαµβ (3.2.15)

Similarly we can define the Lorentz generators

M βα = i (σµ σν) β

α Mµν (3.2.16)

= καµβ − 1

2δβακγµ

γ (3.2.17)

and the dilatation generator

∆ = − i4 (καµα + µακα) (3.2.18)

In this basis the conformal algebra becomes:

[M βα , M δ

γ

]= δδαM

βγ − δβγM δ

α (3.2.19)[Pαβ , M

δγ

]= 2δδ(αPβ)γ − δδγPαβ (3.2.20)[

Kαβ , M δγ

]= −2δ(α

γ Kβ)δ + δδγK

αβ (3.2.21)[Pαβ , K

γδ]

= 4δ(γ(αM

δ)β) + 4iδ(γ

(αδδ)β)∆ (3.2.22)

[∆ , Kαβ

]= −iKαβ ,

[∆ , M β

α

]= 0, [∆ , Pαβ ] = iPαβ (3.2.23)

The conformal group Sp(4,R) in three dimensions admits extensions to supergroupsOSp(N |4,R) with even subgroups Sp(4,R) × O(N). We review the minimal unitaryrealization of OSp(N |4,R) in Appendix A.2.

1Note that the 3d twistor variables defined in [132] look slightly different from the ones defined herebecause the oscillators ai, ai are linear combinations of the ones used in [132].

51

3.3 Conformal and superconformal algebras in four dimensions

In this section, we present two different realizations of the conformal algebra and itssupersymmetric extensions in d = 4. We start by reviewing the doubleton oscillatorrealization [32, 37, 38] and its reformulation in terms of Lorentz covariant twistorialoscillators [37,132]. We then present a novel formulation of the quasiconformal realizationof the minimal unitary representation and its deformations first studied in [23] in termsof deformed twistorial oscillators.

3.3.1 Covariant twistorial oscillator construction of the doubletons of SO(4, 2)

The covering group of the conformal group SO(4, 2) in four dimensions is SU(2, 2). Denotingits generators as MAB the commutation relations in the canonical basis are

[MAB , MCD] = i(ηBCMAD − ηACMBD − ηBDMAC + ηADMBC)

where ηAB = diag(−,+,+,+,+,−) and A,B = 0, . . . , 5. The spinor representation ofSO(4, 2) can be realized in terms in four-dimensional gamma matrices γµ that satisfy

γµ , γν = −2ηµν (3.3.1)

where ηµν = diag(−,+,+,+) (µ, ν = 0, . . . , 3) as follows:

Σµν = − i4 [γµ , γν ] , Σµ4 = − i2γµγ5, Σµ5 = −12γµ, Σ45 = −1

2γ5 (3.3.2)

Consider now two pairs of bosonic oscillators ai, a†j (i, j = 1, 2) and br, b†s (r, s = 1, 2)

that satisfy [ai , a

†j

]= δij ,

[br , b

†s

]= δrs (3.3.3)

We form a twistorial Dirac spinor Ψ and its conjugate Ψ = Ψ†γ0 in terms of theseoscillators:

Ψ =

a1

a2

−b†1−b†2

, Ψ =(a†1 a

†2 b1 b2

)(3.3.4)

Then the bilinears MAB = ΨΣABΨ (A,B = 0, . . . , 5) generate the Lie algebra of SO(4, 2):

[ΨΣABΨ , ΨΣCDΨ

]= Ψ [ΣAB , ΣCD] Ψ (3.3.5)

52

which was called the doubleton realization [32,37,38]2.The Lie algebra of SU(2, 2) can be given a three-grading with respect to the algebra

of its maximal compact subgroup SU(2)L × SU(2)R × U(1)

su(2, 2) = Lir ⊕ (Lij +Rij + E)⊕ Lir (3.3.6)

which is referred to as the compact three-grading. For the doubleton realizations one has

Lij = aiaj −12δ

ij(akak), Rrs = brbs −

12δ

rs(btbt) (3.3.7)

Lir = aibr, E = 12(aiai + brb

r), Lir = aibr (3.3.8)

where the creation operators are denoted with upper indices, i.e a†i = ai.Under the SU(2)L × SU(2)R subgroup of SU(2, 2) generated by the bilinears Lij and

Rrs, the oscillators ai(a†i ) and br(b†r) transform in the (1/2, 0) and (0, 1/2) representation.Contrasting this to the situation in three dimensions where the Fock space of oscillatorsdecomposes into the Di and Rac irreps, the Fock space of these bosonic oscillatorsdecomposes into an infinite set of positive energy unitary irreducible representations(UIRs), called doubletons of SU(2, 2). These UIRs are uniquely determined by a subset ofstates with the lowest eigenvalue (energy) of the U(1) generator and transform irreduciblyunder the SU(2)L×SU(2)R subgroup. The possible lowest energy irreps of SU(2)L×SU(2)Rfor positive energy UIRs of SU(2, 2) are of the form

a†i1a†i2· · · a†in |0〉 ⇔ (jL, jR) =

(n2 , 0

)E = 1 + n/2 (3.3.9)

b†r1b†r2 · · · b

†rm |0〉 ⇔ (jL, jR) =

(0, m2

)E = 1 +m/2 (3.3.10)

Analogous to the singletons in three dimensions, doubleton representations are massless infour dimensions and their tensor products decompose into an infinite set of massless spinrepresentations in AdS5 [32,37,38]. The tensoring procedure in the oscillator constructionjust amounts to taking multiple copies (colors) of oscillators ai(ξ), ai(ξ), bi(η), bi(η) whereξ, η = 1, . . . , P for a P -fold tensor product. The massless representations correspond toP = 2 and for P > 2, the tensor product decomposes into an infinite set of massiverepresentations in AdS5 which are also multiplicity free [32, 37, 38]. We should stress thatThe resulting representations are multiplicity free and the explicit formulas for the tensorproduct decompositions of two irreducible doubleton representations were given in [135].

2The term doubleton refers to the fact that we are using oscillators that decompose into two irrepsunder the action of the maximal compact subgroup. For SU(2, 2) that is the minimal set required. Forsymplectic groups the minimal set consists of oscillators that form a single irrep of their maximal compactsubgroups.

53

To relate the oscillators transforming covariantly under the maximal compact subgroupSO(4)×U(1) to twistorial oscillators transforming covariantly with respect to the Lorentzgroup SL(2,C) with a definite scale dimension one acts with the intertwining operator[38,132]

T = eπ4M05 (3.3.11)

which intertwines between the compact and the non-compact pictures

MaT = TLa

NaT = TRa

DT = TE (3.3.12)

where La and Ra denote the generators of SU(2)L and SU(2)R, respectively. Ma and Na

are the generators of SU(2)M and SU(2)N given by the following linear combinations ofthe Lorentz group generators Mµν

Ma = −12

(12εabcMbc + iM0a

), Na = −1

2

(12εabcMbc − iM0a

)(3.3.13)

They satisfy

[Ma ,Mb] = iεabcMc, [Na , Nb] = iεabcNc, [Ma , Nb] = 0 (3.3.14)

where a, b, .. = 1, 2, 3.Under the action of the intertwining operator T , the compact subgroup SU(2)L×SU(2)R

gets intertwined with the non-compact Lorentz group SL(2,C). The oscillators ai(ai)and bi(bi) that transform in the (1/2, 0) and (0, 1/2) representation of SU(2)L × SU(2)Rgo over to covariant oscillators transforming as Weyl spinors (1/2, 0) and (0, 1/2) of theLorentz group SL(2,C). Denoting the components of the Weyl spinors with undotted(α, β, . . . = 1, 2) and dotted Greek indices (α, β, . . . = 1, 2) one finds :

ηα = TaiT−1 = 1√

2(bi − ai)

λα = TaiT−1 = 1√2

(bi + ai)

ηα = TbiT−1 = 1√

2(ai − bi)

λα = TbiT−1 = 1√2

(ai + bi) (3.3.15)

where α, β, α, β, .. = 1, 2 and the covariant indices on the left hand side match the indices

54

i, j.. on the right hand side of the equations above. They satisfy

[ηα, λβ ] = δαβ

[ηα, λβ ] = δαβ

(3.3.16)

They lead to the standard twistor relations,3.

Pαβ = −(σµPµ)αβ = 2λαλβ = TaibrT−1 (3.3.17)

Kαβ = −(σµKµ)αβ = 2ηαηβ = TaibrT−1 (3.3.18)

The dilatation generator in terms of covariant twistorial oscillators takes the form:

∆ = i

2(λαη

α + ηαλα)

(3.3.19)

The Lorentz generators Mµν in a spinorial basis can also be written as bilinears ofLorentz covariant twistorial oscillators:

M βα = − i2 (σµσν) β

α Mµν = λαηβ − 1

2δβα λγη

γ (3.3.20)

M αβ

= − i2 (σµσν)βαMµν = −(ηαλβ −

12δ

αβηγ λγ

)(3.3.21)

In this basis the conformal algebra becomes:

[M βα , M δ

γ

]= δ β

γ M δα − δ δ

α Mβγ ,

[M α

β, M γ

δ

]= δγ

βM α

δ− δα

δM γ

β(3.3.22)[

Pαβ , Mδγ

]= −δδαPγβ + 1

2δδγPαβ ,

[Pαβ , M

γ

δ

]= δγ

βPαδ −

12δ

γ

δPαβ (3.3.23)[

Kαβ , M δγ

]= δβγK

αδ − 12δ

δγK

αβ ,[Kαβ , M γ

δ

]= −δα

δK γβ + 1

2δγ

δKαβ (3.3.24)[

∆ , Kαβ]

= −iKαβ ,[∆ , M β

α

]=[∆ , M α

β

]= 0,

[∆ , Pαβ

]= iPαβ (3.3.25)[

Pαβ , Kγδ]

= 4(δδαM

γ

β− δγ

βM δα + i∆

)(3.3.26)

The linear Casimir operator Z = Na − Nb = aiai − brbr when expressed in terms ofSL(2,C) covariant oscillators becomes

Z = Na −Nb = λαηα − λαηα (3.3.27)

which shows that 12Z is the helicity operator.

3 Note the overall minus sign in these expressions compared to [132]. This is due to the fact that weare using a mostly positive metric in this thesis.

55

Denoting the lowest energy irreps in the compact basis as |Ω(jL, jR, E)〉 one can showthat the coherent states of the form

e−ixµPµT |Ω(jL, jR, E)〉 ≡ |Φ`jM,jN

(xµ)〉 (3.3.28)

transform exactly like the states created by the action of conformal fields Φ`jM,jN

(xµ)acting on the vacuum vector |0〉

Φ`jM,jN(xµ)|0〉 ∼= |Φ`jM,jN

(xµ)〉

with exact numerical coincidence of the compact and the covariant labels (jL, jR, E) and(jM, jN ,−l), respectively, where l is the scale dimension [38]. The doubletons correspondto massless conformal fields transforming in the (jL, jR) representation of the Lorentzgroup SL(2,C) whose conformal (scaling dimension) is ` = −E where E is the eigenvalueof the U(1) generator which is the conformal Hamiltonian (or AdS5 energy) [32,37,38].

3.3.2 Quasiconformal approach to the minimal unitary representation ofSO(4, 2) and its deformations

The construction of the minrep of the 4d conformal group SO(4, 2) by quantization ofits quasiconformal realization and its deformations were given in [23], which we shallreformulate in this section in terms of what we call deformed twistorial oscillators whichtransform nonlinearly under the Lorentz group.

The quasiconformal realization of SO(4, 2) leaves invariant a light cone defined by aquartic distance function in five dimensions. Upon quantization, this realization leads tothe nonlinear realization of the minrep in terms of two ordinary bosonic oscillators d, d†

and g, g†, and a singlet coordinate x, its conjugate momentum p satisfying [23]:

[x , p] = i,[d , d†

]= 1,

[g , g†

]= 1 (3.3.29)

The nonlinearities can be absorbed into certain “singular” oscillators which are functionsof the coordinate x, momentum p and the oscillators d, g, d†, g†:

AL = a− L√2x

A†L = a† − L√2x

(3.3.30)

where

a = 1√2

(x+ ip)

a† = 1√2

(x− ip)

56

L = Nd −Ng −12 = d†d− g†g − 1

2 (3.3.31)


[AG , AK] = − (G − K)2x2[

A†G , A†K

]= +(G − K)

2x2[AG , A

†K

]= 1 + (G +K)

2x2

(3.3.32)

assuming that [G,K] = 0.The realization of the minrep of SO(4, 2) obtained by the quasiconformal approach

is nonlinear and “interacting” in the sense that they involve operators that are cubicor quartic in terms of the oscillators in contrast to the covariant twistorial oscillatorrealization, reviewed in section 3.3.1 [32], which involves only bilinears. The algebra so(4, 2)can be given a 3-graded decomposition with respect to the conformal Hamiltonian, whichis referred to as the compact 3-grading and the generators in this basis are reproduced inAppendix A.3 following [23].

The Lie algebra of SO(4, 2) also has a non-compact (conformal) three graded decom-position determined by the dilatation generator ∆ as well

so(4, 2) = N− ⊕N0 ⊕N+ (3.3.33)

= Kµ ⊕ (Mµν ⊕∆)⊕ Pµ (3.3.34)

One can write the generators of quantized quasiconformal action of SO(4, 2) as bilinearsof deformed twistorial oscillators Zα, Zα, Yα, Yα (α, α = 1, 2) which are defined as:

Z1 = AL√2− i g†, Y 1 = −A

†L√2

+ i g (3.3.35)

Z1 = A†L√2

+ i g, Y 1 = AL√2

+ i g† (3.3.36)

Z2 = −A†−L√

2− i d, Y 2 = −A−L√

2− i d† (3.3.37)

Z2 = −A−L√2

+ i d†, Y 2 =A†−L√

2− i d (3.3.38)

Using (σµ)αα = (12, ~σ) and (σµ)αα = (−12, ~σ) one finds that the generators of translations

57

and special conformal transformations can be written as follows4:

Pαβ = (σµPµ)αβ = −ZαZβ (3.3.39)

Kαβ = (σµKµ)αβ = −Y αY β (3.3.40)

We see that the operators Z and Y in the quasiconformal realization play similar rolesas covariant twistorial oscillators λ and η in the doubleton realization. However, theytransform nonlinearly under the Lorentz group and their commutations relations aregiven in Appendix A.4

The dilatation generator in terms of deformed twistorial oscillators takes the form:

∆ = i

4

(ZαY

α + Y αZα

)(3.3.41)

The Lorentz group generators Mµν in a spinorial basis can also be written as bilinears ofthese deformed twistorial oscillators:


α Mµν = 12

(ZαY

β − 12δ

βα ZγY

γ

)(3.3.42)

M αβ

= − i2 (σµσν)βαMµν = −12

(Y αZβ −

12δ

αβY γZγ

)(3.3.43)

We should stress the important point that even though the deformed twistorial oscillatorstransform nonlinearly under the Lorentz group, their bilinears Pαβ ,Kαβ ,M β

α and M αβ

transform covariantly and satisfy the commutation relations given in equations 3.3.22.

3.3.3 Deformations of the minimal unitary representation of SU(2, 2)

The minrep corresponds to a conformal massless scalar field but we know from theprevious section that there are also doubleton representations corresponding to masslessconformal fields with non-zero helicity h. These representations can be recovered in thequasiconformal formalism by introducing a one-parameter family of deformations of theminrep as was done in [23]. The continuous deformation parameter ζ is related to thehelicity h of the conformal massless fields as h = ζ

2 .The generators of the deformed minrep take the same form as given in section 3.3.2

with the simple replacement of the singular oscillators AL and A†L by “deformed” singularoscillators:

ALζ = a− Lζ√2x

A†Lζ= a† − Lζ√

2x(3.3.44)

4Note that in our conventions P 0 is positive definite.

58

whereLζ = L+ ζ = Nd −Ng + ζ − 1

2 (3.3.45)

Since ζ/2 labels the helicity we define “helicity deformed twistors” as follows:

Z1(ζ) =ALζ√

2− i g†, Y 1(ζ) = −

A†Lζ√2

+ i g (3.3.46)

Z1(ζ) =A†Lζ√

2+ i g, Y 1(ζ) =

ALζ√2

+ i g† (3.3.47)

Z2(ζ) = −A†−Lζ√

2− i d, Y 2(ζ) = −

A−Lζ√2− i d† (3.3.48)

Z2(ζ) = −A−Lζ√

2+ i d†, Y 2(ζ) =

A†−Lζ√2− i d (3.3.49)

The realization of the minimal unitary representation in terms of deformed twistors carryover to realization in terms of helicity deformed twistors:

Pαβ = (σµPµ)αβ = −Zα(ζ)Zβ(ζ) (3.3.50)

Kαβ = (σµKµ)αβ = −Y α(ζ)Y β(ζ) (3.3.51)

The dilatation generator then takes the form:

∆ = i

4

(Zα(ζ)Y α(ζ) + Y α(ζ)Zα(ζ)

)(3.3.52)

and the Lorentz generators Mµν also take the same form in terms of helicity deformedtwistors:


α Mµν = 12

(Zα(ζ)Y β(ζ)− 1

2δβα Zγ(ζ)Y γ(ζ)

)(3.3.53)

M αβ

= − i2 (σµσν)βαMµν = −12

(Y α(ζ)Zβ(ζ)− 1

2δαβY γ(ζ)Zγ(ζ)

)(3.3.54)

The realization of the generators of SU(2, 2) in terms of helicity deformed twistors describepositive energy unitary irreducible representations which can best be seen by going overto the compact three-grading reviewed in Appendix A.3.

Since the quasiconformal realization of the minrep and its deformations are nonlinearthe tensoring procedure in quasiconformal framework is a non-trivial and open problem.However since the representations with integer values of ζ are isomorphic to the doubletonrepresentations, the result of tensoring is already known and was discussed in section3.3.1.

59

3.3.4 Minimal unitary supermultiplet of SU(2, 2|4), its deformations and de-formed twistors

The construction of the minimal unitary representations of non-compact Lie algebras byquantization of their quasiconformal realizations extends to non-compact Lie superalgebras[22–24, 56]. In particular, the minimal unitary supermultiplets of SU(2, 2|N) and theirdeformations were studied in [23] using quasiconformal methods. In this section we shallreformulate the minimal unitary realization of 4d superconformal algebra SU(2, 2|4) andits deformations in terms of deformed twistorial oscillators 5.

The superconformal algebra su(2, 2|4) can be given a (non-compact) 5-graded decom-position with respect to the dilatation generator ∆:

su(2, 2|4) = N−1 ⊕N−1/2 ⊕N0 ⊕N+1/2 ⊕N+1 (3.3.55)

= Kαβ ⊕ S αI , SIα ⊕ (M β

α ⊕ M αβ⊕∆⊕RIJ)⊕QIα, QIα ⊕ Pαβ ,

(I, J = 1, 2, 3, 4)

where the grade zero subspace N0 consists of the Lorentz algebra so(3, 1) (M βα , M α

β),

the dilatations (∆) and R-symmetry su(4) (RIJ) generators, grade +1 and −1 subspacesconsist of translation (Pαβ) and special conformal generators (Kαβ) and the grade +1/2and -1/2 subspaces consist of Poincaré supersymmetry (QIα, QIα) and special conformalsupersymmetry generators (S α

I , SIα) respectively.The helicity deformed twistors for the superalgebra SU(2, 2|4) are obtained from the

deformed twistors of SU(2, 2) by replacing Lζ in the corresponding deformed singularoscillators ALζ with Lsζ :

Lζ −→ Lsζ = Nd −Ng +Nξ + ζ − 52 (3.3.56)

where Nξ = ξJξJ is the number operator of four fermionic oscillators ξI (ξJ ) (I, J = 1, 2, 3, 4)that satisfy

ξI , ξJ

= δJI (3.3.57)

The expressions for the generators of SU(2, 2) given in 3.3.2 get modified as follows ingoing over to SU(2, 2|4):

Pαβ = −Zsα(ζ)Zsβ(ζ) (3.3.58)

Kαβ = −Y sα(ζ)Y sβ(ζ) (3.3.59)5 PSU(2, 2|4) is the symmetry superalgebra of IIB supergravity compactified over AdS5×S5 ymmetry

[32].

60

∆ = i

4

(Zsα(ζ)Y sα(ζ) + Y sα(ζ)Zsα(ζ)

)(3.3.60)

M βα = 1

2

(Zsα(ζ)Y sβ(ζ)− 1

2δβα Zsγ(ζ)Y sγ(ζ)

)(3.3.61)

M αβ

= −12

(Y sα(ζ)Zs

β(ζ)− 1

2δαβY sγ(ζ)Zsγ(ζ)

)(3.3.62)

where the “supersymmetric” helicity deformed twistors are defined as:

Zs1(ζ) =ALs

ζ√2− i g†, Y s1(ζ) = −

A†Lsζ√2

+ i g

Zs1(ζ) =A†Ls

ζ√2

+ i g, Y s1(ζ) =ALs

ζ√2

+ i g†

Zs2(ζ) = −A†−Ls

ζ√2− i d, Y s2(ζ) = −

A−Lsζ√

2− i d†

Zs2(ζ) = −A−Ls

ζ√2

+ i d†, Y s2(ζ) =A†−Ls

ζ√2− i d

The supersymmetry generators of SU(2, 2|4) are given by the bilinears of deformedtwistorial oscillators and fermionic oscillators:

QIα = Zsα(ζ)ξI , QIα = −ξI Zsα(ζ) (3.3.63)

S αI = −ξIY sα(ζ), SIα = Y sα(ζ)ξI (3.3.64)

The generators RIJ of R-symmetry group SU(4) are given by:

RIJ = ξIξJ −14δ

IJξKξK (3.3.65)

They satisfy the following anti-commutation relations:QIα , QJβ

= δIJPαβ (3.3.66)

SIα , S βJ

= δIJK

αβ (3.3.67)

QIα , S

βJ

= −2δIJM β

α + 2δβαRIJ + δIJδβα (i∆ + C) (3.3.68)

SIα , QJβ

= 2δIJM α

β− 2δβαRIJ + δIJδ

βα (i∆− C) (3.3.69)

61

where C = ζ2 is the central charge.

The commutators of conformal group generators with supersymmetry generators areas follows:

[Pαβ , S

γI

]= 2δγαQIβ ,

[Kαβ , Q I

γ

]= −2δβγ SIα (3.3.70)[

Pαβ , SIγ]

= 2δγβQ Iα ,

[Kαβ , QIγ

]= −2δαγ S

βI (3.3.71)

[M βα , QIγ

]= δβγQ

Iα −

12δ

βαQ

Iγ ,

[M βα , S γ

I

]= −δαγ S

βI + 1

2δβαS

γI (3.3.72)[

M αβ, QIγ

]= −δαγ QIβ + 1

2δαβQIγ ,

[M α

β, SIγ

]= δγ

βSIα − 1

2δαβQIγ (3.3.73)

[∆ , QIα

]= i

2QIα,

[∆ , QIα

]= i

2 QIα (3.3.74)

[∆ , S αI ] = − i2S

αI ,

[∆ , SIα

]= − i2 S

Iα (3.3.75)

The su(4)R generators satisfy the following commutation relations:

[RIJ , R

KL

]= δKJ R

IL − δILRKJ (3.3.76)

They act on the R-symmetry indices I, J of the supersymmetry generators as follows:

[RIJ , Q

Kα

]= δKJ Q

Iα −

14δ

IJQ

Kα,

[RIJ , QKα

]= −δIKQJα + 1

4δIJQKα (3.3.77)[

RIJ , SαK

]= −δIKS α

J + 14δ

IJS

αK ,

[RIJ , S

Kα]

= δKJ SIα − 1

4δIJ S

Kα (3.3.78)

The minimal unitary representation of PSU(2, 2|4) is obtained when the deformationparameter ζ, which is also the central charge, vanishes. The resulting minimal unitarysupermultiplet of massless conformal fields in d = 4 is simply the N = 4 Yang-Millssupermultiplet [23]. For each value of the deformation parameter ζ one obtains anirreducible unitary representation of SU(2, 2|4). For integer values of the deformationparameter these unitary representations are isomorphic to doubleton supermultipletsstudied in [32,37,38].

The unitarity of the representations of SU(2, 2|N) may not be manifest in the Lorentzcovariant non-compact five grading. It is however manifestly unitary in compact threegrading with respect to the subsupergroup SU(2|N −M)× SU(2|M)× U(1) as was shownfor the the doubletons in [32, 37] and for the quasiconformal construction in [23]. TheLie algebra of SU(4) can be given a 3-graded structure with respect to the Lie algebraof its subgroup SU(2)× SU(2)× U(1). Similarly the Lie superalgebra SU(2, 2|4) can be

62

given a 3-graded decomposition with respect to its subalgebra SU(2|2)× SU(2|2)× U(1).This is the basis that was originally used by Gunaydin and Marcus [32] in constructingthe spectrum of IIB supergravity over AdS5 × S5 using twistorial oscillators. In thisbasis, choosing the Fock vacuum as the lowest weight vector leads to CPT-self-conjugatesupermultiplets and it is also the preferred basis in applications to integrable spin chains.The corresponding compact 3-grading of the quasiconformal realization of SU(2, 2|4) wasgiven in [23], to which we refer for details.

3.4 Higher spin (super-)algebras, Joseph ideals and their defor-mations

In this section we start by reviewing Eastwood’s results [130, 136] on defining HS(g)algebras as the quotient of universal enveloping algebra U(g) by its Joseph ideal J(g). Wewill then explicitly compute the Joseph ideal for SO(3, 2), SO(4, 2) and its deformations,using the Eastwood formula [136] and recast it in a Lorentz covariant form.

The universal enveloping algebra U(g), g = so(d− 1, 2) is defined as follows:

U(g) = G/I (3.4.1)

where G is the associative algebra freely generated by elements of g, and I is the ideal ofG generated by elements of form gh− hg − [g , h] (g, h ∈ g).

The enveloping algebra U(g) can be decomposed into standard adjoint action of gwhich by Poincare-Birkhoff-Witt theorem is equivalent to computing symmetric productsMAB ∼ . In particular, ⊗2

so(d− 1, 2) decomposes as:

⊗ = ⊕ ⊕ ⊕ • (3.4.2)

where • is the quadratic Casimir C2 ∼ M AB M B

A . It was already noted in [80] that thehigher spin algebra HS(g) must be a quotient of U(g) because the higher spin fields inAdSd are described by traceless two row Young tableaux. Thus the relevant ideal shouldquotient out all the diagrams except the first one in the above decomposition. This idealwas identified in [130] to be the Joseph ideal or the annihilator of the minimal unitaryrepresentation (scalar doubleton). The uniqueness of this quadratic ideal in U(g) wasproved in [136] and an explicit formula for the generator JABCD of the ideal was given as :

JABCD = MABMCD −MAB MCD −12 [MAB , MCD] + n− 4

4(n− 1)(n− 2) 〈MAB ,MCD〉1

63

= 12MAB ·MCD −MAB MCD + n− 4

4(n− 1)(n− 2) 〈MAB ,MCD〉1 (3.4.3)

where the dot · denotes the symmetric product

MAB ·MCD ≡MABMCD +MCDMAB (3.4.4)

of the generators and 〈MAB ,MCD〉 is the Killing form of SO(n−2, 2). ηAB is the SO(n−2, 2)invariant metric and the symbol denotes the Cartan product of two generators, whichfor SO(n− 2, 2), can be written in the form [137]:

MAB MCD = 13MABMCD + 1

3MDCMBA + 16MACMBD

−16MADMBC + 1

6MDBMCA −16MCBMDA

− 12(n− 2)

(MAEM

EC ηBD −MBEM

EC ηAD +MBEM

EDηAC −MAEM

EDηBC

)− 1

2(n− 2)(MCEM

EA ηBD −MCEM

EB ηAD +MDEM

EB ηAC −MDEM

EA δBC

)+ 1

(n− 1)(n− 2)MEFMEF (ηACηBD − ηBCηAD) (3.4.5)

The Killing term is given by

〈MAB ,MCD〉 = hMEFMGH(ηEGηFH − ηEHηFG)(ηACηBD − ηADηBC) (3.4.6)

where h = 2(n−2)n(4−n) is a c-number fixed by requiring that all possible contractions of JABCD

with the metric vanish. We shall refer to the operator JABCD as the generator of theJoseph ideal.

The generator of the Joseph ideal defined in equation 3.4.3 contains exactly theoperators that correspond to the “unwanted” diagrams described in equation 3.4.2 andquotienting the enveloping algebra by this ideal guarantees that the resulting algebra willcontain only two row traceless diagrams and thus correctly describe massless higher spinfields.

In the following sections we will compute the generator JABCD for d = 3 and 4 conformalalgebras SO(3, 2) and SO(4, 2) in various realizations discussed in previous sections. Weshall also decompose the generators of the Joseph ideal of SO(3, 2) and SO(4, 2) withrespect to the corresponding Lorentz groups SO(2, 1) and SO(3, 1), respectvely. TheLorentz covariant decomposition makes the massless nature of the minimal unitaryrepresentations explicit along with certain other identities that must be satisfied withinthe representation in order for it to be annihilated by the Joseph ideal. This will alsoallow us to define the annihilators of the deformations of the minrep of SO(4, 2) and

64

the corresponding deformations of the Joseph ideal. These deformations define a oneparameter family of AdS5/CFT4 HS algebras.

3.4.1 Joseph ideal for SO(3, 2) singletons

We will now use the twistorial oscillator realization for SO(3, 2) described in section 3.2.1.For Sp(4,R) = SO(3, 2), the generator JABCD of the Joseph ideal is

JABCD = MABMCD −MAB MCD −12 [MAB , MCD]− 1

40 〈MAB ,MCD〉

= 12MAB ·MCD −MAB MCD −

140 〈MAB ,MCD〉 (3.4.7)

Substituting the realization of Sp(4,R) = SO(3, 2) in terms of a twistorial Majorana spinorΨ one finds that the operator JABCD vanishes identically.

Considered as the the three dimensional conformal group the minreps of Sp(4,R) (Diand Rac) correspond to massless scalar and spinor fields which are known to be the onlymassless representations of the Poincaré group in three dimensions [44].

If instead of a twistorial Majorana spinor one considers a twistorial Dirac spinorcorresponding to taking two copies (colors) of the Majorana spinor one finds that thegenerators JABCD of the Joseph ideal do not vanish identically and hence they do notcorrespond to minimal unitary representations. The corresponding Fock space decomposesinto an infinite set of irreducible unitary representations of Sp(4,R), which correspond tothe massless fields in AdS4 [134]. Taking more than two colors in the realization of theLie algebra of Sp(4,R) as bilinears of oscillators leads to representations corresponding tomassive fields in AdS4 [35].

3.4.1.1 Joseph ideal of SO(3, 2) in Lorentz covariant basis

To get a more physical picture of what the vanishing of the Joseph ideal means we shallgo to the conformal 3-graded basis defined in equation 3.2.6. Evaluating the Joseph idealin this basis, we find that the vanishing ideal is equivalent to the linear combinations ofcertain quadratic identities, full set of which hold only in the singleton realization. Firstwe have the masslessness conditions:

P 2 = PµPµ = 0 , K2 = KµKµ = 0 (3.4.8)

The remaining set of quadratic relations that define the Joseph ideal are

6∆ ·∆ + 2Mµν ·Mµν + Pµ ·Kµ = 0 (3.4.9)

Pµ · (Mµν + ηµν∆) = 0 (3.4.10)

65

Kµ · (Mνµ + ηνµ∆) = 0 (3.4.11)

ηµνMµρ ·Mνσ − P(ρ ·Kσ) + ηρσ = 0 (3.4.12)

∆ ·Mµν + P[µ ·Kν] = 0 (3.4.13)

M[µν · Pρ] = 0 (3.4.14)

M[µν ·Kρ] = 0 (3.4.15)

The ideal generated by these relations is completely equivalent to equation 3.4.7 butit sheds light on the massless nature of these representations. The scalar and spinorsingleton modules for SO(3, 2) are the only minreps and there are no other deformations.This is a general phenomenon for all symplectic groups Sp(2n,R) (n > 2) and within thequasiconformal approach it can be explained by the fact that the corresponding quarticinvariant that enters the minimal unitary realization vanishes for symplectic groups.

The Casimir invariants for the singleton or the minrep of SO(3, 2) are as follows:

C2 = M AB M B

A = 52 (3.4.16)

C4 = M AB M B

C M CD M D

A = −358 (3.4.17)

Computing the products of the generators of the above singleton realization corresponding

to the Young tableaux and one finds that they vanish identically and the resulting

enveloping algebra contains only the operators whose Young tableaux have two rows.

3.4.2 Joseph ideal of SO(4, 2)

In this subsection we shall first evaluate the generator JABCD of the Joseph ideal ofSO(4, 2) in the covariant twistorial operator realization and then in the quasiconformalrealization of its minrep to highlight the essential differences.

3.4.2.1 Joseph ideal in the covariant twistorial oscillator or doubleton real-ization

We will use the doubleton realization [32,37] reviewed in section 3.3.1 to compute thegenerator JABCD of the Joseph ideal for SO(4, 2) :

JABCD = MABMCD −MAB MCD −12 [MAB , MCD]− 1

60 〈MAB ,MCD〉

= 12MAB ·MCD −MAB MCD −

160 〈MAB ,MCD〉 (3.4.18)

66

Substituting the expressions for the generators in the covariant twistorial realizationone finds that it does not vanish identically as an operator in contrast to the situationwith the singletonic realization of SO(3, 2). However one finds that JABCD has only 15independent non-vanishing components which turn out to be equal to one of the followingexpressions (up to an overall sign):

((a1b2 + a2b1)± (a†2b

†1 + a†1b

†2))Z (3.4.19)(

(a1b2 − a2b1)± (a†2b†1 − a

†1b†2))Z (3.4.20)(

(a1b1 + a2b2)± (a†2b†2 + a†1b

†1))Z (3.4.21)(

(a1b1 − a2b2)± (a†2b†2 − a

†1b†1))Z (3.4.22)(

a†1a2 + a1a†2 ± b

†1b2 ± b1b

†2

)Z (3.4.23)(

a†1a2 − a1a†2 ± b

†1b2 ∓ b1b

†2

)Z (3.4.24)(

(Na1 −Na2)± (Nb1 −Nb2))Z (3.4.25)

(Na +Nb + 2)Z (3.4.26)

Similarly, computing the products of the generators corresponding to the the Young

tableaux explicitly in the covariant twistorial realization one finds that they do not

vanish but we have the following relation:

= Z (3.4.27)

Thus all non-vanishing four-row diagrams can be dualized to two-row diagrams andthe resulting generators of the universal enveloping algebra in doubleton realization aredescribed by two-row diagrams.

The operator Z = (Na − Nb) commutes with all the generators of SU(2, 2) and itseigenvalues label the helicity of the corresponding massless representation of the conformalgroup [23,32,37]. All the components of the generator JABCD of Joseph ideal vanish onthe states that form the basis of an UIR of SU(2, 2) whose lowest weight vector is theFock vacuum |0〉 since

Z|0〉 = 0 (3.4.28)

The corresponding unitary representation describes a conformal scalar field in fourdimensions (zero helicity) and is the true minrep of SU(2, 2) annihilated by the Josephideal [23].

67

The Casimir invariants for SO(4, 2) in the doubleton representation are as follows:

C2 = M AB M B

A = 32(4−Z2) (3.4.29)

C ′3 = εABCDEFMABMCDMEF = 4ZC2 = 8C2

√1− C2

6 (3.4.30)

C4 = M AB M B

C M CD M D

A = C22

6 − 4C2 (3.4.31)

Thus we see that all the higher order Casimir invariants are functions of the quadraticCasimir C2 which itself is given in terms of Z = Na −Nb.

3.4.2.2 Joseph ideal and the quasiconformal realization of the minrep ofSO(4, 2)

To apply Eastwood’s formula to the generator of Joseph ideal in the quasiconformalrealization it is convenient to go from the conformal 3-graded basis to the SO(4, 2)covariant canonical basis where the generators MAB satisfy the following commutationrelations:


where the metric ηAB = diag(−,+,+,+,+,−) is used to raise and lower the indicesA,B = 0, 1, . . . , 5 etc. In addition to Lorentz generators Mµν (µ, ν, . . . = 0, 1, 2, 3) , wehave the following linear relations between the generators in the canonical basis and theconformal 3-graded basis

Mµ4 = 12 (Pµ −Kµ) (3.4.33)

Mµ5 = 12 (Pµ +Kµ) (3.4.34)

M45 = −∆ (3.4.35)

Substituting the expressions for the quasiconformal realization of the generators of SO(4, 2)given in subsection 3.3.2 into the generator of the Joseph ideal in the canonical basis:

JABCD = 12MAB ·MCD −MAB MCD −

160 〈MAB ,MCD〉 (3.4.36)

one finds that it vanishes identically as an operator showing that the correspondingunitary representation is indeed the minimal unitary representation.

We should stress the important point that the tensor product of the Fock spaces ofthe two oscillators d and g with the state space of the singular oscillator AL (A†L) form the

68

basis of a single UIR which is the minrep. In contrast, the Fock space of the covarianttwistorial oscillators reviewed in section 3.3.1 decomposes into infinitely many UIRs(doubletons) of which only the irreducible representation whose lowest weight vector isthe Fock vacuum, which is annihilated by JABCD, is the minimal unitary representationof SO(4, 2).

3.4.2.3 4d Lorentz Covariant Formulation of the Joseph ideal of SO(4, 2)

Above we showed that the generator of the Joseph ideal given in equation (3.4.36)vanishes identically as an operator for the quasiconformal realization of the minrep ofSO(4, 2) in the canonical basis. To get a more physical picture of what the vanishingof generator JABCD means we shall go to the conformal three grading defined by thedilatation generator ∆. Evaluating the generator of the Joseph ideal in this basis, wefind that the vanishing condition is equivalent to linear combinations of certain quadraticidentities, full set of which hold only in the quasiconformal realization of the minrep.First we have the conditions:

P 2 = PµPµ = 0 , K2 = KµKµ = 0 (3.4.37)

which hold also for the twistorial oscillator realization given in section 3.3.1. The remainingset of quadratic relations that define the Joseph ideal are

4∆ ·∆ +Mµν ·Mµν + Pµ ·Kµ = 0 (3.4.38)

Pµ · (Mµν + ηµν∆) = 0 (3.4.39)

Kµ · (Mνµ + ηνµ∆) = 0 (3.4.40)

ηµνMµρ ·Mνσ − P(ρ ·Kσ) + 2ηρσ = 0 (3.4.41)

Mµν ·Mρσ +Mµσ ·Mνρ +Mµρ ·Mσν = 0 (3.4.42)

∆ ·Mµν + P[µ ·Kν] = 0 (3.4.43)

M[µν · Pρ] = 0, M[µν ·Kρ] = 0 (3.4.44)

In four dimensions, using the Levi-Civita tensor one can define the Pauli-Lubanski vector,Wµ and its conformal analogue, V µ as follows:

Wµ = 12εµνρσPνMρσ, V µ = 1

2εµνρσKνMρσ (3.4.45)

69

where ε0123 = +1, ε0123 = −1 and the indices are raised and lowered by the Minkowskimetric. For massless fields, Wµ and V µ are proportional to Pµ and Kµ respectively withthe proportionality constant related to helicity of the fields [138, 139]. Equations (3.4.44)imply that for the minrep both the Wµ and V µ vanish implying that it describes a zerohelicity (scalar) massless field 6.

Computing the products of the generators in the above quasiconformal realization

corresponding to Young tableaux and explicitly one finds that they vanish

identically and the resulting enveloping algebra contains only operators with two rowYoung tableaux.

The Casimir operators of the minrep of SO(4, 2) are as take on the following values(using the definitions given in equations (3.4.29) - (3.4.31)):

C2 = 6, C3 = 0 C4 = −18. (3.4.46)

3.4.3 Deformations of the minrep of SO(4, 2) and their associated ideals

As was shown in reference [23], the minimal unitary representation of SU(2, 2) thatcorresponds to a conformal scalar field admits a one-parameter (ζ) family of deformationscorresponding to massless conformal fields of helicity ζ

2 in four dimensions, which can becontinuous. For non-integer values of the deformation parameter ζ they correspond, ingeneral, to unitary representations of an infinite covering of the conformal group7.

The generators of the deformed minreps were reformulated in terms of deformedtwistorial oscillators in section 3.3.3. Substituting the expressions for the generators of thedeformed minreps of SO(4, 2) into the generator JABCD of the Joseph ideal one finds that itdoes not vanish for non-zero values of the deformation parameter ζ. One might thereforeask if there exists deformations of the Joseph ideal that annihilate the deformed minimalunitary representations labelled by the deformation parameter ζ. Remarkably, this isindeed the case. The quadratic identities that define the Joseph ideal in the conformalbasis discussed in the previous section go over to identities involving the deformationparameter ζ and define the deformations of the Joseph ideal. One finds that the helicityconditions are modified as follows:

12εµνρσMνρ · Pσ = ζPµ (3.4.47)

6 We should note that a similar set of identities (constraints) were discussed in [139] in the context ofderiving field equations for particles of all spins (acting on field strengths) where they arise as conformallycovariant forms of massless particles.

7Recently, such a continuous helicity parameter was introduced as a spectral parameter for scatteringamplitudes in N=4 super Yang-Mills theory in [140].

70

12εµνρσMνρ ·Kσ = −ζKµ (3.4.48)

The identities (3.4.41), (3.4.42) and (3.4.43) get also modified as follows:

ηµνMµρ ·Mνσ − P(ρ ·Kσ) + 2ηρσ = ζ2

2 ηρσ (3.4.49)

Mµν ·Mρσ +Mµσ ·Mνρ +Mµρ ·Mσν = ζεµνρσ∆ (3.4.50)

∆ ·Mµν + P[µ ·Kν] = −ζ2 εµνρσMρσ (3.4.51)

The other quadratic identities remain unchanged in going over to the deformed minimalunitary representations.

The Casimir invariants for the deformations of the minrep of SO(4, 2) depend only thedeformation parameter ζ and are given as follows (using the definitions given in equations3.4.29 - 3.4.31):

C2 = 6− 3ζ2

2 (3.4.52)

C ′3 = 6ζ(ζ2 − 4

)= −8C2

√1− C2

6 (3.4.53)

C4 = 38(ζ4 + 8ζ2 − 48

)= C2

26 − 4C2 (3.4.54)

The quartic Casimir in [141] is defined using the epison tensor as follows:

Cepsilon4 = εABCDEFMCDMEF εABC′D′E′F ′MC′D′ME′F ′

= −24ζ2 (ζ2 − 4)

(3.4.55)

which is just a linear combination of C4 and C4 defined above.

The products of generators corresponding to the Young tableaux do not vanish for

the deformed minimal unitary representations and depend on the deformation parameterζ as follows:

= ζ (3.4.56)

Hence all non-vanishing four-row diagrams can be dualized to two-row diagrams andthe resulting generators of the universal enveloping algebra in deformed realizationare described by two-row diagrams. For ζ = 0 all four row diagrams vanish and oneobtains the standard high spin algebra of Vasiliev type whose generators transform inrepresentations of the underlying AdS group corresponding to Young tableaux containing

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only two row traceless diagrams. For non-zero ζ the corresponding enveloping algebrasdescribe deformed higher spin algebras as discussed in the next subsection.

We saw earlier in section 3.4.2.1 that the generator JABCD of the Joseph ideal didnot vanish identically as an operator for the covariant twistorial oscillator realization ofSO(4, 2). It annihilates only the states belonging to the subspace that form the basis ofthe true minrep of SO(4, 2). By going to the conformal three grading, one finds that thegenerator JABCD of the Joseph ideal can be written in a form similar to the deformedquadratic identities above with the deformation parameter replaced by the linear Casimiroperator Z = Na −Nb

ηµνMµρ ·Mνσ − P(ρ ·Kσ) + 2ηρσ = Z2

2 ηρσ (3.4.57)

Mµν ·Mρσ +Mµσ ·Mνρ +Mµρ ·Mσν = −Zεµνρσ∆ (3.4.58)

∆ ·Mµν + P[µ ·Kν] = Z2 εµνρσM

ρσ (3.4.59)12εµνρσMνρ · Pσ = −ZPµ (3.4.60)

12εµνρσMνρ ·Kσ = ZKµ (3.4.61)

The Fock space of the oscillators decompose into an infinite set of unitary irreduciblerepresentations of SU(2, 2) corresponding to massless conformal fields of all integer andhalf-integer helicities labelled by the eigenvalues of Z/2.

3.4.4 Higher spin algebras and superalgebras and their deformations

We shall adopt the definition of the higher spin AdS(d+1)/CFTd algebra as the quotient ofthe enveloping algebra U(SO(d, 2)) of SO(d, 2) by the Joseph ideal J(SO(d, 2)) and denoteit as HS(d, 2) [130]:

HS(d, 2) = U(SO(d, 2))J(SO(d, 2)) (3.4.62)

We shall however extend it to define deformed higher spin algebras as the envelopingalgebras of the deformations of the minreps of the corresponding AdSd+1/Confd algebras.For these deformed higher spin algebras the corresponding deformations of the Joseph idealvanish identically as operators in the quasiconformal realization as we showed explicitlyabove for the conformal group in four dimensions. We expect a given deformed higherspin algebra to be the unique infinite dimensional quotient of the universal envelopingalgebra of an appropriate covering8 of the conformal group by the deformed ideal aswas shown for the undeformed minrep in [142]. Similarly, we define the higher spin

8In d = 4 deformed minreps describe massless fields with the helicity ζ2 . For non-integer values of ζ

one has to go to an infinite covering of the 4d conformal group.

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superalgebras and their deformations as the enveloping algebras of the minimal unitaryrealizations of the underlying superalgebras and their deformations, respectively9.

In four dimensions (d = 4) we have a one parameter family of higher spin algebraslabelled by the helicity ζ/2:

HS(4, 2; ζ) = U(SO(4, 2))Jζ(SO(4, 2)) (3.4.63)

where Jζ(SO(4, 2)) denotes the deformed Joseph ideal of SO(4, 2) defined in section 3.4.310

On the AdSd+1 side the generators of the higher spin algebras correspond to higher spingauge fields, while on the Confd side they are related to conserved tensors including theconserved stress-energy tensor. The charges associated with the generators of conformalalgebra SO(d, 2) are defined by conserved currents constructed by contracting the stress-energy tensor with conformal Killing vectors. Similarly, the higher conserved currentsare obtained by contracting the conformal Killing tensors with the stress-energy tensor.These higher conformal Killing tensors are obtained simply by tensoring the conformalKilling vectors with themselves. Though implicit in previous work on the subject [87,145],explicit use of the language of conformal Killing tensors in describing higher spin algebrasseems to have first appeared in the paper of Mikhailov [91]11. This connection wasput on a rigorous foundation by Eastwood in his study of the higher symmetries of theLaplacian [130] who showed that the undeformed higher spin algebra can be obtainedas the quotient of the enveloping algebra of SO(d, 2) generated by the conformal Killingtensors quotiented by the Joseph ideal.

The connection between the doubleton realization of SO(4, 2) in terms of covarianttwistorial oscillators and the corresponding conformal Killing tensors was also studied byMikhailov [91]. He pointed out that the higher conformal Killing tensors correspond tothe products of bilinears of oscillators that generate SO(4, 2) in the doubleton realization

9We should note that the universal enveloping algebra of a Lie group as defined in the mathematicsliterature is an associative algebra with unit element. Under the commutator product inherited from theunderlying Lie algebra it becomes a Lie algebra.

10After the main results of this chapter was announced at the GGI Conference on higher spin theoriesin May 2013, we became aware of the work of [143] where possible deformations of purely bosonic higherspin algebras in arbitrary dimensions were studied and it was shown that the deformations can dependat most on one parameter. In a subsequent work [144] it was shown that the AdSd higher spin algebrais unique in d = 4 and d > 6 under their assumptions on the spectrum of generators. They also implythat the one parameter family of deformations of [143] must be the same as the one parameter familydiscussed in this chapter which is based on earlier work [23] that they cite. The results of [143] are basedon Young tableaux analysis of gauge fields in AdS5. Whether and how their deformation parameter isrelated to helicity in 4d is not known at this point.

11The generators of the higher spin algebra shsE(8|4) of reference [146] in terms of Killing spinors inAdS4 were obtained in the work of [147] on higher spin N = 8 supergravity in d = 4.

73

of [32, 37, 38], which are elements of the enveloping algebra. The undeformed higher spinalgebra HS(4, 2; ζ = 0) was also studied by the authors of [89] who used the doubletonicconstruction as well12. The undeformed higher spin algebra corresponding to 3.4.63describes fields of all spins which are multiplicity-free. The model studied in detail in [89]is based on the minimal infinite dimensional subalgebra of HS(4, 2; ζ = 0) that describesonly even spins. For the purely bosonic higher spin algebras in AdSd the factorization ofthe ideal generated by the bilinear operators corresponding to the last three irreps in3.4.2 was discussed in great detail, at the level of abstract enveloping algebra, in [148]without reference to explicit oscillator realization. They corroborated that in d = 5 thereis an equivalence between factoring out this ideal and the oscillator construction in [89].

The supersymmetric extension of the higher spin algebras HS(4, 2; ζ) is given bythe enveloping algebra of the deformed minimal unitary realization of the N-extendedconformal superalgebras SU(2, 2|N)ζ with the even subalgebras SU(2, 2)⊕U(N). We shalldenote the resulting higher spin algebra as HS[SU(2, 2|N); ζ]. These supersymmetricextensions involve odd powers of the deformed twistorial oscillators and the identitiesthat define the Joseph ideal get extended to a supermultiplet of identities obtained by therepeated actions of Q and S supersymmetry generators on the generators of Joseph ideal.On the conformal side the odd generators correspond to the products of the conformalKilling spinors with conformal Killing vectors and tensors. If we denote the resultingdeformed super-ideal as Jζ [SU(2, 2|N)] we can formally write

HS[SU(2, 2|N); ζ] = U(SU(2, 2|N)Jζ [SU(2, 2|N)] (3.4.64)

As discussed in previous sections, the bosonic higher spin gauge fields are describedby two-row Young tableaux of SO(4, 2). However in order to extend the SO(4, 2) Youngtableaux to super Young tableaux , one needs to represent the corresponding represen-tations in terms of SU(2, 2) Young tableaux since the relevant superconformal algebrasare SU(2, 2|N) with the even subalgebra SU(2, 2) ⊕ U(N) and not superalgebras of theorthosymplectic type. The identifications of Young tableaux can be easily checked by cal-culating the dimensions of corresponding irreps. In going from SU(2, 2) to SU(2, 2|N) onesimply replaces the Young tableaux of SU(2, 2) by super Young tableaux of SU(2, 2|N)13.The Young diagram of the adjoint representation of SO(4, 2) = SU(2, 2) goes over tothe following super tableau of SU(2, 2|N) which involves one dotted and one undotted

12 We should note that the infinite dimensional ideal that appears in the work of [89] is not the Josephideal.

13For a review of super Young tableaux we refer to [149] and a study of the relations between Superyoung tableaux and Kac-Dynkin labelling see [150].

74

“superboxes” :SO(4,2)︷︸︸︷

⇐⇒

SU(2,2)︷︸︸︷supersymmetrize−−−−−−−−−−→

SU(2,2|N)︷︸︸︷• (3.4.65)

Notice that the of SU(2, 2) is represented as • . This is because supersymmetric

extensions of SU(2, 2) do not have an invariant super Levi-Civita tensor. The adjointrepresentation • os SU(2, 2|N) can be decomposed with respect to SU(2, 2)× U(N)

as follows:

• = ( • , 1)⊕ ( • , )⊕ ( , • )⊕ (1, • )⊕ (1, 1) (3.4.66)

The window diagram appearing in the tensor product of two adjoint representationscan be supersymmetrized as follows:

SO(4,2)︷︸︸︷⇐⇒

SU(2,2)︷︸︸︷supersymmetrize−−−−−−−−−−→

SU(2,2|N)︷︸︸︷•• (3.4.67)

Thus the higher spin gauge fields are described by the following SU(2, 2) diagram andcan be consequently supersymmetrized in a straightforward manner:

· · ·

two-row diagram︷︸︸︷⇐⇒

SU(2,2)︷︸︸︷· · ·

· · ·supersymmetrize−−−−−−−−−−→

SU(2,2|N)︷︸︸︷•• · · · • · · ·

(3.4.68)Thus the generators of the supersymmetric extension of the undeformed (ζ = 0)

bosonic higher spin algebra decompose as

HS[SU(2, 2|N); 0] =∑⊕

SU(2,2|N)︷︸︸︷•• · · · • · · · (3.4.69)

For the decomposition of the deformed higher spin algebras for integer values of thedeformation parameter one can also use the super tableaux to represent their generators.However for non-integer values of the deformation parameter it may be necessary touse Kac-Dynkin or other labellings. We should stress the important point that themaximal finite dimensional subalgebra of HS[SU(2, 2|N); ζ] is the superconformal algebraSU(2, 2|N)ζ with ζ labelling the central charge. The decomposition of the generators of

75

the higher spin superalgebra with respect to the SU(2, 2|N)ζ subalgebra is multiplicityfree.

Furthermore, the minimal unitary realization of the supersymmetric extensionsSU(2, 2|N) of SU(2, 2) the enveloping algebra of SU(2, 2|N) has as subalgebras the envelop-ing algebras of different unitary representations of SU(2, 2) that form the minimal unitarysupermultiplet modded out by the corresponding deformed Joseph ideals. These differentirreps of SU(2, 2) are deformations of the minrep where the deformation is driven by thefermionic oscillators and the deformation parameter ζ is given by the eigenvalue of thenumber operator of these fermionic oscillators Nξ in the minimal unitary supermultiplet.In the deformations of the minimal unitary supermultiplet labelled by ζ the envelopingalgebra of SU(2, 2|N) has a similar decomposition in which the irreps making of thesupermultiplet are labelled by a linear combination of ζ and the eigenvalue of Nξ.

We should note that the undeformed super algebra HS[SU(2, 2|4); ζ = 0] for theparticular value of N = 4 was studied in [151] by using the doubletonic realization. As inthe purely bosonic case the higher spin theory studied in [151] is based on a subalgebraof HS[SU(2, 2|4); ζ = 0] and the resulting linearized gauging was shown to correspond tomassless PSU(2, 2|4) multiplets whose maximal spins are even integers 2, 4, ....14

The situation is much simpler for AdS4/Conf3 higher spin algebras. There are onlytwo minreps of SO(3, 2), namely the scalar and spinor singletons. The higher spinalgebra HS(3, 2) is simply given by the enveloping algebra of the singletonic realization ofSp(4,R) [40, 146]. Singletonic realization of the Lie algebra of Sp(4,R) describes both thescalar and spinor singletons. They form a single irreducible supermultiplet of OSp(1|4,R)generated by taking the twistorial oscillators as the odd generators [40]. The oddgenerators correspond to conformal Killing spinors , which together with conformal Killingvectors in d = 3 , generate the Lie super-algebra of OSp(1|4,R). Its enveloping algebra leadsto the higher spin super-algebra of Fradkin-Vasiliev type involving all integer and halfinteger spin fields in AdS4. One can also construct N-extended higher spin superalgebrasin AdS4 as enveloping algebras of the singletonic realization of OSp(N |4,R) [40, 146].

14 We should stress again that the infinite dimensional ideal that appears in the work of [151] is notthe Joseph ideal.

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Chapter 4 |AdS7/CFT6 Higher spin algebrasin d = 6: hs(6, 2; t)

4.1 Introduction

The conformal group in six dimensions is SO∗(8) ∼ SO(6, 2) and its minrep describes aconformal massless scalar field in six dimensions. The “discrete” deformations of theminrep describe massless conformal fields that are symmetric tensors in the spinorialrepresentation of the six dimensional Lorentz group, SO(5, 1) ∼ Sl(2,H. The minrep andits deformations for SO(6, 2) were first studied in [24,56] using the quasiconformal methods.Following the approach of the previous chapter, we shall reformulate the minrep anddeformations thereof in terms of deformed twistorial oscillators such that the generatorsare bilinears of twistors that transform nonlinearly under the six dimensional Lorentzgroup. Subsequently we show that the Joseph ideal vanishes explicitly for the minrep andwe can define a discrete one-parameter family of ideals. The universal enveloping algebraof SO(6, 2) in evaluated in the minrep is then the AdS7 higher spin algebra hs(6, 2) andthe corresponding enveloping algebra evaluated in the deformations of the minrep is theone-parameter discrete family of deformations, hs(6, 2; t) of the AdS7 higher spin algebra.

At this time we should explain the use of the term “deformations” to label a discretefamily of higher spin algebras. As we saw in the previous chapter, there exists an infinitefamily of inequivalent higher spin algebras in four dimensions which are labeled by thecontinuous helicity of conformal massless fields in four dimensions. It was established thatthis helicity is nothing but the eigenvalue of the little group U(1) of massless particlesin four dimensions which is Abelian and takes on continuous values. However, the littlegroup in six dimensions is SO(4) = SU(2)T × SU(2)A whose representations are labeled byeigenvalues (jT , jA) which are discrete. It turns out that for unitary conformal masslessrepresentations, the representations are of the form (jT , 0) or (0, jA). This means that the

77

conformally massless representations in six dimensions are labeled by a discrete parameterin contrast with the continuous helicity labels for the analogous conformally masslessrepresentations in four dimensions. Hence we refer to the higher spin algebras basedon these representations as discrete deformations of hs(6, 2). It should be noted thatthese discrete deformations admit supersymmetric extensions for arbitrary number ofsupersymmetry generators. These results were published in [152] which we follow closelyin our review in this section.

The plan of the chapter is as follows: We start by reviewing the covariant twistorial(doubleton) construction in section 4.2.1 following [33, 39, 153] and its reformulationin terms of Lorentz covariant twistorial oscillators [132, 153]. Next in sections 4.2.2-4.2.4 we present a novel reformulation of the quasiconformal realization of the minimalunitary representation (minrep) of SO(6, 2) and its supersymmetric extensions and theirdeformations [24,56] in terms of deformed twistors that transform nonlinearly under the6d Lorentz group. In section 4.3 we review the Eastwood’s formula for the generators J

of the annihilator of the minrep (Joseph ideal) and show by explicit calculations thatit vanishes identically as an operator for the quasiconformal realization of the minrep.Then in section 4.3.2 we use the 6d Lorentz covariant formulation of the Joseph idealto identify the deformed generators Jt that are the annihilators of the deformations ofthe minrep. These discrete deformations are labelled by the eigenvalues of an SU(2)Gsymmetry realized as bilinears of fermionic oscillators. Interestingly, the 6d analog ofthe Pauli-Lubansky vector does not vanish for the deformed minreps and becomes an(anti-)self-dual operator of rank three. Next we compare the generators of the Josephideal computed for doubleton realization and identify the analog of the deformationSU(2)G in the quasiconformal realization. For the doubleton realization the generators ofthe Joseph ideal do not vanish as operators. They annihilate only the subspace of theFock space of the covariant oscillators corresponding to the minrep. In section 4.3.4 wedefine the AdS7/CFT6 higher spin algebra and its deformations as the enveloping algebraof the minrep and its deformations in the quasiconformal framework, respectively. Wealso extend these results to corresponding higher spin superalgebras.

78

4.2 Realizations of the 6d conformal algebra SO(6, 2) ∼ SO∗(8)and its supersymmetric extension OSp(8∗|4)

4.2.1 Covariant twistorial oscillator construction of the massless representa-tions (doubletons) of 6d conformal group SO(6,2)

In this subsection we shall review the construction of positive energy unitary repre-sentations of SO(6, 2) that correspond to massless conformal fields in six dimensionsfollowing [33,39,153]. The Lie algebra of the conformal group SO(6, 2) in six dimensionsis isomorphic to that of SO∗(8) with the maximal compact subgroup U(4). Commutationrelations of the generators MAB of SO(6, 2) in the canonical basis have the form


where ηAB = diag(−,+,+,+,+,+,+,−) and A,B = 0, . . . , 7. Chiral spinor representationof SO(6, 2) can be written in terms in six-dimensional gamma matrices Γµ that satisfy

Γµ , Γν = −2ηµν

where ηµν = diag(−,+,+,+,+,+) and µ, ν = 0, . . . , 5 as follows1:

Σµν := − i4 [Γµ , Γν ] , Σµ6 := 12ΓµΓ7, Σµ7 := −1

2Γµ, Σ67 := −12γ7 (4.2.2)

We adopt the conventions of [153] for gamma matrices:

Γ0 = σ3 ⊗ 12 ⊗ 12

Γ1 = iσ1 ⊗ σ2 ⊗ 12

Γ2 = iσ1 ⊗ σ1 ⊗ σ2

Γ3 = iσ1 ⊗ σ3 ⊗ σ2

Γ4 = iσ2 ⊗ 12 ⊗ σ2

Γ5 = iσ2 ⊗ σ2 ⊗ σ1

Γ7 = −Γ0Γ1Γ2Γ3Γ4Γ5 (4.2.3)

Consider the bosonic oscillators ci, dj and their hermitian conjugates ci, dj respectively(i, j = 1, 2, 3, 4) that satisfy

[ci , c

j]

= δji ,[di , d

j]

= δji (4.2.4)1Opposite chirality spinor representation is obtained by taking Σµ7 := + 1

2 Γµ and Σ67 = + 12γ7.

79

and form a twistorial spinor operator Ψ and its Dirac conjugate Ψ = Ψ†Γ0 as :

Ψ =(ci

di

), Ψ =

(ci, −di

)(4.2.5)

Then the bilinears MAB = ΨΣABΨ provide a realization of the Lie algebra of SO(6, 2):

[ΨΣABΨ , ΨΣCDΨ

]= Ψ [ΣAB , ΣCD] Ψ (4.2.6)

and the Fock space of the oscillators decompose into an infinite set of positive energyunitary irreducible representations of SO(6, 2) corresponding to massless conformal fieldsin six dimensions. The resulting representations for one pair (color) of oscillators werecalled doubletons of SO(6, 2) in [33,39,153].

The Lie algebra of the conformal group SO(6, 2) has a three-graded decomposition(referred to as compact three-grading) with respect to its maximal compact subalgebraL0 = SU(4)× U(1)E,

SO(6, 2) = L− ⊕ L0 ⊕ L+, (4.2.7)

where the three-grading is determined by the conformal Hamiltonian E = 12 (P0 +K0). To

construct positive energy unitary representations of SO∗(8), one realizes the generatorsas following billnears:

Aij = cidj − cjdi ∈ L−, Aij = cidj − djdi ∈ L+

M ij = cicj + djd

i ∈ L0 (4.2.8)

where i, j = 1, 2, 3, 4.M i

j generate the maximal compact subgroup U(4). The conformal Hamiltonian isgiven by the trace M i

i

QB = 12M

ii = 1

2 (NB + 4) , (4.2.9)

where NB ≡ cici + didi is the bosonic number operator. We shall denote the eigenvaluesof QB as E. The hermitian linear combinations of Aij and Aij are the non-compactgenerators of SO(6, 2) [33, 39,153]. Each lowest weight (positive energy) UIR is uniquelydetermined by a set of states transforming in the lowest energy irreducible representation|Ω〉 of SU(4)×U(1)E that are annihilated by all the elements of L− [33,39].2 The possiblelowest weight vectors for one pair of oscillators (doubletons) in this compact basis have

2 Equivalently, the lowest weight vector of the lowest energy irreducible representation of SU(4)determines the UIR. Hence, by an abuse of terminology, we shall use interchangeably the terms “lowestweight vector” and “lowest energy irreducible representation”.

80

SU(4) Young tableaux with one row [39], They are of the form

|0〉 ,

ci1 |0〉 =∣∣ ⟩

,

c(i1ci2) |0〉 =∣∣ ⟩

,

...

c(i1ci2 . . . cin) |0〉 = | · · ·

n︷︸︸︷〉, (4.2.10)

plus those obtained by interchanging c-type oscillators with d-type oscillators and thestate

c(idj) |0〉 =∣∣ ⟩

. (4.2.11)

The positive energy UIR’s of SO∗(8) can be identified with conformal fields in sixdimensions transforming covariantly under the six-dimensional Lorentz group with definiteconformal dimension. The Lorentz covariant spinorial oscillators are obtained from theSU(4) covariant oscillators by the action of an intertwining operator T = e

π2M06 . We will

use Greek letters for the Lorentz group SO(5, 1) ∼ SU∗(4) spinorial indices – α, β = 1, 2, 3, 4.Withour convention of gamma matrices, the Lorentz covariant spinorial oscillators λ i

α , ηαj

are given by:

λ 1α =

c3 + d1

−c4 + d2

−c1 + d3

c2 + d4

, λ 2α =

−d3 + c1

d4 + c2

d1 + c3

−d2 + c4

(4.2.12)

ηα1 =

−c1 − d3

−c2 + d4

−c3 + d1

−c4 − d2

, ηα2 =

d1 − c3d2 + c4

d3 + c1

d4 − c2

(4.2.13)

They satisfy the following commutation relations:[ηαi , λ j

β

]= −2δαβ εij (4.2.14)

where i, j = 1, 2 (ε12 = ε21 = +1) and α, β = 1, 2, 3, 4. One finds that [132]

(ΣµPµ)αβ = Pαβ = λ iα λ

jβ εij ,

(ΣµKµ

)αβ = Kαβ = −ηαiηβjεij (4.2.15)

81

where Σ-matrices in d = 6 are the analogs of Pauli matrices σµ in d = 4. The explicit formof Σµ, Σµ is given in Appendix B.1. Note that the form of λ i

α , ηαj is slightly different

from [132,153] as we are in the mostly positive signature and also the form of intertwiningoperator is slightly different.

Similarly we can define Lorentz generators with spinor indices as follows:

M βα = − i2

(ΣµΣν

) β

αMµν (4.2.16)

In terms of spinors λ iα , η

αj, they are given as follows:

M βα = −1

2

(λ iα η

βj − 14δ

βαλ

iγ η

γj

)εij (4.2.17)

The dilatation generator is given by:

∆ = i

8(ηαiλ j

α − λ iα η

αj)εij (4.2.18)

The conformal algebra in terms of these generators is as follows:

[M βα , M δ

γ

]= δβγM

δα − δδαM β

γ (4.2.19)

[Pαβ , M

δγ

]= 2δδ[αPβ]γ + 1

2δδγPαβ ,

[Kαβ , M δ

γ

]= −2δ[α

γ Kβ]δ − 1

2δδγK

αβ (4.2.20)

[Pαβ , K

γδ]

= 16(δ

[γ[αM

δ]β] −

i

2δ[γ[αδ

δ]β] ∆

)(4.2.21)

[∆ , Pαβ ] = iPαβ ,[∆ , Kαβ

]= −iKαβ ,

[∆ , M β

α

]= 0 (4.2.22)

The doubletons correspond to massless conformal fields in six dimensions that transformas symmetric tensors Ψαβγ... ≡ Ψ(αβγ...) in their spinor indices

4.2.2 Quasiconformal approach to minimal unitary representation of SO(6, 2)

The construction of the minimal unitary representation of the 6d conformal group SO(6, 2)by quantization of its quasiconformal action and its deformations were given in [24,56].In this section we will reformulate the generators of these representations in terms ofdeformed twistorial oscillators as was done for 4d conformal group SU(2, 2) in 3

The group SO(6, 2) ∼ SO∗(8) can be realized as a quasiconformal group that leaveslight-like separations with respect to a quartic distance function in nine dimensions

82

invariant. The quantization of this geometric action leads to a nonlinear realization ofthe generators of SO(6, 2) in terms of a singlet coordinate x, its conjugate momentum p

and four bosonic oscillators am, am and bm, bm, (m.n = 1, 2) satisfying [24]:

[x , p] = i, [am , an] = δnm, [bm , bn] = δnm (4.2.23)

The semisimple component of the little group of massless particles in six dimensionsis SO(4) which can be written as SU(2)S × SU(2)A. Its normalizer inside SO(6, 2) isSO(2, 2) which also decomposes as SU(1, 1)K × SU(1, 1)N . The generators of SU(2)S andof SU(2)A subgroups of SO∗(8) are realized as bilinears of a and b type oscillators withinthe quasiconformal approach as follows:

S+ = ambm, S− = bmam, S0 = 12 (Na −Nb) (4.2.24)

A+ = a1a2 + b1b2, A− = a1a2 + b1b

2, A0 = 12(a1a1 − a2a2 + b1b1 − b2b2

)(4.2.25)

where Na = amam and Nb = bmbm are the respective number operators. They satisfy

[S0 , S±] = ±S±, [S+ , S−] = 2S0 (4.2.26)

[A0 , A±] = ±A±, [A+ , A−] = 2A0 (4.2.27)

In the previous section we followed conventions given in [153] for the 6d sigma matrices(Σµ, Σµ) within the covariant twistorial construction of doubletons. In order to makecontact with the spinor-helicity formalism used in the amplitudes literature, we will followthe Clifford algebra conventions of [154], summarized in Appendix B.2, for the minimalunitary representation and its deformations within the quasiconformal approach. Toavoid confusion, we will denote these 6d sigma matrices as σµ, ¯σµ (µ, ν = 0, 1, . . . 5)

To express the momentum generators of SO(6, 2) of the minrep we shall introduce twosets of deformed twistors Z i

α and Z iα (α, β = 1, 2, 3, 4, i, j = 1, 2) that transform nonlinearly

under the Lorentz group (their commutation relations are given in Appendix B.4):

Z 11 = b1 −

12 (x− ip) + 1

x

(S0 + 3

4

), Z 2

1 = a1 −S−x

(4.2.28)

Z 12 = b2 −

S+

x, Z 2

2 = a2 −12 (x− ip)− 1

x

(S0 −

34

)(4.2.29)

Z 13 = −a2 + 1

2 (x+ ip) + 1x

(S0 + 3

4

), Z 2

3 = b2 − S−x

(4.2.30)

Z 14 = a1 − S+

x, Z 2

4 = −b1 + 12 (x+ ip)− 1

x

(S0 −

34

)(4.2.31)

83

Z 11 = b1 −

12 (x− ip) + 1

x

(S0 −

34

), Z 2

1 = a1 −S−x

(4.2.32)

Z 12 = b2 −

S+

xZ 2

2 , = a2 −12 (x− ip)− 1

x

(S0 + 3

4

)(4.2.33)

Z 13 = −a2 + 1

2 (x+ ip) + 1x

(S0 −

34

), Z 2

3 = b2 − S−x

(4.2.34)

Z 14 = a1 − S+

x, Z 2

4 = −b1 + 12 (x+ ip)− 1

x

(S0 + 3

4

)(4.2.35)

In terms of these deformed twistors the momentum generators can then be written asbilinears:

Pαβ = Z iα Z

jβ εij (4.2.36)

We should stress the fact that even though the above deformed twistors transformnonlinearly under the Lorentz group the bilinears Pαβ transform covariantly as anti-symmetric tensors in spinorial indices.

In order to realize the special conformal generators, we need another set of deformedtwistors Y αi and Y αi (their commutation relations are given in Appendix B.4):

Y 11 = a1 + S+

x, Y 12 = −b1 − 1

2 (x+ ip) + 1x

(S0 −

34

)(4.2.37)

Y 21 = a2 + 12 (x+ ip) + 1

x

(S0 + 3

4

), Y 22 = −b2 − S−

x(4.2.38)

Y 31 = −b2 −S+

x, Y 32 = −a2 −

12 (x− ip)− 1

x

(S0 −

34

)(4.2.39)

Y 41 = b1 + 12 (x− ip)− 1

x

(S0 + 3

4

), Y 42 = a1 + S−

x(4.2.40)

Y 11 = a1 + S+

x, Y 12 = −b1 − 1

2 (x+ ip) + 1x

(S0 + 3

4

)(4.2.41)

Y 21 = a2 + 12 (x+ ip) + 1

x

(S0 −

34

), Y 22 = −b2 − S−

x(4.2.42)

Y 31 = −b2 −S+

x, Y 32 = −a2 −

12 (x− ip)− 1

x

(S0 + 3

4

)(4.2.43)

Y 41 = b1 + 12 (x− ip)− 1

x

(S0 −

34

), Y 42 = a1 + S−

x(4.2.44)

that transform nonlinearly under the Lorentz group. The special conformal generatorscan then be written as bilinears:

Kαβ = Y αiY βjεij (4.2.45)

which transform covariantly under the Lorentz group.

84

The Lorentz subgroup SO(5, 1) ∼ SU∗(4) ∼ Sl(2,H) generators of the minrep of SO(6, 2)with spinorial indices take the form

M βα = − i2

(σµ ¯σν

) β

αMµν (4.2.46)

which in terms of deformed twistorial oscillators Y,Z can be written as:

M βα = −1

2

(Z iα Y

βj − 14δ

αβZ

iγ Y

γj

)εij (4.2.47)

= 12

(Y βiZ j

α −14δ

αβY

γiZ jγ

)εij (4.2.48)

The dilatation generator ∆ takes the form:

∆ = i

8

(Z iα Y

αj − Y αiZ jα

)εij (4.2.49)

The commutation relations of the generators of the minrep of the conformal algebraSO(6, 2) given above are as follows:

[M βα , M δ

γ

]= δδαM

βγ − δβγM δ

α (4.2.50)

[Pαβ , M

δγ

]= −2δδ[αPβ]γ −

12δ

δγPαβ ,

[Kαβ , M δ

γ

]= 2δ[α

γ Kβ]δ + 1

2δδγK

αβ (4.2.51)

[Pαβ , K

γδ]

= 16(δ

[γ[αM

δ]β] + i

2δ[γ[αδ

δ]β] ∆

)(4.2.52)

[∆ , Pαβ ] = iPαβ ,[∆ , Kαβ

]= −iKαβ ,

[∆ , M β

α

]= 0 (4.2.53)

The algebra so(6, 2) can be given a 3-graded decomposition with respect to theconformal Hamiltonian, which is referred to as the compact 3-grading and the generatorsin this basis are reproduced in Appendix B.3 following [24].

4.2.3 Deformations of the minimal unitary representation of SO(6, 2)

It was shown in [24] that the minrep of SO∗(8) that was studied in previous section issimply isomorphic to the scalar doubleton representation that describes a conformalmassless scalar field in six dimensions. However we have seen in section 4.2.1 that SO∗(8)admits infinitely many doubleton representations corresponding to massless conformalfields transforming as symmetric tensors in the spinorial indices. As in the case of 4dconformal group SU(2, 2), it was shown in [24] that there exists a discrete infinity of

85

deformations to the minrep of SO(6, 2) labeled by the spin t of an SU(2) symmetry groupwhich is the 6d analog of helicity in 4d. Allowing this spin t to take all possible values,one obtains a discretely infinite set of deformations of the minrep which are isomorphic tothe doubleton representations. In this section we will show that the generators of theserepresentations can be recast as bilinears of deformed twistorial operators as was done inthe previous subsection for the true minrep that corresponds to t = 0.

Following [24], let us introduce an arbitrary number P pairs of fermionic oscillatorsρx and χx and their hermitian conjugates ρx = (ρx)† and χx = (χx)†, (x = 1, 2, . . . , P ) thatsatisfy the anti-commutation relations:

ρx , ρy = χx , χy = δyx, ρx , ρy = ρx , χy = χx , χy = 0 (4.2.54)

and refer to them as “deformation fermions”. The following bilinears of these fermionicoscillators

G+ = ρxχx G− = χxρx G0 = 12 (Nρ −Nχ) (4.2.55)

,where Nρ = ρxρx and Nχ = χxχx are the respective number operators, generate an su(2)Galgebra:

[G+ , G−] = 2G0, [G0 , G±] = ±G0 (4.2.56)

The fermionic oscillators ρx and χx form a doublet of SU(2)G. We choose the Fock vacuumof these fermionic oscillators such that

ρx |0〉 = χx |0〉 = 0 (4.2.57)

for all x = 1, 2, . . . , P . The states of the form

χ[x1χx2 . . . ρx(n−1)ρxn] |0〉

with definite eigenvalue n ≤ P of the total number operator NT = Nχ + Nρ transformirreducibly in the spin j = n/2 representation of SU(2)G 3. Among the irreduciblerepresentations of SU(2)G in the Fock space of P pairs of deformation fermions themultiplicity of the highest spin ( j = P/2 ) representation is one.

Now to deform the minimal unitary realization of so∗(8), one extends the subalgebrasu(2)S to the diagonal subalgebra su(2)T of su(2)S and su(2)G [24]. In other words, thegenerators of su(2)S receive contributions from the ρ- and χ-type fermionic oscillators as

3 Note that square bracketing of indices implies complete anti-symmetrization of weight one.

86

follows:

T+ = S+ +G+ = ambm + ρxχx

T− = S− +G− = bmam + χxρx

T0 = S0 +G0 = 12 (Na −Nb +Nρ −Nχ)

(4.2.58)

The quadratic Casimir of this subalgebra su(2)T is given by

C2 [su(2)T ] = T 2 = T0T0 + 12 (T+T− + T−T+) . (4.2.59)

In order to obtain the generators for the deformations of the minrep all we need to do isreplace S0, S± by T0, T± respectively in equations (4.2.28) - (4.2.35) and equations (4.2.37)- (4.2.44). We will denote the resulting deformed twistors as (Zt) i

α , (Zt) iα and (Yt)αi, (Yt)αi.

The generators can then be written as:

Pαβ = (Zt) iα (Zt) j

β εij , Kαβ = (Yt)αi(Yt)βjεij (4.2.60)

M βα = −1

2

((Zt) i

α (Yt)βj −14δ

αβ (Zt) i

γ (Yt)γj)εij (4.2.61)

= 12

((Yt)βi(Zt) j

α −14δ

αβ (Yt)γi(Zt) j

γ

)εij (4.2.62)

∆ = i

8

((Zt) i

α (Yt)αj − (Yt)αi(Zt) jα

)εij (4.2.63)

The Casimir invariants for SO(6, 2) for the deformed minreps depend only on the quadraticCasimir of SU(2)G involving deformation fermions. For one set of fermions (P = 1) onefinds:

C2 = MABM

BA = 16− 6 (Nρ −Nχ)2 (4.2.64)

C4 = MABM

BCM

CDM

DA = −69

12C2 − 20 (4.2.65)

C ′4 = εABCDEFGHMABMCDMEFMGH = 60 (C2 − 16) (4.2.66)

C6 = MABM

BCM

CDM

DEM

EFM

FA = 63

48C2 + 475 (4.2.67)

which shows clearly that the deformations are driven by fermionic oscillators.

4.2.4 Minimal unitary supermultiplet of OSp(8∗|4) and its deformations

The construction of the minimal unitary representations of non-compact Lie algebras byquantization of their quasiconformal realizations extends to non-compact Lie superalgebras

87

[22–24, 56]. In this section we will reformulate the minimal unitary realization of 6dsuperconformal algebra OSp(8∗|4) with the even subgroup SO∗(8) × USp(4) given in[56] in terms of deformed twistors. Extension to general superalgebras OSp(8∗|2N) isstraightforward.

Consider the superconformal (non-compact) 5-graded decomposition of the Lie super-algebra osp(8∗|4) with respect to the dilatation generator ∆:

osp(8∗|4) = N−1 ⊕N−1/2 ⊕N0 ⊕N+1/2 ⊕N+1 (4.2.68)

= Kαβ ⊕ Sαa ⊕ (M βα ⊕∆⊕ Uab)⊕Q a

α ⊕ Pαβ , (a, b = 1, 2, 3, 4)

where the grade zero space consists of the Lorentz algebra so(5, 1) (M βα ), the dilatations

(∆) and R-symmetry algebra usp(4) (Uab), grade +1 and −1 spaces consist of translations(Pαβ) and special conformal transformations (Kαβ) respectively, and the +1/2 and -1/2spaces consist of Poincaré supersymmetries (Q a

α ) and conformal supersymmetries (Sαa)respectively.

We introduce fermionic oscillators ξai where a, b = 1, 2, 3, 4 are the USp(4) ∼ SO(5)indices which are raised and lowered by the antisymmetric symplectic metric

Ωab =(

0 12

−12 0

)

and i, j = 1, 2 are the SU(2) indices raised and lowered by εij (ε12 = ε21 = +1). Theseoscillators satisfy:

ξai , ξbj

= Ωabεij (4.2.69)

and they will be referred to as supersymmetry fermions. The following bilinears of thesefermions

F+ = 12ξ

a1ξb1Ωba, F− = 12ξ

a2ξb2Ωab, F0 = 14(ξa1ξb2 + ξa2ξb1

)Ωba (4.2.70)

generate a su(2)F algebra:

[F+ , F−] = 2F0, [F0 , F±] = ±F0 (4.2.71)

To obtain the supersymmetric extensions of the deformations of the minrep of SO∗(8),one extends the su(2)T subalgebra to su(2)T which is the diagonal subalgebra of su(2)Tand su(2)F . The generators of su(2)T are then given by:

T+ = S+ +G+ + F+ = aibi + ρxχx + 12ξ

a1ξb1Ωba (4.2.72)

88

T− = S− +G− + F− = biai + χxρx + 12ξ

a2ξb2Ωab (4.2.73)

T0 = S0 +G0 + F0 = 12(Na −Nb +Nρ −Nχ) + 1

4(ξa1ξb2 + ξa2ξb1

)Ωba (4.2.74)

The generators of the even subgroup SO∗(8) of the deformations of the minrep ofOSp(8∗|4) are then obtained simply by replacing S0, S± by T0, T± respectively in equations4.2.28 - 4.2.35 and equations 4.2.37 - 4.2.44. We will denote the resulting deformedtwistors as (Zst ) i

α , (Zst ) iα and (Y st )αi, (Y st )αi. The generators of SO∗(8) can then be written

as bilinears of these deformed twistors:

Pαβ = (Zst ) iα (Zst ) j

β εij , Kαβ = (Y st )αi(Y st )βjεij (4.2.75)

M βα = −1

2

((Zst ) i

α (Y st )βj − 14δ

αβ (Zst ) i

γ (Y st )γj)εij (4.2.76)

= 12

((Y st )βi(Zst ) j

α −14δ

αβ (Y st )γi(Zst ) j

γ

)εij (4.2.77)

∆ = i

8

((Zst ) i

α (Y st )αj − (Y st )αi(Zst ) jα

)εij (4.2.78)

The supersymmetry generators Q aα , Sαa can similarly be realized simply as bilinears of

ordinary fermionic oscillators and deformed twistors as follows4:

Q aα = (Zst )

i

αξajεij = ξai(Zst )

j

αεij (4.2.79)

Sαa = (Y st )αiξajεij = ξai(Y st )αjεij (4.2.80)

The supersymmetry generators satisfy

Q aα , Q b

β

= −ΩabPαβ ,

Sαa , Sβb

= −ΩabKαβ (4.2.81)

Sαa , Q b

β

= −2ΩabM α

β − iδαβΩab∆− 2δαβUab (4.2.82)

The R-symmetry group USp(4) ∼ SO(5) can realized as bilinears of fermionic oscillatorsas follows:

Uab =(ξaiξbj − 1

4ΩabΩcdξciξdj)εij (4.2.83)

4To obtain the generators for the true minimal unitary supermultiplet of OSp(8∗|4) one needs only todrop the deformation fermions from the generators of su(2)T and the corresponding deformed twistorswill be denoted as (Zs) i

α , (Zs) iα and (Y s)αi, (Y s)αi.

89


[Uab , U cd

]= 2Ωa(cUd)b + 2Ωb(cUd)a (4.2.84)

The commutators of SO∗(8) generators with the supersymmetry generators are as follows:

[M βα , Q a

γ

]= −δβγQ a

α + 14δ

βαQ

aγ ,

[M βα , Sγa

]= δγαS

βa − 14δ

βαS

γa (4.2.85)[Kαβ , Q a

γ

]= −4δ[α

γ Sβ]a, [Pαβ , Sγa] = −4δγ[αQ

aβ] (4.2.86)[

∆ , Qαaγ]

= i

2Qaα , [∆ , Sαa] = − i2S

αa (4.2.87)

The R-symmetry generators act on USp(4) indices of supersymmetry generators as follows:

[Uab , Q c

α

]= −2Ωc(aQ b)

γ ,[Uab , Sαc

]= −2Ωc(aSγb) (4.2.88)

The generators given above transform covariantly with respect to the subgroupSU∗(4)× SO(1, 1)× USp(4). Unitarity and positive energy nature of the resulting repre-sentations are made manifest by going to the compact three grading of OSp(8∗|4) withrespect to the compact sub-superalgebra SU(4|2)× U(1) [24].

4.3 AdS7/CFT6 higher spin (super-)algebras, Joseph ideals andtheir deformations

As reviewed in Chapter 3, the standard AdSd/CFT(d−1) higher spin algebra of Fradkin-Vasiliev type is simply given by the quotient of the universal enveloping algebra ofSO(d, 2) by a two-sided ideal [40, 80, 130, 136]. This two-sided ideal is the Joseph idealthat annihilates the minimal unitary representation. Denoting the higher spin algebra asHS(g) with g = so(d− 1, 2) and the universal enveloping algebra as U(g) we have

HS(g) = U(g)J(g) (4.3.1)

where J(g) denotes the Joseph ideal.The uniqueness of the Joseph ideal was proved in [136] and an explicit formula for

the generators of this ideal for SO(n− 2, 2) was given as :

JABCD = MABMCD −MAB MCD −12 [MAB , MCD] + n− 4

4(n− 1)(n− 2) 〈MAB ,MCD〉1

= 12MAB ·MCD −MAB MCD + n− 4

4(n− 1)(n− 2) 〈MAB ,MCD〉1 (4.3.2)

90

where the dot · denotes the symmetric product

MAB ·MCD ≡MABMCD +MCDMAB (4.3.3)

〈MAB ,MCD〉 is the Killing form of SO(n− 2, 2) given by

〈MAB ,MCD〉 = hMEFMGH(ηEGηFH − ηEHηFG)(ηACηBD − ηADηBC) (4.3.4)

where h = 2(n−2)n(4−n) chosen such that all possible contractions of JABCD with the the metric

vanish. The symbol denotes the Cartan product of two generators [137]:

MAB MCD = 13MABMCD + 1

3MDCMBA + 16MACMBD

−16MADMBC + 1

6MDBMCA −16MCBMDA

− 12(n− 2)

(MAEM

EC ηBD −MBEM

EC ηAD +MBEM

EDηAC −MAEM

EDηBC

)− 1

2(n− 2)(MCEM

EA ηBD −MCEM

EB ηAD +MDEM

EB ηAC −MDEM

EA δBC

)+ 1

(n− 1)(n− 2)MEFMEF (ηACηBD − ηBCηAD) (4.3.5)

We shall refer to the operator JABCD as the generator of the Joseph ideal which forSO(6, 2) takes the form:

JABCD = 12MAB ·MCD −MAB MCD −

1112 〈MAB ,MCD〉 (4.3.6)

The enveloping algebra U(g) of a Lie algebra g can be decomposed with respect to theadjoint action of g. By Poincare-Birkhoff-Witt theorem this is equivalent to computingsymmetric products of the generators MAB ∼ of g. For so(6, 2) the symmetric productof the adjoint action decomposes as:

⊗ = ⊕ ⊕ ⊕ • (4.3.7)

where the singlet • is the quadratic Casimir C2 ∼M AB M B

A . It was pointed out in [80] thatthe higher spin algebra HS(g) must be a quotient of U(g) since the higher spin fields aredescribed by traceless two row Young tableaux. Thus the relevant ideal should quotientout the all the diagrams except the first one in the above decomposition. The Joseph idealas defined in 3.4.3 includes all the diagrams in the decomposition except the “window”diagram and thus by quotienting U(g) by the ideal generated by JABCD defined in3.4.3, we get rid of all the “unwanted” diagrams and obtain the Fradkin-Vasiliev type

91

higher spin algebra HS(6, 2).

4.3.1 Joseph ideal for minimal unitary representation of SO(6, 2)

Since the Eastwood formula for Joseph ideal is in the canonical basis we define

Mµ6 = 12 (Pµ −Kµ) (4.3.8)

Mµ7 = 12 (Pµ +Kµ) (4.3.9)

M67 = −∆ (4.3.10)

which together with the Lorentz group generators Mµν form the canonical basis MAB

(A,B, .. = 0, 1, ...7). Substituting the expressions for the generators MAB of the minrep ofSO(6, 2) from the quasiconformal realization into the generator of Joseph ideal (equation4.3.6) one finds that it vanishes identically as an operator. To get a better insightinto the physical meaning of the vanishing of the Joseph ideal we write JABCD in theLorentz covariant conformal basis (Kµ,Mµν ,∆, Pµ).which is equivalent to certain quadraticidentities. In addition to the conditions:

PµPµ = KµKµ = 0 or, P 2 = K2 = 0 (4.3.11)

one finds the following identities:

6∆ ·∆ +Mµν ·Mµν + 2Pµ ·Kµ = 0 (4.3.12)

Pµ · (Mµν + ηµν∆) = 0 (4.3.13)

Kµ · (Mνµ + ηνµ∆) = 0 (4.3.14)

ηµνMµρ ·Mνσ − P(ρ ·Kσ) + 4ηρσ = 0 (4.3.15)

Mµν ·Mρσ +Mµσ ·Mνρ +Mµρ ·Mσν = 0 (4.3.16)

∆ ·Mµν + P[µ ·Kν] = 0 (4.3.17)

M[µν · Pρ] = 0 (4.3.18)

M[µν ·Kρ] = 0 (4.3.19)

Defining the generalized Pauli-Lubanski tensor and its conformal analogue in six dimen-sions as:

Aµνρ = 13!εµνρσδτM

[σδ · P τ ] Bµνρ = 13!εµνρσδτM

[σδ ·Kτ ] (4.3.20)

92

we find that they vanish identically for the minrep given above

Aµνρ = 0 Bµνρ = 0 (4.3.21)

Computing the products of the generators of the above minimal unitary realization

corresponding to the Young tableaux and one finds that they vanish identically

and the resulting enveloping algebra contains only the operators whose Young tableauxhave two rows:

· · ·· · ·

︸︷︷︸n boxes

4.3.2 Deformations of the minimal unitary representation of SO(6,2) andthe Joseph ideal

As we saw above the generator JABCD of the Joseph ideal for SO∗(8) vanishes identicallyas an operator for the minrep obtained by quasiconformal techniques. However when onesubstitutes the generators of the deformed minreps one finds that JABCD does not vanishidentically. However as we will show the generators of the deformed minreps satisfycertain quadratic identities which correspond to deformations of the Joseph ideal.

To exhibit the quadratic identities corresponding to deformations of the Josep idealwe decompose the components of JABCD ( A,B, .. = 0, 1, 2, ..., 7) in terms of Lorentz(SU∗(4) ) covariant indices. We find that the totally antisymmetric tensors Aµνρ andBµνρ ( µ, ν, . . . = 0, 1, .., 5) defined in the previous section that vanished identically for theminrep do not vanish for the deformed minreps. Remarkably they become self-dual andanti-self-dual tensorial operators, respectively:

Aµνρ = Aµνρ

Bµνρ = −Bµνρ (4.3.22)

The identities (4.3.16) and (4.3.19) no longer hold separately but they combine and thefollowing identity holds true for deformed generators:

Mµν ·Mρσ +Mµσ ·Mνρ +Mµρ ·Mσν = ε δτµνρσ (P[δ ·Kτ ] +Mδτ ·∆) (4.3.23)

93

The deformation of the identity (4.3.15) is as follows:

ηµνMµρ ·Mνσ − P(ρ ·Kσ) + 4ηρσ = 2G2ηρσ (4.3.24)

where G2 is quadratic Casimir of su(2)G defined in equation 4.2.55 and involves only thedeformation fermions. Note that the quadratic Casimir operator G2 of su(2)G is relatedto the quadratic Casimir operator of the deformed minrep of SO∗(8) as follows [24]:

C2 [so∗(8)]deformed = 8(2− G2) (4.3.25)

The eigenvalues t(t+ 1) (t = 0, 1/2, 1, 3/2, ... ) of G2 label the deformations of the minrep ofSO∗(8). This is to be contrasted with the deformations of the minrep of 4d conformal groupSO(4, 2) which are labelled by a continuous parameter that enters the quadratic identitiesexplicitly in the form of continuous helicity 3. Since the minrep and its deformationsobtained by quasiconformal methods correspond to massless conformal fields we do notexpect any continuous deformations in six dimensions since the little group of masslessparticles is SO(4) = SU(2)T ×SU(2)A whose unitary representations are discretely labelled(jT , jA) where jA and jT are non-negative integers or half integers. Furthermore forconformally massless representations either jT or jA (or both) vanishes. However we donot have a proof that continuous deformations do not exist.

For the discrete deformations of the minrep the operators corresponding to thesymmetric tensor with Young Tableau appearing in the symmetric product of thegenerators as shown in 3.4.2 still vanish On the other hand the operators with the Youngtableau

do not vanish. These operators satisfy an eight dimensional self-duality condition whichcorresponds to a three form gauge field with a self-dual field strength. In AdS7 theycorrespond to three form gauge fields that satisfy odd dimensional self-duality conditionjust like the three form field that descends from eleven dimensional supergravity onAdS7 × S4. In six dimensions they correspond to conformal two form fields with a self-dual field strength, which is simply the tensor field that appears in the (2, 0) conformalsupermultiplet whose interacting theory is believed to be dual to M-theory over AdS7×S4.

The symmetric tensor products of the generators of the discrete deformations ofSO(6, 2) leads to a AdS7/CFT6 higher spin algebra whose generators include Youngtableaux of the form

94

· · ·· · ·

· · ·· · ·︸︷︷︸

m boxes

· · ·· · ·

︸︷︷︸n boxes

This suggests that the theories based on discrete deformations of the minrep describehigher spin theories of Fradkin-Vasiliev type in AdS7 coupled to tensor fields that satisfyself-duality conditions and their higher extensions corresponding to the Young tableaux

· · ·· · ·

· · ·· · ·︸︷︷︸

m boxesThe study of higher spin theories based on discrete deformations of the minrep thatextend Fradkin-Vasiliev type higher spin theories will be the subject of a separate study.

4.3.3 Comparison with the covariant twistorial oscillator realization

Substituting the generators MAB of SO∗(8) realized as bilinears of covariant twistorialoscillators (doubletons) (section 4.2.1) in equation 4.3.6 to compute the generator of theJoseph ideal one finds that it does not vanish identically. However the non-vanishingcomponents of the generator JABCD of the Joseph ideal factorize in a similar fashion asin the doubleton realization of SO(4, 2) 3. Symbolically this factorization takes the form

JABCD = (...)Ba (4.3.26)

where Ba (a = 1, 2, 3) is a generator of an SU(2)B algebra that commutes with SO∗(8). Interms of the covariant twistorial oscillators the generators of SU(2)B are

B− = dici, B+ = cidi, B0 = 12(cici − didi) (4.3.27)

with the quadratic Casimir

B2 = B20 + 1

2(B+B− +B−B+) (4.3.28)

Acting on the subspace of the Fock space of covariant twistorial oscillators that is SU(2)Bsinglet the generator JABCD vanishes. This subspace corresponds to the true minrep ofSO∗(8) and describes a conformal scalar field in six dimensions. In fact the authors of [155]studied a purely bosonic higher spin algebra in AdS7 using the doubletonic realization

95

of SO∗(8). After imposing an infinite set of constraints, restricting to an SU(2) singletsector and modding out by an infinite ideal containing all the traces they obtain an higherspin algebra. They also state that their results can not be extended to SU(2) non-singletsectors.

The algebra su(2)B for the doubleton representations is the analog of su(2)G for thedeformed minreps studied above. The Casimir operator B2 of SU(2)B is related to thequadratic Casimir C2 of the doubletonic realization of SO∗(8) in terms of covarianttwistorial oscillators :

C2 [so∗(8)]doubleton = 8(2− B2) (4.3.29)

which reflects the fact that su(2)B and so∗(8)doubleton form a reductive dual pair insidesp(16,R). However this is not the case with the deformed minreps since su(2)G doesnot commute with the generators of so∗(8)deformed. Another critical difference is thefact that the possible eigenvalues b(b + 1) of SU(2)B span the entire set of irreps, i.eb = 0, 1/2, 1, 3/2, .... On the other hand, for a given number P pairs of deformationFermions possible eigenvalues j(j + 1) of SU(2)G is j = 0, 1/2, 1, .., P/2

For the doubleton realization the quadratic identities satisfied by the generators satisfyformally the same identities as the deformed minrep given in the previous subsectionwith G2 replaced by B2:

ηµνMµρ ·Mνσ − P(ρ ·Kσ) + 4ηρσ = 2B2ηρσ (4.3.30)

The Casimir invariants for SO(6, 2) in the doubleton representation are as follows:

C2 = 8(2− B2) (4.3.31)

C4 = C22

8 − 9C2 (4.3.32)

C ′4 = 96C2 − 6C22 (4.3.33)

C6 = C32

64 −278 C

22 + 81C2 (4.3.34)

4.3.4 AdS7/CFT6 Higher spin algebras and superalgebras and their defor-mations

Following [130] and Chapter 3, we will use the following definition for the standart higherspin algebra in six dimensions:

HS(6, 2) = U(so(6, 2))J(so(6, 2)) (4.3.35)

96

where U(SO(6, 2)) is the universal enveloping algebra and J(so(6, 2)) denotes the Josephideal of so(6, 2). The Joseph ideal vanishes identically for the quasiconformal realizationof the minrep. Therefore to construct HS(6, 2) one needs simply take the envelopingalgebra of the minrep in the quasiconformal construction. Since the minrep of so(6, 2)admits deformations we define deformed AdS7/CFT6 higher spin algebras HS(6, 2; t) asthe enveloping algebras of so(6, 2) quotiented by the deformed Joseph ideal Jt(so(6, 2))

HS(6, 2; t) = U(so(6, 2))Jt(so(6, 2)) (4.3.36)

For these deformed high spin algebras the corresponding deformed Joseph ideal vanishesidentically as operator as we showed explicitly above for the conformal group in sixdimensions. Deformed minreps describe massless conformal fields of higher spin labeledby the spin t of the SU(2)G subgroup which is the analogue of helicity in d = 4.

We saw in section 4.2.3 that deformations of the minrep are driven by fermionicoscillators. For P pairs of deformation fermions the Fock space decomposes as the directsum of the two spinor representations of SO(4P ) generated by all the bilinears of theoscillators. The centralizer of SU(2)G inside SO(4P ) is USp(2P ). Under USp(2P )×SU(2)Gthe Fermionic Fock space decomposes as

22P =P∑r=0

(Rr, t = (P − r)/2) (4.3.37)

where Rr is the symplectic traceless tensor of rank r of USp(2P ) and t is the spin of SU(2)G.The USp(2P ) invariant (singlet) subspace transforms in the spin t = P/2 representationof SU(2)G. Therefore restricting to this invariant subspace we get a deformed minrepcorresponding to a 6d massless conformal field transforming as a totally symmetric tensorof rank P in the spinor indices with respect to the Lorentz group SU∗(4). This wayone can construct all conformally massless representations of SO(6, 2) as deformationsof the minrep by choosing P = 0, 1, 2, .... The enveloping algebras of these deformedirreducible irreps define then a discrete infinity of higher spin algebras labelled by t = P/2.Equivalently one can simply subsitute (P + 1)× (P + 1) irreducible representation matricesfor generators of SU(2)G in place of the bilinears of deformation fermions. The latter isuseful for writing down the irreducible infinite higher spin algebra without reference toits action on a representation space.

We shall define the 2N extended AdS7/CFT6 higher spin superalgebra as the envelopingalgebra of the minimal unitary realization of the super algebra OSp(8∗|2N) obtainedvia the quasiconformal approach. For this algebra there are only 2N supersymmetryfermions and the minimal supermultiplet of OSp(8∗|2N) consists of the following massless

97

conformal fields:

Φ[A1A2...AN ]| ⊕Ψ[A1A2...AN−1]|α ⊕ Φ[A1A2...AN−2]|

(αβ) ⊕ · · · (4.3.38)

where α, β, .. are the spinor indices of the 6d Lorentz group SU∗(4) and Ai denote theUSp(2N) indices. The generator JABCD of the Joseph ideal vanishes when acting on theconformal scalars that are part of the minimal unitary supermultiplet. On the otherfields of the minimal unitary supermultiplet deformed quadratic identities 4.3.2 involvingsupersymmety fermions are satisfied.

Deformed AdS7/CFT6 higher spin superalgebras HS(6, 2|2N ; t) algebras are definedsimply as enveloping algebras of the deformed minimal unitary realizations of the superalgebras OSp(8∗|2N) with the even subalgebra SO∗(8)⊕ USp(2N) involving deformationfermions 4.2.4. As explained above restricting to USp(2P ) invariant sector of the Fockspace of 2P deformation fermions one gets a deformed minimal unitary realization ofOSp(8∗|2N)t for t = P/2. Equivalently one can simply substitute the (P + 1) × (P + 1)representation matrices of SU(2)G in place of the bilinears of deformation fermions. Theirenveloping algebras define a discrete infinite family of higher spin supealgebras labeledby t = 0, 1/2, 1, 3/2, 2, ....

98

Chapter 5 |Discussion

5.1 AdS5/CFT4 higher spin holography

The existence of a one-parameter family of AdS5/Conf4 higher spin algebras and superal-gebras raises the question as to their physical meaning. Before discussing the situation infour dimensions, let us summarize what is known for AdS4/Conf3 higher spin algebras.In 3d, there is no deformation of the higher spin algebra except for the super extensioncorresponding to Sp(4;R)→ OSp(N |4;R). For the bosonic AdS4 higher spin algebras onefinds that the higher spin theories of Vasiliev are dual to certain conformally invariantvector-scalar/spinor models in 3d [15, 84] in the large N limit. More recently, Maldacenaand Zhiboedov [100] studied the constraints imposed on a conformal field theory in threedimensions by the existence of a single conserved higher spin current. They found thatthis implies the existence of an infinite number of conserved higher spin currents. Thiscorresponds simply to the fact that the generators of SO(3, 2) get extended to an infinitespin algebra when one takes their commutators with an operator which is bilinear orhigher order in the generators, except for those operators that correspond to the Casimirelements. They also showed that the correlation functions of the stress tensor and theconserved currents are those of a free field theory in three dimensions, either a theoryof N free bosons or a theory of N free fermions [100], which are simply the scalar andspinor singletons.

The distinguishing feature of 3d is the fact that there exists only two minimal unitaryrepresentations corresponding to massless conformal fields which are simply the Di andRac representations of Dirac. However in 4d, we have a one-parameter, ζ, family ofdeformations of the minimal unitary representation of the conformal group correspondingto massless conformal fields of helicity ζ/2. The same holds true for the minimal unitarysupermultiplet of SU(2, 2|N). From M/superstring point of view the most importantinteracting and supersymmetric CFT in d = 4 is the N = 4 super Yang-Mills theory.

99

It was argued in references [85, 89, 156] that the holographic dual of N = 4 superYang-Mills theory with gauge group SU(N) at g2

YMN = 0 for N → ∞ should be a freegauge invariant theory in AdS5 with massless fields of arbitrarily high spin and this wassupported by calculations in [91]. Moreover the scalar sector of N = 4 super Yang-Millstheory at g2

YMN = 0 with N → ∞ should be dual to bosonic higher spin theories inAdS5 which provides a non-trivial extension of AdS/CFT correspondence in superstringtheory [14,41] to non-supersymmetric large N field theories.The Kaluza-Klein spectrumof IIB supergravity over AdS5 × S5 was first obtained by tensoring of the minimal unitarysupermultiplet (scalar doubleton) of SU(2, 2|4) with itself repeatedly and restricting toCPT self-conjugate sector [32]. The massless graviton supermultiplet in AdS5 sits at thebottom of this infinite tower. In fact all the unitary representations corresponding tomassless fields in AdS5 can be obtained by tensoring of two doubleton representations ofSU(2, 2) which describe massless conformal fields on the boundary of AdS5 [32, 37, 38, 40].As was argued by Mikhailov [91], in the large N limit the correlation functions in theCFT side become products of two point functions which correspond to products of twodoubletons. As such they correspond to massless fields in the AdS5 bulk. At the level ofcorrelation functions the same arguments suggest that corresponding to a one parameterfamily of deformations of the N=4 Yang-Mills supermultiplet there must exists a familyof supersymmetric massless higher spin theories in AdS5. Turning on the gauge couplingconstant on the Yang-Mills side leads to interactions in the bulk and most of the higherspin fields become massive.

The fact that the quasiconformal realization of the minrep of SU(2, 2|4) is nonlinearimplies that the corresponding higher spin theory in the bulk must be interacting. Sincethe same minrep can also be obtained by using the doubletonic realization [32], whichcorresponds to free field realization, suggests that the interacting supersymmetric higherspin theory may be integrable in the sense that its holographic dual is an integrableconformal field theory with infinitely many conserved currents , just like the classicalN = 4 super Yang-Mills theory in four dimensions [157]. There are other deformed higherspin algebras corresponding to non-CPT self-conjugate supermultiplets of SU(2, 2|4) thatcontain scalar fields and are deformations of the minrep. The above arguments suggestthat they too should correspond to interacting but integrable supersymmetric higher spintheories in AdS5. One solid piece of the evidence for this is provided by the fact that thesymmetry superalgebras of interacting (nonlinear) superconformal quantum mechanicalmodels of [115] furnish a one parameter family of deformations of the minimal unitaryrepresentation of the N = 4 superconformal algebra D(2, 1;α) in one dimension. This waspredicted in [125] and shown explicitly in [116].

Most of the work on higher spin algebras until now have utilized the realizations of

100

underlying Lie (super)algebras as bilinears of oscillators which correspond to free fieldrealizations. The quasiconformal approach allows one to give a natural definition of superJoseph ideal and leads directly to the interacting realizations of the superextensions ofhigher spin algebras. The next step in this approach is to reformulate these interactingquasiconformal realizations in terms of covariant gauge fields and construct Vasiliev typenonlinear theories of interacting higher spins in AdS5.

Another application of our results will be to reformulate the spin chain modelsassociated with N = 4 super Yang-Mills theory in terms of deformed twistorial oscillatorsand study the integrability of corresponding spin chains non-perturbatively. In fact, aspectral parameter related to helicity and central charge was introduced recently forscattering amplitudes in N = 4 super Yang-Mills theory [158]. This spectral parametercorresponds to our deformation parameter which is helicity and appears as a centralcharge in the quasiconformal realization of the super algebra SU(2, 2|4).

5.2 AdS7/CFT6 higher spin holography

The main result of Chapter four is the reformulation of the minimal unitary representationof SO∗(8) and its deformations in terms of deformed twistors that transform nonlinearlyunder the Lorentz group in six dimensions. Their enveloping algebras lead to a discreteinfinite family of AdS7/CFT6 higher spin algebras labelled by the spin of an SU(2)symmetry for which certain deformations of the Joseph ideal vanish. Remarkably thesedeformations involve (anti-)self-duality of the 6d tensorial operator which is the analog ofPauli-Lubanski vector in four dimensions. These results carry to superalgebras OSp(8∗|2N)and one finds a discrete infinity of AdS7/CFT6 higher spin superalgebras. As we argued forfor AdS5/CFT4 algebras, our results suggest the existence of a family of (supersymmetric)higher spin theories in AdS7 that are dual to free (super) CFT’s or to interacting butintegrable (supersymmetric) CFT’s in six dimensions.

Of particular interest are the higher spin superalgebras based on OSp(8∗|4) andOSp(8∗|8) whose minimal unitary supermultiplets reduce to N = 4 Yang-Mills super-multiplet and N = 8 supergravity multiplet under dimensional reduction to four dimen-sions [132]. The minimal unitary supermultiplet of OSp(8∗|4) is the 6d (2,0) conformaltensor multiplet [33, 39] whose interacting theory is believed to be dual to M-theoryover AdS7 × S4 [14]. Whether there exists a limit of this interacting theory that is dualto a higher spin theory in AdS7 is an interesting open problem. Our results in section3.2 suggest that such a limit should exist. On the other hand it is not known if thereexists an interacting non-metric (4, 0) supergravity theory based on the minimal unitarysupermultiplet of OSp(8∗|8) [132,159].

101

Appendix A|

A.1 Spinor conventions for SO(2, 1)

We follow [160] for the spinor conventions in d = 3 and thus all the 3d spinors are Majoranawith ηµν = diag(−,+,+). The gamma-matrices in Majorana representation terms of thePauli matrices σi are as follows:

γ0 = −iσ2, γ1 = σ1, γ2 = σ3 (A.1.1)

and they satisfyγµ, γναβ = 2ηµνδαβ . (A.1.2)

Thus the matrices (γµ)αβ are real and the Majorana condition on spinors imply that theyare real two component spinors. Spinor indices are raised/lowered by the epsilon symbolswith ε12 = ε12 = 1 and choosing NW-SE conventions

εαγεβγ = δαβ , λα := εαβλβ ⇔ λβ = λαεαβ . (A.1.3)

Introducing the real symmetric matrices (σµ)αβ := (γµ)ρβ ερα and (σµ)αβ := (ε·σµ·ε)αβ =−εβρ (γµ)αρ, a three vector in spinor notation writes as a symmetric real matrix as

Vαβ := (σµVµ)αβ ⇒ V µ = 12 (σµ)αβ Vαβ . (A.1.4)

A.2 Minimal unitary supermultiplet of OSp(N |4,R)

In this section we will formulate the minimal unitary representation of OSp(N |4,R)which is the superconformal algebra with N supersymmetries in three dimensions. Thesuperalgebra osp(N |4) can be given a five graded decomposition with respect to the

102

noncompact dilatation generator ∆ as follows:

osp(N |4) = Kαβ ⊕ SIα ⊕ (OIJ ⊕M βα ⊕∆)⊕Q I

α ⊕ Pαβ (A.2.1)

We shall call this superconformal 5-grading. The bosonic conformal generators arethe same as given in previous section. In order to realize the R-symmetry algebraSO(N) and supersymmetry generators, we introduce Euclidean Dirac gamma matrices γI

(I, J = 1, 2, . . . , N) which satisfy γI , γJ

= δIJ (A.2.2)

The R-symmetry generators are then simply given as follows:

OIJ = 12(γIγJ − γJγI

)(A.2.3)

and supersymmetry generators are the bilinears of 3d twistorial oscillators and γI :

Q Iα = καγ

I , SIα = γIµα (A.2.4)


Q Iα , Q J

β

= δIJPαβ ,

SIα , SJβ

= δIJKαβ (A.2.5)

Q Iα , SJβ

= M β

α δIJ + 2OIJδβa + i

(∆− i

2

)δIJδβα (A.2.6)

The action of conformal group generators on supersymmetry generators is as follows:

[M βα , Q I

γ

]= −δβγQ I

α + 12Q

Iγ ,

[M βα , SIγ

]= δαγ S

Iβ − 12S

Iγ (A.2.7)

[∆ , Q I

α

]= i

2QIα ,

[∆ , SIα

]= − i2S

Iα (A.2.8)

[Pαβ , S

Iγ]

= −2δγ(αQIβ),

[Kαβ , Q I

γ

]= 2δ(α

γ Sβ)I (A.2.9)

[Pαβ , Q

Iγ

]=[Kαβ , SIγ

]= 0 (A.2.10)

The R-symmetry generators act only on the I, J indices and rotate them as follows:

[Q Iα , OJK

]= 1

2δI[JQ K]

α ,[SIα , OJK

]= 1

2δI[JSK]α (A.2.11)

103

For even N the singleton supermultiplet consists of conformal scalars transforming ina chiral spinor representation of SO(N) and conformal space-time spinors transformingin the opposite chirality spinor representation of SO(N) [35,161]. There exists anothersingleton supermultiplet in which the roles of two spinor representations of SO(N) areinterchanged. For odd N both the conformal scalars and conformal space-time spinorstransform in the same spinor representation of SO(N).

A.3 The quasiconformal realization of the minimal unitary rep-resentation of SO(4, 2) in compact three-grading

In this appendix we provide the formulas for the quasiconformal realization of generatorsof SO(4, 2) in compact 3-grading following [23]. Consider the compact three gradeddecomposition of the Lie algebra of SU(2, 2) determined by the conformal Hamiltonian

so(4, 2) = C− ⊕ C0 ⊕ C+

where C0 = so(4) ⊕ so(2). Folllowing [23] we shall label the generators in C± and C0

subspaces as follows:

(B1, B2, B3, B4) ∈ C− (A.3.1)

L±,0 ⊕H ⊕R±,0 ∈ C0 (A.3.2)

(B1, B2, B3, B4) ∈ C+ (A.3.3)

where L±,0 and R±,0 denote the generators of SU(2)L×SU(2)R and H is the U(1) generator.The generators of so(4, 2) in the compact 3-grading take on very simple forms when

expressed in terms of the singular oscillators introduced in section 3.3.2:

H = 12

(A†L+1AL+1 + L+ 1

2(Nd +Ng) + 52

)(A.3.4)

L+ = − i2ALd†, L− = i

2dA†L, L3 = Nd −

12 (H − 1) (A.3.5)

R+ = i

2g†A−L, R− = − i2A

†−Lg, R3 = Ng −

12 (H + 1) (A.3.6)

B1 = −iALA−L, B2 = −i√

2dA−L, B3 = −i√

2ALg, B4 = −2igd (A.3.7)

104

B1 = iA†−LA†L, B2 = i

√2A†−Ld

†, B3 = i√

2g†A†L, B4 = 2ig†d† (A.3.8)

A.4 Commutation relations of deformed twistorial oscillators forSO(4, 2)

[Y 1 , Y 1

]= 1

2x (Y 1 − Y 1),[Y 2 , Y 2

]= − 1

2x (Y 2 − Y 2) (A.4.1)[Y 1 , Y 2] = 1

2x (Y 1 + Y 2),[Y 1 , Y 2

]= 1

2x (Y 1 + Y 2) (A.4.2)[Y 1 , Y 2

]= − 1

2x (Y 1 + Y 2),[Y 1 , Y 2

]= − 1

2x (Y 1 + Y 2) (A.4.3)

[Z1 , Z1

]= − 1

2x (Z1 + Z1),[Z2 , Z2

]= − 1

2x (Z2 + Z2) (A.4.4)

[Z1 , Z2] = − 12x (Z1 − Z2),

[Z1 , Z2

]= 1

2x (Z1 − Z2) (A.4.5)[Z1 , Z2

]= − 1

2x (Z1 + Z2),[Z1 , Z2

]= 1

2x (Z1 + Z2) (A.4.6)

[Y 1 , Z1

]= 2 + 1

2x (Y 1 − Z1),[Y 1 , Z1

]= 2− 1

2x (Y 1 + Z1) (A.4.7)[Y 1 , Z2

]= 1

2x (Y 1 − Z2),[Y 1 , Z2

]= − 1

2x (Y 1 + Z2) (A.4.8)[Y 1 , Z1

]= 1

2x (Y 1 + Z1),[Y 1 , Z1

]= − 1

2x (Y 1 − Z1) (A.4.9)[Y 1 , Z2

]= 1

2x (Y 1 + Z2),[Y 1 , Z2

]= − 1

2x (Y 1 − Z2) (A.4.10)

[Y 2 , Z1

]= 1

2x (Y 2 + Z1),[Y 2 , Z1

]= − 1

2x (Y 2 − Z1) (A.4.11)[Y 2 , Z2

]= 2 + 1

2x (Y 2 + Z2),[Y 2 , Z2

]= 2− 1

2x (Y 2 − Z2) (A.4.12)[Y 2 , Z1

]= 1

2x (Y 2 − Z1),[Y 2 , Z1

]= − 1

2x (Y 2 + Z1) (A.4.13)[Y 2 , Z2

]= 1

2x (Y 2 − Z2),[Y 2 , Z2

]= − 1

2x (Y 2 + Z2) (A.4.14)

105

Appendix B|

B.1 Clifford algebra conventions for doubleton realization

In this appendix, we will give the 6d analogs of σµ matrices (in mostly positive metric)used in section 4.2.1 as defined in [39] for the doubleton representations of SO∗(8). Paulimatrices

σ0 =(

1 00 1

), σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

). (B.1.1)

satisfyσµσν + σν σµ = 2ηµν . (B.1.2)

Their 6d counterparts are defined as [39]

Σ0 = −iσ2 ⊗ σ3 Σ0 = −iσ2 ⊗ σ3 (B.1.3a)

Σ1 = iσ2 ⊗ σ0 Σ1 = −iσ2 ⊗ σ0 (B.1.3b)

Σ2 = iσ1 ⊗ σ2 Σ2 = −iσ1 ⊗ σ2 (B.1.3c)

Σ3 = iσ3 ⊗ σ2 Σ3 = −iσ3 ⊗ σ2 (B.1.3d)

Σ4 = σ0 ⊗ σ2 Σ4 = σ0 ⊗ σ2 (B.1.3e)

Σ5 = σ2 ⊗ σ1 Σ5 = σ2 ⊗ σ1. (B.1.3f)

with the convention that the six dimensional Σµ have lower spinorial indices while the Σµ

have upper spinorial indices.

106

B.2 Clifford algebra conventions for deformed twistors

In order to make contact with the spinor helicity literature in 6d, we will use the mostlypositive metric and follow the conventions of [154] for 6d analogs of Pauli matrices in theformulation of deformed minreps in terms of deformed twistors. We use a hat over the6d sigma matrices in order to avoid confusion with the standard Pauli matrices.

σ0 = iσ1 ⊗ σ2 ¯σ0 = −iσ1 ⊗ σ2 (B.2.1a)

σ1 = iσ2 ⊗ σ3 ¯σ1 = iσ2 ⊗ σ3 (B.2.1b)

σ2 = −σ2 ⊗ σ0 ¯σ2 = σ2 ⊗ σ0 (B.2.1c)

σ3 = −iσ2 ⊗ σ1 ¯σ3 = −iσ2 ⊗ σ1 (B.2.1d)

σ4 = −σ3 ⊗ σ2 ¯σ4 = σ3 ⊗ σ2 (B.2.1e)

σ5 = iσ0 ⊗ σ2 ¯σ5 = iσ0 ⊗ σ2. (B.2.1f)

They satisfy :σµ ¯σν + σν ¯σµ = −2ηµν . (B.2.2)

Again we adopt the convention that the six dimensional σµ have lower spinorial indiceswhile the ¯σµ have upper spinorial indices. With these conventions, we define:

Pαβ = (σµPµ)αβ =

0 iP4 + P5 P1 + iP2 P0 − P3

−iP4 − P5 0 −P0 − P3 −P1 + iP2

−P1 − iP2 P0 + P3 0 −iP4 + P5

−P0 + P3 P1 − iP2 iP4 − P5 0

(B.2.3)

Kαβ = (¯σµKµ)αβ =

0 −iK4 +K5 K1 − iK2 −K0 −K3

iK4 −K5 0 K0 −K3 −K1 − iK2

−K1 + iK2 −K0 +K3 0 iK4 +K5

K0 +K3 K1 + iK2 −iK4 −K5 0

(B.2.4)

B.3 The quasiconformal realization of the minimal unitary rep-resentation of SO(6, 2) in compact 3-grading

In this appendix we provide the formulas for the quasiconformal realization of generatorsof SO(6, 2) and their deformations in compact 3-grading following [24]. We shall give theformulas with the deformation fermions included. The generators for the true minrep

107

can be obtained simply by setting the deformation fermions to zero.Consider the compact three graded decomposition of the Lie algebra of SO∗(8)

determined by the conformal Hamiltonian H

so∗(8) = C− ⊕ C0 ⊕ C+ (B.3.1)

where C0 = su(4)⊕ u(1). We shall label the generators in C± and C0 as follows:

(Wm, Xm, N−, B−) ∈ C− (B.3.2)

(Dm, Em, Dm, Em, T±,0, A±,0, J,H) ∈ C0 (B.3.3)

(Wm, Xm, N−, B+) ∈ C+ (B.3.4)

where m,n, .. = 1, 2. The generators of su(4) algebra in C0 subspace has a 3-gradeddecomposition with respect to its su(2)T ⊕ su(2)A ⊕ u(1)J subalgebra where the U(1)Jgenerator J determines the 3-grading of su(4). The generators for su(2)T were given inequation 4.2.58 (for the true minrep without deformation fermions su(2)T reduces simplyto su(2)S whose generators were given in equation 4.2.24) and those of su(2)A are asfollows:

A+ = a1a2 + b1b2, A− = a1a2 + b1b

2, A0 = 12(a1a1 − a2a2 + b1b1 − b2b2

)and they satisfy:

[A+ , A−] = 2A0, [A0 , A±] = ±A± (B.3.5)

The generators Dm.Em, Dm, Em belonging to the coset SU(4)/SU(2)× SU(2)× U(1) are

realized as bilinears of the oscillators am, bm and the following “singular" oscillators :

AL± = 1√2

(x+ ip− L±

x

), AK± = 1√

2

(x+ ip− K±

x

)(B.3.6)

whereL± = 2

(T0 ± T− −

34

), K± = −2

(T0 ± T+ + 3

4

)(B.3.7)

They satisfy the commutation relations:

[L+ , L−] = 2(L+ − L−), [K+ , K−] = 2(K+ −K−) (B.3.8)

[L± , K±] = 2(L∓ −K∓), [L± , K∓] = −2(L± −K∓) (B.3.9)

In general for two singular oscillators defined in terms of operators L1 and L2that commute

108

with the singlet coordinate x but not with each other we have we have

[AL1 , AL2

]= 1

2

(L2 − L1

x2 + [L1 , L2]x2

)(B.3.10)

[A†L1

, A†L2

]= 1

2

L†1 − L†2x2 +

[L†1 , L

†2

]x2

(B.3.11)

[AL1 , A

†L2

]= 1 + 1

2

L1 + L†2x2 +

[L1 , L†2

]x2

(B.3.12)

where A†L± = 1√2

(x+ ip−

L†±x

). The commutation relations (B.3.9) lead to the commu-

tation relations for AL± and AK± as follows:

[AL+ , AL−

]= (L+ − L−)

2x2 ,[AK+ , AK−

]= (K+ −K−)

2x2 (B.3.13)[AL+ , AK−

]= −3(L+ −K−)

2x2 ,[AL− , AK+

]= −3(L− −K+)

2x2 (B.3.14)

The generator that determines the compact 3-grading of SO∗(8) is given as follows:

H = Ha +Hb +H (B.3.15)

whereH = 1

4

(AL−A

†L−

+AK+A†K+

+ L− +K+ − 1)

(B.3.16)

andHa = 1

2 (Na + 2) , Hb = 12 (Nb + 2) (B.3.17)

are simply the Hamiltonians of standard bosonic oscillators of a- and b-type (Na =a1a1 + a2a2, Nb = b1b1 + b2b2). This u(1) generator is the AdS energy or the conformalHamiltonian when SO∗(8) ' SO(6, 2) is taken as the seven dimensional AdS group or thesix dimensional conformal group, respectively.

The generator that determines the 3-grading of su(4) is as follows:

J = Ha +Hb −H (B.3.18)

The SU(4)/SU(2)S × SU(2)A × U(1)J coset generators can be written as

Dm = 1√2

(amA

†L+

+ bmA†K−

), Dm = 1√

2

(AL+a

m +AK−bm)

(B.3.19)

Em = 1√2

(amA

†L−− bmA†K+

), Em = 1√

2

(AL−a

m −AK+bm)

(B.3.20)

109

where m,n = 1, 2. They close into the generators of the subgroup SU(2)S ×SU(2)A×U(1)Jgiven in (4.2.58) and (B.3.5), which do not involve singular oscillators. Then the su(4)algebra can be rewritten in a fully SU(2)S × SU(2)A covariant form[

Sm′n′ , S

k′l′

]= δk

′n′ S

m′l′ − δ

m′l′ S

k′n′

[Amn , A

kl

]= δknA

ml − δml Akn[

Cm′m , Cn′n

]= δmn Sm

′n′ + δm

′n′ A

mn + δm

′n′ δ

mn J[

Sm′n′ , C

k′m]

= δk′n′ C

m′m − 12δ

m′n′ C

k′m[Amn , C

m′k]

= δkn Cm′m − 1

2δmn Cm

′k .

(B.3.21)

where we have labeled the generators of su(2)S and su(2)A as Sm′n′ (m′, n′ = 1, 2) and Amn,

respectively:

S11 = −S2

2 = S0

A11 = −A2

2 = A0

S12 = S+

A12 = A+

S21 =

(S1

2)† = S−

A21 =

(A1

2)† = A−

(B.3.22)

and defined

C1m = Dm + Em, C2m = Dm − Em (B.3.23)

C1m = Dm + Em, C2m = Dm − Em (B.3.24)

The generators belonging to C− are given as follows:

Wm = 1√2

(AK+am +AL−bm

), Xm = 1√

2

(AK−am −AL+bm

)(B.3.25)

N− = a1b2 − a2b1, B− = 14

(AK+A−K†+

+AL−A−L†−

)(B.3.26)

and the generators in C+ are given by their hermitian conjugates:

Wm = 1√2

(amA†K+

+ bmA†L−

), Xm = 1√

2

(amA†K− − b

mA†L+

)(B.3.27)

N+ = a1b2 − a2b1, B+ = 14

(A†−K†+

A†K++A†

−L†−A†L−

)(B.3.28)

110

The commutators [C− , C+] close into C0:

[Wm , Wn] = 2 (δnmH +Anm) + δnm (T− + T+)

[Wm , Xn] = δnm (2T0 − T− + T+)

[Wm , N+] = εmnEn

[Wm , B+] = Dm

[N− , N+] = H + J

[Xm , Xn] = 2 (δnmH +Anm)− δnm (T− + T+)

[Xm , Wn] = δnm (2T0 + T− − T+)

[Xm , N+] = εmnDn

[Xm , B+] = Em

[B− , B+] = H − J(B.3.29)

B.4 Commutation relations of deformed twistorial oscillators forSO(6, 2)

We should note that the deformed twistorial operators transform nonlinearly underthe Lorentz group. However their bilinears that enter the generators of the SO∗(8)transform covariantly with respect to the Lorentz group SU∗(4). We give below some ofthe commutators of deformed twistorial oscillators:

[Z 1

1 , Z 21]

= − 32xZ

21 ,[

Z 11 , Z 1

2]

= 12xZ

12 ,[

Z 11 , Z 2

2]

= 12x(Z 1

1 − Z 22),[

Z 11 , Z 1

3]

= − 12x(Z 1

1 − Z 13),[

Z 11 , Z 2

3]

= − 12x(2Z 2

1 + Z 23),[

Z 11 , Z 1

4]

= 12xZ

14 ,[

Z 11 , Z 2

4]

= 12x(Z 1

1 − Z 24),

[Z 1

1 , Z 11

]= 1

2x

(Z 1

1 − Z 11

),[

Z 11 , Z 2

1

]= − 3

2xZ2

1 ,[Z 1

1 , Z 12

]= 1

2xZ1

2 ,[Z 1

1 , Z 22

]= 1

2x

(Z 1

1 − Z 22

),[

Z 11 , Z 1

3

]= − 1

2x

(Z 1

1 − Z 13

),[

Z 11 , Z 2

3

]= − 1

2x

(2Z 2

1 + Z 23

),[

Z 11 , Z 1

4

]= 1

2xZ1

4 ,[Z 1

1 , Z 24

]= 1

2x

(Z 1

1 − Z 24

)

(B.4.1)

111

[Z 1

1 , Y 11] = 12xY

11,[Z 1

1 , Y 12] = − 12x(Z 1

1 + Y 12 + 2),[

Z 11 , Y 21] = − 1

2x(Z 1

1 − Y 21),[Z 1

1 , Y 22] = − 12x(2Z 2

1 + Y 22),[Z 1

1 , Y 31] = 12xY

31,[Z 1

1 , Y 32] = 12x(Z 1

1 − Y 32),[Z 1

1 , Y 41] = 12x(Z 1

1 + Y 41),[Z 1

1 , Y 42] = 12x(2Z 2

1 − Y 42),

[Z 1

1 , Y 11]

= 12xY

11,[Z 1

1 , Y 12]

= − 12x

(Z 1

1 + Y 12 + 2),[

Z 11 , Y 21

]= − 1

2x

(Z 1

1 − Y 21),[

Z 11 , Y 22

]= − 1

2x

(2Z 2

1 + Y 22),[

Z 11 , Y 31

]= 1

2xY31,[

Z 11 , Y 32

]= 1

2x

(Z 1

1 − Y 32),[

Z 11 , Y 41

]= 1

2x

(Z 1

1 + Y 41),[

Z 11 , Y 42

]= 1

2x

(2Z 2

1 − Y 42)

(B.4.2)

[Y 11 , Y 12] = − 3

2xY11,[

Y 11 , Y 21] = − 12xY

11,[Y 11 , Y 22] = − 1

x

(Y 12 + Y 21),[

Y 11 , Y 31] = 0,[Y 11 , Y 32] = 1

2x(Y 11 − 2Y 31),[

Y 11 , Y 41] = 12xY

11,[Y 11 , Y 42] = 1

x

(Y 12 − Y 41),

[Y 11 , Y 11

]= 0,[

Y 11 , Y 12]

= − 32xY

11,[Y 11 , Y 21

]= − 1

2xY11,[

Y 11 , Y 22]

= − 1x

(Y 12 + Y 21

),[

Y 11 , Y 31]

= 0,[Y 11 , Y 32

]= 1

2x

(Y 11 − 2Y 31

),[

Y 11 , Y 41]

= 12xY

11,[Y 11 , Y 42

]= 1x

(Y 12 − Y 41

),

(B.4.3)

112

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Vita

Karan Govil

Born in Gurgaon, India, Karan Govil received the Bachelor and Master of Technologydegrees in Aerospace Engineering from the Indian Institute of Technology Madras,Chennai, India, in July of 2009. He enrolled in the Ph.D. program at the PennsylvaniaState University in August 2009.

algebraic aspects of higher spin symmetries

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