Supersymmetric mechanics vol 1 supersymmetry noncommutativity matrix models 2006 (LNP 698 bellucci s (ed)))(ISBN 3540333134)(239s)

1 A Journey Through Garden Algebras 31.2 GRd, N Algebras 1.2.1 Geometrical Interpretation ofGRd, N Algebras In a field theory, boson and fermions are to be regarded as diffeomorphismsgener

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Stefano Bellucci (Ed.)

Supersymmetric

Mechanics – Vol 1

Supersymmetry, Noncommutativity and Matrix Models

ABC

Stefano Bellucci

Istituto Nazionale di Fisica Nucleare

Via Enrico Fermi, 40

00044 Frascati (Rome), Italy

ISBN-10 3-540-33313-4 Springer Berlin Heidelberg New York

ISBN-13 978-3-540-33313-5 Springer Berlin Heidelberg New York

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To Annalisa, with fatherly love and keen anticipation

This is the first volume in a series of books on the general theme of metric Mechanics, which are based on lectures and discussions held in 2005and 2006 at the INFN-Laboratori Nazionali di Frascati These schools orig-inated from a discussion among myself, my long-time foreign collaborators,and my Italian students We intended to organize these schools as an intenseweek of learning around some specific topics reflecting our current and “tra-ditional” interests In this sense, the choice of topics was both rather specificand concrete, allowing us to put together different facets related to the mainfocus (provided by Mechanics) The selected topics include Supersymmetryand Supergravity, Attractor Mechanism, Black Holes, Fluxes, Noncommuta-tive Mechanics, Super-Hamiltonian Formalism, and Matrix Models

Supersym-All lectures were meant for beginners and covered only half of each day.The rest of the time was dedicated to training, solving of problems proposed

in the lectures, and collaborations One afternoon session was devoted to shortpresentations of recent original results by students and young researchers Theinterest vigorously expressed by all attendees, as well as the initiative of theEditors at Springer Verlag, Heidelberg, prompted an effort by all lecturers,helped in some cases and to various degrees by some of the students, includ-ing myself, to write down the content of the lectures The lecturers made asubstantial effort to incorporate in their write-ups the results of the animateddiscussion sessions that followed their lectures In one case (i.e for the lec-tures delivered by Sergio Ferrara) the outgrowth of the original notes duringthe subsequent reworking, for encompassing recent developments, as well astaking into account the results of the discussion sessions, yielded such a largecontribution as to deserve a separate volume on its own This work is published

as the second volume in this series, Lect Notes Phys 701 “Supersymmetric

Mechanics – Vol 2: The Attractor Mechanism” (2006), ISBN: 3-540-34156-0

A third volume on related topics is in preparation

In spite of the heterogeneous set of lecturers as well as topics, the resultingvolumes have reached a not so common unity of style and a homogeneouslevel of treatment This is in part because of the abovementioned discussions

VIII Preface

that have been taken into account in the write-ups, as well as due to thepedagogical character that inspired the school on the whole In practice, noprevious knowledge by attendees was assumed on the treated topics

As a consequence, these books will be suitable for academic instructionand research training on such topics, both at the postgraduate level, as well

as for young postdoctoral researchers wishing to learn about supersymmetry,supergravity, superspace, noncommutativity, especially in the specific context

of Mechanics

I warmly thank both lecturers and students for their collective work andstrenuous efforts, which helped shaping up these volumes Especially, I wish tomention Professors Ferrara, Gates, Krivonos, Nair, Nersessian and Sochichiufor their clear teaching, enduring patience, and deep learning, as well as inparticular the students Alessio Marrani and Emanuele Orazi for their relent-less questioning, sharp curiosity, and thorough diligence Last, but not least,

I wish to express my gratitude to Mrs Silvia Colasanti, at INFN in Frascati,for her priceless secretarial work and skilled organizing efforts Finally, I amgrateful to my wife Gloria, and my daughters Costanza and Eleonora, for pro-viding me a peaceful and favorable environment for the long hours of workneeded to complete these contributions

December 2005

1 A Journey Through Garden Algebras

S Bellucci, S.J Gates Jr., and E Orazi 1

1.1 Introduction 1

1.2 GR(d, N) Algebras 3

1.2.1 Geometrical Interpretation ofGR(d, N) Algebras 3

1.2.2 Twisted Representations 6

1.2.3GR(d, N) Algebras Representation Theory 7

1.3 Relationships Between Different Models 13

1.3.1 Automorphic Duality Transformations 13

1.3.2 Reduction 17

1.4 Applications 18

1.4.1 Spinning Particle 18

1.4.2 N = 8 Unusual Representations 27

1.5 Graphical Supersymmetric Representation Technique: Adinkras 28

1.5.1 N = 1 Supermultiplets 29

1.5.2 N = 2 Supermultiplets 31

1.5.3 Adinkras Folding 33

1.5.4 Escheric Supermultiplets 34

1.5.5 Through Higher N 37

1.5.6 Gauge Invariance 43

1.6 Conclusions 44

References 46

2 Supersymmetric Mechanics in Superspace S Bellucci and S Krivonos 49

2.1 Introduction 49

2.2 Supersymmetry in d = 1 49

2.2.1 Super-Poincar´e Algebra in d = 1 50

2.2.2 Auxiliary Fields 52

2.2.3 Superfields 54

2.2.4 N = 4 Supermultiplets 56

X Contents

2.3 Nonlinear Realizations 63

2.3.1 Realizations in the Coset Space 63

2.3.2 Realizations: Examples and Technique 65

2.3.3 Cartan’s Forms 67

2.3.4 Nonlinear Realizations and Supersymmetry 71

2.4 N = 8 Supersymmetry 76

2.4.1 N = 8, d = 1 Superspace 77

2.4.2 N = 8, d = 1 Supermultiplets 78

2.4.3 Supermultiplet (4, 8, 4) 85

2.4.4 Supermultiplet (5, 8, 3) 88

2.4.5 Supermultiplet (6, 8, 2) 90

2.4.6 Supermultiplet (7, 8, 1) 92

2.4.7 Supermultiplet (8, 8, 0) 92

2.5 Summary and Conclusions 94

References 94

3 Noncommutative Mechanics, Landau Levels, Twistors, and Yang–Mills Amplitudes V.P Nair 97

3.1 Fuzzy Spaces 97

3.1.1 Definition and Construction ofH N 97

3.1.2 Star Products 99

3.1.3 Complex Projective Space CP k 101

3.1.4 Star Products for Fuzzy CPk 103

3.1.5 The Large n-Limit of Matrices 105

3.2 Noncommutative Plane, Fuzzy CP1, CP2, etc 107

3.3 Fields on Fuzzy Spaces, Schr¨odinger Equation 109

3.4 The Landau Problem on R2 N C and S2 F 111

3.5 Lowest Landau Level and Fuzzy Spaces 113

3.6 Twistors, Supertwistors 115

3.6.1 The Basic Idea of Twistors 115

3.6.2 An Explicit Example 118

3.6.3 Conformal Transformations 118

3.6.4 Supertwistors 119

3.6.5 Lines in Twistor Space 120

3.7 Yang–Mills Amplitudes and Twistors 121

3.7.1 Why Twistors Are Useful 121

3.7.2 The MHV Amplitudes 123

3.7.3 Generalization to Other Helicities 128

3.8 Twistor String Theory 130

3.9 Landau Levels and Yang–Mills Amplitudes 131

3.9.1 The General Formula for Amplitudes 131

3.9.2 A Field Theory on CP1 133

References 135

Contents XI

4 Elements of (Super-)Hamiltonian Formalism

A Nersessian 139

4.1 Introduction 139

4.2 Hamiltonian Formalism 140

4.2.1 Particle in the Dirac Monopole Field 143

4.2.2 K¨ahler Manifolds 145

4.2.3 Complex Projective Space 146

4.2.4 Hopf Maps 147

4.3 Hamiltonian Reduction 149

4.3.1 Zero Hopf Map: Magnetic Flux Tube 151

4.3.2 1st Hopf Map: Dirac Monopole 152

4.3.3CIN +1 →CPI N and T ∗CIN +1 → T ∗CPI N 155

4.3.4 2nd Hopf Map: SU (2) Instanton 156

4.4 Generalized Oscillators 160

4.4.1 Relation of the (Pseudo)Spherical Oscillator and Coulomb Systems 163

4.5 Supersymplectic Structures 167

4.5.1 Odd Super-Hamiltonian Mechanics 171

4.5.2 Hamiltonian Reduction:CIN +1.M →CPI N M , ΛCIN +1 → ΛCPI N 172 4.6 Supersymmetric Mechanics 175

4.6.1N = 2 Supersymmetric Mechanics with K¨ahler Phase Space 177

4.6.2N = 4 Supersymmetric Mechanics 179

4.6.3 Supersymmetric K¨ahler Oscillator 183

4.7 Conclusion 186

References 186

5 Matrix Models C Sochichiu 189

5.1 Introduction 189

5.2 Matrix Models of String Theory 190

5.2.1 Branes and Matrices 190

5.2.2 The IKKT Matrix Model Family 191

5.2.3 The BFSS Model Family 192

5.3 Matrix Models from the Noncommutativity 193

5.3.1 Noncommutative String and the IKKT Matrix Model 193

5.3.2 Noncommutative Membrane and the BFSS Matrix Model 199

5.4 Equations of Motion: Classical Solutions 201

5.4.1 Equations of Motion Before Deformation: Nambu–Goto–Polyakov String 201

5.4.2 Equations of Motion After Deformation: IKKT/BFSS Matrix Models 203

5.5 From the Matrix Theory to Noncommutative Yang–Mills 206

5.5.1 Zero Commutator Case: Gauge Group of Diffeomorphisms 206

5.5.2 Nonzero Commutator: Noncommutative Yang–Mills Model 209

XII Contents

5.6 Matrix Models and Dualities

of Noncommutative Gauge Models 215

5.6.1 Example 1: The U (1) −→ U(n) Map 217

5.6.2 Example 2: Map Between Different Dimensions 220

5.6.3 Example 3: Change of θ 220

5.7 Discussion and Outlook 223

References 223

Index 227

141980 Dubna, Moscow Reg., Russiakrivonos@thsun1.jinr.ru

Alessio Marrani

Centro Studi e Ricerche “E Fermi”Via Panisperna 89A,

00184 Roma, Italyand

INFN-Laboratori Nazionali diFrascati

Via E Fermi 40, C.P 13

00044 Frascati, ItalyAlessio.Marrani@lnf.infn.it

V.P Nair

City College of the CUNYNew York, NY 10031, USAvpn@sci.ccny.cuny.edu

Armen Nersessian

Artsakh State UniversityStepanakert and Yerevan StateUniversity

Alex Manoogian St., 1Yerevan, 375025, Armeniaarnerses@yerphi.am

XIV List of Contributors

Emanuele Orazi

Dipartimento di Fisica

Universit`a di Roma “Tor Vergata”

Via della Ricerca Scientifica 1

andBogoliubov Laboratory ofTheoretical PhysicsJoint Institute for Nuclear Research

141980 Dubna, Moscow Reg., Russiasochichi@mppmu.mpg.de

A Journey Through Garden Algebras

S Bellucci,1S.J Gates Jr.,2 and E Orazi1,3

Scientifica 1, 00133 Roma, Italy

orazi@lnf.infn.it

Abstract The main purpose of these lectures is to give a pedagogical overview on

the possibility to classify and relate off-shell linear supermultiplets in the context

of supersymmetric mechanics A special emphasis is given to a recent graphicaltechnique that turns out to be particularly effective for describing many aspects ofsupersymmetric mechanics in a direct and simplifying way

1.1 Introduction

Sometimes problems in mathematical physics go unresolved for long periods

of time in mature topics of investigation During this World Year of Physics,which commemorates the pioneering efforts of Albert Einstein, it is perhapsappropriate to note the irreconcilability of the symmetry group of Maxwellequations with that of Newton’s equation (via his second law of motion) wasone such problem The resolution of this problem, of course, led to one of thegreatest revolutions in physics This piece of history suggests a lesson on whatcan be the importance of problems that large numbers of physicists regard asunimportant or unsolvable

In light of this episode, the presentation which follows hereafter is focused

on a problem in supersymmetry that has long gone unresolved and seems erally regarded as one of little importance While there is no claim or preten-sion that this problem has the importance of the one resolved by the brilliantgenius of Einstein, it is a problem that perhaps holds the key to a more math-ematically complete understanding of the area known as “supersymmetry.”The topic of supersymmetry is over 30 years old now It has been vigor-ously researched by both mathematicians and physicists During this entiretime, this subject has been insinuated into a continuously widening array of

gen-S Bellucci et al.: A Journey Through Garden Algebras, Lect Notes Phys 698, 1–47 (2006)

DOI 10.1007/3-540-33314-2 1  Springer-Verlag Berlin Heidelberg 2006c

2 S Bellucci et al.

increasingly sophisticated mathematical models At the end of this stream ofdevelopment lies the mysterious topic known as “M-theory.” Accordingly, itmay be thought that all fundamental issues regarding this area have already

One fact that hides this situation is that much of the language used todescribe supersymmetrical theories appears to utilize the superspace formal-ism However, this appearance is deceiving Most often what appears to be

a superspace presentation is actually a component presentation in disguise

A true superspace formulation of a theory is one that uses “unconstrained”superfields as their fundamental variables This is true of a tiny subset ofthe discussions of supersymmetrical theories and is true of none of the mostinteresting such theories involving superstrings

This has led us to the belief that possibly some important fundamentalissues regarding supersymmetry have yet to be properly understood Thisbelief has been the cause of periodic efforts that have returned to this issue.Within the last decade this investigation has pointed toward two new tools

as possibly providing a fresh point of departure for the continued study (andhopefully ultimate resolution) of this problem One of these tools has relied on

a totally new setting in which to understand the meaning of supersymmetry.This has led to the idea that the still unknown complete understanding of therepresentation theory of supersymmetry lies at the intersection of the study

of Clifford algebras and K-theory In particular, a certain class of Cliffordalgebras (to which the moniker GR(d, N) have been attached) provides a

key to making such a connection Within the confines of an interdisciplinaryworking group that has been discussing these problems, the term “gardenalgebra” has been applied to the symbolic nameGR(d, N) It has also been

shown that these Clifford algebras naturally lead to a graphical representationsomewhat akin to the root and weight spaces seen in the classification ofcompact Lie algebras These graphs have been given the name “Adinkras.”The topic of this paper will be introducing these new tools for the study ofsupersymmetry representation theory

1 A Journey Through Garden Algebras 3

1.2 GR(d, N) Algebras

1.2.1 Geometrical Interpretation ofGR(d, N) Algebras

In a field theory, boson and fermions are to be regarded as diffeomorphismsgenerating two different vector spaces; the supersymmetry generators arenothing but sets of linear maps between these spaces Following this pic-ture we can include a supersymmetric theory in a more general geometricalframework defining the collection of diffeomorphisms

φ i : R → R d L , i = 1, , d L (1.1)

ψ αˆ: R → R d R , i = 1, , d R , (1.2)where the one-dimensional dependence reminds us that we restrict our atten-tion to mechanics The free vector spaces generated by{φ i } d L

i=1 and{ψ αˆ} d R

ˆ

α=1

are respectively V L and V R , isomorphic to R d L and R d R For matrix

repre-sentations in the following, the two integers are restricted to the case d L =

d R = d Four different linear mappings can act on V L andV R

M L:V L → V R , M R:V R → V L

U L:V L → V L , U R:V R → V R , (1.3)with linear maps space dimensions

dimM L= dimM R = d R d L = d2,

dimU L = d L2= d2, dimU R = d R2= d2, (1.4)

as a consequence of linearity To relate this construction to a general real(≡ GR) algebraic structure of dimension d and rank N denoted by GR(d, N),

two more requirements need to be added

1 Let us define the generators of GR(d, N) as the family of N + N linear

1

4 S Bellucci et al.

2 After equipping V L and V R with euclidean inner products ·, · V L and

·, · V R, respectively, the generators satisfy the property

The role of{U L } and {U R } maps is to connect different representations once

a set of generators defined by conditions (1.6) and (1.7) has been chosen

Notice that (R I L J)i j ∈ U L and (L I R J)αˆβˆ ∈ U R Let us consider A ∈ {U L }

andB ∈ {U R } such that

In general (1.6) and (1.7) do not identify a unique set of generators Thus,

an equivalence relation has to be defined on the space of possible sets ofgenerators, say{L I , R I } ∼ {L 

I , R  I } if and only if there exist A ∈ {U L } and

i=1and{ψ αˆ} d R

ˆ

α=1can be interpreted as bosonicand fermionic respectively The fermionic nature attributed to the V R ele-ments implies that M and M generators, together with supersymmetry

1 A Journey Through Garden Algebras 5

transformation parameters  I, anticommute among themselves Introducing

the d L + d R dimensional spaceV L ⊕ V R with vectors

Ψ =

φ ψ

It is important to stress that components of (1.23) can be interpreted as perfield components, so it is as if we were working with a particular superfieldmultiplet containing only these physical bosons and fermions From (1.16) it

su-is clear that δ  acts as a supersymmetry generator, so that we can set

δ Q Ψ := δ  Ψ = i I Q I Ψ (1.17)which is equivalent to writing

is not unique: one can check that

δ Q ξ αˆ =  I (L I)αˆi F i ,

δ Q F i=−i I (R I)i αˆ∂ τ ξ αˆ , (1.21)

is another proposal linked to ordinary supersymmetry as the previous one Inthis case we will refer to the supermultiplet defined by (1.21) as the spinorialone

6 S Bellucci et al.

1.2.2 Twisted Representations

The construction outlined above suffers from an ambiguity in the definition of

superfield components (φ i , ψ αˆ) and (ξ αˆ, A i) due to the possibility of

exchang-ing the role of R and L generators, givexchang-ing rise to the new superfields (φ αˆ, ψ i)

and (ξ i , A αˆ) with the same supersymmetric properties of the previous ones.The variations associated to these twisted versions are, respectively

δ Q φ αˆ = i I (L I)αˆi ψ i

δ Q ψ i =− I (R I)i αˆ∂ τ φ αˆ, (1.22)and

δ Q ξ i =  I (R I)i αˆF αˆ,

δ Q F αˆ =−i I (L I)αˆi ∂ τ ξ i (1.23)The examples mentioned above are just some cases of a wider class of in-equivalent representations, referred to as “twisted” ones The possibility topass from a supermultiplet to its twisted version is realized by the so called

“mirror maps.” Moreover, it is possible to define superfields in a completelydifferent manner by parameterizing the supermultiplet using component fieldswhich take value in the algebra vector space We will refer to these objects

as Clifford algebraic superfields An easy way to construct this kind of sentations is tensoring the superspace{V L } ⊕ {V R } with {V L } or {V R } For

repre-instance, if we multiply from the right by{V L } then we have

({V L } ⊕ {V R }) ⊗ {V L } = {U L } ⊕ {M L } (1.24)whose fields content is

to{U R } ⊕ {M L }, {U L } ⊕ {M R }, and {U R } ⊕ {M R } type superspaces Even

in these cases, twisted versions can be constructed applying considerationssimilar to those stated above The important difference between the Cliffordalgebraic superfields approach and the V R ⊕ V L superspace one, resides inthe fact that in the latter case the number of bosonic fields (which actuallydescribe coordinates) increases with the number of supersymmetric charges,while in the first case there is a way to make this not happen, allowing for adescription of arbitrary extended supersymmetric spinning particle systems,

as it will be shown in the third section

1 A Journey Through Garden Algebras 7

1.2.3 GR(d, N) Algebras Representation Theory

It is time to clarify the link with real Clifford Γ -matrices of Weyl type ( ≡

block skew diagonal) space which is easily seen to be

η AB = diag(1, , 1

  

N

In the following we assume that A, B indices run from 1 to N + 1 while

I, J run from 1 to N The generator (1.29), that has the interpretation of a

fermionic number, allow us to construct the following projectors on bosonicand fermionic sectors:

8 S Bellucci et al.

In this way, we have just demonstrated that representations ofGR(d, N) are

in one-to-one correspondence with the real-valued representations of Cliffordalgebras, which will be classified in the following using considerations of [3]

To this end, let M be an arbitrary d × d real matrix and let us consider

Equation (1.39) tells us that for all Γ A ∈ C(p, q) there exists at least one S

such that [Γ A , S] = 0 Thus, by Shur’s lemma, S has to be invertible (if not

vanishing) It follows that any set of such M matrices defines a real division

algebra As a consequence of a Frobenius theorem, three possibilities existthat we are going to analyze

1 Normal representations (N) The division algebra is generated by the

identity only

2 Almost complex representations (AC) There exists a further

divi-sion algebra real matrix J such that J2=−I and we have

S = µI + νJ, µ, ν ∈ R (1.42)

3 Quaternionic representations (Q) Three elements E1, E2, and E3

satisfying quaternionic relations

S = µI + νE1+ ρE2+ σE3, µ, ν, ρ, σ ∈ R (1.44)

The results about irreducible representations obtained in [3] for C(p, q)

are summarized in Table 1.1

The dimensions of irreducible representations are referred to faithful ones

except the p − q = 1, 5 cases where exist two inequivalent representations of

the same dimension, related to each other by ¯Γ A =−Γ A To obtain faithfulrepresentations, the dimensions of those cases should be doubled defining

1 A Journey Through Garden Algebras 9

Table 1.1 Representation dimensions for C(p, q) algebras

to values of p − q from 0 to 7 As mentioned in [4], the dimensions reported

in Table 1.1 can be expressed as functions of the signature (p, q) introducing integer numbers k, l, m, and n such that

q = 8k + m, 0≤ m ≤ 7 ,

p = 8l + m + n, 1≤ n ≤ 8 , (1.49)

where n fix p − q up to l − k multiples of eight as can be seen from

p − q = 8(l − k) + n , (1.50)

while m encode the p, q choice freedom keeping p − q fixed Obviously, k and

l take into account the periodicity properties The expression of irreducible

representation dimensionalities reads

d = 2 4k+4l+m F (n) , (1.51)

where F (n) is the Radon–Hurwitz function defined by

F (n) = 2 r , [log2n] + 1 ≥ r ≥ [log2n], r ∈ N (1.52)Turning back to GR(d, N) algebras, from (1.31) we deduce that we have

to deal only with C(N, 1) case which means that irreducible representation dimensions depend only on N in the following simple manner:

of Table 1.1 to the C(N, 1) case are summarized in Table 1.2.

In what follows we focus our attention to the explicit representation’sconstruction First of all we enlarge the set of linear mappings acting between

V L and V R, namely M L ⊕ M R (i.e., GR(d, N)), to U L ⊕ U R defining theenveloping general real algebra

EGR(d, N) = M L ⊕ M R ⊕ U L ⊕ U R (1.54)

As noticed before, we have the possibility to construct elements ofU LandU R

as products of alternating elements ofM Land M R so that

L I R J , L I R J L K R L , ∈ U R ,

R I L J , R I L J R K L L , ∈ U L , (1.55)

but L I and R J come from C(N, 1) through (1.35) so all the ingredients are

present to develop explicit representation ofEGR(d, N) starting from Clifford

algebra

We focus now on the building of enveloping algebras’ representations ing from Clifford algebras Indeed, we need to divide the representations intothree cases

start-1 Normal representations In this case basic definition of Clifford algebra

(1.30) suggests a way to construct a basis {Γ } by wedging Γ matrices {Γ } = {I, Γ I , Γ IJ , Γ IJ K , , Γ N +1 }, I < J < K , (1.56)

where Γ I, ,J are to be intended as the antisymmetrization of Γ I · · · Γ J

matrices otherwise denoted by Γ [I · · · Γ J ] or Γ [N ] if the product involve

N elements Dividing into odd and even products of Γ we obtain the sets

{Γ e } = {I, Γ N +1 , Γ IJ , Γ IJ Γ N +1 , Γ IJ KL , } , {Γ o } = {Γ I , Γ I Γ N +1 , Γ IJ K , Γ IJ K Γ N +1 } , (1.57)respectively related to {M} and {U} spaces Projectors (1.32) have the

key role to separate left sector from right sector In fact, for instance, wehave

1 A Journey Through Garden Algebras 11

P+Γ IJ P+= P+Γ [I Γ J ] P+P+= P+Γ [I P − Γ J ] P+

= P+Γ [I P − P − Γ J ] P+ = R [I L J ] ∈ {U L } , (1.58)

and in a similar way P − Γ IJ P − = L [I R J ] ∈ {U R }, P+Γ IJ K P − =

R [I L J R K] ∈ {M R }, and so on Those remarks provide the following

solution

{U R } = { P − , P − Γ IJ P − , , P − Γ [N ] P − } ≡ { I V R , L [I R J ] , } , {M R } = {P+Γ I P − , , P+Γ [N −1] P

− } ≡ { R I , R [I L J R K] , }, {U L } = { P+, P+Γ IJ P+, , P+Γ [N ] P+} ≡ { I V L , R [I L J ] , } , {M L } = {P − Γ I P+, , P − Γ [N −1] P+} ≡ { L I , L [I R J L K] , } ,

(1.59)which we will denote as ∧GR(d, N) to remember that it is constructed

by wedging L I and R J generators Clearly enough, from each Γ [I, ,J ]

matrix we get two elements of EGR(d, N) algebra as a consequence of

the projection Thus, we can say that in the normal representation case,

C(N, 1) is in 1–2 correspondence with the enveloping algebra which can be

identified by∧GR(d, N) By the wedging construction in (1.59) naturally

arise p-forms that is useful to denote

The superfield components for the {U L } ⊕ {M L } type superspace

intro-duced in (1.25), can be expanded in terms of this normal basis as follows

φ i j = φ δ i j + φ IJ (f IJ)i j+· · · ∈ {U L } ,

ψ αˆi = ψ I (f I)αˆi + ψ IJ K (f IJ K)αˆi+· · · ∈ {M L } (1.61)

according to the fact that f[even]∈ {U L } and f[odd]∈ {M L } We will refer

to this kind of supefields as bosonic Clifford algebraic ones because of thebosonic nature of the level zero field Similar expansion can be done for the

{U R } ⊕ {M R } type superspace where ˆ f[even]∈ {U R } and ˆ f[odd]∈ {M R }

φ αˆβˆ= φ δ αˆβˆ+ φ IJ( ˆf IJ)αˆβˆ+· · · ∈ {U R } ,

ψ i αˆ= ψ I( ˆf I)i αˆ+ ψ IJ K( ˆf IJ K)i αˆ+· · · ∈ {M R } (1.62)

In the (1.62) case we deal with a fermionic Clifford algebraic superfield

because the component φ is a fermion For completeness we include the

remaining cases, namely{U } ⊕ {M } superspace

12 S Bellucci et al.

φ αˆβˆ= φ δ αˆβˆ+ φ IJ( ˆf IJ)αˆβˆ+· · · ∈ {U R } ,

ψ αˆi = ψ I (f I)αˆi + ψ IJ K (f IJ K)αˆi+· · · ∈ {M L } , (1.63)and{U L } ⊕ {M R } superspace

φ i j = φ δ i j + φ IJ (f IJ)i j+· · · ∈ {U L } ,

ψ i αˆ= ψ I( ˆf I)i αˆ+ ψ IJ K( ˆf IJ K)i αˆ+· · · ∈ {M R } (1.64)

2 Almost complex representations As already pointed out, those kind

of representations contain one more generator J with respect to normal

representations so that to span all the space, the normal part, which isgenerated by wedging, is doubled to form the basis for the Clifford algebra

{Γ } = {I, J, Γ I , Γ I J, Γ IJ , Γ IJ J, , Γ N +1 , Γ N +1 J } (1.65)Starting from (1.65) it is straightforward to apply considerations from(1.57) to (1.59) to end with aEGR(d, N) almost complex representation

in 1–2 correspondence with the previous Concerning almost complex ford algebra superfields, it is important to stress that we obtain irreduciblerepresentations only restricting to the normal part

Clif-3 Quaternionic representations Three more generators E α satisfying

Even in this case only the normal part gives irreducible representationsfor the Clifford algebra superfields

Notice that from the group manifold point of view, the presence of the

generator J for the almost complex case and generators E α for the nionic one, separate the manifold into sectors which are not connected byleft or right group elements multiplication giving rise to intransitive spaces.Division algebra has the role to link those different sectors

quater-Finally we explain how to produce an explicit matrix representation using

a recursive procedure mentioned in [1] that can be presented in the following

manner for the case N = 8a + b with a ≥ 1:

L1= iσ2⊗ I b ⊗ I 8a = R1,

L I = σ3⊗ (L b)I ⊗ I 8a = R I , 1 ≤ I ≤ b − 1 ,

L J = σ1⊗ I b ⊗ (L 8a)J = R J , 1 ≤ I ≤ 8a − 1 ,

1 A Journey Through Garden Algebras 13

where I n stands for the n-dimensional identity matrix while L b and L 8a are

referred respectively to the cases N = b and N = 8a Expressions for the cases where N ≤ 7 which are the starting points to apply the algorithm in (1.68),

can be found in appendix A of [5]

1.3 Relationships Between Different Models

It turns out that apparently different supermultiplets can be related to eachother using several operations

1 Leaving N and d unchanged, one can increase or decrease the number

of physical bosonic degrees of freedom (while necessarily and ously to decrease or increase the number of auxiliary bosonic degrees offreedom) within a supermultiplet by shifting the level of the superfield

simultane-θ-variables expansion by mean of an automorphism on the superalgebra

representation space, commonly called automorphic duality (AD)

2 It is possible to reduce the number of supersymmetries maintaining fixedrepresentation dimension (reduction)

3 The space-time coordinates can be increased preserving the tries (oxidation)

supersymme-4 By a space-time compactification, supersymmetries can be eventually creased

in-These powerful tools can be combined together to discover new tiplets or to relate the known ones The first two points will be analyzed inthe following paragraphs while for the last two procedures, we remind to [6, 7]and references therein

supermul-1.3.1 Automorphic Duality Transformations

Until now, we encountered the following two types of representations: the firstone defined onV LandV Rsuperspace complemented with the second one, Clif-ford algebraic superfields In the latter case we observed that in order to obtainirreducible representations, it is needed a restriction to normal representations

or to their normal parts If we consider irreducible cases of Clifford algebraicsuperfields then there exists the surprising possibility to transmute physicalfields into auxiliary ones changing the supermultiplet degrees of freedom dy-namical nature The best way to proceed for an explanation of the subject is

to begin with the N = 1 example which came out to be the simplest In this

case only two supermultiplets are present:

• the scalar supermultiplet (X, ψ) respectively composed of one bosonic and

one fermionic field arranged in the superfield

X(τ, θ) = X(τ ) + iθψ(τ ) , (1.69)

14 S Bellucci et al.

with transformation properties

δ Q X = iψ ,

• the spinor supermultiplet (ξ, A) respectively composed of one bosonic and

one fermionic field arranged in the superfield

Y (τ, θ) = ξ(τ ) + θA(τ ) , (1.71)with transformation properties

δ Q A = i∂ τ ξ ,

The invariant Lagrangian for the scalar supermultiplet transformations (1.70)

L = ˙ X2+ igψ ˙ ψ , (1.73)

gives to the fields X and ψ a dynamical meaning and offers the possibility to

perform an automorphic duality map that at the superfield level reads

Y (τ, θ) = −iDX(τ, θ) , (1.74)whereD = ∂ θ + iθ∂ τ is the superspace covariant derivative At the componentlevel (1.74) corresponds to the map upon bosonic components

X(τ ) = ∂ τ −1 A(τ ) , (1.75)and identification of fermionic ones The mapping (1.96) is intrinsically notlocal but it can be implemented in a local way both in the transformations(1.70) and in the Lagrangian (4.220) producing respectively (1.72) and theLagrangian

L = A2+ igψ ˙ ψ (1.76)

As a result we get that automorphic duality transformations map N = 1

supermultiplets into each other in a local way, changing the physical

mean-ing of the bosonic field X from dynamical to auxiliary A (not propagatmean-ing)

as is showed by the Lagrangian (1.76) invariant for (1.72) transformations

Note that the auxiliary meaning of A is already encoded into (1.72)

trans-formations that enlighten on the nature of the fields and consequently of thesupermultiplet

Let us pass to the analysis of the N = 2 case making a link with the siderations about representation theory discussed above At the N = 2 level,

con-we deal with an AC representation so, in order to implement AD tions, we focus on the normal part, namely∧GR(d, N), defining the Clifford

transforma-algebraic bosonic superfield

1 A Journey Through Garden Algebras 15

φ j i = φδ j i + φ IJ (f IJ)j i ,

constructed with the forms (1.60) Notice that if we work in an N -dimensional space then the highest rank for the forms is N This is the reason why, writing (1.77), we stopped at f IJ level Some comments about transformation prop-erties By comparing each level of the expansion, it is straightforward to provethat superfields (1.61) transform according to (1.26) if the component fieldtransformations are recognized to be

δ Q φ I1···I peven =−i [I1ψ I2···I peven]+ i(peven+ 1) J ψ I1···I peven J ,

δ Q ψ I1···I podd =− [I1φ˙I2···I podd]+ i(podd+ 1) J φ˙I1···I podd J (1.78)

Therefore (1.78) for the N = 2 case read

of [8], we adhere to a convention that list three numbers (P B, P F, AB) where

P B denotes the number of “propagating” bosonic fields, AB denotes the

num-ber of “auxiliary” bosonic fields, and P F denotes the numnum-ber of “fermionic”

16 S Bellucci et al.

• Finally, if both redefinitions (1.80) and (1.82) are adopted, then we are

left with (0, 2, 2) spinor supermultiplet whose components behave as

From (1.87), one may argue that C is auxiliary while D is physical The point

is which is the meaning of the fields we started from? An invariant action from(1.87) is

L = C2+ igξ ˙ ξ + ˙ D IJ D˙IJ , (1.88)

so that going backward, we can deduce the initial action

L = ¨φ2+ ig ˙ ψ ¨ ψ + ¨ φ IJ φ¨IJ (1.89)The examples above should convince any reader that Clifford superfieldsare a starting point to construct a wider class of representation by means of

AD maps Following this idea, one can identify each supermultiplet with a

correspondent root label (a1, , a k)± where a i ∈ Z are defined according to

type For instance, the last supermultiplet (1.87), corresponds to the case

(a0, a1, a2)± = (2, −1, 1)+ We name base superfield the one with all zero in

the root label (0, , 0) ±, underling that in the plus (minus) case, this multiplet has to be intended as the one with all bosons (fermion) differentiated

super-in the r.h.s of variations They are of particular super-interest super-in the plets whose root labels involve only 0 and 1 All these supermultiplets formwhat we call root tree

supermulti-1 A Journey Through Garden Algebras 17

1.3.2 Reduction

It is shown in Table 1.2 that N = 8, 7, 6, 5 irreducible representations have the same dimension The same happens for the N = 4, 3 cases This fact re-

flects the possibility to relate those supermultiplets via a reduction procedure

To explain how this method works, consider a form f I1···I K and notice that

the indices I1, , I K run on the number of supersymmetries: reducing this

number corresponds to diminishing the components contained in the rank k

form The remaining components have to be rearranged into another form

For instance, if we consider a 3-rank form for the N = 8 case then the number

of components is given by2 8

3



= 56 but, reducing to the N = 7 case and

leaving invariant the rank, we get7

3



= 35 components The remaining onescan be rearranged in a 5-rank form This means that the maximum rank ofClifford superfield expansion is raised until the irreducible representation di-mension is reached However, the right way to look at this rank enhancing isthrough duality An enlightening example will be useful By a proper count-ing of irreducible representation dimension for theEGR(8, 8), we are left with {U L } ⊕ {M L } type Clifford algebraic superfield

φ ij = φδ ij + φ IJ (f IJ)ij + φ IJ KL (f IJ KL)ij

ψ αiˆ = ψ I (f I)αiˆ + ψ IJ K (f IJ K)αiˆ , (1.91)where the 4-form has definite duality or, more precisely, the sign in the equa-tion

 IJ KLM N P Q f M N P Q=±f IJ KL , (1.92)has been chosen, halving the number of independent components To reduce

to the N = 7 case, we need to eliminate all “8” indices and this can be done

by exploiting the duality For instance, f I8can disappear if transformed into

 IJ KLM N P 8 f P 8=±f IJ KLM N (1.93)This trick adds the 6-rank to the expansion manifesting the enhancing phe-nomenon previously discussed Once the method is understood, it is straight-

forward to prove that for the N = 7 case, the proper superfield expression

is

φ ij = φδ ij + φ IJ (f IJ)ij + φ IJ KL (f IJ KL)ij + φ IJ KLM N (f IJ KLM N)ij

ψ iˆ j = ψ I (f I)iˆ j + ψ IJ K (f IJ K)iˆ j + ψ IJ KLM (f IJ KLM)iˆ j

+ψ IJ KLM P Q (f IJ KM N P Q)iˆ j (1.94)

The explicit reduction procedure for N ≤ 8 can be found in [2] and

summa-rized in Tables 1.3 and 1.4

chiral multiplet has been used in this connection [10], as well as in related tasks[11]

18 S Bellucci et al.

Table 1.3.EGR(4, 4) and its reduction: algebras representation in terms of forms

and division algebra

EGR(d, N) Λ GR(d, N) basis Division Structure

EGR(4, 4) {U

L } = {I, f IJ , E µˆ, f IJ E µˆ} {M L } = {f I , f I Eˆµˆ} E

ˆ

µ , ˆ E µˆ

EGR(4, 3) {U

L } = {I, f IJ , E µˆ, f IJ E µˆ} {M L } = {f I , f I Eˆµˆ, f IJ K , f IJ K Eˆµˆ} E

Before we begin a detailed analysis of spinning particle system it is important

to understand what a spinning particle is Early models of relativistic particlewith spin involving only commuting variables can be divided into the twofollowing classes:

• vectorial models, based on the idea of extending Minkowski space-time by

vectorial internal degrees of freedom;

• spinorial models, characterized by the enhancing of configuration space

using spinorial commuting variables

These models lack the following important requirement: after first tion, they never produce relativistic Dirac equations Moreover, in the spino-rial cases, a tower of all possible spin values appear in the spectrum Furtherprogress in the development of spinning particle descriptions was achieved bythe introduction of anticommuting variables to describe internal degrees of

quantiza-freedom [12] This idea stems from the classical limit (h → 0) formulation of

Fermi systems [13], the so called “pseudoclassical mechanics” referring to thefact that it is not an ordinary mechanical theory because of the presence ofGrassmannian variables By means of pseudoclassical approach, vectorial andspinorial models can be generalized to “spinning particle” and “superparticle”

models, respectively In the first case, the extension to superspace (x , θ , θ )

1 A Journey Through Garden Algebras 19

Table 1.4.EGR(8, 8) and its reductions EGR(d, N) Λ GR(d, N) Basis Division Structure

EGR(8, 8) {U

L } = {I, f IJ , f IJ KL } {M L } = {f I , f IJ K }

I

EGR(8, 7) {U

L } = {I, f IJ , f IJ KL , f IJ KLM N } {M L } = {f I , f IJ K , f[5], f[7]}

I

EGR(8, 6) {U

L } = {I, f I7 , f IJ , f IJ K7 , f IJ KL , f[5]7, f[6]} {M L } = {f7, f I , f IJ 7 , f IJ K , , f[4]7, f[5], f[6]7}

D = f7

EGR(8, 5)

{U L } = {I, f67, f I6 , f I7 , f IJ , f IJ 67 , f IJ K7 ,

f IJ KL , f IJ KL67 , f[5]7, f[5]6} {M L } = {f7, f6, f I , f I67 , f IJ 7 , f IJ 6 , f IJ K ,

f IJ k67 , f[4]7, f[4]6, f[5], f[5]67, }

E µˆ = (f67, f[5]6, f[5]7)ˆ

E µˆ = (f7, f6, f[5]67)

Here the subscript [n] is used in place of n anticommuting indices.

is made possible by a pseudovector θ µ and a pseudoscalar θ5[14, 15] The

pres-ence of vector index associated with θ-variables implies the vectorial character

of the model

In the second case, spinorial coordinates are considered, giving rise to nary superspace approach whose underlying symmetry is the super-Poincar´egroup (eventually extended) [13] The superparticle is nothing but a general-ization of relativistic point particle to superspace

ordi-It turns out that after first quantization, the spinning particle model duced Dirac equations and all Grassmann variables are mapped into Cliffordalgebra generators Superfields that take values on this kind of quantized su-perspace are precisely Clifford algebraic superfields described in the previousparagraphs On the other side, a superspace version of Dirac equation arises

pro-from superparticle quantization Moreover, θ-variables are still present in the

quantized version

20 S Bellucci et al.

To have a more precise idea, we spend a few words discussing the Barducci–Casalbuoni–Lusanna model [14], which is one of the first works on pseudo-classical model As already mentioned, it is assumed that the configuration

space to be described by (x µ , θ µ , θ5) The Lagrangian of the system

after a canonical analysis These (1.97) are classical limits of Klein–Gordon

and Dirac equations, respectively Moreover, the first quantization maps

θ-variables into Clifford algebra generators

1 A Journey Through Garden Algebras 21

δψ µ = (τ )P µ , δφ µ = i(τ )ψ µ , δP µ = 0 ,

δe = i(τ )χ, δχ = 2 ˙(τ ) , (1.101)corresponding to pure supergravity transformations as is shown by calculatingthe commutators

[δ 1, δ 2] X µ = ξ ˙ X µ + i˜ ψ µ ,

[δ 1, δ 2] ψ µ = ξ ˙ ψ µ+ ˜P µ ,

[δ 1, δ 2] e = ξ ˙e + ˙ ξe + i˜ χ ,

[δ 1, δ 2] χ = ξ ˙ χ + ˙ ξχ + 2 ˙˜  , (1.102)where

forwardly from actions (1.106) and (1.100) eliminating the P fields using their

equations of motion

An advance on this line of research yielded the on-shell N-extension [17] However, a satisfactory off-shell description with arbitrary N requires the

GR(d, N) approach In the paragraphs below we describe in detail how this

construction is worked out

Second-Order Formalism for Spinning Particle

with Rigid N -Extended Supersymmetry

The basic objects of this model are Clifford algebraic bosonic superfields ued in{U L } ⊕ {M L } superspace with transformations (1.26) One can easily

val-check that the action

(ψ1)i = ψ I (L I)i+ ˜ψ i = ψ I (L I)i + µ i , (1.114)

1 A Journey Through Garden Algebras 23

S =



dτ {d(∂ τ X∂ τ X − iψ I ∂ τ ψ I) +F j F j

+ iµ i αˆ∂ τ µ αˆi } (1.123)

24 S Bellucci et al.

that represent the second-order approach to the spinning particle problemwith global supersymmetry A remarkable difference between the AD pre-sented in Subsect 1.3.1 and the AD used to derive USPM resides in the factthat in the latter case we map ˜φ i j and ˜ψ i

ˆ

α which are Clifford algebraic perfield while in the previous we work at the component level Finally, it isimportant to keep in mind that the superfields ˜φ i j and ˜ψ i

su-ˆ

α take values on thenormal part of the enveloping algebra which is equivalent to say that they can

be expanded on the basis (1.60)

First-Order Formalism for Spinning Particle

with Rigid N -Extended Supersymmetry

To formulate a first-order formalism, one more fermionic supermultiplet is

required This time the superfields ((φ2)i j , (ψ2)αˆ

i ), valued in{U L } ⊕ {M R }

superspace, transform according to

δ Q (φ2)i j =−i I (L I)αˆj ∂ τ (ψ2)i αˆ

δ Q (ψ2)i αˆ =  I (R I)j αˆ(φ2)i j (1.124)The expansions needed turns out to be

α) has to be treated in the

fol-lowing different way: the off-trace superfield µ i

1 A Journey Through Garden Algebras 25

The action can be thought as the sum of two separated pieces

second-i , ˆ G i j), which is fermionic in nature

i plays a different role with respect to theother off-trace component because it is responsible, by its equation of motion,

for setting the mass equal to M It can be easily recognized the resemblance

between the Sherk–Shwartz method [18] and the above way to proceed if weinterpret the mass multiplet as a (D + 1)th Minkowski momentum compo-nent without coordinate analogue We underline that the “mass multiplet”(1.133) is crucial if we want to insert the mass and preserve the preexisting

symmetries, as it happens for the ψ field (1.105)

26 S Bellucci et al.

First- and Second-Order Formalism for Spinning Particle Coupled

to Minimal N -Extended Supergravity

For completeness, we include the coupling of the above models to minimal dimensional supergravity The supergravity multiplet escape from GR(d, N)

one-embedding because its off-shell fields content (e, χ I),

pled to N -extended supergravity on the worldline Equation of motion ciated to P field reads

1 A Journey Through Garden Algebras 27

that, substituted in (1.139), give us the second-order formulation

1.4.2 N = 8 Unusual Representations

There exists also some “unusual” representations in this approach to dimensional supersymmetric quantum mechanics As an illustration of these,the discussion will now treat such a case forGR(8, 8) It may be verified that

one-a suitone-able representone-ation is provided by the 8× 8 matrices

where I, J, K, etc now take on the values 1, 2, ,8 Proper closure of the

supersymmetry algebra requires in addition to (1.6) also the fact that

(R N)KJ (L N)IM + (R N)KI (L N)J M =− 2δ IJ δKM, (1.145)which may be verified for the representation in (1.143) This is the fact thatidentifies the representation in (1.144) as being an “unusual” representation