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Stefano Bellucci (Ed.)
Supersymmetric
Mechanics – Vol. 1
Supersymmetry, Noncommutativity
and Matrix Models
ABC
Editor
StefanoBellucci
IstitutoNazionalediFisicaNucleare
ViaEnricoFermi,40
00044Frascati(Rome),Italy
E-mail:[email protected]
S.Bellucci,SupersymmetricMechanics–Vol.1,Lect.NotesPhys.698(Springer,Berlin
Heidelberg2006),DOI10.1007/b11730286
LibraryofCongressControlNumber:2006926534
ISSN0075-8450
ISBN-10 3-540-33313-4SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-33313-5SpringerBerlinHeidelbergNewYork
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To Annalisa, with fatherly love and keen anticipation
Preface
ThisisthefirstvolumeinaseriesofbooksonthegeneralthemeofSupersym-
metric Mechanics, which are based on lectures and discussions held in 2005
and 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 intense
week of learning around some specific topics reflecting our current and “tra-
ditional” interests. In this sense, the choice of topics was both rather specific
and concrete, allowing us to put together different facets related to the main
focus (provided by Mechanics). The selected topics include Supersymmetry
and Supergravity, Attractor Mechanism, Black Holes, Fluxes, Noncommuta-
tive Mechanics, Super-Hamiltonian Formalism, and Matrix Models.
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
inthelectures,andcollaborations.Oneafternoonsessionwasdevotedtoshort
presentationsofrecentoriginalresultsbystudentsandyoungresearchers.The
interest vigorously expressed by all attendees, as well as the initiative of the
Editors 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 a
substantialefforttoincorporateintheirwrite-upstheresultsoftheanimated
discussion sessions that followed their lectures. In one case (i.e. for the lec-
tures delivered by Sergio Ferrara) the outgrowth of the original notes during
the subsequent reworking, for encompassing recent developments, as well as
taking into account the results of the discussion sessions, yielded such a large
contributionastodeserveaseparatevolumeonitsown.Thisworkispublished
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.
Inspiteoftheheterogeneoussetoflecturersaswellastopics,theresulting
volumes have reached a not so common unity of style and a homogeneous
level 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 the
pedagogical character that inspired the school on the whole. In practice, no
previous knowledge by attendees was assumed on the treated topics.
As a consequence, these books will be suitable for academic instruction
and 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,especiallyinthespecificcontext
of Mechanics.
I warmly thank both lecturers and students for their collective work and
strenuousefforts,whichhelpedshapingupthesevolumes.Especially,Iwishto
mention Professors Ferrara, Gates, Krivonos, Nair, Nersessian and Sochichiu
for their clear teaching, enduring patience, and deep learning, as well as in
particular 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 am
gratefultomywifeGloria,andmydaughtersCostanzaandEleonora,forpro-
viding me a peaceful and favorable environment for the long hours of work
needed to complete these contributions.
Frascati, Italy Stefano Bellucci
December 2005
Contents
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
1.2.1Geometrical Interpretation of GR(d, N) Algebras ............ 3
1.2.2Twisted Representations.................................. 6
1.2.3GR(d, N) Algebras Representation Theory.................. 7
1.3 Relationships Between Different Models ......................... 13
1.3.1Automorphic Duality Transformations...................... 13
1.3.2Reduction .............................................. 17
1.4 Applications ................................................. 18
1.4.1Spinning Particle ........................................ 18
1.4.2 N =8 Unusual Representations........................... 27
1.5 Graphical Supersymmetric Representation Technique: Adinkras .... 28
1.5.1N =1 Supermultiplets ................................... 29
1.5.2N =2 Supermultiplets ................................... 31
1.5.3Adinkras Folding ........................................ 33
1.5.4Escheric Supermultiplets.................................. 34
1.5.5Through Higher N ....................................... 37
1.5.6Gauge 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
2.2.1Super-Poincar´e Algebra in d=1........................... 50
2.2.2Auxiliary Fields ......................................... 52
2.2.3Superfields.............................................. 54
2.2.4N = 4 Supermultiplets ................................... 56
X Contents
2.3 Nonlinear Realizations ........................................ 63
2.3.1Realizations in the Coset Space............................ 63
2.3.2Realizations: Examples and Technique...................... 65
2.3.3Cartan’s Forms.......................................... 67
2.3.4Nonlinear Realizations and Supersymmetry ................. 71
2.4 N = 8 Supersymmetry ........................................ 76
2.4.1N = 8, d = 1 Superspace ................................. 77
2.4.2N = 8, d = 1 Supermultiplets ............................. 78
2.4.3Supermultiplet (4, 8, 4).................................. 85
2.4.4Supermultiplet (5, 8, 3).................................. 88
2.4.5Supermultiplet (6, 8, 2).................................. 90
2.4.6Supermultiplet (7, 8, 1).................................. 92
2.4.7Supermultiplet (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
3.1.1Definition and Construction of H ......................... 97
N
3.1.2Star Products ........................................... 99
3.1.3Complex Projective Space CPk............................101
3.1.4Star Products for Fuzzy CPk .............................103
3.1.5The Large n-Limit of Matrices ............................105
3.2 Noncommutative Plane, Fuzzy CP1, CP2, etc....................107
3.3 Fields on Fuzzy Spaces, Schro¨dinger Equation....................109
3.4 The Landau Problem on R2 and S2 ..........................111
NC F
3.5 Lowest Landau Level and Fuzzy Spaces .........................113
3.6 Twistors, Supertwistors .......................................115
3.6.1The Basic Idea of Twistors................................115
3.6.2An Explicit Example.....................................118
3.6.3Conformal Transformations ...............................118
3.6.4Supertwistors ...........................................119
3.6.5Lines in Twistor Space ...................................120
3.7 Yang–Mills Amplitudes and Twistors ...........................121
3.7.1Why Twistors Are Useful .................................121
3.7.2The MHV Amplitudes....................................123
3.7.3Generalization to Other Helicities..........................128
3.8 Twistor String Theory ........................................130
3.9 Landau Levels and Yang–Mills Amplitudes ......................131
3.9.1The General Formula for Amplitudes.......................131
3.9.2A 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
4.2.1Particle in the Dirac Monopole Field .......................143
4.2.2Ka¨hler Manifolds ........................................145
4.2.3Complex Projective Space ................................146
4.2.4Hopf Maps..............................................147
4.3 Hamiltonian Reduction .......................................149
4.3.1Zero Hopf Map: Magnetic Flux Tube.......................151
4.3.21st Hopf Map: Dirac Monopole ............................152
4.3.3CI N+1 →CI PN and T∗CI N+1 →T∗CIPN .....................155
4.3.42nd Hopf Map: SU(2) Instanton ..........................156
4.4 Generalized Oscillators........................................160
4.4.1Relation of the (Pseudo)Spherical Oscillator
and Coulomb Systems....................................163
4.5 Supersymplectic Structures ....................................167
4.5.1Odd Super-Hamiltonian Mechanics.........................171
4.5.2Hamiltonian Reduction: CI N+1.M →CI PN.M, ΛCI N+1 →ΛCI PN .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
4.6.3Supersymmetric Ka¨hler 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
5.2.1Branes and Matrices .....................................190
5.2.2The IKKT Matrix Model Family ..........................191
5.2.3The BFSS Model Family .................................192
5.3 Matrix Models from the Noncommutativity......................193
5.3.1Noncommutative String and the IKKT Matrix Model ........193
5.3.2Noncommutative Membrane and the BFSS Matrix Model.....199
5.4 Equations of Motion: Classical Solutions ........................201
5.4.1Equations of Motion Before Deformation:
Nambu–Goto–Polyakov String.............................201
5.4.2Equations of Motion After Deformation:
IKKT/BFSS Matrix Models...............................203
5.5 From the Matrix Theory
to Noncommutative Yang–Mills ................................206
5.5.1Zero Commutator Case: Gauge Group of Diffeomorphisms ....206
5.5.2Nonzero Commutator: Noncommutative Yang–Mills Model....209
