D-Branes, Guage-String Duality and Noncommutative Theories [thesis]

293 Pages·2004·1.844 MB·English

by  T. Mateos

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Ph. D. Thesis on
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v D-branes, gauge/string duality
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N and noncommutative theories

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Toni Mateos
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Advisor: Joaquim Gomis Torn´e
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h Departament d’Estructura i Constituents de la Mat`eria
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v Universitat de Barcelona
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Barcelona, April 2004
Thesis defended on June 19th 2004

This thesis is mainly based on the following published articles:
1. J.Gomis, K.KamimuraandT.Mateos, “GaugeandBRSTgenerators
for space-time non-commutative U(1) theory,” JHEP 0103(2001)010
[arXiv:hep-th/0009158].
2. T.MateosandA.Moreno,“Anoteonunitarityofnon-relativisticnon-
commutative theories,” Phys. Rev. D 64 (2001) 047703 [arXiv:hep-
th/0104167].
3. J. Gomis and T. Mateos, “D6 branes wrapping Kaehler four-cycles,”
Phys. Lett. B 524 (2002) 170 [arXiv:hep-th/0108080].
4. J. Brugues, J. Gomis, T. Mateos and T. Ramirez, “Supergravity duals
of noncommutative wrapped D6 branes and supersymmetry without
supersymmetry,” JHEP 0210 (2002) 016 [arXiv:hep-th/0207091].
5. T. Mateos, J. M. Pons and P. Talavera, “Supergravity dual of non-
commutative N = 1 SYM,” Nucl. Phys. B 651 (2003) 291 [arXiv:hep-
th/0209150].
6. J. Brugues, J. Gomis, T. Mateos and T. Ramirez, “Commutative and
noncommutative N= 2SYMin2+1fromwrappedD6-branes,” Class.
Quant. Grav. 20 (2003) S441 [arXiv:hep-th/0212179].
7. J. Gomis, T. Mateos, P. J. Silva and A. Van Proeyen, “Supertubes
in reduced holonomy manifolds,” Class. Quant. Grav. 20 (2003) 3113
[arXiv:hep-th/0304210].
8. D. Mateos, T. Mateos and P. K. Townsend, “Supersymmetry of ten-
sionless rotating strings in AdS S5, and nearly-BPS operators,”
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JHEP 0312 (2003) 017 [arXiv:hep-th/0309114].
9. D. Mateos, T. Mateos and P. K. Townsend, “More on supersymmetric
tensionless rotating strings in AdS S5,” arXiv:hep-th/0401058.
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ACKNOWLEDGEMENTS
Here comes the most pleasant part of the writing of this thesis, a part
for which I have written down mentally so many little notes throughout all
these years, trying not to forget anyone.
The first person I would like to thank is my advisor Joaquim Gomis.
I still remember that it was him who wrote for me the first field theory
action I had seen in my life. He said that thanks to the fact that it was
two-dimensional, it enjoyed a symmetry called ’conformal’ which happened
to be infinite-dimensional, and that the absence of a certain anomaly called
’Weyl’ implied that the world had to have 26 dimensions. At that moment
I just wondered how long would it take for me to start distinguishing String
Theory from Chinese. Thanks Quim for having helped me so much with
this enterprize, putting pressure on me in the right moments. Thanks as
well for having been a friend and for creating such a good atmosphere in
the department.
The second person I would like to thank is Paul Townsend, with whom I
am also indebted. Thank you so much for hosting me in Cambridge and for
all those ’sobremesas’ with dissertations about life, the huge damage caused
by the prehistorical agriculture or the role of Kings in modern democracies.
It has been really fascinating to get to know your human side. From an
academical point of view, I had the feeling that my learning of string theory
speeded up every time we discussed in the blackboard, be sure that your
way of viewing physics has left a deep fingerprint on me. I really hope to
have the chance to keep learning from you in the future.
Next I would like to thank some other persons with whom I had the
opportunity to collaborate. I would like to thank Antoine Van Proeyen
for those two concentrated weeks in which, together with Quim and Pedro,
we ran against time to finish a project. Thank you too for helping me
every time I needed it, and for all those suggestions and improvements
on the manuscript of this thesis. Thanks Pedro for always being full of
projects and for always listening to my crazy ideas. Thank you too for your
friendship, I hope we manage to coincide more than two weeks together in
the future! A special mention goes to the meson formed by Josep M. Pons
and Pere Talavera. Sharing our first steps in string theory was a wonderful
experience. I hope that the next time we collaborate we will know a little
bit of what these guys are talking about! It is also a pleasure to thank
Alfonso Ramallo for sharing his mythical notebooks with me and for so
many ’tertulias’ at lunches and dinners. Thanks as well for sharing those
early days to Alex Moreno, to whom I seem to have scared to the point of
quitting physics! Thanks too to the Jedi knights Jan and Tonir, who have
just started to feel the Force.
Thanks to Jos´e Edelstein, Roberto Emparan, Javier Mas, Carlos Nu´n˜ez,
Prem Kumar andJorgeRusso for many discussions andvaluable comments,
and to the professors of my department Dom`enec Espriu, Josep I. Latorre,
Josep Taron, Joan Soto, Rolf Tarrach and Enric Verdaguer for always being
available to solve my doubts in four dimensions.
I am also in debt with the Persian Gang (Saman, Ahmed, Amir, Ali,
Hussein, Nazdereh...) whomademystayinIranunforgettable. SinceIcame
I have been planning to travel back there again every year, still without
success. I wish you good luck with your lifes and with your country.
A huge hug to the Parisian Gang (Antonio, Nicco, Aldino and Aldina,
Fabio, Martina, crazy Paskal and even more crazy Tasos, Steffi, my fairy
godmother Liattina...). You know that I do not exaggerate when I say that
those were possibly the best three months of my life. It is great that we all
remain in Europe and that we keep meeting every now and then.
Thanks as well to the Cambridge Gang (Rub´en, Sean, Guishermito,
Marta, Christophe, Maruxa) for making my stay there so good too. Thanks
to all the students I have met in my department: Aleix, A`lex, Luca, Juli´an,
Enrique, Diego, Ernest, Miriam, Roman, Xavi, Jaume, David, Dani, Joan,
Majo andvery specially to Lluis, Toninho Ac´ın, Enric Jan´e andAdamLove.
I am sure we will manage to keep our friendship in the future. Indeed, I
have gone with Lluis through so many extreme circumstances that it seems
unbelievable that we both will finish the PhD alive. I hope our life will be
easier in England!
Last and most important, thanks to my family, specially to my brother
David, to whom I love and admire and with whom I have had the pleasure
to share both life and physics.
CONTENTS
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I.1 AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
I.2 Beyond AdS/CFT: the gauge/string duality . . . . . . . . . 7
I.3 Noncommutative theories in string theory . . . . . . . . . . 9
I.4 Linking NC theories, AdS/CFT and gauge/string duality . 11
I.5 Map of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 13
II. Physics of D-branes . . . . . . . . . . . . . . . . . . . . . . . 15
II.1 Perturbative definition and spectrum of a single D-brane . . 15
II.1.1 Low energy effective action for a single D-brane . . . 16
II.1.2 Multiple D-branes . . . . . . . . . . . . . . . . . . . 18
II.1.3 = 4 SYM . . . . . . . . . . . . . . . . . . . . . . 20
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II.2 D-branes as solutions of closed string theory . . . . . . . . . 22
II.3 An example of brane dynamics: supertubes . . . . . . . . . 25
II.3.1 Generalities of D-brane stabilization . . . . . . . . . 25
II.3.2 Preliminaries for the construction of the supertube
in the open string picture . . . . . . . . . . . . . . . 26
II.3.3 Plan and summary of the results . . . . . . . . . . . 29
II.3.4 Probe worldvolume analysis . . . . . . . . . . . . . . 30
II.3.4.1 The setup . . . . . . . . . . . . . . . . . . . 30
II.3.4.2 Proof of worldvolume supersymmetry . . . . 32
II.3.5 Hamiltonian analysis . . . . . . . . . . . . . . . . . . 35
II.3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . 37
II.3.6.1 Supertubes in ALE spaces: 4 supercharges . 37
II.3.6.2 Supertubes in CY spaces: 1 supercharge . . 39
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vi Contents
II.3.7 Supergravity analysis . . . . . . . . . . . . . . . . . 41
II.3.7.1 Supersymmetry analysis . . . . . . . . . . . 41
II.3.7.2 Equations of motion . . . . . . . . . . . . . 44
II.3.7.3 Constructing the supertube . . . . . . . . . 46
II.3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 46
III. AdS/CFT beyond supergravity and supersymmetry . . 49
III.1 The AdS/CFT correspondence . . . . . . . . . . . . . . . . 49
III.1.1 Pre-BMN ranges of validity and comparability . . . 52
III.2 The BMN limit of AdS/CFT . . . . . . . . . . . . . . . . . 55
III.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . 58
III.3 The GKP simplification . . . . . . . . . . . . . . . . . . . . 59
III.3.1 Twist two operators . . . . . . . . . . . . . . . . . . 62
III.3.2 BMN operators . . . . . . . . . . . . . . . . . . . . . 63
III.4 Trying to check AdS/CFT beyond supersymmetry . . . . . 63
III.4.1 Rotating strings in spheres . . . . . . . . . . . . . . 65
III.4.2 Strings with 3 angular momenta . . . . . . . . . . . 67
III.4.3 BPS Bound from the Superalgebra . . . . . . . . . . 71
III.4.4 Supersymmetry from κ-symmetry . . . . . . . . . . 73
III.4.5 Physics of the large angular momentum limit . . . . 75
III.4.6 Nearly-BPS Operators . . . . . . . . . . . . . . . . . 77
III.4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . 79
III.5 Stable non-BPS AdS branes . . . . . . . . . . . . . . . . . . 83
III.5.1 Stability of AdS-branes . . . . . . . . . . . . . . . . 85
III.5.2 Applications to string/M-theory . . . . . . . . . . . 89
III.5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . 92
IV. Engineering the gauge/string duality . . . . . . . . . . . . 95
IV.1 More general dualities involving flat D-branes . . . . . . . . 95
IV.2 Phase diagrams for flat D5 and D6 branes . . . . . . . . . . 97
IV.2.1 Flat D5 Branes . . . . . . . . . . . . . . . . . . . . . 97
IV.2.2 Flat D6 branes . . . . . . . . . . . . . . . . . . . . . 99
IV.3 Moving away from flatness . . . . . . . . . . . . . . . . . . 101
Contents vii
IV.4 Twisting gauge theories . . . . . . . . . . . . . . . . . . . . 103
IV.5 D-branes wrapping cycles in special holonomy manifolds . . 108
IV.5.1 Special holonomy manifolds . . . . . . . . . . . . . . 108
IV.5.2 Calibrations . . . . . . . . . . . . . . . . . . . . . . 113
IV.5.2.1 Definitions and properties of calibrations . . 113
IV.5.2.2 Calibrations of special holonomy manifolds . 114
IV.5.2.3 Calibrated cycles are supersymmetric cycles 116
IV.5.2.4 A caveat on homology and homotopy . . . . 117
IV.6 The geometrical twisting . . . . . . . . . . . . . . . . . . . 118
IV.6.1 A problem common to (almost all) supergravity so-
lutions . . . . . . . . . . . . . . . . . . . . . . . . . 122
IV.7 How to find supergravity solutions of wrapped branes . . . 123
IV.7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . 123
IV.7.2 Using gauged supergravities to find the solutions . . 124
IV.8 Supergravity duals using D6 Branes . . . . . . . . . . . . . 126
IV.8.1 D6 branes and M-theory . . . . . . . . . . . . . . . . 126
IV.8.2 Twisting to get = 2 in 2+1 dimensions . . . . . . 127
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IV.8.3 BPS equations in D=8 gauged supergravity . . . . . 128
IV.8.4 Solutions of the BPS equations . . . . . . . . . . . . 131
IV.9 Non-perturbative physics of = 2 in 2+1 from its super-
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gravity dual . . . . . . . . . . . . . . . . . . . . . . . . . . 134
IV.9.1 Supersymmetry without supersymmetry . . . . . . . 134
IV.9.2 A non-supersymmetric compactification and a zero-
dimensional moduli space . . . . . . . . . . . . . . . 136
IV.9.3 A supersymmetric compactification and an all-loops
perturbative moduli space . . . . . . . . . . . . . . . 138
IV.9.3.1 The IIA solution . . . . . . . . . . . . . . . 138
IV.9.3.2 The moduli space from supergravity . . . . 139
IV.9.4 Comparison with the field theory results . . . . . . . 141
V. From D-branes to NC Field Theories . . . . . . . . . . . . 143
V.1 The interest of NC field theories per s´e . . . . . . . . . . . 143
V.1.1 The Landau Problem . . . . . . . . . . . . . . . . . 145
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V.1.2 Projecting to the first Landau level . . . . . . . . . . 146
V.1.3 Weyl Quantization . . . . . . . . . . . . . . . . . . . 147
V.1.4 A few properties of the Weyl-Moyal product . . . . . 150
V.2 From D-branes to NC theories . . . . . . . . . . . . . . . . 151
V.2.1 The low energy limit for magnetic backgrounds . . . 154
V.2.2 The effective action from the S-matrix . . . . . . . . 155
V.2.3 A look at the NC Yang-Mills action and NC gauge
invariance . . . . . . . . . . . . . . . . . . . . . . . . 157
V.2.4 The Seiberg-Witten map . . . . . . . . . . . . . . . 159
V.2.5 Electric Backgrounds . . . . . . . . . . . . . . . . . 160
V.3 Quantum NC Field Theories . . . . . . . . . . . . . . . . . 163
V.3.1 Perturbative NC φ4 . . . . . . . . . . . . . . . . . . 163
V.3.2 The 1-loop correction to the self energy and UV/IR
mixing . . . . . . . . . . . . . . . . . . . . . . . . . 165
V.3.3 Optical theorem and unitarity . . . . . . . . . . . . 167
V.3.4 Trying to restore unitarity. The χ-particles. . . . . . 169
V.4 Unitarity of non-relativistic NC theories . . . . . . . . . . . 170
V.4.1 Four Points Function and Unitarity . . . . . . . . . 171
V.4.1.1 Magnetic Case . . . . . . . . . . . . . . . . 172
V.4.1.2 Electric Case . . . . . . . . . . . . . . . . . 172
V.4.2 Two Points Function and the failure of χ-particles. . 173
VI. Supergravity duals of Noncommutative field theories . . 175
VI.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 175
VI.2 Constructing solutions dual to NC theories with less than
16 supercharges . . . . . . . . . . . . . . . . . . . . . . . . 177
VI.2.1 Method one: brute force . . . . . . . . . . . . . . . . 177
VI.2.2 Method two: T-dualities . . . . . . . . . . . . . . . . 178
VI.3 The supergravity dual of the NC = 1 SYM in 3+1 . . . . 182
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VI.3.1 The NC deformation of the Maldacena-Nu´n˜ez back-
ground . . . . . . . . . . . . . . . . . . . . . . . . . 183
VI.3.1.1 Validity of Supergravity and KK states . . . 185
VI.3.1.2 Properties of the solution and UV/IR mixing 186