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Aspects of Brane Dynamics in String Theory Aspects of Brane Dynamics in String Theory John Ward A report presented for the examination for the transfer of status to the degree of Doctor of Philosophy of the University of London Thesis Supervisor Prof. Steven Thomas Center for Research in String Theory Department of Physics Queen Mary, University of London Mile End Road London E1 4NS United Kingdom June 2007 To my family - for all their support and encouragement Declaration I declare that the material in this thesis is a representation of my own work, unless otherwise stated, and resulted from collaborations with Steven Thomas, Burin Gumjudpai, Tapan Naskar, Sudhakar Panda, Mohammad Sami and Shinji Tsujikawa. Much of the material in the thesis is based upon the papers [1] - [9]. John Ward 3 Acknowledgements First and foremost I must thank my supervisor Steve Thomas for all his support and assistance over the past few years. It would not be an exaggeration to say that I would not be in the position to write a thesis without his supervision and encouragement. I must also acknowledge my long suffering office mates, friends and collaborators Costis Papageorgakis and James Bedford. Their opinions, comments and advice have been invalu- able, and made the office such a fantastic place to work. Additionally I wish to thank all the members of the string theory group for making Queen Mary such a great place to be. It has been an amazing experience for me on so many levels. I also gratefully acknowledge the non-academic staff for all their assistance during this time, Kate, Kathy, Denise and Sarah. Many thanks must also go to my collaborators and colleagues; Daniel Grumiller, Burin Gumjudpai, Tapan Naskar, Sudhakar Panda, Narit Pidokrajt, Diego Rodriguez-Gomez, Mohammad Sami, Shinji Tsujikawa and Jan Aman.˚ I also wish to acknowledge the Euro- pean Union for giving me the opportunity to spend a great six months at CERN, and also wish to thank Diana de Toth, Ignatios Antoniadis, Malcolm Fairbairn and my office mates Cedric Delaunay and Ninh le Duc for making my time there so memorable. On a similar note I also wish to thank Larus Thorlacius and the University of Iceland for supporting me during my visit. I must also mention the following people for their comments on the material in this thesis; Magdalena Larfors, Linus Wulff, Lotta Mether, Jaydeep Majumder, Niko Jokela, Nic Toumbas, Robert de Mello Koch, Paulo Moniz, Claudia de Rham, Subodh Patil, Alexander Morisse, Jessica Barrett, Gianna Pappa, Simon McNamara, David Mulryne and Cecilia Albertsson. Special thanks must also go to Mark Matsen and everyone in the physics department at the University of Reading who played an important role in my early training. There are also many other people who have contributed, in their own way, to the final production of this thesis. Firstly many thanks go to my long(er) suffering housemates, Andrew, Derek, Joseph, Mike and Chris for putting up with me all these years. Also Tom, ODB, Michael, Colleen, Mitch, Catie, Sanddie, Claire, Fumika, Pernilla, Keljay, Concetta, Manu, Sarah, Natalie, Vicky, Katla, Kate G and Kristen for keeping me sane during my 4 studies. The path to this thesis has been far from straightforward, but the following people are partially responsible for me following a path in physics. Many thanks to Kevin, Marcus, Sam and everyone who supported the Natural Disasters. Thanks also to Mike, Gary C, Gaz, Rich, Magic Matt, Tom, Mowgli, Chips and Mark B for teaching me to skate, and to Vert Streetsports for supporting me in this endeavour. Finally thanks to the Bus Station Loonies, The Filth and Oi Polloi for leading the way. 5 Abstract This thesis investigates the dynamics and potential application of D-brane systems in Type II String Theory. In the first section we investigate the dynamics of a single brane in the NS5-brane ring geometry, and further develop the tachyon-radion correspondance which maps dynamical brane solutions to open string tachyon condensation on unstable branes. We show that the resulting Geometrical Tachyon exhibits many of the properties of the open string tachyon. In the second section we investigate the dynamics of N coincident Dp-branes in various brane backgrounds. This involves the use of matrix degrees of freedom and non-commutative geometries known as fuzzy spheres. We study the collapse of these fuzzy spheres in Dp-brane and NS5-brane backgrounds, before generalising the results. We also examine the D1 D3 − intersection in a general background, from the macroscopic and microscopic viewpoints. This section closes with the microscopic description of (p,q) strings in the Warped Deformed Conifold. We calculate the tension spectrum and find agreement with the macroscopic solution in the limit of large flux. Using the finite q prescription for the Myers action, we conjecture a form for the string tension when there are a finite number of D-strings. The final section emphasises the cosmological aspects of such dynamics. We being by using the Geometrical Tachyon as a toy model of tachyonic inflation. We then construct a hybrid inflation scenario using the Geometrical Tachyon coupled to the open string tachyon. Both models compare well to the experimental data. Finally we develop a new model of DBI inflation using multiple branes at the IR tip of a warped throat. We study the theory in the large N limit and show how it is similar to the single brane models. We then extend the analysis to the case of finite N, which we expect to be more phenomenologically viable. In the large N limit we find that inflation is possible, but the amplitude of gravitational waves is large. This is not the case in the finite N limit, which can allow for small perturbations unusual levels of non-gaussianities making it a testable prediction of string theory. 6 Table of contents DECLARATION 3 ACKNOWLEDGEMENTS 4 ABSTRACT 6 1 INTRODUCTION. 12 1.0.1 Introducing the Abelian DBI action . 15 2 THE TACHYON-RADION CORRESPONDANCE. 18 2.1 Introduction. ................................... 18 2.2 The NS5-braneringsolution. .. .. .. .. .. .. .. 20 2.2.1 Dynamicsoftheprobebrane.. 21 2.2.2 Probemotionintheringplane.. 24 2.2.3 Probe motion transverse to the ring plane. 26 2.2.4 Motion in the ring plane with L˜ =0 ................. 30 θ 6 2.2.5 Motion transverse to the ring plane with L˜ =0............ 31 φ 6 2.3 Tachyonmap.................................... 33 2.3.1 Insidethering. .............................. 35 2.3.2 Outsidethering. ............................. 36 2.3.3 Transversetothering.. 37 2.4 Geometrical Tachyon kinks. 38 2.5 DynamicsofNon-BPSbranes.. 44 2.5.1 Onespatialdirection. 46 2.5.2 Two (or more) spatial directions. 46 2.5.3 Spaceandtimedependence.. 48 2.6 Compactification in a transverse direction. ......... 49 2.7 Discussion..................................... 51 7 3 COINCIDENT D-BRANES AND THE NON-ABELIAN DBI ACTION 53 3.1 Introduction.................................... 53 3.1.1 The Non-Abelian DBI action. 53 3.1.2 Fuzzy spheres as non-commutative geometry. ..... 56 3.2 Dynamics in Dp and NS5-braneBackgrounds. 58 3.2.1 D-BraneBackgrounds.. .. .. .. .. .. .. .. 58 3.2.2 RadialCollapse. ............................. 59 3.2.3 Dynamics in the p = p′ case. ...................... 60 3.2.4 Dynamics in the p = p case. ...................... 71 6 ′ 3.2.5 Corrections from the symmetrized trace. ..... 73 3.2.6 Remarks on the D6-D0solution. .................... 74 3.2.7 Inclusion of Angular Momentum. 78 3.2.8 Alternativeansatz. 80 3.2.9 Non-BPSbranes.............................. 82 3.2.10 Decomposition of charge. 85 3.2.11 NS5-branebackground. 87 3.2.12 Correction from symmetrized trace. 92 3.2.13 Non-Abelian tachyon map. 92 3.2.14 Non-BPS branes in fivebrane backgrounds. 94 3.2.15 Higher (even) dimensional fuzzy spheres. ....... 99 3.3 Dynamicsinmoregeneralbackgrounds. 101 3.3.1 Minkowskispacedynamics. 104 3.3.2 1/N Corrections in Minkowski space. 105 3.3.3 Curvedspacedynamics. 107 3.3.4 1/N corrections in Curved space. 111 3.3.5 Brane Intersections in Curved Space. 112 3.3.6 Funnelsolutions. ............................. 114 3.3.7 1/N Corrections to the Fuzzy Funnel. 117 3.3.8 Time Dependence and Dualities. 119 3.3.9 The Dual Picture - D3 world-volume theory. 121 3.3.10 Higher Dimensional Fuzzy Funnels. 127 3.4 Application: (p,q) strings in the Warped Deformed Conifold. 130 3.4.1 TheWarpedDeformedConifold. 131 3.4.2 Macroscopic (p,q)-strings......................... 134 8 3.4.3 Microscopic (p,q)-strings. ........................ 136 3.5 Discussion. .................................... 143 4 BRANE DYNAMICS AND COSMOLOGY. 145 4.1 Introduction. ................................... 145 4.2 Geometrical Tachyon Inflation. 148 4.2.1 Cosmologyintheringplane. 149 4.2.2 Cosmology in the plane transverse to the ring . 165 4.3 D3-brane dynamics in the D5-branebackground . 173 4.4 DBIInflationintheIR. ............................. 186 4.4.1 The large N limit ............................ 189 4.4.2 Inflation at finite N ........................... 205 4.4.3 Twobraneinflation. ........................... 207 4.4.4 Threebraneinflation.. .. .. .. .. .. .. .. 215 4.5 Discussion..................................... 222 5 CONCLUSIONS 224 A COSMOLOGICAL PERTURBATIONS. 228 9 List of Figures 2.1 Plot of the radial coordinate ρ vs t, for E˜ = 0.6,Lθ = Lφ = 0 and taking electric flux F = 0, 0.8. ............................. 26 ρ ˜ 2.2 Plot of the radial coordinate R vs t, for E = 1.5,Lθ = Lφ = 0 and taking electric flux F = 0, 0.8. ............................. 27 ρ ˜ 2.3 Plot of the effective potential Veff vs R , for E = (0, 1.5),Lθ = Lφ = 0 and taking electric flux F = 0.6, 0.8. ........................ 27 σ ˜ 2.4 Plot of the distance R of the probe from the ring plane vs t, for E = 0.98,Lθ = Lφ = 0 and taking electric flux F = 0. This describes motion of the probe through the centre of the ring at ρ =0 ...................
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