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Exact Conformal Field Theory Durham E-Theses Aspects of plane waves and Taub-NUT as exact string theory solutions Svendsen, Harald Georg How to cite: Svendsen, Harald Georg (2004) Aspects of plane waves and Taub-NUT as exact string theory solutions, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/2947/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk Aspects of plane waves and Taub-NUT as exact string theory solutions Harald Georg Svendsen A copyright of this thesis rests with the author. No quotation from it should be published without his prior written consent and information derived from it should be acknowledged. A thesis presented for the degree of Doctor of Philosophy - 6 DEC 2004 Centre for Particle Theory Department of Mathematical Sciences University of Durham England September 2004 Aspects of plane waves and Taub-NUT as exact string theory solutions Harald Georg Svendsen Submitted for the degree of Doctor of Philosophy September 2004 Abstract This thesis is a study of some aspects of string theory solutions that are exact in the inverse string tension a/, and thus are valid beyond the low-energy limit. I investigate D-brane interactions in the maximally supersymmetric plane wave solution of type lib string theory, and study the fate of the stringy halo surrounding D-branes. I find that the halo is like in flat space for Lorentzian D-branes, while it receives a non-trivial modification for Euclidean D-branes. I also comment on the connection between Hagedorn temperature and T-duality, which motivates a more general study of T-duality in null directions. I consider such transformations in a spinning D-brane solution of supergravity, and find that divergences in the field components associated with null T-dualities are invisible to string and brane probes. I also observe that there are closed timelike curves in all the T-dual solutions, but that none of them are geodesics. The second half of the thesis is an investigation of the fate of closed timelike curves and of cosmological singularities in an exact stringy Taub-NUT solution of heterotic string theory, and in a rotating generalisation of it. I compute the exact spacetime fields, using a description in terms of a gauged Wess-Zumino-Novikov­ Witten model, and find that the a/ corrections are mild. The key features of the Taub-NUT geometry persist, together with the emergence of a new region of space with Euclidean signature. Closed timelike curves are still present, which is inter­ preted as a sign that they might be a natural ingredient in string theory, for instance in pre-Big-Bang cosmological scenarios. Declaration The work in this thesis is based on research carried out at the Centre for Particle Theory, the Department of Mathematical Sciences, University of Durham, England. No part of this thesis has been submitted elsewhere for any other degree or qualifi­ cation and it is all my own work unless referenced to the contrary in the text. Parts of the work presented in this thesis have been published as research papers written in collaboration with my supervisor, Prof. Clifford V. Johnson. These are refs. [1,2], and correspond to chapters 3 and 6. Chapters 2 and 5 are mostly reviews of already known material, while chapters 4 and 7 contain new unpublished results based on my own research. Copyright© 2004 by Harald G. Svendsen. The copyright of this thesis rests with the author. No quotations from it should be published without the author's prior written consent and information derived from it should be acknowledged. iii Acknowledgements First of all, I am very grateful to Prof. Clifford V. Johnson who has been my super­ visor, and a tremendous source of inspiration and knowledge throughout my time as his student. Thank you. I also owe many thanks to my colleagues, Emily Hackett-Jones and Anton Ilder­ ton for answering many of my odd questions, and also Matthew Killeya and Gavin Probert for enlightening discussions on both scientific and other topics. You have all made office days very enjoyable. Many thanks to Profs. Hallstein H0gasen and Ulf Lindstrom for helping me to get started with my doctoral studies, and to Drs. Chong-Sun Chu, Douglas Smith, Simon Ross, Laur Jiirv, and James Gray for help at various stages. Thanks also to all other members of the maths department at Durham University for providing a very stimulating and friendly atmosphere. I thank Norway for a doctoral student fellowship without which I could not have pursued these studies, and Britain for an Overseas Research Student Award. I am also grateful for financial support from the University of Durham, through a Nick Brown memorial bursary and other travel grants. Thanks to the high-energy physics group at the University of Southern California for hospitality. I have enjoyed my time in Durham - the hills, the trees, the paths and buildings surrounding me have kept me happy. But more so because of my friends. Thank you all. Especially Grace (for all the fun) and the other "UN house" people: Aris, Karen, Manuel, and Maria. Thanks also to Durham University Fencing Club and Graduate Society Boat Club for many happy times. The work towards the completion of this thesis started in many ways long before I came to Durham. And ever since my first day in primary school, I have had excellent teachers who have inspired me and encouraged me. A great part of the achievement manifested by this thesis belongs to you. Finally, huge thanks to my parents, Gerd Marit and Johan, and to my siblings. Through your persistent encouragement and faith in me you have helped me much more than you know. iv Contents Abstract ii Declaration iii Acknowledgements IV 1 Introduction 1 1.1 State of string theory . 1 1.2 Context and motivation 3 1.3 Outline and summary of results 4 2 Plane wave solutions 6 2.1 Strings in plane waves 7 2.1.1 Quantisation . 7 2.1.2 Plane waves as a Penrose-Giiven limit. 8 2.2 The AdS/CFT correspondence and its BMN limit 9 2.2.1 AdS/CFT correspondence 9 2.2.2 Dictionary .. 12 2.2.3 BMN limit .. 13 2.3 D-brane interactions 14 2.3.1 D-branes in plane waves 14 2.3.2 Interactions in fiat space 16 2.3.3 Interactions in plane waves . 18 2.4 Hagedorn temperature and T-duality 21 3 D-brane-anti-D-brane interactions in plane waves 26 3.1 Introduction . 26 3.1.1 Superstrings in fiat space . 27 3.1.2 Plane waves 30 3.2 The Interaction .. 30 v Contents vz 3.2.1 The Amplitude and Potential .... 31 3.2.2 Divergences, Tachyons, and the Halo 32 3.3 Summary .................. 36 4 T-dualities in null directions, and closed timelike curves 37 4.1 Introduction 37 4.2 T-dualities . 40 4.2.1 T-dual along z direction 41 4.2.2 T-dual along ¢ 1 direction . 41 4.2.3 T-dual along both (h and z directions. 42 4.3 Probe calculations in T q,1 z-dual solution . 44 4.3.1 Fundamental string coupled to B-field 44 4.3.2 D1-D5-brane probe 46 4.3.3 D3-brane probe .. 47 4.4 Closed timelike curves . 48 4.4.1 CTCs and geodesics 50 4.5 Discussion . 52 5 Wess-Zumino-Novikov-Witten models 54 5.1 WZNW models ............ 54 5.2 Coset models as gauged WZNW models 57 5.3 Exact spacetime fields from coset models 59 5.3.1 The Exact Effective Action ... 60 5.3.2 Computation of the Exact Effective Action . 61 5.3.3 Extracting the Exact Geometry 64 5.3.4 Example: A 2D black hole 67 5.4 Heterotic coset models ..... 69 5.4.1 Abelian bosonisation . 70 5.4.2 Non-Abelian bosonisation 71 5.4.3 Re-fermionisation ..... 72 5.5 A deformation of a charged 2D black hole 74 6 High energy corrections in a stringy Taub-NUT spacetime 78 6.1 Introduction and Motivation . 79 6.2 Stringy Taub-NUT . 82 6.3 Exact Conformal Field Theory . 87 6.3.1 The Definition. 87 6.3.2 Writing The Full Action 88 6.3.3 Extracting the Low Energy Metric 90 Contents vn 6.4 Exact spacetime fields . 91 6.4.1 Computations . 92 6.4.2 Properties of the Exact Metric . 96 6.5 Discussion . 98 7 High energy corrections in a stringy Kerr-Taub-NUT solution 102 7.1 Exact conformal field theory . 102 7.2 Low-energy limit . 104 7.3 Exact metric and dilaton . 105 7.3.1 Properties of metric . 106 7.4 Discussion 109 8 Conclusion 111 Appendices 114 A Definition of functions 114 A.1 Generalised Jacobi functions 114 A.2 Deformed theta functions .
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