View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Queen Mary Research Online T-duality invariant approaches to string theory Thompson, Daniel C. The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the author For additional information about this publication click this link. https://qmro.qmul.ac.uk/jspui/handle/123456789/390 Information about this research object was correct at the time of download; we occasionally make corrections to records, please therefore check the published record when citing. For more information contact [email protected] T-duality Invariant Approaches to String Theory Daniel C. Thompsony A thesis submitted for the degree of Doctor of Philosophy y Centre for Research in String Theory, Department of Physics Queen Mary University of London Mile End Road, London E1 4NS, UK Abstract This thesis investigates the quantum properties of T-duality invariant formalisms of String Theory. We introduce and review duality invariant formalisms of String Theory including the Doubled Formalism. We calculate the background field equations for the Doubled Formalism of Abelian T-duality and show how they are consistent with those of a conventional String Theory description of a toroidal compactification. We generalise these considerations to the case of Poisson{Lie T-duality and show that the system of renormalisation group equations obtained from the duality invariant parent theory are equivalent to those of either of the T-dual pair of sigma-models. In duality invariant formalisms it is quite common to loose manifest Lorentz invariance at the level of the Lagrangian. The lack of manifest invariance means that at the quantum level one might anticipate Lorentz anomalies and we show that such anomalies cancel non-trivially. These represent important and non-trivial consistency checks of the duality invariant approach to String Theory. 2 Acknowledgements The research presented in this thesis was conducted under the support of an STFC studentship. I wish to thank my supervisor David Berman to whom I owe a deep debt of gratitude. Over the course of my studies he has shared his insight, wisdom and enthusiasm for the subject and has been a tremendous teacher and collaborator. His advice and guidance were essential in helping me navigate the tortuous path of research. I also wish to thank those others with whom I have collaborated during the course of these studies: Neil B. Copland; Malcolm J. Perry; Ergin Sezgin; Kostas Sfetsos; Kostas Siampos and Laura C. Tradrowski. I thank my fellow colleagues at Queen Mary with whom I have enjoyed many prof- itable discussions during the course of this work, in particular: Ilya Bakhmatov; An- dreas Brandhuber; Tom Brown; Vincenzo Calo; Andrew Low; Moritz McGarrie; Sanjaye Ramgoolam; Rudolfo Russo; Steve Thomas; David Turton and Gabriele Travaglini. Lastly, I thank my family and most of all my wife Jennifer for unflinching belief and unfailing support. 3 Declaration Except where specifically acknowledged in the text the work in this thesis is the original work of the author. Much of the research presented in this thesis has appeared in the following publications by the author: [1] together with D. Berman and N. Copland; [2] with D. Berman; [3] and [4] with K. Sfetsos and K. Siampos. Additionally, during the course of the preparation of this thesis the author has published several other articles [5{8] the results of which are not included in this thesis. 4 Table of contents ABSTRACT 2 ACKNOWLEDGEMENTS 3 DECLARATION 4 1 INTRODUCTION 8 1.1 Unification and Duality . 8 1.2 String Theory and its Dualities . 9 1.3 Duality Invariance . 11 1.4 Structure of Thesis . 11 2 T-DUALITY 13 2.1 Twenty-five Years of T-duality . 13 2.2 Radial Inversion Duality . 15 2.2.1 Path Integral (Buscher) Approach . 16 2.2.2 Partition Function Equivalence . 20 2.2.3 Spectral Approach to T-duality . 22 2.2.4 T-duality as a Canonical Transformation . 24 2.3 Canonical Transformations in Ramond Backgrounds . 26 2.4 Fermionic T-duality as a Canonical Transformation . 30 2.5 Further Perspectives . 35 3 O(d; d; Z), T-FOLDS AND THE DOUBLED FORMALISM 38 3.1 Toroidal Compactification and O(d; d) Duality Group . 38 3.1.1 Example of O(2; 2; Z) ........................ 41 3.2 T-folds . 42 5 3.3 The Doubled Formalism . 45 3.3.1 Lagrangian and Constraint . 46 3.3.2 Action of O(n; n; Z) and Polarisation . 49 3.3.3 Recovering the standard sigma model . 50 3.3.4 Geometric vs. Non-Geometric . 52 3.3.5 Extensions to the Doubled Formalism . 53 4 QUANTUM ASPECTS OF DOUBLED STRING THEORY 56 4.1 Overview . 56 4.2 Implementing the Constraint . 57 4.3 The Background Field Expansion . 59 4.4 Simplification Strategy . 62 4.5 Wick Contraction . 64 (2) 4.5.1 Single contraction terms for hSinti . 65 (2) (2) 4.5.2 Double contraction terms for hSintSinti . 66 4.5.3 The Weyl Divergence . 67 4.5.4 Relation to the Doubled Ricci tensor . 68 4.6 Doubled Renormalisation . 69 4.7 Comparison with Standard Sigma Model . 70 4.8 A Brief Recapitulation . 73 4.9 Inclusion of the Dilaton . 74 4.9.1 Using a constant Doubled dilaton . 74 4.9.2 Dilaton contributions to existing beta functions . 75 4.9.3 Dilaton Beta Function . 76 4.10 Target Space Interpretation . 80 4.11 Discussion and conclusions . 82 5 POISSON-LIE T-DUALITY BEYOND THE CLASSICAL LEVEL 84 5.1 Overview . 84 5.2 Drinfeld Doubles . 86 5.2.1 Examples of Drinfeld Doubles . 87 5.2.2 Subspaces of the Double . 87 5.3 Poisson Lie T-duality . 88 6 5.3.1 Geometry of the Double . 91 5.4 Renormalisation of the Poisson-Lie Duality Invariant Theory . 94 5.4.1 Weyl Anomaly . 98 5.4.2 Lorentz Anomaly . 99 5.5 Some Examples and a general proof of equivalence . 99 5.5.1 Renormalisation of T-dual pairs . 100 5.5.2 General Equivalence . 106 5.6 On-shell Lorentz invariant actions . 108 5.6.1 Recovering on-shell Lorentz invariance . 111 5.6.2 Solving the Conditions . 113 5.7 Discussion and Conclusions . 117 6 CONCLUSIONS AND OUTLOOK 118 A CONVENTIONS 120 A.1 String σ-model Conventions . 120 A.2 Notation . 122 B CHIRAL FIELDS 123 C BACKGROUND FIELD METHOD 126 D PROPAGATORS AND WICK CONTRACTIONS 131 E DIMENSIONAL REDUCTION 135 7 CHAPTER 1 INTRODUCTION 1.1 Unification and Duality Two important themes in theoretical physics are Unification and Duality. At first these themes seem to be, if not contradictory, at least competing ideas; Unification is the reduction of multiple theories to a single all-encompassing theory, whereas Duality is the notion that there may be several distinct and complementary frameworks that describe the same physics. Nonetheless, both of these ideas have lead to tremendous advances in our theoretical understanding of the universe. The older idea of Unification, whose roots can be traced at least as far back as Maxwell's theory of Electro-Magnetism, underpins the development of what is perhaps the pinnacle of scientific endeavour { the Standard Model of particle physics. Time and time again Unification has proven to be a guiding light for theoretical physicists. The ultimate objective of Unification would be a theory that unites Gravity with Quantum Mechanics and with the other forces of nature encapsulated in the Standard Model. To this end, String Theory [9] is widely accepted as the leading candidate for a Quantum Theory of Gravity and offers the tantalising possibility of Unification of all the forces. Duality is a comparatively newer concept and not only is it an interesting and deep theoretical property in its own right, it can also help make physical predictions. For instance, a question which may be very hard to answer within the context of one framework may be addressed much more easily by means of a dual description. One of the most important examples of duality, the AdS − CFT correspondence [10] does just this; questions asked in a strongly coupled gauge theory can be answered by means of a calculation performed in a weakly coupled gravity theory. The full power of this sort of duality is only now being unlocked and the past few years have seen its application to a range of physical systems including: the Quark Gluon Plasma [11, 12] thought to have been observed at the RHIC experiment; fluid mechanics in general [13] and condensed 8 CHAPTER 1. INTRODUCTION matter systems such as high Tc superconductors [14]. There are many other fascinating examples of duality, particularly in the context of Gauge Theories [15{18], but the focus of this thesis is on Duality within the context of String Theory. Nowhere is Duality more pronounced than in String theory and indeed the development of both go hand-in-hand to an extent. Certainly, a fuller understanding of String Theory necessitates a deeper understanding of Duality. 1.2 String Theory and its Dualities Although born out of an ultimately unsuccessful attempt to formulate a theory of the Strong Nuclear Force, Superstring Theory was first developed as a Quantum Theory of Gravity during the late 1970's and early 1980's. In String Theory the fundamental objects are not point particles but string-like extended objects whose characteristic length is of the Planck scale (1:6 × 10−33 cm). The primary reason for the success of String Theory as a Quantum Theory of Gravity is that the spectrum of closed strings includes a massless, spin-two excitation corresponding to the graviton.