DUALITY AND GEOMETRY OF STRING THEORY Nipol Chaemjumrus, Department of Physics The Blackett Laboratory Imperial College London, UK Supervisors Prof. Chris Hull March 31, 2019 Submitted in part fulfilment of the requirements for the degree of Philosophy in Physics of Imperial College London and the Diploma of Imperial College 1 Declaration of Authorship Unless otherwise referenced, the work presented in this thesis is my own - Nipol Chaemjumrus. The main results of the project are outlined in chapters 5, chapter 6, chapter 7 and chapter 8. These follow the work published in [1] N. Chaemjumrus and C. M. Hull, \Finite Gauge Transformations and Geometry in Extended Field Theory," Phys. Rev. D 93 (2016) no.8, 086007 [arXiv:1512.03837 [hep-th]], [2] N. Chaemjumrus and C. M. Hull, \Degenerations of K3, Orientifolds and Exotic Branes," in preparation, [3] N. Chaemjumrus and C. M. Hull, \Special Holonomy Domain Walls, Intersecting Branes and T-folds," in preparation, and [4] N. Chaemjumrus and C. M. Hull, \The Doubled Geometry of Nilmanifold Reductions," in preparation. The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 2 Abstract String theory possesses duality symmetries that relate different string backgrounds. One symmetry is known as T-duality symmetry. In general, when n dimensions are toroidally compactified, the T-duality group is O(n; n; Z). String theory has another duality symmetry known as S-duality, which does not commute with T-duality. The full S-duality group is SL(2; Z). The last duality symmetry is U-duality symmetry, which is En+1;(n+1)(Z) for type II string theory on T n. Duality symmetries tell us that strings experience geometry differently from particles. In order to understand string theory, a new way to understand string geometry is required. In this thesis, first we introduce some basic ideas on duality symmetries in string theory, namely, T-duality, S-duality, and U-duality. Next, we review string field theory. We, then, provide the basic constructions of DFT and EFT. Next, we consider the finite gauge trans- formations of DFT and EFT. The expressions for finite gauge transformations in double field theory with duality group O(n; n) are generalized to give expressions for finite gauge transfor- mations for extended field theories with duality groups SL(5; R), SO(5; 5) and E6. Another topic is the T-duality chain of special holonomy domain wall solutions. This example can arise in string theory in solutions in which these backgrounds appear as fibres over a line. The cases with 3-torus with H-flux over a line were obtained from identifications of suitable NS5-brane solutions, and are dual to D8-brane solutions. This T-duality chain implies that K3 should have a limit in which it degenerates to a long neck of the form N × R capped off by suitable smooth geometries. A similar result applies for the higher dimensional analogues of the nilfold. In each case, the space admits a multi-domain wall type metric that has special holonomy, so that taking the product of the domain wall solution with Minkowski space gives a supersymmetric solution. 3 Acknowledgements First of all, I am extremely grateful to Prof. Chris Hull, my advisor, for teaching and inspiring me to love and take interest in theoretical physics, and giving me advices on my projects during the past five years. His guidance helped me in all the time of research and writing of this thesis. Futhermore, I would like to thank Prof. Daniel Waldram and Ms. Graziela De Nadai for their help, especially during my final year of PhD. Next I would like to thank Mr. Apimook Watcharangkool, Mr. Chakrit Pongkitivanichkul, and Miss Premyuda Ontawong for their help in everything since I was here. I am grateful to Mr. Simon Nakach, Mr. Santiago M´arquez,Mr. Edward Tasker and Miss Hiu Yan Samantha for their support, friendship during my PhD life. I would like to thank Miss Somsinee Wisitpitthaya, Mr. Monchai Kooakachai, Mr. Thada Udomprapasup, Mr. Teeratas Kijpornyongpan, and Mr. Waruj Akaraworatikamporn for their support along the way. Finally I would like to thank the Queen Sirikit scholarship for financial support. 4 Contents 1. Introduction 9 2. String Theories, M-theory, and Dualities 18 2.1. String theory . 18 2.2. Toroidal compactification . 19 2.3. Hamiltonian and level-matching condition . 22 2.4. T-duality and O(n; n; Z)............................... 24 2.5. Example of O(n; n; Z) transformation . 27 2.6. Superstring theories and M-theory . 29 2.6.1. Open superstring . 29 2.6.2. Closed string theory: type II . 30 2.6.3. Type I . 32 2.6.4. Heterotic string theory . 32 2.6.5. M-theory . 33 2.7. U-duality . 35 2.7.1. T-duality of type IIA and IIB . 35 2.7.2. S-duality of type IIB . 36 2.7.3. U-duality . 36 3. String Field Theory 37 3.1. Conformal field theory . 37 3.2. BPZ conjugate . 40 3.3. Closed string field action and gauge transformation . 41 3.4. 1st-massive open string state . 42 3.5. The first massive closed string state . 45 3.5.1. T-duality transformation of the first massive level . 47 4. Double Field Theory and Extended Field Theory 49 4.1. Double field theory . 49 4.1.1. Free Theory action . 49 4.1.2. Cubic interaction action . 53 5 4.1.3. Double field theory action . 53 4.2. Extended field theory . 56 5. Finite Transformation for Double Field Theory and Extended Field theory 58 5.1. Finite transformations for double field theory . 58 5.2. Finite Transformations for Extended Field Theory . 61 5.2.1. SL(5; R) Extended field theory . 61 5.2.2. SO(5; 5) Extended field theory . 64 5.2.3. E6 Extended field theory . 68 5.3. Generalized tensors . 72 5.3.1. Generalized tensors of double field theory . 72 5.3.2. Generalized Tensors of Extended Field Theory . 74 5.4. The generalized metric in DFT and EFT . 77 6. Degenerations of K3, Orientifolds and Exotic Branes 82 6.1. The Nilfold and its and T-duals . 82 6.2. Domain wall solution from NS5-brane solution . 84 6.2.1. Smeared NS5-brane solution . 84 6.2.2. Nilfold background . 88 6.2.3. T-fold and R-fold backgrounds . 89 6.2.4. Exotic Branes and T-folds . 90 6.2.5. Single-sided domain walls and the Tian-Yau space . 91 6.3. Embedding in string theory . 92 6.3.1. The D8-brane and its duals . 92 6.3.2. The type I0 string . 93 6.4. Duals of the type I0 string . 95 6.4.1. Moduli spaces and dualities . 97 6.5. Degenerations . 98 6.6. A degeneration of K3. ................................ 99 6.7. Duals of the degenerate K3.............................. 102 6.8. Non-geometric string theory . 104 7. Special Holonomy Domain Walls, Intersecting Branes and T-folds 106 7.1. Nilmanifolds as torus bundles over tori . 106 7.1.1. S1 bundle over T 4 ............................... 107 7.1.2. T 2 bundle over T 3 .............................. 108 7.1.3. T 2 bundle over T 4 .............................. 110 7.1.4. T 3 bundle over T 3 .............................. 111 6 7.1.5. T 3 bundle over T 4 .............................. 112 7.1.6. S1 bundle over T 6 ............................... 113 7.2. Supersymmetric domain wall solutions . 114 7.2.1. 6-dimensional domain wall solutions with SU(3) holonomy . 115 7.2.2. 7-dimensional domain wall solutions with G2 holonomy . 115 7.2.3. 8-dimensional domain wall solution with Spin(7) holonomy . 116 7.2.4. 8-dimensional domain wall solution with SU(4) holonomy . 116 7.3. Special holonomy domain walls and intersecting branes . 117 7.3.1. S1 fibred over T 4 ............................... 117 7.3.2. T 2 fibred over T 3 ............................... 119 7.3.3. T 2 fibred over T 4 ............................... 120 7.3.4. T 3 fibred over T 3 ............................... 122 7.3.5. T 3 fibred over T 4 ............................... 124 7.3.6. S1 fibred over T 6 ............................... 126 7.4. T-folds fibred over line . 128 7.4.1. T-fold from S1 bundle over T 4 fibred over a line . 128 7.4.2. T-fold form T 3 bundle over T 3 fibred over a line . 129 7.4.3. T-fold form S1 bundle over T 3 fibred over a line . 129 8. The Doubled Geometry of Nilmanifold Reductions 131 8.1. The doubled formalism . 131 8.2. The doubled nilmanifold . 134 8.2.1. Sigma model . 135 8.2.2. Polarisation . 135 8.2.3. Recovering the physical space . 136 8.2.4. Quotienting by the discrete group . 138 8.3. The nilfold example . 138 8.3.1. The nilfold . 139 8.3.2. T 3 with H-flux . 139 8.3.3. T-fold . 140 8.3.4. R-flux ..................................... 140 8.4. Higher dimensional nilmanifolds . 141 8.4.1. S1 bundle over T 4 ............................... 141 8.4.2. T 2 bundle over T 3 .............................. 144 8.4.3. T 2 bundle over T 4 .............................. 147 8.4.4. T 3 bundle over T 3 .............................