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Signature Redacted Department of Physics May 3, 2018 Classification of base geometries in F-theory by Yinan Wang B. S., Peking University (2013) Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2018 Massachusetts Institute of Technology 2018. All rights reserved. A uth or ............................... Signature redacted Department of Physics May 3, 2018 Signature redacted Certified by.........................Signature red ated *ashington Taylor Professor Thesis Supervisor Signature redacted Accepted by ..................... MASSACHUSETTS INSTITUTE Scott A. Hughes Interim ssociate Department Head JUN 2 5 2018 LIBRARIES ARCHIVES 2 Classification of base geometries in F-theory by Yinan Wang Submitted to the Department of Physics on May 3, 2018, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Abstract F-theory is a powerful geometric framework to describe strongly coupled type JIB supcrstring theory. After we compactify F-theory on elliptically fibercd Calabi-Yau manifolds of various dimensions, we produce a large number of minimal supergravity models in six or four spacetime dimensions. In this thesis, I will describe a current classification program of these elliptic Calabi-Yau manifolds. Specifically, I will be focusing on the part of classifying complex base manifolds of these elliptic fibrations. Besides the usual algebraic geometric description of these base manifolds, F-theory provides a physical language to characterize them as well. One of the most important physical feature of the bases is called the "non-Higgsable gauge groups", which is the minimal gauge group in the low energy supergravity model for any elliptic fibration on a specific base. I will present the general classification program of complex base surfaces and threefolds using algebraic geometry machinery and the language of non- Higgsable gauge groups. While the complex base surfaces can be completely classified in principle, the zoo of generic complex threefolds is not well understood. However, I will present an exploration of the subset of toric threefold bases. I will also describe examples of base manifolds with non-Higgsable U(1)s, which lead to supergravity models in four and six dimensions with a U(1) gauge group but no massless charged matter. Thesis Supervisor: Washington Taylor Title: Professor 3 4 Acknowledgments First, I would like to thank my research advisor Prof. Wati Taylor for his continuous support, guidance and inspiration. I feel extremely lucky that my computer program- ming skills match his enthusiasm in computational mathematics. This is crucial for our string geometric classification projects to succeed. I was also influenced by his persistence in one subfield that is considered too complicated to work out by many people. I have benefited a lot from the group meetings organized by Wati. The discussions with my fellow students Samuel Johnson, Nikhil Raghuram, Yu-Chien Huang and Andrew Turner are very helpful and I want to thank them as well. I would also like to thank Prof. Barton Zwiebach and my collaborator Olaf Hohm for their early research guidance in the field of double/exceptional field theory, which is an indispensable starting point for my graduate level research. I would like to thank them for the recommendations for various purposes as well. I have also benefited from other faculty members at CTP, from their teaching and guidance on the oral exam. They include Prof. Hong Liu, Tracy Slatyer, William Detmold, Jesse Thaler and Robert Jaffe. I would also like to thank other graduate students for valuable discussions, including Usman Naseer, Nick Rodd, Srivatsan Rajagopal and Yifan Wang. I would like to thank the CTP staff members Scott Morley, Charles Suggs and Joyce Berggren for their support as well. I have really enjoyed my life at CTP and I will cherish this part of memory forever. Of course, I would like to thank Prof. Hong Liu again and Prof. Xiao-Gang Wen for their parts on my thesis committee. Besides the people at MIT, I would like to thank my colleagues at Northeastern University: Prof. James Halverson, Dr. Cody Long, Jiahua Tian and Ben Sung. Our shared interest always pushes us further. I would also like to thank Harvard people including Prof. Xi Yin, Prof. Cumrun Vafa, Dr. Dan Xie and Shuheng Shao for teaching and discussions. Finally, I would like to thank my family for their love and support and my friends 5 for regular dining activities. 6 Contents 1 Introduction and summary 13 2 Physics and Mathematics background 27 2.1 Type IIB superstring theory and F-theory . .. 27 2.1.1 JIB superstring theory and 7-branes . .. 27 2.1.2 F-theory and M/F-duality .... .... 31 2.1.3 Gauge group and matter in F-theory . .33 2.2 Basics of complex algebraic geometry .. .... .35 2.2.1 Divisor and line bundles . ........ 35 2.2.2 Birational geometry .... ........ 40 2.3 Mathematics of elliptic fibration .. ....... 42 2.4 Toric geometry ....... ........... 47 3 Classification of complex base surfaces 53 3.1 6D F-theory and non-Higgsable clusters ..... ....... .... 53 3.2 Classification of toric and semi-torie base surfaces ... .... ... 60 3.3 Classification of general non-toric bases . .... ...... ..... 67 3.3.1 General strategy ............. ... .. ... .. 67 3.3.2 The algorithm .. ............. .. .. .. .. 74 3.3.3 R esults ....... ....... ..... .. .. .. .. 79 4 Classification of complex base threefolds 83 4.1 Calabi-Yau fourfolds and 4D F-theory ... .... .... .... .. 83 7 4.2 Classification of base geometry in 4D F-theory . ..... ..... 91 4.3 Random walk on the set of toric threefold bases ..... ..... 99 4.3.1 M ethodology .. ......... ......... ..... 99 4.3.2 Unbounded random walk ... ......... .. ... ... 102 4.3.3 Estimate the total number of bases in C ..... ..... 106 4.4 Random blow up on the set of toric threefold bases .. ... .. 110 4.4.1 M ethodology ... .............. ... ..... 110 4.4.2 Results about the end points ..... ....... ..... 116 4.4.3 Estimating the total number of resolvable and good bases ... 121 4.4.4 Global structure of the set of bases ........ ...... 124 4.5 The geometry with most flux vacua ............ ...... 129 5 Bases with non-Higgsable U(1)s 135 5.1 Weierstrass models with additional rational sections .. .... .. 135 5.2 Counting the Weierstrass moduli ............. ...... 142 5.2.1 The minimal tuning of U(1) ............ ...... 146 5.2.2 6D F-theory examples .............. .. .... 149 5.3 Conditions on a base with a non-Higgsable U(1) ..... ...... 153 5.4 Examples of non-Higgsable U(1)s ............. ...... 160 5.4.1 A universal class ............. ..... .. .... 160 5.4.2 Semi-toric generalized Schoen constructions .. .. .... 161 6 Conclusions and outlook 173 A Notation for supersymmetry 181 8 List of Figures 1-1 Loop summation in quantum field theory ... ........ ..... 13 1-2 Loop summation in string theory .... ...... ...... .... 14 1-3 String duality web in mid-1990s ...... ......... ...... 18 1-4 String duality web with F-theory .. ........ ......... 22 2-1 2-torus as a quotient of the complex plane .... ...... ..... 31 2-2 Graphic presentation of an elliptic fibration .... ...... .... 32 2-3 A singular fiber with A 5 topology .... ...... ...... .... 34 2-4 The toric fan of p3 ..... ....... ...... ...... .... 51 3-1 Picture of Green-Schwarz mechanism in 6D ...... ...... ... 54 3-2 Geometry of Hirzebruch surface ..... ..... ..... ..... 61 3-3 The toric fan of Hirzebruch surface .... ...... ...... ... 61 3-4 The Hodge numbers from Kreuzer-Skarke database and generic elliptic threefolds over toric bases ...... ........ ....... ... 66 3-5 Sem i-toric surface . ....... ...... ...... ...... ... 67 3-6 Geometric ambiguity on a base surface .. ...... ...... ... 78 3-7 An example of Pappus's theorem ... ..... ..... ..... .. 78 3-8 Hodge numbers of generic elliptic CY3 over non-toric bases with h' 1 (S) < 8 ....... ... ...................................... 80 3-9 Hodge numbers of generic elliptic CY3 with h2,1 (x) > 150 ....... 81 4-1 Distribution of Hodge numbers for a large set of Calabi-Yau fourfolds 85 4-2 Blow ups on a toric threefold . ........ ........ ..... 94 9 4-3 Flop on a toric threefold .. ...... ..... .... ..... ... 95 4-4 Distribution of hl'1 (B) in the random walk approach .... ..... 102 4-5 Average values of (h1'"(X), h3 '1(X)) in the random walk approach . 103 4-6 Estimation of the number of bases in the random walk approach .. 108 4-7 Estimation of logio(N) in the random walk approach ......... 108 4-8 Average number of flops in the random walk approach ........ 109 4-9 The demonstration of random blow up approach ........... .111 4-10 Underestimation of the total number of bases from multiple starting p oints ................................... 114 4-11 A systematic error in the random blow up approach due to the uneven probability distribution ................. ......... 116 4-12 The dropping of h3 '1 (X) as a function of hl"'(B) in two random blow up sequences....... ................................ 117 4-13 Logarithm of the number of resolvable bases from P3 ......... 121 4-14 Logarithm of the number of good bases from P3 ............ 122 3 4-15 Logarithm of the number of resolvable bases starting from P , F 2 and P 1 x P1 x P 1 . ................ 123 4-16 Logarithm of the number of good bases starting from P3, F 2 and P' x P ' x P . .. ....................... ......... 124 4-17 A rough picture of the set of toric threefold bases ........... 128 5-1 A fictional configuration of 2D toric base with monomials in g aligned along the y-axis ....... ....................... 157 5-2 Geometric configuration of a generalized del Pezzo surface gdPst .. 161 5-3 The construction of a base surface gdP8 t with non-Higgsable U(1)s: step 1 ......... ................................... 162 5-4 The construction of a base surface gdPgot with non-Higgsable U(1)s: step 2 ......... ................................... 164 5-5 The construction of a base surface gdP8 t with non-Higgsable U(1)s: step 3 ......... ................................... 165 10 List of Tables 2.1 Kodaira's classification of elliptic fiber ................
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