Extending the Geometric Tools of Heterotic String Compactification and Dualities Mohsen Karkheiran Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Lara B. Anderson, Chair James Gray Eric Sharpe Uwe Täuber May 13, 2020 Blacksburg, Virginia Keywords: Heterotic string, F-theory, Fourier-Mukai, Spectral cover Copyright 2020, Mohsen Karkheiran Extending the Geometric Tools of Heterotic String Compactification and Dualities Mohsen Karkheiran (ABSTRACT) In this work, we extend the well-known spectral cover construction first developed by Friedman, Morgan, and Witten to describe more general vector bundles on ellipti- cally fibered Calabi-Yau geometries. In particular, we consider the case in which the Calabi-Yau fibration is not in Weierstrass form but can rather contain fibral divisors or multiple sections (i.e., a higher rank Mordell-Weil group). In these cases, gen- eral vector bundles defined over such Calabi-Yau manifolds cannot be described by ordinary spectral data. To accomplish this, we employ well-established tools from the mathematics literature of Fourier-Mukai functors. We also generalize existing tools for explicitly computing Fourier-Mukai transforms of stable bundles on elliptic Calabi-Yau manifolds. As an example of these new tools, we produce novel examples of chirality changing small instanton transitions. Next, we provide a geometric for- malism that can substantially increase the understood regimes of heterotic/F-theory duality. We consider heterotic target space dual (0,2) GLSMs on elliptically fibered Calabi-Yau manifolds. In this context, each half of the “dual” heterotic theories must, in turn, have an F-theory dual. Moreover, the apparent relationship between two heterotic compactifications seen in (0,2) heterotic target space dual pairs should, in principle, induce some putative correspondence between the dual F-theory geome- tries. It has previously been conjectured in the literature that (0,2) target space duality might manifest in F-theory as multiple K3-fibrations of the same elliptically fibered Calabi-Yau manifold. We investigate this conjecture in the context of both 6-dimensional and 4-dimensional effective theories and demonstrate that in general, (0,2) target space duality cannot be explained by such a simple phenomenon alone. In all cases, we provide evidence that non-geometric data in F-theory must play at least some role in the induced F-theory correspondence while leaving the full de- termination of the putative new F-theory duality to the future work. Finally, we consider F-theory over elliptically fibered manifolds, with a general conic base. Such manifolds are quite standard in F-theory sense, but our goal is to explore the extent of the heterotic/F-theory duality over such manifolds. Extending the Geometric Tools of Heterotic String Compactification and Dualities Mohsen Karkheiran (GENERAL AUDIENCE ABSTRACT) String theory is the only physical theory that can lead to self-consistent, effective quantum gravity theories. However, quantum mechanics restricts the dimension of the effective spacetime to ten (and eleven) dimensions. Hence, to study the con- sequences of string theory in four dimensions, one needs to assume the extra six dimensions are curled into small compact dimensions. Upon this “compactification,” it has been shown (mainly in the 1990s) that different classes of string theories can have equivalent four-dimensional physics. Such classes are called dual. The advan- tage of these dualities is that often they can map perturbative and non-perturbative limits of these theories. The goal of this dissertation is to explore and extend the geometric limitations of the duality between heterotic string theory and F-theory. One of the main tools in this particular duality is the Fourier-Mukai transformation. In particular, we consider Fourier-Mukai transformations over non-standard geome- tries. As an application, we study the F-theory dual of a heterotic/heterotic duality known as target space duality. As another side application, we derive new types of small instanton transitions in heterotic strings. In the end, we consider F-theory compactified over particular manifolds that if we consider them as a geometry dual to a heterotic string, can lead to unexpected consequences. Dedication I humbly dedicate this dissertation to Zoroaster. iv Acknowledgments Firstly, I would like to express my sincere gratitude to my advisor Lara Anderson for the continuous support of my Ph.D. study and related research and her patience and motivation. Besides my advisor, I would like to thank the rest of my dissertation committee: Eric Sharpe, James Gray, and Uwe Täuber, for their insightful comments. My sincere thanks also go to Xin Gao, Paul Oehlmann, Nikhil Raghuram and He Feng, for their collaboration and discussions. Without them, it wouldn’t be possible to finish the projects. I thank my fellow office mates, Wei Cui, Juntao Wang, Hao Zou, Zhou Chen, Jirui Guo, and especially Wei Gu, for the stimulating and intuitive discussions, and for all the fun we have had in the last five years. Lastly, I should mention the projects in this dissertation were supported in part by NSF grant PHYS-1720321. v Contents List of Figures xi List of Tables xiii 1 Introduction 1 1.1 Motivation and overview ......................... 1 1.2 Heterotic string compactification .................... 6 1.3 Heterotic M-theory ............................ 9 1.4 F-theory compactification ........................ 12 1.4.1 F-theory as the strong limit of type IIB string theory ..... 13 1.4.2 Elliptic fibration ......................... 15 1.4.3 Compactifying to lower dimensions. ............... 18 1.4.4 Higgs bundle ........................... 19 1.4.5 F-theory as the singular limit of M-theory ........... 23 1.5 Heterotic/F-theory duality ........................ 26 1.5.1 Basic duality ........................... 26 1.5.2 Stable degeneration and Heterotic/F-theory duality in eight dimensions ............................ 27 1.5.3 Heterotic/F-theory duality in lower dimensions ........ 29 vi 2 Heterotic Spectral Cover Constructions and Generalization 32 2.1 A review of vector bundles over Weierstrass elliptic fibrations and Fourier-Mukai transforms ........................ 33 2.1.1 Irreducible smooth elliptic curve ................ 33 2.1.2 Weierstrass elliptic fibration ................... 36 2.1.3 Spectral cover ........................... 37 2.1.4 Topological data ......................... 38 2.2 Elliptically fibered manifolds with fibral divisors ............ 41 2.2.1 (Semi) stable vector bundles over I2 elliptic curves ...... 42 2.2.2 Topological data ......................... 44 2.2.3 What is missing in the singular limit .............. 47 2.3 Non-trivial Mordell-Weil group with a holomorphic zero section ... 50 2.3.1 Topological data ......................... 51 2.3.2 Rank one Mordell-Weil group .................. 53 2.4 Non-trivial Mordell-Weil group with rational generators ........ 53 2.4.1 Flop transitions .......................... 54 2.4.2 Comment on the chirality of the effective theory ........ 58 2.5 Examples of explicit Fourier-Mukai transforms ............. 59 2.5.1 Bundles defined by extension on Weierstrass CY threefolds .. 60 2.5.2 FM transforms of monad bundles over 3-folds ......... 64 2.5.3 An extension bundle defined on an elliptic fibration with fibral divisors .............................. 71 2.5.4 A bundle defined via extension on a CY threefold with rk(MW)=1 73 2.6 Small instanton transitions and spectral covers ............. 74 2.6.1 New chirality changing small instanton transitions ...... 75 vii 2.7 Reducible spectral covers and obstructions to smoothing ....... 79 3 Heterotic/Heterotic and Heterotic/F-theory Duality 84 3.1 A brief review of (0,2) target space duality ............... 88 3.2 A target space dual pair with elliptically fibered Calabi-Yau threefolds 92 3.2.1 A tangent bundle deformation .................. 93 3.2.2 More general vector bundles ................... 97 3.3 Inducing a duality in F-theory ...................... 99 3.3.1 Essential tools for Heterotic/F-theory duality ......... 99 3.3.2 Spectral cover construction ................... 101 3.3.3 Warm up: Heterotic/F-theory duality in 6-dimensions .... 104 3.3.4 Spectral cover of monads ..................... 105 3.3.5 Counterexamples of the conjecture ............... 110 3.3.6 F-theory duals of 4-dimensional Heterotic TSD pairs ..... 118 4 F-theory on General Conic Bundle Bases 124 4.1 P1-fibrations in a nutshell ........................ 126 4.1.1 P1-bundles ............................ 126 4.1.2 Conic bundles ........................... 129 4.2 P1-fibered bases in 6-dimensional .................... 131 4.2.1 P1-bundles and P1-fibrations over P1 .............. 131 4.2.2 Dual Heterotic/F-theory geometry ............... 132 4.2.3 Jumping phenomena ....................... 134 4.3 General P1-bundle bases for 4-dimensional F-theory compactifications 136 4.3.1 Examples: Type I ........................ 140 4.3.2 Examples: Type II ........................ 142 viii 4.4 Conic bundles without monodromy ................... 144 4.4.1 Sections .............................. 146 4.4.2 Degenerations ........................... 147 4.5 Conic bundles with monodromy ..................... 149 4.5.1 Degenerations ........................... 150 4.5.2 Monodromy ............................ 154 4.6 Heterotic dual