Aspects of string model building and heterotic/F-theory duality
Callum R. Brodie · Oxford University Research Archive (ORA) (University of Oxford) · 2019
In this thesis, we develop tools for model building in heterotic string theory and F-theory. We focus in particular on heterotic line bundle models and their F-theory duals. The first reason for this is that line bundle models are particularly fruitful models for phenomenology, and the second reason is that they provide a convenient background in which to make progress in studying heterotic/F-theory duality for more general features. In the first two parts of this thesis, we focus on transferring unique model building advantages between the heterotic and F-theory descriptions, via heterotic/F-theory duality. In the first part, we lay groundwork by analysing the interplay between phenomenological constraints and the requirement of an F-theory dual, for heterotic line bundle models. We describe the required geometries in detail and construct many examples. We also describe the constraints on the line bundle sums, and perform a systematic search for phenomenologically interesting models, finding several hundred. In the second part, we first develop in detail the F-theory dual picture of heterotic line bundle models. The spectral cover description is inappropriate, so we first conjecture the details of the duality. We then verify various aspects, including gauge groups, spectra, supersymmetry conditions, and anomaly conditions, and we treat some interesting subtleties. Using knowledge of this duality, we then develop the F-theory perspective on a well-known but little-utilised aspect of heterotic model building: heterotic NS5-branes, including stacks and intersections. We describe the dual geometry to various NS5-brane configurations, including singular transitions between configurations. We make heavy use of toric geometry and find many pleasing correspondences between the toric data and the NS5-brane configuration. In the third part, we develop methods to determine closed-form expressions for line bundle cohomology. We focus on the simpler case of complex surfaces as a step towards understanding the case of Calabi-Yau manifolds. These surfaces can also be used as bases for elliptic three- folds, and their cohomology lifted via the Leray spectral sequence. We derive index expressions for all line bundle cohomology on del Pezzo and Hirzebruch surfaces, and give methods to derive similar formulae on any compact tori surface, using a form of Zariski decomposition. We also discuss general methods to determine formulae on generic spaces, and in a companion paper we construct machine learning networks for this purpose.