TASI Lectures on F-theory
Timo Weigand
TL;DR
The notes present a comprehensive, pedagogical treatment of F-theory as a geometric framework for non-perturbative string compactifications, emphasizing the elliptic fibration as a dictionary between physics (7-branes, gauge algebras, and matter) and geometry (Kodaira fibers, Tate models, and the Mordell–Weil group). Central techniques include the M-theory duality, Tate's algorithm, and the Shioda map for abelian sectors, together with Deligne cohomology and Chow groups for gauge backgrounds and flux counting. The work systematically develops the correspondence from codimension-one to higher-codimension singularities, detailing how localized charged matter, Yukawa couplings, and discrete symmetries arise from fiber enhancements and global global structure. It also highlights the role of fluxes in generating chirality, moduli stabilization, and the Green–Schwarz mechanism, linking geometry to phenomenology and to broader mathematical structures in algebraic geometry. The overarching aim is to enable robust model building, non-perturbative QFT insights, and mathematical developments through a unified geometric lens.
Abstract
F-theory is perhaps the most general currently available approach to study non-perturbative string compactifications in their geometric, large radius regime. It opens up a wide and ever-growing range of applications and connections to string model building, quantum gravity, (non-perturbative) quantum field theories in various dimensions and mathematics. Its computational power derives from the geometrisation of physical reasoning, establishing a deep correspondence between fundamental concepts in gauge theory and beautiful structures of elliptic fibrations. These lecture notes, which are an extended version of my lectures given at TASI 2017, introduce some of the main concepts underlying the recent technical advances in F-theory compactifications and their various applications. The main focus is put on explaining the F-theory dictionary between the local and global data of an elliptic fibration and the physics of 7-branes in Type IIB compactifications to various dimensions via duality with M-theory. The geometric concepts underlying this dictionary include the behaviour of elliptic fibrations in codimension one, two, three and four, the Mordell-Weil group of rational sections, and the Deligne cohomology group specifying gauge backgrounds.