Modularity of strings on F-theory backgrounds

Abstract

This thesis regards topological string theory on elliptically and genus one fibered Calabi-Yau compactifications to study counting indexes over states in a dual six-, five-, and four-dimensional theory realized by F-theory.<br /> The first part focuses on studying several quantum gravity constraints that should fulfill a six-dimensional theory with minimal supersymmetry. We exploit the modular properties of the topological string partition function, equivalently the elliptic genera of six-dimensional strings, to argue the consistency of such constraints—including the absence of quantum anomalies. In particular, we manage to prove the sublattice non-Abelian weak gravity conjecture for F-/M-theory compactified on elliptically and genus one fibered Calabi-Yau 3-folds admitting a K3 fibration.<br /> In the next part, we focus on a subclass of K3 fibered geometries to analyze their K3 fiber reduced Gromov-Witten theory, equivalently their Noether-Lefschetz’s theory: We find a correspondence between the elliptic genera of Heterotic strings and the enumerative geometry in the reduced K3 fibers. Through Noether-Lefschetz symmetries, we derive that this information encodes the massless spectrum of six-dimensional supergravity theories, the data necessary to compute conjectured refined BPS invariants in compact geometries, and a supersymmetric index for four-dimensional theories with N = 2 supersymmetry. Using the latter result, we examine Heterotic theories with a CHL orbifold construction that possess a 5d M-theory dual interpretation compactified on a genus one fibered Calabi-Yau admitting a K3 fibration. We conclude that the CHL-Heterotic strings elliptic genera must be meromorphic vector-valued lattice Jacobi forms.<br /> In the last part, we turn to study elliptically fibered Calabi-Yau 4-folds and the modularity of their quantum periods. By turning on gauge fluxes, we achieve F-theory compactifications leading to four-dimensional theories with minimal supersymmetry. Moreover, the quantum periods of gauge fluxes interpret as elliptic genera of four-dimensional strings, described by quasi-Jacobi forms. We exemplify these objects and realize that they follow conjectural holomorphic anomaly equations.