Enhanced gauge symmetry in 6D F-theory

Abstract

This thesis reports on progress in understanding the set of 6D F-theory vacua. F-theory provides a strikingly clean correspondence between physics and physical quantities and mathematics and geometrical quantities, which allows us to make precise mathematical statements using well defined and understood methods. We present two related results that both serve the following principal goal: to understand the set of 6D F-theory vacua using geometrical methods, and then to compare these to low-energy supergravities. In doing so, we find a near-perfect correspondence between low-energy supergravities that can be obtained from F-theory and field theories that satisfy known low-energy consistency conditions, e.g. anomaly cancellation. However, we will also isolate several cases that we prove can never arise in F-theory yet have no visible lowenergy inconsistencies. The results are presented in two chapters. First, we describe a complete, systematic enumeration of all elliptically fibered Calabi-Yau threefolds (EF CY3s) with Hodge number h²,¹ >/= 350; physically, this classifies all F-theory models that lead to low-energy supergravities with >/= 351 neutral hypermultiplets. This result is obtained using global geometric calculations in finitely many, specific geometries. Second, we classify which local geometrical structures, corresponding to combinations of gauge algebras and (potentially shared) matter, can arise in F-theory. This classification is performed using local geometric calculations. This investigation reveals an exceedingly tight correspondence between F-theory models and consistent low-energy supergravities. Indeed, this near-perfect agreement provides a backdrop against which discrepancies between F-theory and low-energy supergravities stand out in sharp contrast. We describe in detail these discrepancies, in which seemingly consistent field theories cannot be described in F-theory. This work has several implications. First, it further refines the understanding of 6D supergravity models in F-theory, which has implications for string universality in 6D. It adds a level of mathematical precision to the study of 6D superconformal field theories (SCFTs) begun in [4, 3], which is a conjecturally complete classification of all 6D SCFTs. Our analysis confirms many of their results, but also explicitly shows that some of their proposed models cannot in fact be realized through their construction. Since our results can be phrased in terms of geometry, they also have implications for the study of EF CY3s. Finally, we discuss the subset of our results that hold in 4D F-theory as well, where they provide additional structure in a still difficult-to-constrain landscape.