Gauge symmetries and charged matter in F-theory

Abstract

F-theory has emerged as a powerful tool for string compactification, as it can realize a number of string vacua with varying gauge groups, charged matter, and other properties. However, some aspects of F-theory are still not completely understood, particularly those involving charged matter. There is currently no complete classification of all the charged matter representations that can occur in F-theory. Moreover, constructing models with a desired charge spectrum can be difficult. Some simple representations, such as the fundamental representations of nonabelian gauge groups, appear automatically in F-theory constructions. But representations beyond these simple types appear only in algebraically intricate constructions, and few techniques exist to systematically construct models with these more unusual representations. This thesis therefore focuses on gauge groups and charged matter in F-theory models. On the one hand, we examine whether models with various representations can be constructed in F-theory and discuss systematic methods for deriving them. We also describe interesting physical phenomena that occur in theories with less common representations. After briefly reviewing string theory and F-theory, we begin by describing a novel kind of transition between string vacua that changes the matter content of the model without changing the gauge group or other parts of the spectrum. These "matter transitions" are realized by passing through superconformal theories at the transition point. We first discuss how such transitions can be motivated from supergravity considerations. We then illustrate how the transitions manifest themselves in dual F-theory and heterotic models. We also explicitly analyze the duality between the heterotic and F-theory constructions, as the models we consider serve as nontrivial examples of heterotic/F-theory duality in their own right. We then turn to the topic of higher-genus representations in F-theory. Certain representations occur only on gauge divisors with a genus greater than zero. In particular, some representations, such as the symmetric and three-index symmetric representations of SU (N), occur at singular loci of gauge divisors. We present a systematic method for constructing models with these representations that relies on the normalization of curves from algebraic geometry. With this technique, we can construct, from scratch, SU(N) models with matter in the symmetric representation and SU(2) models with matter the 4 representation. Our constructions generalize the previous isolated examples in the literature. We also describe how the matter transitions mentioned previously appear in models with higher-genus representations. Finally, we argue that certain representations and matter spectra cannot be realized in F-theory, even though the corresponding models seem to satisfy the known low-energy considerations. Such models are therefore interesting candidates for the F-theory swampland, the set of seemingly consistent supergravity theories that cannot be realized in F-theory. In particular, these results suggest that F-theory, and possibly string theory more broadly, can only realize a limited number of nonabelian representations; they therefore may represent fundamental string theory constraints on the types of matter that can be coupled to quantum gravity. The final part of this thesis focuses on abelian F-theory models with matter having large charges. While U(1) F-theory models with charges less than 2 (in appropriately quantized units) are easily constructed in F-theory, models with charges 3 and above are less well understood. For instance, the previously known examples with charge-3 matter were found somewhat by chance. These models involve complicated algebraic expressions with intricate structures that hint at a deeper logic. In this part of the thesis, we show that charge-3 models can be systematically constructed using techniques similar to those used for models with higher-genus representations. The resulting construction generalizes the previous charge-3 examples known in the literature. We also obtain a class of charge-4 models, which, to the author's knowledge, are the first F-theory examples with charge-4 matter. We conclude by presenting some conjectures regarding models with charges larger than 4.