Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Physics & Astronomy

First Advisor

Ron Y. Donagi


F-theory offers a compelling, nonperturbative framework in which to construct string vacua in even dimensional space-time. These theories are described by an elliptically fibered Calabi-Yau manifold over a base of complex dimension $d$ for a low energy theory in $10-2d$ dimensional space-time. The structure of the gauge group in the low-energy theory is captured principally in the singularities of the elliptic fibration. In the parlance of type IIB string theory, the gauge group is reflected in 7-branes which wrap on topologically nontrivial cycles of the base manifold.

The geometry of the gauge groups and matter representations that arise in F-theory has been worked out by Taylor-Morrison. While there is a finite set of gauge groups and matter content that arise in F-theory constructions of 6D supergravity theories, a given base may yield a wide range of both. Models that have different spectra over a common F-theory base can be configured by tuning the coefficients in the Weierstrass description of the model such that certain codimension-one and -two singularities materialize in the elliptic fibration. As the structure of singularities in the fibration becomes more sophisticated, the accompanying matter representations become more exotic.

Taylor-Morrison systematically classified the smooth toric bases that support elliptically fibered Calabi-Yau threefolds. By analyzing the intersection structure of irreducible effective divisors on the base, they found that there are 61,539 distinct toric bases. Many Calabi-Yau manifolds can be realized as hypersurfaces in toric varieties of one higher dimension which, following Batyrev, can be constructed with reflexive polyhedra. Kreuzer-Skarke systematically identified the 473,800,776 reflexive polytopes in 4D. This suggests another approach to understanding elliptically fibered Calabi-Yau threefolds for 6D theories: by analyzing polytopes in 4D, one can understand all toric elliptic fibrations for F-theory from this list. Braun identified the polytopes corresponding to elliptic fibrations over base $C\mathbb{P}^2$.

In this thesis, I systematically identify the polytopes that correspond to elliptic fibrations of the first Hirzebruch surfaces $F_1$ as well as $F_{12}$, the final Hirzebruch surface admitted in an F-Theoretic context. I study the hodge structure of all admitted 4D polytopes; as well as the Gauge groups and Kodaira fibers.


Available to all on Monday, June 22, 2020