Date of Award

2016

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Physics & Astronomy

First Advisor

Mirjam Cvetic

Abstract

In this dissertation, we focus on important physical and mathematical aspects, especially

abelian gauge symmetries, of F-theory compactifications and its dual formulations

within type IIB and heterotic string theory.

F-theory is a non-perturbative formulation of type IIB string theory which enjoys important

dualities with other string theories such as M-theory and E8 × E8 heterotic string

theory. One of the main strengths of F-theory is its geometrization of many physical problems

in the dual string theories. In particular, its study requires a lot of mathematical tools

such as advanced techniques in algebraic geometry. Thus, it has also received a lot of interests

among mathematicians, and is a vivid area of research within both the physics and

the mathematics community.

Although F-theory has been a long-standing theory, abelian gauge symmetry in Ftheory

has been rarely studied, until recently. Within the mathematics community, in 2009,

Grassi and Perduca first discovered the possibility of constructing elliptically fibered varieties

with non-trivial toric Mordell-Weil group. In the physics community, in 2012, Morrison

and Park first made a major advancement by constructing general F-theory compactifications

with U(1) abelian gauge symmetry. They found that in such cases, the ellipticallyfibered

Calabi-Yau manifold that F-theory needs to be compactified on has its fiber being a

generic elliptic curve in the blow-up of the weighted projective space P(1;1;2) at one point.

Subsequent developments have been made by Cvetiˇc, Klevers and Piragua extended the works of Morrison and Park and constructed general F-theory compactifications with U(1)

U(1) abelian gauge symmetry. They found that in the U(1) × U(1) abelian gauge symmetry

case, the elliptically-fibered Calabi-Yau manifold that F-theory needs to be compactified

on has its fiber being a generic elliptic curve in the del Pezzo surface dP2. In chapter 2 of

this dissertation, I bring this a step further by constructing general F-theory compactifications

with U(1) × U(1) × U(1) abelian gauge symmetry. I showed that in the case with three

U(1) factors, the general elliptic fiber is a complete intersection of two quadrics in P3, and

the general elliptic fiber in the fully resolved elliptic fibration is embedded as the generic

Calabi-Yau complete intersection into Bl3P3, the blow-up of P3 at three generic points.

This eventually leads to our analysis of representations of massless matter at codimension

two singularities of these compactifications. Interestingly, we obtained a tri-fundamental

representation which is unexpected from perturbative Type II compactifications, further

illustrating the power of F-theory.

In chapter 1 of this dissertation, I proved finiteness of a region of the string landscape in

Type IIB compactifications. String compactifications give rise to a collection of effective

low energy theories, known as the string landscape. However, it is not known whether the

number of physical theories we can derive from the string landscape is finite. The vastness

of the string landscape also poses a serious challenge to attempts of studying it. A

breakthrough was made by Douglas and Taylor in 2007 when they studied the landscape of

intersecting brane models in Type IIA compactifications on a particular Z2× Z2 orientifold.

They found that two consistency conditions, namely the D6-brane tadpole cancellation

condition, and the conditions on D6-branes that were required for N = 1 supersymmetry in

four dimensions, only permitted a finite number of D6-brane configurations. These finite

number of allowed D6-brane configurations thus result in only a finite number of gauge

sectors in a 4D supergravity theory, allowing them to be studied explicitly. Douglas and

Taylor also believed that the phenomenon of using tadpole cancellation and supersymmetry consistency conditions to restrict the possible number of allowed configurations to a

finite one is not a mere coincidence unique to their construction; they conjectured that this

phenomenon also holds for theories with magnetised D9- or D5-branes compactified on

elliptically fibered Calabi-Yau threefolds. Indeed, this was what my collaborators and I

also felt. To this end, I showed, using a mathematical proof, that their conjecture is indeed

true for elliptically fibered Calabi-Yau threefolds p X B whose base B satisfy a

few easily-checked conditions (summarized in chapter 1 of this dissertation). In particular,

these conditions are satisfied by, although not limited to, the almost Fano twofold bases

B given by the toric varieties associated to all 16 reflexive two-dimensional polytopes and

the del Pezzo surfaces dPn for n = 0;1; :::; 8. This list, in particular, also includes the Hirzebruch

surfaces F0 = P1 ×P1;F1 = dP1;F2. My proof also allowed us to derive the explicit

and computable bounds on all flux quanta and on the number of D5-branes. These bounds

only depends on the topology of the base B and are independent on the continuous moduli

of the compactification, in particular the Kahler moduli, as long as the supergravity approximation

is valid. Physically, my proof showed that these compactifications only give rise

to a finite number of four-dimensional N = 1 supergravity theories, and that these theories

only have finitely many gauge sectors with finitely many chiral spectra. Such finiteness

properties are not observed in generic quantum field theories, further fortifying superstring

theory as a more promising theory.

In chapter 3 of this dissertation, I study abelian gauge symmetries in the duality

between F-theory and E8 × E8 heterotic string theory. It is conjectured that F-theory, when

compactified on an elliptic K3-fibered (n + 1)-dimensional Calabi-Yau manifold X B,

and heterotic string theory when compactified on an elliptically fibered n-dimensional

Calabi-Yau manifold Z B with the same base B, are dual to each other. Thus under such

duality, in particular, if the F-theory compactification admits abelian gauge symmetries,

the dual heterotic string theory must admit the same abelian gauge symmetry as well. However, how abelian gauge symmetries can arise in the dual heterotic string theory has

never been studied. The main goal of this chapter is to study exactly this. We start with

F-theory compactifications with abelian gauge symmetry. With the help of a mathematical

lemma as well as a computer code that I came up with, I was able to construct a rich list of

specialized examples with specific abelian and nonabelian gauge groups on the F-theory

side. The computer code also directly computes spectral cover data for each example

constructed, allowing us to further analyze how abelian gauge symmetries arise on

heterotic side. Eventually, we found that in general, there are three ways in which U(1)-s

can arise on the heterotic side: the case where the heterotic theory admits vector bundles

with S(U(1) ×U(m)) structure group, the case where the heterotic theory admits vector

bundles with SU(m)×Zn structure group, as well as the case where the heterotic theory

admits vector bundles with structure groups having a centralizer in E8 which contains a

U(1) factor. Another important achievement was my discovery of the non-commutativity

of the semi-stable degeneration map which splits a K3 surface into two half K3 surfaces,

and the map to Weierstrass form, which was not previously known in the literature.

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