Date of Award:

12-2020

Document Type:

Thesis

Degree Name:

Master of Science (MS)

Department:

Mathematics and Statistics

Committee Chair(s)

Andreas Malmendier

Committee

Andreas Malmendier

Committee

Ian Anderson

Committee

Nathan Greer

Abstract

K3 surfaces are an important tool used to understand the symmetries in physics that link different string theories, called string dualities. For example, heterotic string theory compactified on an elliptic curve describes a theory physically equivalent to (dual to) F-theory compactified on a K3 surface. In fact, M-theory, the type IIA string, the type IIB string, the Spin(32)/Z2 heterotic string, and the E8 x E8 heterotic string are all related by compactification on Calabi-Yau manifolds.

We study a special family of K3 surfaces, namely a family of rank sixteen K3 surfaces polarized by the lattice H‚äïE7(-1)‚äïE7(-1). A generic member of this family is a K3 surface defined by resolving the singularities of a specific quartic surface. Intersecting this quartic with a pencil of planes containing a particular line or conic corresponds with a Jacobian elliptic fibration on the resulting K3 surface. We show that a generic member of this family of K3 surfaces admits exactly four inequivalent Jacobian elliptic fibrations; i.e., there are four non-isomorphic ways a pencil of planes containing a line or conic can intersect the quartic surface defining a member of this special family. We construct explicit Weierstrass models for these Jacobian elliptic fibration whose coefficients are modular forms on a suitable bounded symmetric domain of type IV . Finally, we explain how this construction provides a geometric interpretation for the F-theory/heterotic string duality in eight dimensions with two Wilson lines.

Checksum

8d9a05dd2bf17098d051a8d1d821032c

Included in

Mathematics Commons

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