Date of Award:
Master of Science (MS)
Mathematics and Statistics
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⊕E7(-1)⊕E7(-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.
Hill, Thomas, "Classification of Jacobian Elliptic Fibrations on a Special Family of K3 Surfaces of Picard Rank Sixteen" (2020). All Graduate Theses and Dissertations, Spring 1920 to Summer 2023. 7921.
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