On the Geometry of the Moduli Space of Certain Lattice Polarized K3 Surfaces and Their Picard-Fuchs Operators

K3 surfaces have a long and rich study in mathematics, and more recently in physics via string theory. Often, K3 surfaces come in multiparameter families - the parameters describing these surfaces fit together to form their own geometric space, a so-called moduli space. In particular, the moduli spaces of K3 surfaces equipped with a lattice polarization can sometimes be constructed explicitly, which subsequently reveals important information about the original K3 surface. In this work, we construct such families explicitly from certain rational elliptic surfaces via the so-called mixed-twist construction of Doran & Malmendier, which in turn produces the moduli space. After identifying the lattice polarization by computing Jacobian elliptic fibrations, we find a rich differential geometric content imparted to the moduli space - an integrable holomorphic conformal structure - via quadratic relations satisfied by the period integrals of the K3 surface. This geometry allows one to compute crucial data about the K3 surface family, the Picard-Fuchs operators, by applying a general programme on uniformizing differential equations discovered by Sasaki & Yoshida. In physics, this differential geometric data is known as a flat special geometry, and has implications for a type of supersymmetric quantum field theory associated with the K3 surface. Via the mixed-twist construction, this is related to Nf= 4 Seiberg-Witten curves from N = 2 SU(2) super Yang-Mills theory with various mass configurations. We show as well how one can restrict the moduli, leading to subvarieties of the moduli space on which the lattice polarization extends. This can allow one to construct interesting families of Calabi-Yau manifolds, which are of crucial importance in string theory as well. Moreover, we study how other data that governs the complex structure of the elliptic fibres of certain generic fibrations determines global information about a Jacobian elliptic K3 surface in terms of string theoretic and index theoretic terms via holomorphic anomalies