Algebraic Structures on Moduli Spaces of Mirror Geometries

Abstract

This thesis is concerned with algebraic structures appearing on the moduli spaces of mirror geometries enhanced with differential forms. It was shown in \cite{Movasati:2012elliptic} that in the case of elliptic curves, the local structure of such moduli spaces can be constructed from a variation of polarized Hodge structure, giving rise to finitely generated graded differential rings, generalizing the ring of quasi-modular forms. An algebra, named the Gauss-Manin or AMSY Lie algebra, of derivations on these rings was constructed from a suitable combination of vector fields on the moduli space. The program of investigating moduli spaces of Landau-Ginzburg models enhanced with differential forms was named the Gauss-Manin Connection in Disguise (GMCD).

The thesis is split in three independent parts:

In the first part the GMCD construction is carried out for several families of lattice polarized elliptic K3 surfaces. The local structure of moduli spaces of K3 surfaces enhanced with differential forms is identified and an algebra of derivations on the rings of regular functions is found. We show that the ring of regular functions can be identified with the ring of quasi-modular forms in two variables.

In the second part of the thesis we extend the GMCD program to families of non-compact Calabi-Yau threefolds. A definition of families enhanced with differential forms is proposed and the local structure of the moduli spaces of such families is investigated. We show that in the case of mirrors of local $\mathbb{P}^2$ and local $\mathbb{F}^2$ the rings of regular functions are closely related to the rings of quasi-modular forms, arising from the associated mirror curves. We construct the Gauss-Manin Lie algebra in both cases and identify an $\mathrm{sl}_2(\mathbb{C})$ Lie subalgebra.

The third part of the thesis is concerned with extending the GMCD program to families of toric Landau-Ginzburg models. We propose a definition of Landau-Ginzburg models, enhanced with a GKZ local system of solutions to differential equations, generalizing both of the previous constructions. We apply the constructions to Landau-Ginzburg mirrors of $\mathbb{CP}^n$, constructing differential rings associated to the families and giving a first example of a GMCD construction for non-Calabi-Yau Landau-Ginzburg models.