Geometry and Arithmetic of Special Loci in the Moduli Spaces of Type II String Theory

TL;DR

This paper investigates special loci in the moduli spaces of Calabi-Yau manifolds within Type II string theory, using advanced mathematical techniques to analyze their geometric and physical properties, revealing new insights into string vacua and flux compactifications.

Contribution

It introduces a novel approach combining Dwork's deformation method with Hasse-Weil Zeta functions to identify and analyze special subloci in moduli spaces relevant to string theory.

Findings

Identification of codimension-one subslices with split Hodge structures

Calculation of background fluxes and potentials for these subslices

Discovery of integral expansions related to disk instantons

Abstract

We use Dwork's deformation method to calculate the Hasse-Weil Zeta function of multi-parameter families of Calabi-Yau three and fourfolds. This information is used to identify subslices of codimension one in the complex-structure moduli space, where the Hodge structure splits in particular ways and different type IIB flux vacua emerge. We calculate the corresponding background fluxes and their potential that drives the IIB string compactification to these subslices and analyse the properties of the corresponding physical vacua. We address the question whether the subslices correspond to fixed loci of symmetries acting on the original family and whether they can be identified with consistent complex-structure moduli spaces of Picard-Fuchs systems with standard integral monodromy bases for fewer complex deformation parameters. We distinguish between supersymmetric vacua and singular subslices. In the latter case a standard geometrical basis can be expected if a physical transition leads to a smooth type II vacuum. In many cases the differential equations on the subslice are fulfilled by the restricted periods after adding an inhomogeneous term. This suggests that the resolution of the singularity provides three-chains and we indeed find that the corresponding integrals allow an integral expansion compatible with their interpretation as generating functions of disk instantons.