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

TL;DR

This work develops an arithmetic-geometry framework to locate special codimension-one loci in the complex-structure moduli spaces of multi-parameter Calabi–Yau threefolds and fourfolds by extracting Hasse-Weil zeta data via Dwork deformations. By analyzing factorization patterns of Euler factors across primes and employing the Chinese remainder theorem, Chebotarëv density, and modularity considerations, the authors identify rank-two attractor-like loci and SUSY flux vacua, often tied to symmetry fixed points or Hadamard products of base/fibre periods. They introduce and refine a practical reduction to linear subspaces in moduli space, handle apparent singularities in Picard–Fuchs systems, and connect these arithmetic signals to geometric transitions (strong coupling, conifold) with explicit toric and blowup examples. The results reveal new supersymmetric vacua, symmetry-induced modular structures, and instances of motive-splittings (including CM cases), illustrating how arithmetic criteria illuminate the landscape of Type II vacua and their geometric underpinnings. Collectively, the paper provides a concrete, scalable program to probe flux vacua and moduli-space geometry in higher-dimensional CY families, with potential links to RCFT, automorphic forms, and string dualities.

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.

Figures (8)

• Figure 1: Elliptic families embedded in the bi-quadratic in $\mathbb{P}^1\times \mathbb{P}^1$. Polytopes (1,2) and (3,4) form reflexive pairs while 5 and 6 are self-dual. Polytopes (2,4,6) can also be embedded in the quartic family in \ref{['fig:ellembed4']} and (2-6) in the cubic family of \ref{['fig:ellembed3']}.
• Figure 2: Elliptic families embedded in the quartic in $\mathbb{P}_{211}$. Polytopes (7,8) and (9,10) form reflexive pairs while 11 and 12 are self-dual, where (8) and (10) can also be embedded in both the bi-quadratic in \ref{['fig:ellembed22']} and the cubic in \ref{['fig:ellembed3']}.
• Figure 3: Elliptic families embedded in the cubic in $\mathbb{P}^2$. Polytopes (13,14) and (15,16) form reflexive pairs, where (14) and (16) can also be realised in both the bi-quadratic and the quartic of \ref{['fig:ellembed22', 'fig:ellembed4']}, respectively.
• Figure 4: Factorizations for the model $\text{Bl}^{(4)}(X_{21111})$ in various directions $(z_1,z_2)=(nz,z)$. For each direction the average number of factorizations is above one and, apart from points of bad reduction, for each prime there is at least one factorization at the reduction modulo $p$ of $z_1=1/27$.
• Figure 5: Schematic picture of the moduli space of the model $\text{Bl}^{(4)}(X_{21111})$. Shown are the discriminant components and exceptional divisors resolving tangencies. The dashed lines show the location of the supersymmetric vacua discussed in the text and we indicate the flux lattices they support.