Torus-Fibered Calabi-Yau Threefolds with Non-Trivial Fundamental Group

TL;DR

This work constructs torus-fibered Calabi–Yau threefolds with nontrivial fundamental group by forming the fiber product $X=B\times_{\mathbb{P}^1}B'$ of two rational elliptic surfaces and then quotientting by a freely acting $\mathbb{Z}_2\times\mathbb{Z}_2$ to obtain $Z$ with $\pi_1(Z)=\mathbb{Z}_2\times\mathbb{Z}_2$. The authors classify involutions on rational elliptic surfaces, identify parameter-restricted subfamilies that yield commuting involutions, and show that suitable invariant fiber classes emerge to support standard-like model building with Wilson lines. They prove that generic $X$ is Calabi–Yau and that the quotient $Z$ is torus-fibered and Calabi–Yau, providing a geometric framework for stable $SU(4)$ bundles and Wilson-line breaking toward realistic gauge theories. This construction lays the groundwork for a series of papers implementing vector bundles and Wilson lines to achieve three-family, nucleon-decay-suppressed grand unified theories in string compactifications.

Abstract

We construct smooth Calabi-Yau threefolds Z, torus-fibered over a dP_9 base, with fundamental group Z_2 X Z_2. To do this, the structure of rational elliptic surfaces is studied and it is shown that a restricted subset of such surfaces admit at least a Z_2 X Z_2 group of automorphisms. One then constructs Calabi-Yau threefolds X as the fiber product of two such dP_9 surfaces, demonstrating that the involutions on the surfaces lift to a freely acting Z_2 X Z_2 group of automorphisms on X. The threefolds Z are then obtained as the quotient Z=X/(Z_2 X Z_2). These Calabi-Yau spaces Z admit stable, holomorphic SU(4) vector bundles which, in conjunction with Z_2 X Z_2 Wilson lines, lead to standard-like models of particle physics with naturally suppressed nucleon decay.

Figures (13)

• Figure 1: The curves $f_1=0$ and $f_2=0$ intersecting at nine points.
• Figure 2: A double ${\mathbb P}^{1}$ line with four surface $A_1$ singularities. Its resolution is an $I_0^{*}$ Kodaira fiber. The cross-hatch indicates the identification of the two ${\mathbb P}^{1}$ lines.
• Figure 3: The bidegree $(2,4)$ curve $M=T\cup r$ in $Q={\mathbb P}^{1}\times {\mathbb P}^{1}$. The projections $p_i: Q \to \mathbb{P}_i^1, i=1,2$ are explicitly shown.
• Figure 4: The double cover of ${\mathbb P}^{1}$ branched over four points.
• Figure 5: A schematic representation of the curve $T$ on $Q={\mathbb P}^{1}\times {\mathbb P}^{1}$ indicating a simple ramification point $R_i$ over the branch point ${b_i}$ in ${\mathbb P}^{1}$.