Elliptic Calabi-Yau Threefolds with Z_3 x Z_3 Wilson Lines

TL;DR

This work constructs a non-simply connected Calabi–Yau threefold suitable for heterotic string compactifications by quotienting the fiber product of two rational elliptic surfaces ($dP_9$) by a free action of ${f Z}_3 imes{f Z}_3$, yielding $ obreak obreak obreak ( obreakoldsymbol{ obreak ext{pi}}_1(X)= obreak {f Z}_3 imes{f Z}_3)$. The authors detail the required Wilson-line symmetry breaking pattern from ${ m Spin}(10)$ to ${SU(3)_C imes SU(2)_L imes U(1)_Y imes U(1)_{B-L}}$ using two independent ${f Z}_3$ holonomies, with a stable ${SU}(4)$ bundle in the observable sector and a flat ${f Z}_3 imes{f Z}_3$ bundle to realize the quotient. They develop a full geometric framework: classifying ${f Z}_3 imes{f Z}_3$ actions on rational elliptic surfaces via the Mordell–Weil group, constructing explicit Weierstrass and pencil-of-cubics realizations, and computing the induced action on homology. The resulting Calabi–Yau $X$ has $b_2=3$ and $h^{1,1}=h^{2,1}=3$, with three complex structure moduli, and supports heterotic vacua with three generations, a right-handed neutrino, and a gauged ${U(1)}_{B-L}$, thereby naturally suppressing nucleon decay.

Abstract

A torus fibered Calabi-Yau threefold with first homotopy group Z_3 x Z_3 is constructed as a free quotient of a fiber product of two dP_9 surfaces. Calabi-Yau threefolds of this type admit Z_3 x Z_3 Wilson lines. In conjunction with SU(4) holomorphic vector bundles, such vacua lead to anomaly free, three generation models of particle physics with a right handed neutrino and a U(1)_{B-L} gauge factor, in addition to the SU(3)_C x SU(2)_L x U(1)_Y standard model gauge group. This factor helps to naturally suppress nucleon decay. The moduli space and Dolbeault cohomology of the threefold is also discussed.

Figures (12)

• Figure 1: The ${\mathop{\text{Spin}}\nolimits}(10)$ Dynkin diagram.
• Figure 2: ${\mathop{\text{Spin}}\nolimits}(10)$ Dynkin diagram with the root $\alpha^5$ removed.
• Figure 3: ${\mathop{\text{Spin}}\nolimits}(10)$ Dynkin diagram with the root $\alpha^3$ removed.
• Figure 5: Singular fiber of the $I_3$ type.
• Figure 6: An automorphism of the $I_6$ singular fiber.