A Three-Generation Calabi-Yau Manifold with Small Hodge Numbers

TL;DR

The authors construct a Calabi–Yau threefold $Y$ with Euler characteristic $-72$ that admits free actions by groups of order $12$, producing smooth quotients with $igl(\chi,(h^{11},h^{21})\bigr)=\bigl(-6,(1,4)\bigr)$. Starting from the standard embedding, the resulting $E_6$ vacua yield 3 chiral generations, with two routes to gauge-breaking: the Hosotani mechanism and continuous deformations of the gauge bundle; notably, a non-Abelian quotient enables a path to the Standard Model. A conifold transition from a 3-generation model to a manifold with $(h^{11},h^{21})=(2,2)$ suggests a heterotic vacuum continuation, placing these manifolds at the tip of the Calabi–Yau landscape. The work also develops toric and mirror descriptions via Batyrev’s construction, including a toric realization of $Y$ as a hypersurface in a product of del Pezzo surfaces and a detailed analysis of the mirror $Y^{*}$, its singularities, and resolutions. An Abelian quotient by $\mathbb{Z}_{12}$ is shown to be smooth with the same $(h^{11},h^{21})=(1,4)$, while discrete holonomies offer rich yet constrained gauge-breaking possibilities toward the Standard Model. Together, these results illuminate how carefully engineered CY geometries with small Hodge numbers can host realistic gauge structures and hinge on intricate interplays between topology, toric geometry, and string phenomenology.

Abstract

We present a complete intersection Calabi-Yau manifold Y that has Euler number -72 and which admits free actions by two groups of automorphisms of order 12. These are the cyclic group Z_12 and the non-Abelian dicyclic group Dic_3. The quotient manifolds have chi=-6 and Hodge numbers (h^11,h^21)=(1,4). With the standard embedding of the spin connection in the gauge group, Y gives rise to an E_6 gauge theory with 3 chiral generations of particles. The gauge group may be broken further by means of the Hosotani mechanism combined with continuous deformation of the background gauge field. For the non-Abelian quotient we obtain a model with 3 generations with the gauge group broken to that of the standard model. Moreover there is a limit in which the quotients develop 3 conifold points. These singularities may be resolved simultaneously to give another manifold with (h^11,h^21)=(2,2) that lies right at the tip of the distribution of Calabi-Yau manifolds. This strongly suggests that there is a heterotic vacuum for this manifold that derives from the 3 generation model on the quotient of Y. The manifold Y may also be realised as a hypersurface in the toric variety. The symmetry group does not act torically, nevertheless we are able to identify the mirror of the quotient manifold by adapting the construction of Batyrev.

Figures (9)

• Figure 1: The very tip of the distribution of Calabi--Yau manifolds showing the manifolds that have $h^{11}{+}h^{21}\leq 30$. The Euler number $\chi=2(h^{11}{-}h^{21})$ is plotted horizontally, the height $h^{11}{+}h^{21}$ is plotted vertically and the oblique axes bound the region $h^{11}\geq 0,\, h^{21}\geq 0$. The quotients $Y/\text{\frak G}$ with $|\text{\frak G}|=12$, that we discuss here, have $(h^{11},\,h^{21})=(1,4)$ these and their corresponding mirrors are shown. The conifold transition between the manifolds with $(h^{11},\,h^{21})=(1,4)$ and $(h^{11},\,h^{21})=(2,2)$ is indicated by the red arrow. Manifolds with $|\chi|=6$ are distinguished by red dots and there are many of these for $h^{11}+h^{21}>30$. The manifolds indicated in this plot with $h^{11}{+}h^{21}\leq 24$ are identified in SHN.
• Figure 2: A sketch of $\text{dP}_6$, in the centre, showing the hexagon formed by the six (-1)-lines. This surface may be realised in $\mathbb{P}^2{\times}\mathbb{P}^2$ as the locus $p(w,z)=q(w,z)=0$ defined by two bilinear polynomials in the coordinates $w_j$ and $z_j$ of the two $\mathbb{P}^2$'s. If we project to the first $\mathbb{P}^2$ then the image is as in the sketch on the left, in which the three lines $E_i$ project to points. If, instead, we project to the second $\mathbb{P}^2$ then the image is as in the sketch on the right, in which the three lines $L_{ij}$ have been projected to points.
• Figure 3: Two sketches of the surface $\Gamma$, the locus of $\text{Dih}_6$-invariant varieties showing the discriminant of the space of 3-nodal varieties. The components of the discriminant locus are labeled according to \ref{['tab:DiscriminantComponents']}. For the resolved manifold with Hodge numbers $(h^{11},\,h^{21})=(2,2)$ this is the space of complex structures. The second sketch zooms out to show how the components intersect. The four intersections of the pairs of blue and purple lines lie on the dashed line $\Gamma^{\rm (x)}$.
• Figure 4: The divisors $\text{pt}_a{\times}\widetilde{\mathcal{S}}$ and $\mathcal{S}{\times}\widetilde{\text{pt}}_b$. These intersect in the 36 nodes which form three $\text{\frak G}$-orbits that are distinguished by colour.
• Figure 5: The fan and polygon for $\text{dP}_6$. The one-dimensional cones can be taken to correspond to the divisors as indicated.