New Calabi-Yau Manifolds with Small Hodge Numbers

TL;DR

The authors explore the landscape of complete intersection Calabi–Yau threefolds (CICYs) at the low end of $(h^{11},h^{21})$, using conifold transitions and freely acting quotient groups to generate new Calabi–Yau manifolds with small Hodge numbers and nontrivial fundamental groups. They develop and apply a diagrammatic configuration framework, together with transversality checks and parameter counting, to identify and verify smooth quotients and their Euler characteristics. A key focus is the network of webs produced by $\\mathbb{Z}_3$, $\\mathbb{Z}_5$, and $\mathbb{H}$ symmetries, including links to Gross–Popescu manifolds with Euler number zero and several intriguing quotients with $\,\chi=-6$ or other small Euler numbers. They also discuss transgressions of vector bundles across conifold transitions, aiming to connect heterotic vacua across the Calabi–Yau web. The results enrich the catalog of small-Hodge-number Calabi–Yau manifolds and illuminate how symmetries and conifold physics structure the connected components of the Calabi–Yau landscape, with potential phenomenological relevance.

Abstract

It is known that many Calabi-Yau manifolds form a connected web. The question of whether all Calabi-Yau manifolds form a single web depends on the degree of singularity that is permitted for the varieties that connect the distinct families of smooth manifolds. If only conifolds are allowed then, since shrinking two-spheres and three-spheres to points cannot affect the fundamental group, manifolds with different fundamental groups will form disconnected webs. We examine these webs for the tip of the distribution of Calabi-Yau manifolds where the Hodge numbers (h^{11}, h^{21}) are both small. In the tip of the distribution the quotient manifolds play an important role. We generate via conifold transitions from these quotients a number of new manifolds. These include a manifold with χ=-6 and manifolds with an attractive structure that may prove of interest for string phenomenology. We also examine the relation of some of these manifolds to the remarkable Gross-Popescu manifolds that have Euler number zero.

Figures (4)

• Figure 1: The tip of the distribution of Calabi--Yau manifolds showing the manifolds that have nontrivial fundamental group, together with their mirrors. The Euler number $\chi=2(h^{11}{-}h^{21})$ is plotted horizontally, $h^{11}{+}h^{21}$ is plotted vertically and the oblique axes bound the region $h^{11}\geq 0,\, h^{21}\geq 0$. Manifolds with $h^{11}{+}h^{21}\leq 24$ are identified in Table \ref{['tiptab']}.
• Figure 2: The webs of CICY's with fundamental group $\mathbb{Z}_5$, $\mathbb{Z}_5{\times}\mathbb{Z}_2$ and $\mathbb{Z}_5{\times}\mathbb{Z}_5$.
• Figure 3: The webs of CICY's with fundamental group $\mathbb{Z}_3$, $\mathbb{Z}_3{\times}\mathbb{Z}_2$ and $\mathbb{Z}_3{\times}\mathbb{Z}_3$.
• Figure 4: Webs of resolved orbifolds, that are either simply connected or with fundamental group $\mathbb{Z}_3$. The boxes below the main figure identify the manifolds.