Triadophilia: A Special Corner in the Landscape

TL;DR

Triadophilia identifies a sparsely populated tip in the Calabi–Yau landscape where manifolds with small Hodge numbers host three-generation heterotic models, notably a $(h^{11},h^{21})=(3,3)$ manifold closely related to the Tian–Yau family with $ ext{χ}=-6$. The paper systematically builds and connects three-generation manifolds—the Tian–Yau trio and the split bicubic family—via explicit quotients and conifold transitions, showing they inhabit the same irreducible component of moduli space and can be linked through transgressions of bundles. It then analyzes the transfer of heterotic bundles across conifold transitions, presenting necessary conditions and monad constructions that hint at a coherent picture of interconnected vacua in a compactification landscape. The work emphasizes a dynamical perspective where the universe could drift toward the tip through mildly singular manifolds and transgressions, with torsion in cohomology and Brauer groups playing a central role in distinguishing these rare regions and guiding transitions between models.

Abstract

It is well known that there are a great many apparently consistent vacua of string theory. We draw attention to the fact that there appear to be very few Calabi--Yau manifolds with the Hodge numbers h^{11} and h^{21} both small. Of these, the case (h^{11}, h^{21})=(3,3) corresponds to a manifold on which a three generation heterotic model has recently been constructed. We point out also that there is a very close relation between this manifold and several familiar manifolds including the `three-generation' manifolds with χ=-6 that were found by Tian and Yau, and by Schimmrigk, during early investigations. It is an intriguing possibility that we may live in a naturally defined corner of the landscape. The location of these three generation models with respect to a corner of the landscape is so striking that we are led to consider the possibility of transitions between heterotic vacua. The possibility of these transitions, that we here refer to as transgressions, is an old idea that goes back to Witten. Here we apply this idea to connect three generation vacua on different Calabi-Yau manifolds.

Figures (3)

• Figure 1: A plot of the Hodge numbers of the Kreuzer--Skarke list. $\chi=2(h^{11}-h^{21})$ is plotted horizontally and $h^{11}+h^{21}$ is plotted vertically. The oblique axes bound the region $h^{11}\geq 0,\,h^{21}\geq 0$.
• Figure 2: A plot of the 264 distinct pairs of Hodge Numbers for the CICY's.
• Figure 3: The underpopulated corner of the landscape. $\chi=2(h^{11}{-}h^{21})$ is plotted horizontally, $h^{11}{+}h^{21}$ is plotted vertically and the oblique axes bound the region $h^{11}\geq 0,\, h^{21}\geq 0$. In the electronic version of this figure the points are coloured according to provenance and have partial transparency in order to show overlays. The manifolds with $h^{11}{+}h^{21}\leq 22$ are identified in Table \ref{['tiptab']}.