Heterotic Models from Vector Bundles on Toric Calabi-Yau Manifolds

TL;DR

This work develops a systematic program to realize heterotic $E_8\times E_8$ models on Calabi–Yau threefolds constructed as toric hypersurfaces, using monad bundles with $SU(N)$ structure to satisfy anomaly-cancellation constraints. By focusing on 101 simple toric CYs (with smooth ambient spaces and simplicial Kähler cones), the authors prove finiteness and perform an exhaustive classification of positive monads, finding 2190 bundles on 11 base spaces, of which 21 yield three families (all on manifolds that are hypersurfaces in products of projective spaces). A substantial semi-positive extension is then performed, producing roughly 44,000 semi-positive monads and 280 three-family candidates, highlighting a rich set of promising heterotic models beyond the strictly positive case. The results provide a concrete, data-driven foundation for constructing realistic heterotic vacua on toric Calabi–Yau manifolds and point to further abundant model-building opportunities in the semi-positive sector. The methodology combines toric geometry, monad bundle techniques, Chern-class constraints, and Mori/Kähler-cone analyses to connect geometric data with physical consistency and generation-counting criteria.

Abstract

We systematically approach the construction of heterotic E_8 X E_8 Calabi-Yau models, based on compact Calabi-Yau three-folds arising from toric geometry and vector bundles on these manifolds. We focus on a simple class of 101 such three-folds with smooth ambient spaces, on which we perform an exhaustive scan and find all positive monad bundles with SU(N), N=3,4,5 structure groups, subject to the heterotic anomaly cancellation constraint. We find that anomaly-free positive monads exist on only 11 of these toric three-folds with a total number of bundles of about 2000. Only 21 of these models, all of them on three-folds realizable as hypersurfaces in products of projective spaces, allow for three families of quarks and leptons. We also perform a preliminary scan over the much larger class of semi-positive monads which leads to about 44000 bundles with 280 of them satisfying the three-family constraint. These 280 models provide a starting point for heterotic model building based on toric three-folds.

Figures (5)

• Figure 1: The number of positive monads as a function of ${\rm ind}(V)$. Fig. (a) contains all models, Fig. (b) only those which satisfy the three-familiy constraint ${\rm ind}(V)=3k$, $k\; |\;\chi(X)$. The three colours blue, red, and green correspond to SU(3), SU(4) and SU(5) models, respectively.
• Figure 2: The number of semi-positive monads as a function of ${\rm ind}(V)$. Fig. (a) contains all models, Fig. (b) only those which satisfy the "strong" three-familiy constraint, Eq. \ref{['constraints']}. The three colours blue, red, and green correspond to SU(3), SU(4) and SU(5) models, respectively.
• Figure 3: The fan for $\mathbb{P}^2$ (left) and the 2-dimensional dual cones (right).
• Figure 4: A polytope $\Delta \subset M_\mathbb{R}$(left) and its dual polytope $\Delta^\circ \subset N_\mathbb{R}$(right).
• Figure 5: The in/out-put screen in PALP palp. The first input 4 and 6 denote the lattice rank and the number of the vertices in $\Delta^\circ$, respectively, and the second input is the list of those vertices, $\rho$-th column being $\bold v_\rho$ for $\rho=1, \cdots, 6$. The output includes two Hodge numbers and Euler character of $X$, which are denoted by H in the middle, as well as the incidence information of the normal fan $\Sigma$. The latter is expressed in binary notation: for instance, the first entry in the last row, 111010, represents a four-cone generated by the four edge vectors $\bold v_2, \bold v_4, \bold v_5$ and $\bold v_6$.