Toric Resolutions of Heterotic Orbifolds

TL;DR

The paper develops a toric-geometric framework to resolve heterotic orbifolds, translating curvature and gauge-flux data into divisor integrals of $(1,1)$-forms. It starts from explicit blowups of $\mathbb{C}^n/\mathbb{Z}_n$, showing that the integrals determining the integrated Bianchi identities reproduce topologically from toric divisor data and identifying the blowup flux with orbifold gauge shifts up to lattice vectors. It then generalizes to orbifolds with multiple exceptional divisors, using toric diagrams and triangulations to construct resolutions, classify consistent flux backgrounds, and match the resulting spectra with their orbifold counterparts, including explicit checks for $\mathbb{C}^2/\mathbb{Z}_3$, $\mathbb{C}^3/\mathbb{Z}_4$, and $\mathbb{C}^3/\mathbb{Z}_2\times\mathbb{Z}_2'$. The study also discusses non-uniqueness of resolutions, showing that only some triangulations yield fully consistent blowups compatible with orbifold data, with implications for model-building and future extensions to more intricate orbifolds.

Abstract

We investigate resolutions of heterotic orbifolds using toric geometry. Our starting point is provided by the recently constructed heterotic models on explicit blowup of C^n/Z_n singularities. We show that the values of the relevant integrals, computed there, can be obtained as integrals of divisors (complex codimension one hypersurfaces) interpreted as (1,1)-forms in toric geometry. Motivated by this we give a self contained introduction to toric geometry for non-experts, focusing on those issues relevant for the construction of heterotic models on toric orbifold resolutions. We illustrate the methods by building heterotic models on the resolutions of C^2/Z_3, C^3/Z_4 and C^3/Z_2xZ_2'. We are able to obtain a direct identification between them and the known orbifold models. In the C^3/Z_2xZ_2' case we observe that, in spite of the existence of two inequivalent resolutions, fully consistent blowup models of heterotic orbifolds can only be constructed on one of them.

Figures (2)

• Figure 1: The left graph displays the toric diagram of $\text{Res}(\mathbb{C}^2/\mathbb{Z}_2)\,$. The right picture displays a projected view of the toric diagram of $\text{Res}(\mathbb{C}^3/\mathbb{Z}_3)\,$. Because the latter is a projection, there are no arrows from the origin pointing to the divisors as in the former toric diagram.
• Figure 2: The toric diagram of $\text{Res}(\mathbb{C}^2/\mathbb{Z}_3)$ is displayed. Both exceptional divisors $E_1$ and $E_2$ are compact.