We investigate resolutions of heterotic orbifolds using toric geometry. Our starting point is provided by the recently constructed heterotic models on explicit blowup of ${\mathbb{C}}^n/{\mathbb{Z}}_n$ 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 nonexperts, 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 ${\mathbb{C}}^2/{\mathbb{Z}}_3$, ${\mathbb{C}}^3/{\mathbb{Z}}_4$, and ${\mathbb{C}}^3/{\mathbb{Z}}_2\times {\mathbb{Z}}_2^{\prime }$. We are able to obtain a direct identification between them and the known orbifold models. In the ${\mathbb{C}}^3/{\mathbb{Z}}_2\times {\mathbb{Z}}_2^{\prime }$ 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.

• Received 24 July 2007

DOI:https://doi.org/10.1103/PhysRevD.77.026002