Resolutions of C^n/Z_n Orbifolds, their U(1) Bundles, and Applications to String Model Building

TL;DR

This work develops explicit, SU($n$)–symmetric Calabi–Yau resolutions of ${\mathbb C}^n/\mathbb{Z}_n$ as complex line bundles over ${\mathbb{CP}}^{n-1}$ and provides concrete gauge backgrounds, including a standard embedding, a ${\mathrm{U(1)}}$ bundle, and an ${\mathrm{SU(n-1)}}$ bundle, all obeying Hermitian Yang–Mills equations. The authors show how the integrated Bianchi identity tightly constrains consistent ten-dimensional ${\mathrm{SO(32)}}$ super Yang–Mills compactifications on these backgrounds, yielding a finite set of ${\mathrm{U(1)}}$ models whose blow-down data reproduce corresponding heterotic orbifold models on ${\mathbb C}^2/\mathbb{Z}_2$ and ${\mathbb C}^3/\mathbb{Z}_3$; they verify this via anomaly polynomials and compute the resulting chiral spectra on the resolutions. The paper also analyzes the matching with heterotic orbifolds in four and six dimensions, discusses implications for Type I duality, and points to extensions to more general orbifolds and torsionful backgrounds. Overall, the results provide a concrete, calculable connection between orbifold singularities and their smooth resolutions in string model building, with clear criteria that couple geometry, gauge backgrounds, and anomaly consistency.

Abstract

We describe blowups of C^n/Z_n orbifolds as complex line bundles over CP^{n-1}. We construct some gauge bundles on these resolutions. Apart from the standard embedding, we describe U(1) bundles and an SU(n-1) bundle. Both blowups and their gauge bundles are given explicitly. We investigate ten dimensional SO(32) super Yang-Mills theory coupled to supergravity on these backgrounds. The integrated Bianchi identity implies that there are only a finite number of U(1) bundle models. We describe how the orbifold gauge shift vector can be read off from the gauge background. In this way we can assert that in the blow down limit these models correspond to heterotic C^2/Z_2 and C^3/Z_3 orbifold models. (Only the Z_3 model with unbroken gauge group SO(32) cannot be reconstructed in blowup without torsion.) This is confirmed by computing the charged chiral spectra on the resolutions. The construction of these blowup models implies that the mismatch between type-I and heterotic models on T^6/Z_3 does not signal a complication of S-duality, but rather a problem of type-I model building itself: The standard type-I orbifold model building only allows for a single model on this orbifold, while the blowup models give five different models in blow down.

Figures (2)

• Figure 1: The first picture gives the cross section profiles of the regularized delta function defined in \ref{['regdelta']} for various values of $r\,$. The second picture gives a three dimensional impression of its shape.
• Figure 2: To obtain a realization the $(L,L)$ model of Honecker:2006qz as a blowup of $T^4/\mathbb{Z}_2\,$ equal Wilson lines $A$ have to be put in both directions of the first torus. As can be seen from the local shift vectors $V$, $V+A$ and $V+2\,A\,$, the fixed points are not treated democratically on the blowup. In the orbifold limit these Wilson lines are irrelevant and all fixed points are equivalent.