Heterotic SO(32) model building in four dimensions

TL;DR

This work provides a comprehensive, systematic classification of four-dimensional heterotic SO(32) orbifold models, delivering all $\mathbb{Z}_3$, $\mathbb{Z}_7$, and $\mathbb{Z}_{2N}$ constructions from vectorial gauge shifts and revealing a unifying gauge-group structure $SO(2n_0)\times U(n_1)\times\cdots\times SO(2n_N)$. A key advance is the simultaneous, parametric determination of the complete twisted spectrum in terms of $n_0,\dots,n_N$ using a standardized framework of irreducible twisted sectors, spectral-flow relations via $S_p$ matrices, and a structured representation theory based on $U(n)$ and $SO(2n)$ factors. The paper also provides explicit 4D spectra, anomaly checks, and notable model-spin-offs, including an $SO(10)$ GUT with four generations from $\mathbb{Z}_4$, and discusses connections to heterotic/Type-I duality in four dimensions. The approach is designed to facilitate targeted model-building and scans for MSSM-like or GUT-like physics, and it lays groundwork for extensions to more general orbifolds, Wilson lines, and dual theories, with clear procedures to derive full six- and four-dimensional spectra from local twisted data.

Abstract

Four dimensional heterotic SO(32) orbifold models are classified systematically with model building applications in mind. We obtain all Z3, Z7 and Z2N models based on vectorial gauge shifts. The resulting gauge groups are reminiscent of those of type-I model building, as they always take the form SO(2n_0)xU(n_1)x...xU(n_{N-1})xSO(2n_N). The complete twisted spectrum is determined simultaneously for all orbifold models in a parametric way depending on n_0,...,n_N, rather than on a model by model basis. This reveals interesting patterns in the twisted states: They are always built out of vectors and anti--symmetric tensors of the U(n) groups, and either vectors or spinors of the SO(2n) groups. Our results may shed additional light on the S-duality between heterotic and type-I strings in four dimensions. As a spin-off we obtain an SO(10) GUT model with four generations from the Z4 orbifold.