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Exploring the SO(32) Heterotic String

Hans Peter Nilles, Saul Ramos-Sanchez, Patrick K. S. Vaudrevange, Akin Wingerter

TL;DR

This work provides a comprehensive classification of four-dimensional Z_N orbifold compactifications of the heterotic SO(32) string, develops a systematic method to derive admissible shift vectors V from automorphisms of the extended Dynkin diagram, and analyzes the resulting spectra, including spinor representations of SO(2n). It reports 16 inequivalent Z4 shifts and, more broadly, 5141 Z_N models without Wilson lines, with explicit three-family examples in both Z4 and Z6-II illustrating practical model-building potential. The study highlights the appearance of SO(10) and SO(12) spinors in twisted sectors and discusses implications for GUT-like constructions and heterotic–type I duality, suggesting rich phenomenological possibilities within the SO(32) framework. Overall, the results expand the string landscape accessible from the heterotic SO(32) theory and provide publicly accessible data to guide future Wilson-line explorations and duality investigations.

Abstract

We give a complete classification of Z_N orbifold compactification of the heterotic SO(32) string theory and show its potential for realistic model building. The appearance of spinor representations of SO(2n) groups is analyzed in detail. We conclude that the heterotic SO(32) string constitutes an interesting part of the string landscape both in view of model constructions and the question of heterotic-type I duality.

Exploring the SO(32) Heterotic String

TL;DR

This work provides a comprehensive classification of four-dimensional Z_N orbifold compactifications of the heterotic SO(32) string, develops a systematic method to derive admissible shift vectors V from automorphisms of the extended Dynkin diagram, and analyzes the resulting spectra, including spinor representations of SO(2n). It reports 16 inequivalent Z4 shifts and, more broadly, 5141 Z_N models without Wilson lines, with explicit three-family examples in both Z4 and Z6-II illustrating practical model-building potential. The study highlights the appearance of SO(10) and SO(12) spinors in twisted sectors and discusses implications for GUT-like constructions and heterotic–type I duality, suggesting rich phenomenological possibilities within the SO(32) framework. Overall, the results expand the string landscape accessible from the heterotic SO(32) theory and provide publicly accessible data to guide future Wilson-line explorations and duality investigations.

Abstract

We give a complete classification of Z_N orbifold compactification of the heterotic SO(32) string theory and show its potential for realistic model building. The appearance of spinor representations of SO(2n) groups is analyzed in detail. We conclude that the heterotic SO(32) string constitutes an interesting part of the string landscape both in view of model constructions and the question of heterotic-type I duality.

Paper Structure

This paper contains 12 sections, 23 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Classification of Orbifolds
  3. Classification of $\boldsymbol{\text{SO}(32)}$ Orbifold Models
  4. The $\boldsymbol{\mathbb{Z}_4}$ Orbifold
  5. The $\boldsymbol{\mathbb{Z}_N}$ Orbifold
  6. Spinors in SO(32) Orbifold Models
  7. Three-Family Orbifold Models
  8. The $\boldsymbol{\mathbb{Z}_4}$ Orbifold
  9. Model in the $\boldsymbol{\mathbb{Z}_6}$-II Orbifold
  10. Conclusions and Outlook
  11. Table of $\mathbb{Z}_4$ orbifold models
  12. The General Form of a Shift in $\boldsymbol{\mathbb{Z}_N}$ Orbifolds of the SO(32) Heterotic String

Figures (5)

  • Figure 1: Extended Dynkin diagram of $\text{SO}(32)$ and the associated Kač labels.
  • Figure 2: Extended Dynkin diagram of SO(32) corresponding to the breaking due to the shifts V$^{(3)}$ and V$^{(4)}$ of the $\mathbb{Z}_4$ orbifold.
  • Figure 3: Twisted sectors of the $\mathbb{Z}_4$ orbifold.
  • Figure 4: Localization of the three generations with SU(5) gauge group. There are two families in the bulk and one family localized in the origin. The boxes correspond to the degeneracy of the fixed points without Wilson lines. This degeneracy has been lifted by the Wilson lines in the $e_1$, $e_3$, $e_5$ and $e_6$ directions.
  • Figure 5: Localization of the 3 generations with SO(10) gauge group. There are 3 families in the second twisted sector and 6 anti-families in the fourth twisted sector, giving a net number of 3 (anti-)families free to move in two of the six compactified dimensions. The box corresponds to the degeneracy of the fixed points in the SU(3) 2-torus.