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Comment on the Generation Number in Orbifold Compactifications

Jens Erler, Albrecht Klemm

TL;DR

The paper tackles the longstanding issue that the number of $(1,1)$-forms, $h_{1,1}$, in toroidal $Z_N$ orbifolds of the heterotic string depends on the underlying six-dimensional lattice, not just on twist eigenvalues. It develops a comprehensive classification framework for symmetric $(2,2)$ orbifolds with vanishing $B$-field by analyzing crystallographic twists in $GL(6,\tz{Z})$, constructing the one-loop partition function with careful treatment of fixed tori and fixed points, and validating these results through detailed toric resolutions of singularities. The authors report 18 inequivalent $N=1$ models, correct several previously published spectra, and demonstrate how invariant-lattice volumes modify state degeneracies and hence physical spectra. They further connect the spectral data to Calabi–Yau resolutions via Hirzebruch–Jung strings and toric geometry, providing explicit counts of $(1,1)$-forms and the associated Hodge numbers, thereby clarifying how lattice choice shapes low-energy couplings and moduli dynamics. The work emphasizes the practical impact on model building and phenomenology, urging reexamination of couplings, threshold corrections, and non-perturbative effects under lattice-dependent spectra.

Abstract

There has been some confusion concerning the number of $(1,1)$-forms in orbifold compactifications of the heterotic string in numerous publications. In this note we point out the relevance of the underlying torus lattice on this number. We answer the question when different lattices mimic the same physics and when this is not the case. As a byproduct we classify all symmetric $Z_N$-orbifolds with $(2,2)$ world sheet supersymmetry obtaining also some new ones.

Comment on the Generation Number in Orbifold Compactifications

TL;DR

The paper tackles the longstanding issue that the number of -forms, , in toroidal orbifolds of the heterotic string depends on the underlying six-dimensional lattice, not just on twist eigenvalues. It develops a comprehensive classification framework for symmetric orbifolds with vanishing -field by analyzing crystallographic twists in , constructing the one-loop partition function with careful treatment of fixed tori and fixed points, and validating these results through detailed toric resolutions of singularities. The authors report 18 inequivalent models, correct several previously published spectra, and demonstrate how invariant-lattice volumes modify state degeneracies and hence physical spectra. They further connect the spectral data to Calabi–Yau resolutions via Hirzebruch–Jung strings and toric geometry, providing explicit counts of -forms and the associated Hodge numbers, thereby clarifying how lattice choice shapes low-energy couplings and moduli dynamics. The work emphasizes the practical impact on model building and phenomenology, urging reexamination of couplings, threshold corrections, and non-perturbative effects under lattice-dependent spectra.

Abstract

There has been some confusion concerning the number of -forms in orbifold compactifications of the heterotic string in numerous publications. In this note we point out the relevance of the underlying torus lattice on this number. We answer the question when different lattices mimic the same physics and when this is not the case. As a byproduct we classify all symmetric -orbifolds with world sheet supersymmetry obtaining also some new ones.

Paper Structure

This paper contains 9 sections, 5 theorems, 48 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Classification of the $Z_N-$Orbifolds
  3. Concepts in toroidal- and orbifold compactifications
  4. Description of the Classification
  5. Partition Functions
  6. Geometrical Resolution of the Orbifold Singularities
  7. The fixed sets
  8. Description of the desingularisations
  9. Conclusions

Key Result

Lemma 1

Let $M\in GL(6,{\sf Z Z})$. The string theory on $T^6$ remains invariant under the simultaneous transformations $n\rightarrow M\, n$, $m\rightarrow {M^T}^{-1}\, m$ and $g \rightarrow {M^T}^{-1}\, g\, M^{-1}$.

Figures (2)

  • Figure 1: Schematic configuration of the orbifold singularities. Fixed point singularites are depicted by dots, fixed torus singularities by lines. We indicate the maximal order of the group element under which the sets stay fix in parantheses. The numbers on the sets indicates their multiplicity on the torus.
  • Figure :

Theorems & Definitions (11)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Corollary 1
  • Theorem 1
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Theorem 1
  • ...and 1 more