Comment on the Generation Number in Orbifold Compactifications

TL;DR

This paper clarifies the role of the underlying torus lattice in determining the number of (1,1)-forms in heterotic string orbifold compactifications and classifies symmetric Z_N-orbifolds with (2,2) supersymmetry.

Contribution

It highlights the importance of the torus lattice in orbifold compactifications and provides a classification of symmetric Z_N-orbifolds with (2,2) supersymmetry, including new examples.

Findings

Clarified the influence of lattice choice on (1,1)-form count.

Identified when different lattices produce equivalent physics.

Classified all symmetric Z_N-orbifolds with (2,2) supersymmetry, discovering new cases.

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.