Dai-Freed anomalies and level matching in heterotic asymmetric orbifolds

TL;DR

This paper investigates Dai-Freed anomalies in heterotic asymmetric orbifolds, linking anomaly cancellation conditions to level-matching constraints and higher-genus partition function transformations.

Contribution

It provides a novel anomaly-based perspective on the consistency conditions of asymmetric heterotic orbifolds, including level-matching and mod-2 conditions.

Findings

Global anomaly conditions match level-matching constraints.

Additional mod-2 conditions arise for even orbifold order m.

Anomaly matching under bosonization explains standard orbifold consistency.

Abstract

We study asymmetric orbifolds of the $E_8\times E_8$ heterotic string from the perspective of worldsheet Dai-Freed anomalies. Focusing on cyclic symmetries $G = \mathbb{Z}_m$ that act chirally on the fermions and symmetrically on the bosons, we compute the corresponding spin-bordism invariants and derive the conditions for the vanishing of global anomalies from this perspective. In the fermionic description, these conditions are exactly the familiar level-matching constraints, together with the additional mod-2 conditions that appear for even $m$. We then discuss the same conditions from the transformation properties of higher-genus fermion partition functions and explain how the anomaly is matched under bosonization for a large class of inner automorphisms of the $E_8\times E_8$ lattice theory. This gives an interpretation of the standard consistency conditions for asymmetric heterotic orbifolds from the Dai-Freed anomaly perspective.