Heterotic global anomalies and torsion Witten index

TL;DR

This paper investigates anomalies in heterotic string theories through 2D supersymmetric quantum field theories, introducing a torsion Witten index linked to topological invariants, which helps classify perturbative and global anomalies.

Contribution

It generalizes the notion of the B-field and relates target space anomalies to invariants of SQFTs, including a torsion Witten index connected to topological modular forms.

Findings

Perturbative anomalies correspond to the Witten index of SQFTs.

Global anomalies are related to a torsion version of the Witten index.

The torsion index aligns with invariants suggested by topological modular forms.

Abstract

We study the structure of anomalies in general heterotic string theories by considering general 2-dimensional $\mathcal{N}=(0,1)$ supersymmetric quantum field theories (SQFTs), without assuming conformal invariance nor the correct central charges. First we generalize the precise notion of the $B$-field introduced by Witten. Then we express the target space anomalies by invariants of SQFTs. Perturbative anomalies correspond to the Witten index of some class of SQFTs, while global anomalies correspond to a torsion version of the Witten index. The torsion index gives some of the invariants of SQFTs suggested by topological modular forms, and is expected to be zero for the cases that are relevant to actual heterotic string theories.