Revisiting Wess-Zumino-Witten terms

TL;DR

This paper reexamines topological aspects of 4D Wess-Zumino-Witten terms in QCD, emphasizing the role of spin structures and anomalies using modern bordism techniques.

Contribution

It provides a modern bordism-based analysis of 4D WZW terms, clarifying their dependence on spin structures and anomaly matching in QCD.

Findings

WZW terms require spin structures in 4D.

Mixed anomalies are reproduced in low-energy sigma models.

Bordism perspective clarifies topological issues in QCD.

Abstract

We revisit various topological issues concerning four-dimensional ungauged and gauged Wess-Zumino-Witten (WZW) terms for $SU$ and $SO$ quantum chromodynamics (QCD), from the modern bordism point of view. We explain, for example, why the definition of the $4d$ WZW terms requires the spin structure. We also discuss how the mixed anomaly involving the 1-form symmetry of $SO$ QCD is reproduced in the low-energy sigma model.