Discrete Gauge Anomalies and Instantons

TL;DR

This paper explores the relationship between gauge anomalies, instantons, and fermion zero modes, clarifying conditions under which certain chiral theories are anomaly-free and examining implications for different spacetime signatures.

Contribution

It revisits the connection between large gauge anomalies and instantons, and analyzes how anomaly cancellation conditions influence the consistency of chiral gauge theories.

Findings

$SU(N)$ theories avoid anomalous signs due to perturbative anomaly cancellation.

Chiral theories in the studied models are free of anomalies despite unrestricted spectra.

Differences in fermionic zero modes between Euclidean and Lorentzian frameworks are highlighted.

Abstract

We revisit anomalous phases related to large gauge transformations, such as the Witten anomaly. The latter, known to plague $d=4$ $Sp(k)$ theories, is well-understood in terms of $\pi_4(Sp(k))=\mathbb{Z}_2$, but it also has an oblique relation to the instantons, labeled by $\pi_3(G)=\mathbb{Z}$, via the fermion zero mode counting. We revisit this relation and point out how $SU(N)$ theories escape an anomalous sign of the latter type, only thanks to the perturbative anomaly cancelation condition that restricts the chiral fermion spectrum. This leads to the question of what happens if the latter, more mundane anomaly is canceled by an inflow instead. After raising an open question about fractional D3 probe theories, we explore the simplest bottom-up model of such a kind, due to Witten and Yonekura, from which we find the relevant chiral theories to be free of such a disease despite the unrestricted chiral spectra. We close with a simple but often-overlooked observation about how fermionic zero modes enter physics differently between Euclidean and Lorentzian descriptions and point out a related issue in $d=3$.