8d gauge anomalies and the topological Green-Schwarz mechanism

TL;DR

This work identifies eight-dimensional global gauge anomalies for certain algebras in ${\rm N}=1$ SYM and demonstrates that while ${\frak g}_2$ is anomaly-free, ${\frak f}_4$ and ${\rm so}(2N{+}1)$ exhibit $\,\pi_8$-driven anomalies. It develops a new instanton-based method that corroborates the traditional homotopy approach and reveals a subtler anomaly for ${\rm Sp}(N)$ not captured by $\pi_8({\rm Sp})$, suggesting cancellation via a topological Green-Schwarz mechanism. The paper argues that traditional global anomalies cannot be canceled by a TQFT in general, but that certain higher-form/KO-theoretic topological sectors could cancel the subtler eight-dimensional anomalies, with a concrete KO-theory perspective proposed for future realization. Overall, the results sharpen the landscape of anomaly constraints in 8d gauge theories and point toward a KO-homology framework for robust topological cancellations in string-theory-inspired constructions.

Abstract

String theory provides us with 8d supersymmetric gauge theory with gauge algebras $\mathfrak{su}(N)$, $\mathfrak{so}(2N)$, $\mathfrak{sp}(N)$, $\mathfrak{e}_{6}$, $\mathfrak{e}_{7}$ and $\mathfrak{e}_{8}$, but no construction for $\mathfrak{so}(2N{+}1)$, $\mathfrak{f}_4$ and $\mathfrak{g}_2$ is known. In this paper, we show that the theories for $\mathfrak{f}_4$ and $\mathfrak{so}(2N{+}1)$ have a global gauge anomaly associated to $π_{d=8}$, while $\mathfrak{g}_2$ does not have it. We argue that the anomaly associated to $π_d$ in $d$-dimensional gauge theories cannot be canceled by topological degrees of freedom in general. We also show that the theories for $\mathfrak{sp}(N)$ have a subtler gauge anomaly, which we suggest should be canceled by a topological analogue of the Green-Schwarz mechanism.