8d gauge anomalies and the topological Green-Schwarz mechanism

Abstract String theory provides us with 8d supersymmetric gauge theory with gauge algebras suN,so2N,spN,e6,e7ande8 $$ \mathfrak{s}\mathfrak{u}(N),\mathfrak{s}\mathfrak{o}(2N),\mathfrak{s}\mathfrak{p}(N),{\mathfrak{e}}_6,{\mathfrak{e}}_7\kern0.5em \mathrm{and}\kern0.5em {\mathfrak{e}}_8 $$, but no construction for so2N+1 $$ \mathfrak{so}\left(2N+1\right) $$, f4 $$ {\mathfrak{f}}_4 $$ and g2 $$ {\mathfrak{g}}_2 $$ is known. In this paper, we show that the theories for f4 $$ {\mathfrak{f}}_4 $$ and so2N+1 $$ \mathfrak{so}\left(2N+1\right) $$ have a global gauge anomaly associated to π d=8, while g2 $$ {\mathfrak{g}}_2 $$ 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 spN $$ \mathfrak{s}\mathfrak{p}(N) $$ have a subtler gauge anomaly, which we suggest should be canceled by a topological analogue of the Green-Schwarz mechanism