8d gauge anomalies and the topological Green-Schwarz mechanism

Abstract

Abstract String theory provides us with 8d supersymmetric gauge theory with gauge algebras $$ \mathfrak{s}\mathfrak{u}(N),\mathfrak{s}\mathfrak{o}(2N),\mathfrak{s}\mathfrak{p}(N),{\mathfrak{e}}_6,{\mathfrak{e}}_7\kern0.5em \mathrm{and}\kern0.5em {\mathfrak{e}}_8 $$ s u N , so 2 N , s p N , e 6 , e 7 and e 8 , but no construction for $$ \mathfrak{so}\left(2N+1\right) $$ so 2 N + 1 , $$ {\mathfrak{f}}_4 $$ f 4 and $$ {\mathfrak{g}}_2 $$ g 2 is known. In this paper, we show that the theories for $$ {\mathfrak{f}}_4 $$ f 4 and $$ \mathfrak{so}\left(2N+1\right) $$ so 2 N + 1 have a global gauge anomaly associated to π d=8, while $$ {\mathfrak{g}}_2 $$ 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 $$ \mathfrak{s}\mathfrak{p}(N) $$ s p N have a subtler gauge anomaly, which we suggest should be canceled by a topological analogue of the Green-Schwarz mechanism.