8d gauge anomalies and the topological Green-Schwarz mechanism

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}}_7kern0.5em mathrm{and}kern0.5em {mathfrak{e}}_8 , but no construction for mathfrak{so}left(2N+1right) , {mathfrak{f}}_4 and {mathfrak{g}}_2 is known. In this paper, we show that the theories for {mathfrak{f}}_4 and mathfrak{so}left(2N+1right) have a global gauge anomaly associated to πd=8, while {mathfrak{g}}_2 does not have it. We argue that the anomaly associated to πd 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) have a subtler gauge anomaly, which we suggest should be canceled by a topological analogue of the Green-Schwarz mechanism.