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In quantum field theory in \(2n\) spacetime dimensions, gauge anomalies potentially arise from chiral loops with \(n+1\) vertices attached to bosonic lines. Gauge anomalies destroy classical gauge invariance, and must be avoided. Gauge anomalies are called gravitational if the bosonic lines are gravitons.
In standard supersymmetric string theory in 10d, gauge anomalies vanish only for gauge groups of dimension 496, specifically \(\textrm{E}_8 \!\times\! \textrm{E}_8\) or \(\textrm{SO}(32)\) (Alvarez-Gaumé+ 1984, 2022). Supersymmetry demands “Majorana” spinors of just one handedness, and requires a magical cancellation of anomalies between fermion and multivector boson fields. This mathematical miracle (Green & Schwarz 1984) precipitated the first superstring revolution.
In Spin(11,1) string theory on the other hand, vanishing of gauge and gravitational anomalies is almost trivial. Fermions are chiral, not Majorana, occupying spinor representations of \[ \textrm{SU}(8)_R \times \textrm{SU}(8)_L \ , \] and their antifermion partners occupy a matching conjugate representation. A sufficient condition for the fermionic contribution to anomalies to vanish is that fermions of opposite chirality come in matching representations of the gauge group, which is true here. The same applies to bosons.
11/15. Fermion generations