Philip Boyle Smith, Yunqin Zheng
A bstract For a fermionic quantum field theory in d = 1 + 1 dimensions, there is a subtle difference between summing over spin structures and gauging (– 1) F . If the gravitational anomaly vanishes mod 16, then both operations are equivalent and yield a bosonic theory. But if the gravitational anomaly only vanishes mod 8, then only gauging (– 1) F is allowed, and the result is a fermionic theory. Our goal is to understand in detail how this happens, despite the fact (– 1) F is defined in terms of shifting the spin structure, which would naïvely suggest that both operations are equivalent. We do this from three perspectives: an abstract view in terms of anomalies, explicit CFT calculations, and a Symmetry TFT perspective. To conclude, we illustrate our results using the heterotic string and the famous self-triality of 8 Majorana-Weyl fermions.