Backfiring Bosonisation

TL;DR

This paper explores the subtle differences between summing over spin structures and gauging $(-1)^F$ in 1+1 dimensional fermionic quantum field theories, revealing how gravitational anomalies influence the resulting bosonic or fermionic nature of the theory.

Contribution

It provides a detailed analysis from anomaly, CFT, and Symmetry TFT perspectives on how gravitational anomalies determine the equivalence or distinction between two key operations in fermionic theories.

Findings

Gauging $(-1)^F$ yields a fermionic theory when the anomaly vanishes mod 8.

Summing over spin structures results in a bosonic theory if the anomaly vanishes mod 16.

Explicit examples include heterotic string and 8 Majorana-Weyl fermions.

Abstract

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 na\"ively 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.