Galois action on VOA gauge anomalies

Assuming regularity of the fixed subalgebra, any action of a finite group $G$ on a holomorphic VOA $V$ determines a gauge anomaly $\alpha \in $\mathrm{H}$^3(G; \boldsymbol{\mu})$,where$\boldsymbol{\mu} \subset $\mathbb{C}$^\times$ is the group of roots of unity. We show that under Galois conjugation $V \mapsto {^\gamma V}$,thegaugeanomalytransformsas$\alpha \mapsto \gamma^2(\alpha)$\mathrm{.}Thisprovidesanaprioriupperboundof$24$ on the order of anomalies of actions preserving a $\mathbb{Q}$-structure, for example the Monster group $\mathbb{M}$ acting on its Moonshine VOA $V^{\mathrm{♮}}$. We speculate that each field $\mathbb{K}$ should have a “vertex Brauer group” isomorphic to $$\mathrm{H}$^3($\mathrm{G}\mathrm{a}\mathrm{l}$(\bar{$\mathbb{K}$}/$\mathbb{K}$); \boldsymbol{\mu}^{\otimes 2})$. In order to motivate our constructions and speculations, we warm up with a discussion of the ordinary Brauer group, emphasizing the analogy between VOA gauging and quantum Hamiltonian reduction.