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 $α\in \mathrm{H}^3(G; \boldsymbolμ)$, where $\boldsymbolμ \subset \mathbb{C}^\times$ is the group of roots of unity. We show that under Galois conjugation $V \mapsto {^γV}$, the gauge anomaly transforms as $α\mapsto γ^2(α)$. This provides an a priori upper bound of $24$ on the order of anomalies of actions preserving a $\mathbb{Q}$-structure, for example the Monster group $\mathbb{M}$ acting on its Moonshine VOA $V^\natural$. We speculate that each field $\mathbb{K}$ should have a "vertex Brauer group" isomorphic to $\mathrm{H}^3(\mathrm{Gal}(\bar{\mathbb{K}}/\mathbb{K}); \boldsymbolμ^{\otimes 2})$. 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.

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