Galois action on VOA gauge anomalies

Abstract

Assuming regularity of the fixed subalgebra, any action of a finite group $G$ on a holomorphic VOA $V$ determines a gauge anomaly $\alpha \in {\mathrm{H}}^3(G;\mathbf{\mu })$, where $\mathbf{\mu }\subset {\mathbb{C}}^{\times }$ is the group of roots of unity. We show that under Galois conjugation $V\mapsto {}^{\gamma }V$, the gauge anomaly transforms as $\alpha \mapsto {\gamma }^2(\alpha )$. 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 $V^{♮}$. We speculate that each field $\mathbb{K}$ should have a "vertex Brauer group" isomorphic to ${\mathrm{H}}^3(\mathrm{Gal}(\bar{\mathbb{K}}/\mathbb{K});{\mathbf{\mu }}^{\otimes 2})$. 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.

17 pages. v2 is the final form, to appear in the Progress in Mathematics volume in honour of Kolya Reshetikhin