Anomalies of Duality Groups and Extended Conformal Manifolds

Understand

A self-duality group $\cal G$ in quantum field theory can have anomalies.

• In that case, the space of ordinary coupling constants $\cal M$ can be extended to include the space $\cal F$ of coefficients of counterterms in background fields.
• The extended space $\cal N$ forms a bundle over $\cal M$ with fiber $\cal F$, and the topology of the bundle is determined by the anomaly.
• For example, the ${\cal G}=SL(2,\mathbb{Z})$ duality of the 4d Maxwell theory has an anomaly, and the space ${\cal F}=S^1$ for the gravitational theta-angle is nontrivially fibered over ${\cal M}=\mathbb{H}/SL(2,\mathbb{Z})$.