$\mathcal{N}=1$ Liouville SCFT in Four Dimensions

Understand

We construct a four supercharges Liouville superconformal field theory in four dimensions.

• The Liouville superfield is chiral and its lowest component is a log-correlated complex scalar whose real part carries a background charge.
• The action consists of a supersymmetric Paneitz operator, a background supersymmetric $\mathcal{Q}$-curvature charge and an exponential potential.
• It localizes semiclassically on solutions that describe curved superspaces with a constant complex supersymmetric $\mathcal{Q}$-curvature.