$\mathcal{N}=2$ Liouville SCFT in Four Dimensions

We construct a Liouville superconformal field theory with eight real supercharges in four dimensions. The Liouville superfield is an $\mathcal{N}=2$ chiral superfield with sixteen bosonic and sixteen fermionic component fields. Its lowest component is a log-correlated complex scalar field whose real part carries a background charge. The theory is non-unitary with a continuous spectrum of scaling dimensions. We study its quantum dynamics on the supersymmetric 4-sphere and show that the classical background charge is not corrected quantum mechanically. We calculate the super-Weyl anomaly coefficients and find that $c$ vanishes, while $a$ is negative and depends on the background charge. We derive an integral expression for the correlation functions of superfield vertex operators in $\mathcal{N}=2$ superspace and analyze them in the semiclassical approximation by using a quaternionic formalism for the $\mathcal{N}=2$ superconformal algebra.