$N$ = 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 $N$ = 2 chiral superfield with sixteen bosonic and sixteen fermionic component fields. Its lowest component is a logcorrelated 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 $N$ = 2 superspace and analyze them in the semiclassical approximation by using a quaternionic formalism for the $N$ = 2 superconformal algebra.