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

TL;DR

This paper constructs a four-dimensional $ ext{N}=1$ Liouville superconformal field theory with unique features like a chiral superfield, background charge, and non-unitarity, analyzing its classical solutions, anomalies, and correlation functions.

Contribution

It introduces a novel 4D $ ext{N}=1$ Liouville SCFT with a chiral superfield, background charge, and explicit correlation functions, expanding the understanding of superconformal theories in higher dimensions.

Findings

The theory localizes on curved superspaces with constant super-$ ext{Q}$-curvature.

Classical background charge remains uncorrected quantum mechanically.

Derived integral form for vertex operator correlation functions.

Abstract

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. The theory is non-unitary with a continuous spectrum of scaling dimensions. We study the dynamics on the supersymmetric 4-sphere, show that the classical background charge is not corrected quantum mechanically and calculate the super-Weyl anomaly. We derive an integral form for the correlation functions of vertex operators.