On the three-point functions in timelike N=1 Liouville CFT

TL;DR

This paper derives explicit formulas for the structure constants of timelike N=1 Liouville CFT using bootstrap methods, revealing their relation to spacelike counterparts and implications for supersymmetric minimal models and string theory.

Contribution

It provides the first explicit expressions for timelike N=1 Liouville structure constants and clarifies their analytic properties and relation to minimal models.

Findings

Timelike structure constants are inverses of spacelike ones.

Consistency with N=1 minimal models at degenerate momenta.

Potential applications to supersymmetric minimal string theory.

Abstract

We use analytic (super-)conformal bootstrap methods to derive explicit expressions for the structure constants of $\mathcal{N}=1$ Liouville CFT in the `timelike' regime of the superconformal central charge. The obtained expressions take the form of inverses of the appropriate spacelike counterparts, which we explain concretely by elucidating the analytic properties of the corresponding shift relations in the NS- and R-sectors for the normalization-independent bootstrap data on the sphere. In a particular normalization, the timelike structure constants are shown to agree with the OPE coefficients of $\mathcal{N}=1$ Minimal Models when specified at degenerate values of the momenta, exactly as in the non-supersymmetric case. We discuss possible applications of our results, with emphasis on the construction of the $\mathcal{N}=1$ supersymmetric analog of the Virasoro Minimal String.