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

TL;DR

The work constructs explicit timelike 2D $ ext{N}=1$ Liouville CFT data via analytic bootstrap, showing timelike structure constants are inverses of spacelike ones under a Virasoro-Wick Rotation, with momentum-rotated arguments and supersymmetric sector distinctions. It derives closed-form expressions for NS and Ramond sector constants in both spacelike and timelike regimes using meromorphic shift relations and Barnes-type special functions, and reveals a direct connection to $ ext{N}=1$ minimal models at degenerate momenta while noting no truncated fusion in the timelike case. The results illuminate the supersymmetric analogue of the bosonic timelike Liouville story and pave the way for an $ ext{N}=1$ Virasoro minimal string, while raising questions about crossing symmetry proofs and sphere partition functions in the timelike setting. Overall, the paper provides a rigorous bootstrap-based definition of $ ext{N}=1$ timelike Liouville theory and highlights its broad implications for supersymmetric 2D gravity and minimal string constructions.

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.

Figures (10)

• Figure 1: The physical NS-spectrum in the $p-$plane in spacelike $\mathcal{N}=1$ Liouville theory (blue) and the degenerate representations of the NS-sector algebra (magenta) for central charge values $c\geqslant 9$ (or $b\in\mathbb{R}_{(0,1]}$). The corresponding expressions are given in (\ref{['eq:hNS']}) and (\ref{['eq: degenerate prs']}).
• Figure 2: The physical R-spectrum in the $p-$plane in spacelike $\mathcal{N}=1$ Liouville theory (blue) and the degenerate representations of the R-sector algebra (magenta) for central charge values $c\geqslant 9$ (or $b\in\mathbb{R}_{(0,1]}$). The corresponding expressions are given in (\ref{['eq:hR']}) and (\ref{['eq: degenerate prs']}).
• Figure 3: Teschner's trick in the NS-sector of $\mathcal{N}=1$ Liouville theory: the analytic bootstrap problem involving crossing of the sphere four-point function between a NS degenerate field $V_{p_{\langle1,3\rangle}}$ and three general NS fields $V_{p_1},V_{p_2},V_{p_3}$. The analysis (done in App.\ref{['app:NS']}) leads to the shift relations (\ref{['eq: shift equations NS sector1']}), (\ref{['eq: shift equations NS sector1binv']}).
• Figure 4: Teschner's trick in the R-sector of $\mathcal{N}=1$ Liouville theory: the analytic bootstrap problem involving crossing of the sphere four-point function between two NS fields with momenta $p_2,p_3$ and two R fields, one with momentum $p_{\langle1,2\rangle}$ and one with momentum $p_1$. The analysis (done in App.\ref{['app:R']}) leads to the shift relations (\ref{['eq: shiftequations R sector']}), (\ref{['eq: shift equations R sectorbinv']}).
• Figure 5: The expected physical NS-spectrum in timelike $\mathcal{N}=1$ Liouville theory (blue) and the degenerate representations for the NS algebra (magenta) for central charge values $\hat{c}\leqslant 1$ (or $\hat{b}\in\mathbb{R}_{(0,1]}$). The corresponding expressions are given in (\ref{['eq:hTimelike']}), (\ref{['eq:degtimelike']}).