$\mathcal{N}=1$ super complex Liouville string

TL;DR

This work extends the complex Liouville string to an $ ext{N}=1$ supersymmetric setting (S$ ext{C}$LS) and analyzes its dual description via a two-matrix model, showing that sphere NS-NS-NS and NS-R-R amplitudes reproduce the bosonic C$ ext{LS}$ results. A parallel theory ${ ext{hat S$ ext{C}$LS}}$ with a different worldsheet supercurrent is studied, yielding distinct three-point data yet a similarly constrained four-point structure; both theories admit dual two-matrix models with specific spectral curves. By leveraging SUSY SLFT HEOM and ground-ring constructions, the authors derive reductions of higher-point amplitudes to lower-point data and verify these relations against the proposed matrix-model answers, including explicit numerical checks for moduli-space integrals. The results provide substantial evidence that the S$ ext{C}$LS and ${ ext{hat S$ ext{C}$LS}}$ admit dual matrix-model descriptions closely related to the bosonic C$ ext{LS}$, illuminating a SUSY extension of the 2D string/matrix-model correspondence and suggesting avenues toward SUSY de Sitter holography. The work also sets the stage for non-perturbative tests and higher-genus generalizations, with potential links to SUSY SYK models.

Abstract

We study the (type 0B) $\mathcal{N}=1$ supersymmetric complex Liouville string ($\text{S}\mathbb{C}\text{LS}$), a supersymmetric extension of the bosonic complex Liouville string ($\mathbb{C}\text{LS}$). We compute the sphere three-point amplitudes (including NS-NS-NS and NS-R-R types) and find they share the same form as the sphere three-point amplitude of the bosonic $\mathbb{C}\text{LS}$. Analysis of the analytic structure of the NS-NS-NS-NS four-point amplitude and the higher equations of motion also yields results identical to the bosonic case. Based on these findings, we propose that the dual matrix model for the $\text{S}\mathbb{C}\text{LS}$ is the same as that for the bosonic $\mathbb{C}\text{LS}$. We also investigate a related theory $\widehat{\text{S}\mathbb{C}\text{LS}}$, which differs in the gauged worldsheet supersymmetry. A parallel analysis is performed for $\widehat{\text{S}\mathbb{C}\text{LS}}$, and a candidate for its dual matrix model is proposed. We then carry out a partial numerical evaluation of the moduli space integral, which provides further evidence for both the proposals of the dual matrix model regarding $\text{S}\mathbb{C}\text{LS}$ and $\widehat{\text{S}\mathbb{C}\text{LS}}$.

Figures (1)

• Figure 1: The dots represent the numerical results for ${\mathsf{A}}^{(b)-}_{0,4,0}(\boldsymbol{P})$ under specific choices of external momenta, while the solid curves correspond to the exact results.