The complex Liouville string

TL;DR

The complex Liouville string provides a solvable 2d quantum gravity model built from two complex-conjugate Liouville CFTs with central charges $c^+ \in 13+i\mathbb{R}_+$ and $c^- \in 13-i\mathbb{R}_+$, plus ghosts. Its amplitudes are computed by analytic bootstrap and it has a dual description as a double-scaled two-matrix model with a non-algebraic spectral curve, enabling a topological-recursion treatment that mirrors and extends minimal string theory. A sine dilaton gravity reformulation yields AdS2 and dS2 vacua and enables a gravitational path integral interpretation of the amplitudes, including a detailed analysis of boundaries and non-perturbative (ZZ) effects with oscillatory, rather than exponentially decaying, corrections. The work also develops a complete non-perturbative framework, including trumpet boundary states, stable-graph expansions, and multiple non-perturbative completions on the matrix side, linking to questions about dS quantum gravity in three dimensions. Overall, the paper provides a comprehensive, solvable realization of holography that encompasses both AdS and dS sectors and couples worldsheet, matrix-model, and 2d dilaton-gravity formalisms in a consistent, non-perturbatively controlled setting.

Abstract

We introduce the complex Liouville string, a solvable string theory defined by coupling two Liouville theories with complex conjugate central charges $c \in 13+i \mathbb{R}$ on the worldsheet. We compute its amplitudes from first principles and establish a duality with a double-scaled two-matrix integral. We also analyze general worldsheet boundaries and non-perturbative effects in the genus expansion. By expressing the complex Liouville string as a 2d dilaton gravity theory with a sine potential, we show that it admits both AdS$_2$ and dS$_2$ vacua.