The complex Liouville string

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.