The complex Liouville string: the matrix integral

1- The paper presents an interesting instance of holographic duality.
2- It includes a useful review of topological recursion for two-matrix models, which may be of independent interest.
3- The manuscript is well structured and presents the results clearly, discussing them within a broader context. It strikes a good balance in providing background information and relevant references.

1- The relation between the string amplitudes and the two-matrix model is supported by strong evidence, but a full analytic proof is still missing.

In this paper, the authors propose a two-matrix model whose perturbative expansion reproduces the perturbative string amplitudes of the complexified Liouville string theory. This model is defined by coupling two Liouville theories with complex central charges $c = 13 \pm i\lambda$ on the worldsheet. The duality presented here is framed as an example of holography, and one of the key achievements of the paper is to demonstrate that, while more intricate than simpler cases, the setup remains analytically controllable.

More precisely, the authors identify the relevant spectral curve and perform calculations using topological recursion to obtain the amplitudes from the matrix model side. These results are then compared with amplitudes derived independently via the worldsheet formulation. Among the tests discussed are symmetry constraints on the central charge and momenta, analytic properties of the amplitudes, and the dilaton equation.

A particularly interesting feature of the paper is the spectral curve itself. To the best of my knowledge, this is the first example of a meaningful spectral curve with infinitely many ramification points. The associated cohomological field theory is also of infinite rank—a rare property shared by only a few objects in the mathematical literature, such as the double ramification cycle. A more geometric understanding of this cohomological field theory (along the lines of the Chern characters of the Verlinde bundle) would be valuable, and I would be interested to hear the authors' thoughts on the possible existence of such a description.

The exposition is technically solid, and the results are presented clearly. The paper also includes a helpful review of topological recursion for two-matrix models. Overall, this is a very interesting and well-executed paper that consolidates and extends the authors' previous work.

I recommend the manuscript for publication.

A few minor corrections and clarifications:

1- Equation (2.51): add that $\sigma_m \neq \mathrm{id}$.
2- Page 18: “Corrolary” should be corrected to “Corollary” (appears twice).
3- Page 18, $x$–$y$ symmetry: the original proof from [46] only holds under specific conditions. In general, symplectic invariance does not hold. The general relation between $x$–$y$ dual free energies has been recently worked out by Hock in “Symplectic (non-)invariance of the free energy in topological recursion”, following substantial recent progress on $x$–$y$ duality by Hock, and Alexandrov–Bychkov–Dunin-Barkowski–Kazarian–Shadrin. See, for instance, “A universal formula for the $x$–$y$ swap in topological recursion”.
4- Page 29, first sentence: missing comma.
5- Page 35, equation (3.46): I assume $Q = b + 1/b$. It would be helpful to remind the reader of this convention here, as you do after (4.23).
6- Page 36, above equation (3.50): please specify “linear maps” for precision.
7- Page 37, equation (3.53): the differentials also depend on the choice of basis element; it should read $d\eta_{m_i,k_i}(z_i)$. Same applies to the line below.
8- Page 38, footnote 17: the statement is valid only for semisimple CohFTs.
9- Page 55, “Modified brackets” paragraph, second sentence: “get rid off” should be corrected to “get rid of”.

Publish (surpasses expectations and criteria for this Journal; among top 10%)