The complex Liouville string: worldsheet boundaries and non-perturbative effects

This paper offers a detailed introduction to and first examination of the worldsheet boundaries of the recently proposed complex Liouville string theory, and of the non-perturbative contributions to closed string amplitudes associated with the subset of instanton boundaries in this model. After establishing the expressions of the relevant contributions on the string theory side of the previously proposed string theory / matrix integral duality, the paper presents a detailed match of these contributions with their analogs in the matrix integral dual and therefore provides strong evidence for establishing the proposed duality. As written, the paper displays a pedagogical style with detailed expressions, a clear line of reasoning, and well-formulated arguments, and contributes to progress in the very interesting field of string/matrix duality. I therefore recommend the paper for publication after minor revision.

1-In the complex Liouville string, the single-instanton boundary state, eqn. (3.1), which is later argued to correspond to a single-instanton sector in the matrix integral, consists of the (r, s) ZZ in the "+" sector and the (1, 1) ZZ in the "-" sector. This statement parallels the one in the minimal string, where the (1, 1) ZZ in the "-" sector is replaced with the (1, 1) matter Cardy state. The discussion of Seiberg-Shih equivalence suggests that not all ZZ boundary states should have simple analogs on the matrix integral side of the duality, but it does not seem to immediately single out the ((r, s), (1, 1)) states in particular. If the authors have some intuition for why the ((r, s), (1, 1)) boundary condition is precisely the one matching the matrix integral contribution, details would be most welcome. In lieu of this, a short statement that this is not yet understood would also be welcome.

2- The eigenvalue density of the dual matrix acquiring a negative value on certain subsets of its support might appear somewhat avant-garde. While I see no obvious reason to doubt the proposed duality with the matrix integral introduced by the authors given the evidence the paper provides in matching the non-perturbative contributions in the two systems, it would be helpful to understand the authors' physical interpretation of the phenomenon and their vision for how the data obtained through perturbation theory that is studied in these systems might be used to construct a non-perturbative completion of the model in future work. An example of this would be adding a statement about physical consistency conditions they might expect the completion to satisfy (e.g., what is the expectation for the asymptotic behavior of the partition function in the limit of small / large string coupling?). Another example might be including a statement about their expectation for the matrix integral to be able (or not) to cure the instability caused by the negativity of the eigenvalue density in the single-cut case by exhibiting a stable multi-cut leading saddle with a density that is everywhere positive on its support. Some inspiration might be taken from the (2, 4k) minimal superstring theory which exhibits a gap-closing transition in the support of the eigenvalue density. The string coupling in the perturbative string theory description of the gapped phase is related to the one in the string theory description of the ungapped phase by analytic continuation, but the perturbative data in the two phases looks very different. In this example, some unpublished work following 2412.08698 suggests one can obtain a description of the gapped phase by starting with a version of the ungapped phase with an eigenvalue density which is not everywhere positive. It would be helpful to understand if the authors have something similar in mind for the example of the complex Liouville string.

3-In eqn. (5.4), I believe the sphere diagram with 3 insertions along with (n-3) disks, each with one insertion, should also contribute at order e^{-S_0}.
 
4-Minor typos which are inconsequential to the content but can prove to be mildly distracting: In the first paragraph on page 2, "was leveraged" should be changed to "were leveraged." On page 16, the paragraph before eqn. (2.38) requires revision. Right before this, there appears to be a run-on sentence. On page 18, the reference to eqn. (2.46) was likely meant to be to eqn. (2.48). On page 30, in the line below eqn. (3.17), a "to" is missing. In the second line of page 31, there is an "s" missing in "multi-instanton". In footnote 18, in the second line, "carry" should be replaced by "carries."

Ask for minor revision

This work studies worldsheet boundaries and non-perturbative effects in the newly discovered complex Liouville string theory. The results in this work are novel and significant. However, there are a few minor issues that need to be addressed. Thus, I recommend that this paper be published after a minor revision.

(1) After eq. (3.5), the tensions $T^{(b)}_{r,s}$ are said to be imaginary for purely imaginary $b^2$. Although this is the correct mathematical result, this calls into question the whole framework of the instanton expansion. Typically, the instanton effects are non-perturbatively suppressed because the tensions are positive. This means that we are expanding about the correct saddle-point for the leading eigenvalue density. However, in this case, the non-perturbative effects are as large as the perturbative answer because the suppression is missing. It is plausible that there is a better saddle-point about which the instanton expansion exhibits its usual features. A similar feature can be observed for minimal superstring theory, where the strong and weak coupling instantons look completely different; see e.g., ref. [11].

This is even more relevant here because the leading density of states goes negative, as shown in fig. 8. Since the non-perturbative effects are large, they can significantly change the full non-perturbative density of states and even make it positive over the whole range.

This is an extremely important issue and the authors should examine this in more detail.

(2) In eq. (2.9), the non-degenerate character has a superscript $^{(b)}$, but the right hand side does not depend on $b$. Thus, the role of this superscript should be clarified or it should be deleted.

(3) In eq. (2.40), the final answer comes from a particular choice of contour. Although footnote 9 describes how this contour is chosen generically, some details about the exact contour in this particular case would be beneficial.

(4) In eq. (2.48), the lower bound $E=2$ seems physically significant as it cannot be shifted/rescaled away. Some clarification about how this arises outside of the mathematical justification would be beneficial.

(5) In sec. 2.2.3, the authors claim that any pair of positive integers $(r,s)$ represent distinct boundary conditions. This might only hold if $b^2$ is irrational; some clarification on this is required.

(6) In sec. 2.3, the FZZT$(u)\times$ZZ boundary condition is described. No specifics are provided about the range of $u$. This should be clarified. Relatedly, after eq. (2.59), it is assumed that $u$ and $iu'$ have the same phase. This seems unjustified given the lack of details about the range of $u$. Some details about this are given in Appendix A, but they are missing from the main discussion.

(7) In eq. (2.59), the coefficients of the linear combination have an explicit dependence on $p$ as $\rho_{b}(p)$ appears in the denominator. Thus, the right hand side is not a naive linear combination of eq. (2.58). This should be explained.

(8) After eq. (2.61), it is claimed that $\Gamma$ can be any vertical contour to the right of the singularities. However, this is not correct. For instance, for the result in eq. (2.63), the folding shown in fig. 5 can only be done if the contour is to the left of the origin, otherwise, the $e^{\beta x}$ picks up a divergent piece at $\infty$ and the contour cannot be shifted. This should be clarified.

Ask for minor revision

In this work the authors continue their exposition of a solvable model of string theory. This work consists of a treatment of worldsheet boundaries and non-perturbative effects. One of the novelties in recent years since ref. 6 has been to reinterpret string worldsheet calculations in terms of 2d spacetime baby universe correlators. This interpretation raises various new puzzles.
The work itself is very nicely and pedagogically written and contains many detailed discussions and derivations of the resuls.

The main interpretational issue I see in this (and previous) works is about the spectral density turning negative at some energy eigenvalue.
The authors state on p41 this is "clearly unphysical". I would go beyond that and say that it is a puzzle how a matrix integral can get a negative eigenvalue density at all. How do the authors counter this naive argument? (It suggests that the disc spectral density is not the leading large N (or e^S0) limit of the matrix integral with one length beta boundary to begin with.) The authors state the analogy with SSS at that stage, but the initial type of problem seems very different (support at negative energies compared to negative density of states). The authors then perform an investigation into a problematic region of eigenvalue space causing a divergence of the naive matrix integral. They propose (in the spirit of SSS) a contour deformation for the matrix integral to cure this behavior. This can be done in many ways, leaving open the precise non-perturbative completion of the model. I understand this argument, but I am also left somewhat unsatisfied with this final state of affairs as follows. If we cannot trust the full thermal disc partition function (2.48), then what precisely can we trust? Should we expect deviations then also from all other quantities mentioned in this work (even those that don't have any apparent issue)? The authors mention on page 41 that this negative region "only affects non-perturbative quantities". How can one see this? Usually, the full spectral density at disc level is a physically relevant quantity. I think the work would benefit from some more discussion on how the authors think about these matters. Finally a typo: Figure 8 on page 42: part of the caption is missing at the end.

After clarifying how the authors think about the interpretation of the negativity in the spectral density, I would recommend the paper for publication.

Ask for minor revision