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The complex Liouville string: the matrix integral
by Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann, Victor A. Rodriguez
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Submission summary
| Authors (as registered SciPost users): | Scott Collier · Lorenz Eberhardt · Beatrix Mühlmann · Victor Rodriguez |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2410.07345v2 (pdf) |
| Date accepted: | May 1, 2025 |
| Date submitted: | Feb. 11, 2025, 2:43 p.m. |
| Submitted by: | Mühlmann, Beatrix |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We propose a duality between the complex Liouville string and a two-matrix integral. The complex Liouville string is defined by coupling two Liouville theories with complex central charges $c = 13 \pm i \lambda$ on the worldsheet. The matrix integral is characterized by its spectral curve which allows us to compute the perturbative string amplitudes recursively via topological recursion. This duality constitutes a controllable instance of holographic duality. The leverage on the theory is provided by the rich analytic structure of the string amplitudes that we discussed in arXiv:2409.18759 and allows us to perform numerous tests on the duality.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Published as SciPost Phys. 18, 154 (2025)
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2025-4-27 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2410.07345v2, delivered 2025-04-27, doi: 10.21468/SciPost.Report.11090
Strengths
1-Fleshes out a concrete two matrix model theory dual to a novel critical worldsheet string theory built from two complex Liouville CFTs. 2- Gives strong evidence for the duality by matching low point string amplitudes with corresponding matrix model observables. In addition, certain symmetries on both sides are matched and most nontrivially the analytic structure of the string amplitudes is reproduced from the topological recursion relations of the matrix model resolvents. 3-Clearly organised and gives very helpful background on the two matrix model including its topological recursion relations.
Report
I had a couple of comments to offer to the authors with regard to connections to earlier work:
1-The minimal string theory and related backgrounds also have a formulation in terms of a twisted topological string theory. Thus in
https://arxiv.org/abs/hep-th/0312085
the authors considered the spectral curve as part of a non-compact Calabi- Yau geometry and the B-model topological string. It would be interesting to see if there is a similar connection for the present case, even though the spectral curve is not algebraic.
2-Regarding the discussion on p.49 of the landscape of minimal string theories: while the c=1 string theory with non-compact target space is indeed dual to a matrix QM and not a matrix model, this is not true when the theory is on a circle at special radii. Thus, for instance, at the self dual radius there is a dual description of the string amplitudes in terms of a matrix model. It might therefore be fruitful to connect the minimal strings to the c=1 string theory through this special point on the moduli space.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Report #1 by Anonymous (Referee 1) on 2025-4-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2410.07345v2, delivered 2025-04-19, doi: 10.21468/SciPost.Report.11043
Strengths
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.
Weaknesses
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.
Report
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.
Requested changes
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”.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)