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The complex Liouville string: worldsheet boundaries and non-perturbative effects
by Scott Collier, Lorenz Eberhardt, Beatrix Mühlmann, Victor A. Rodriguez
Submission summary
| Authors (as registered SciPost users): | Scott Collier · Lorenz Eberhardt · Beatrix Mühlmann |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2410.09179v2 (pdf) |
| Date submitted: | 2025-02-11 14:46 |
| 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 investigate general observables of the complex Liouville string with worldsheet boundaries. We develop a universal formalism that reduces such observables to ordinary closed string amplitudes without boundaries, applicable to any worldsheet string theory, but particularly simple in the context of 2d or minimal string theories. We apply this formalism to the duality of the complex Liouville string with the matrix integral proposed in arXiv:2409.18759 and arXiv:2410.07345 and showcase the formalism by finding appropriate boundary conditions for various matrix model quantities of interest, such as the resolvent or the partition function. We also apply this formalism towards the computation of non-perturbative effects on the worldsheet mediated by ZZ-instantons. These are known to be plagued by extra subtleties which need input from string field theory to resolve. These computations probe and uncover the duality between the complex Liouville string and the matrix model at the non-perturbative level.
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- Present a breakthrough on a previously-identified and long-standing research stumbling block
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Report
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.
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