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Les Houches lectures on non-perturbative topological strings
by Marcos Mariño
Submission summary
| Authors (as registered SciPost users): | Marcos Mariño |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2411.16211v2 (pdf) |
| Date accepted: | June 24, 2025 |
| Date submitted: | Feb. 26, 2025, 7:36 p.m. |
| Submitted by: | Marcos Mariño |
| Submitted to: | SciPost Physics Lecture Notes |
| for consideration in Collection: |
| Ontological classification | |
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| Academic field: | Physics |
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Abstract
In these lecture notes for the Les Houches School on Quantum Geometry I give an introductory overview of non-perturbative aspects of topological string theory. After a short summary of the perturbative aspects, I first consider the non-perturbative sectors of the theory as unveiled by the theory of resurgence. I give a self-contained derivation of recent results on non-perturbative amplitudes, and I explain the conjecture relating the resurgent structure of the topological string to BPS invariants. In the second part of the lectures I introduce the topological string/spectral theory (TS/ST) correspondence, which provides a non-perturbative definition of topological string theory on toric Calabi-Yau manifolds in terms of the spectral theory of quantum mirror curves
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Editorial decision:
For Journal SciPost Physics Lecture Notes: Publish
(status: Editorial decision fixed and (if required) accepted by authors)
Reports on this Submission
Report
The presentation is often terse, but usually clearly written. I'd expect it to be useful for young researchers or PhD students looking for a fast track into this field of research.
I'd certainly like to recommend these lecture notes for publication.
Requested changes
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P.2, bottom, "...and a non-rigorous ... is typical in QFT". Confusing or misleading formulation
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P.12, "...is obtained as a global function on the moduli space...": This is very confusing, as just a little later the dependence of the pre-potential on choices such as a symplectic basis is observed, playing a very important role later on. On page 29 it is written that "The free energies in a given frame F_g are not globally defined on the moduli space, and they are analytic only on a region...". While it may well be true that local coordinates defining a particular frame may admit an analytic continuation to a much larger domain, it seems better to stress the frame dependence rather than the "global" nature.
P. 23, "...that t is "quantised" in units of g_s.", and similar bottom of P.32: I can't help observing a tension with the fact that later it plays an important role to have functions that are entire as function of the moduli, and not just defined on a lattice. To my mind it remains obscure what the precise meaning of the "quantisation" of t in this context is supposed to be. Having finite shifts alone can hardly be enough to justify such far-reaching proposals. A continuation away from the integer multiples of g_s does exist, and it plays an important role in the story.
P. 34, "In this correspondence, the dual of the string theory is not a field theory, but a one-dimensional quantum-mechanical model." -- A very confusing statement. The duality discussed here is of very different nature than other dualities in QFT and string theory such as S-duality of N=4 SYM, or AdS-CFT. Such dualities claim dual descriptions of a QFT or string theory as a whole in terms of either another QFT, or another string theory. One certainly does not expect the ST to capture all of the string theory compactified on CY.
P. 35, "different canonical forms of the curve": Shouldn't the form be unique if it is canonical? And functional-difference -> finite difference
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Report
Requested changes
I have only one minor comment concerning Remark 2.4. To the best of my knowledge, the proof in [44] applies only for \epsilon_1, \epsilon_2 > 0. The convergence in the topological string phase was established prior to [44] in [arXiv:1608.02566], Proposition 3.1.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Thank you very much for your positive feedback. You are right about ref. [44], I will definitely add the relevant reference that you point out in the revised version.
Author: Marcos Mariño on 2025-06-26 [id 5602]
(in reply to Report 2 on 2025-06-20)Thank you very much for your feedback and the detailed reading. Let me make some comments on your observations.
1) Concerning your observation on my statements in p.2: what I mean when I say "...and a non-rigorous ... is typical in QFT" is explained in the paragraph immediately after this sentence. Would it be possible to have a more explicit explanation of what is confusing or misleading in this paragraph?
2) I agree that this requires clarification. What I mean here is that in the B model one can write expressions for the genus g free energies in terms of propagators and global functions which are really global and universal, frame-independent. Expressions in different frames are then obtained by different choices of the form of the propagator.
3) I also agree that this requires further clarification. One important comment that could be clarifying is that the expressions obtained for the multi-instantons in the topological string are very similar to the multi-instantons in matrix models obtained by eigenvalue tunneling. The latter involve an actual quantization since one has to "tunnel" integer numbers of eigenvalues. The matrix model realization of the free energies explained later in the lectures also leads to a realization of this "quantization". I will try to clarify these issues and the existing tension in the revised version.
4) I'm not sure why one should be so restrictive concerning the use of the word "duality". In addition, there is evidence that in practice the full topological string in the toric case can be described by this quantum mechanical model. The observables of topological string are closed string amplitudes F_g and open string amplitudes involving toric D-branes. For the first one, there is clearly a complete description in terms of spectral traces. For the second one, there is also evidence that they can be described by wavefunctions in the quantum mechanics. Perhaps where I concur with the referee is that this duality can not be explained right now by a more fundamental string theory construction, for example. This could be added as a qualification.
5) For this last comment, I refer in the lectures to the paper [15] for explaining what is meant by "canonical" forms, but I agree that the phrasing is confusing. I can definitely change functional-> finite.