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Large N instantons from topological strings
by Marcos Mariño, Ramon Miravitllas
Submission summary
| Authors (as registered SciPost users): | Marcos Mariño · Ramon Miravitllas |
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| Preprint Link: | scipost_202403_00027v1 (pdf) |
| Date submitted: | 2024-03-19 13:36 |
| Submitted by: | Miravitllas, Ramon |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The $1/N$ expansion of matrix models is asymptotic, and it requires non-perturbative corrections due to large $N$ instantons. Explicit expressions for large $N$ instanton amplitudes are known in the case of Hermitian matrix models with one cut, but not in the multi-cut case. We show that the recent exact results on topological string instanton amplitudes provide the non-perturbative contributions of large $N$ instantons in generic multi-cut, Hermitian matrix models. We present a detailed test in the case of the cubic matrix model by considering the asymptotics of its $1/N$ expansion, which we obtain at relatively high genus for a generic two-cut background. These results can be extended to certain non-conventional matrix models which admit a topological string theory description. As an application, we determine the large $N$ instanton corrections for the free energy of ABJM theory on the three-sphere, which correspond to D-brane instanton corrections in superstring theory. We also illustrate the applications of topological string instantons in a more mathematical setting by considering orbifold Gromov-Witten invariants. By focusing on the example of $\mathbb{C}^3/\mathbb{Z}_3$, we show that they grow doubly-factorially with the genus and we obtain and test explicit asymptotic formulae for them.
Current status:
Reports on this Submission
Report 1 by Min-xin Huang on 2024-4-28 (Invited Report)
Strengths
1. Non-perturbative formulation is probably the most outstanding problem in string/M-theory. Non-perturbative effects are also important in many topics in mathematics and physics, in particular in situations involving strong coupling dynamics. The author’s approach is a promising direction where many concrete precise results can be obtained.
2. The numerical tests have some quite impressive precision.
3. These examples establish the universality of the general formalism proposed in the previous papers.
Weaknesses
1. The main novelty of general formalism has been proposed in the previous papers. The current manuscript is just an application to more though still non-trivial examples.
2. The current tests are mostly numerical. It would be better have some analytic derivation or proof of some of the results.
Report
The authors apply the results in the previous recent papers [22, 23] to some more non-trivial examples. In the previous papers, in the context of topological string theory on Calabi-Yau 3-folds, some novel conjectures were proposed concerning the relation between the instanton action and the periods of Calabi-Yau geometry, as well as the formulas for non-perturbative trans-series free energy. The manuscript summarized the developments in Section 2, e.g. formulas (2.4), (2.9), then test the formula (2.9) in three more examples related to matrix models using the large order asymptotics of the perturbative free energy. Such tests often involve some heroic efforts of computations to very large order.
As in the previous papers, the manuscript provides some impressive numerical tests. As can be seen in the various figures, the dots extracted from the large order asymptotics lie well on the lines from the proposed trans-series formula, providing convincing evidence for its validity. We should note that the computations of perturbative free energy are by themselves quite non-trivial, as one needs to use various boundary conditions to fix the holomorphic ambiguity. But a surprising aspect in [22, 23] is that once the perturbative part is determined, the boundary conditions for trans-series parts are relatively quite simple. Using previous works on the holomorphic anomaly equation for the trans-series parts, simple formulas at least for low orders can be written, as e.g. in (2.9).
I think the manuscript contains some solid research results. I recommend publication of the manuscript.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)