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Peacock patterns and new integer invariants in topological string theory
by Jie Gu, Marcos Marino
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Jie Gu |
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| Preprint Link: | https://arxiv.org/abs/2104.07437v3 (pdf) |
| Date accepted: | 2022-02-04 |
| Date submitted: | 2021-12-13 04:16 |
| Submitted by: | Gu, Jie |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Topological string theory near the conifold point of a Calabi-Yau threefold gives rise to factorially divergent power series which encode the all-genus enumerative information. These series lead to infinite towers of singularities in their Borel plane (also known as "peacock patterns"), and we conjecture that the corresponding Stokes constants are integer invariants of the Calabi-Yau threefold. We calculate these Stokes constants in some toric examples, confirming our conjecture and providing in some cases explicit generating functions for the new integer invariants, in the form of q-series. Our calculations in the toric case rely on the TS/ST correspondence, which promotes the asymptotic series near the conifold point to spectral traces of operators, and makes it easier to identify the Stokes data. The resulting mathematical structure turns out to be very similar to the one of complex Chern-Simons theory. In particular, spectral traces correspond to state integral invariants and factorize in holomorphic/anti-holomorphic blocks.
Published as SciPost Phys. 12, 058 (2022)
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022-1-16 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2104.07437v3, delivered 2022-01-16, doi: 10.21468/SciPost.Report.4188
Report
This paper studies resurgent structures of topological string theory.
The starting point is the partition function of closed topological strings on toric Calabi-Yau threefolds near conifold points. Adopting the gauge/string large N duality dictionary, the Calabi-Yau Kahler moduli are interpreted as 't Hooft parameters for a gauge theory $\lambda_i\sim N_i g_s$.
For each value of $N_i$, the paper considers the Borel summation of the partition function, and studies its Stokes phenomena. A general lesson that emerges, is that singularities in the Borel plane occur in towers resembling `peacock patterns', whose number depends on $N_i$.
The paper presents precise conjectures on the type of `minimal resurgent structures' appearing in this context, concerning the number and types of asymptotic series associated to each singularity, and the Stokes matrices.
A remarkable conjecture formulated in this paper states that Stokes coefficients are integer. This is verified with rather nontrivial computations for local surfaces $\mathbb{F_0}$ and $\mathbb{P}^2$, for different values of $N_i$.
The findings of this paper are genuinely new, and suggest the existence of extensive new structures governing nonperturbative sectors of topological strings. This picture is supported by solid evidence, at least for a few examples.
These results and conjectures raises several interesting questions for future studies, concerning the interpretation of these structures and their integrality properties.
Report #1 by Anonymous (Referee 1) on 2022-1-11 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2104.07437v3, delivered 2022-01-11, doi: 10.21468/SciPost.Report.4169
Strengths
1. This paper presents interesting novel results in an area of intense current research.
2. The authors define a new family of intrinsically non-perturbative numerical invariants for CY3-folds, conjectured to be integers and verified in examples.
3. The authors show in detail how to derive these invariants in examples using the TS/ST correspondence, discussing the resulting mathematical structure and relating this to a counterpart in complex Chern-Simons theory.
4. The paper is well structured and presents results clearly, nicely discussing them in a broader context. It furthermore strikes a good balance when providing background information and references.
Report
This paper builds on results from a series of recent works by the authors and their collaborators. Although rather technical, it is very well structured and presents interesting novel results clearly.
The authors introduce a new family of numerical invariants for Calabi-Yau 3-folds X, which they define in Section 2 as Stokes constants relying only on the resurgence structure of certain associated formal power series. These series are the conifold free energies $\mathcal{F}_g(\boldsymbol{\lambda})$ which are determined by topological string theory on X. They enter the partition function $\Phi_{\mathbf{N}}(g_s)$, which is a formal power series in the string coupling constant $g_s$ labeled by a tuple of integers $\mathbf{N}$ when viewing $\boldsymbol{\lambda}$ as 't Hooft parameters. From $\Phi_{\mathbf{N}}(g_s)$, the authors define an associated family of formal power series called the "minimal resurgent structure" $\mathcal{B}_{\Phi_{\boldsymbol{N}}}$, whose Borel transforms in turn define the Stokes constants associated to singularities in the Borel plane. The theory of resurgence allows to determine these invariants, which the authors conjecture to be integers and naturally organised into q-series.
This conjecture is verified in examples in Section 3, which presents a detailed analysis of these invariants in cases where X is toric, describing an efficient computational method through the TS/ST correspondence. The authors provide in this section a short review of this correspondence and discuss interesting parallels to results obtained in complex Chern Simons theory. The paper concludes with a concise summary in Section 4, which also provides a list of open problems . The three appendices collect the technical background for relevant special functions and computational techniques, while the introduction provides an excellent overview which sets this work in the broader context and ties together the different sections.
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