SciPost Submission Page
$x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm
by Alexander Hock
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Alexander Hock |
| Submission information | |
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| Preprint Link: | scipost_202312_00015v2 (pdf) |
| Date accepted: | 2024-08-15 |
| Date submitted: | 2024-06-21 15:46 |
| Submitted by: | Hock, Alexander |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
For a given spectral curve, the theory of topological recursion generates two different families $\omega_{g,n}$ and $\omega_{g,n}^\vee$ of multi-differentials, which are for algebraic spectral curves related via the universal $x-y$ duality formula. We propose a formalism to extend the validity of the $x-y$ duality formula of topological recursion from algebraic curves to spectral curves with exponential variables of the form $e^x=F(e^y)$ or $e^x=F(y)e^{a y}$ with $F$ rational and $a$ some complex number, which was in principle already observed in \cite{Dunin-Barkowski:2017zsd,Bychkov:2020yzy}. From topological recursion perspective the family $\omega_{g,n}^\vee$ would be trivial for these curves. However, we propose changing the $n=1$ sector of $\omega_{g,n}^\vee$ via a version of the Faddeev's quantum dilogarithm which will lead to the correct two families $\omega_{g,n}$ and $\omega_{g,n}^\vee$ related by the same $x-y$ duality formula as for algebraic curves. As a consequence, the $x-y$ symplectic transformation formula extends further to important examples governed by topological recursion including, for instance, Gromov-Witten invariants of $\mathbb{C}^3$ (or, equivalently, triple Hodge integrals), orbifold Hurwitz numbers, and stationary Gromov-Witten invariants of $\mathbb{P}^1$. The proposed formalism is related to the issue topological recursion encounters for specific choices of framings for the topological vertex curve.
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
Author comments upon resubmission
List of changes
All points raised by the second referee are corrected or specified. I will list the corrections that are not straightforward by the referees comments:
p.4: the reference which proves in invariance under $y\to y+R(x)$ is just the https://arxiv.org/abs/math-ph/0702045 Thm 7.1
p.5: I have specified for the integration that it is local, with base point close to $z_i$ and I am also just considering genus zero spectral curves
p.24-25,Sec.4.3: I have added; "We will now discuss that taking this curve and the (conjectured) application of the $x-y$ formula together with the proposition of this article is compatible, also with taking singular limits, for instance, of colliding ramification points. This will further support the proposed structure of Sec. 3.2."
This explains the purpose of Sec. 4.3.
p.26: I have mentioned just before Example 4.4 that the order of integration is relevant, and it has to be the same for each summand in the sum over all $\sigma\in S_n$
Throughout the article: I have tried to make it clearer whenever a formula is conjectured (like for x-y duality and TR for higher order ramification points, or including logarithmic singularities for x,y).
I have added a few footnotes stating that the proposed extension of TR in this article was further developed, extended and proved in https://arxiv.org/abs/2312.16950. These are the footnote numbers: 2,5,7
Published as SciPost Phys. 17, 065 (2024)
Reports on this Submission
Strengths
I can only repeat the points from the initial report:
1. New innovative idea of deforming the topological recursion in a way that makes it compatible with the x-y duality formula beyond the known algebraic case.
2. Many new and known formulas reproduced as a direct application of this new idea.
3. The paper has already enriched the realm of topological recursion and inspired new research in this area.
Weaknesses
None
Report
This paper addresses questions of absolute importance on the edge between integrability, topological string theory, enumerative geometry, and matrix models. It features a new idea that has crucial importance for the field of topological recursion, and this idea immediately generates a huge number of applications; many of them worked out in detail in this paper.
I strongly believe that all the journal criteria are met; moreover, thanks to a revision, the paper was improved qua presentation and qua accuracy of statements. The paper has my strongest recommendation for being accepted.
Requested changes
None
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)