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$x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm
by Alexander Hock
Submission summary
| Authors (as registered SciPost users): | Alexander Hock |
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| Preprint Link: | scipost_202312_00015v2 (pdf) |
| Date submitted: | 2024-06-21 15:46 |
| Submitted by: | Hock, Alexander |
| Submitted to: | SciPost Physics |
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| Academic field: | Physics |
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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.
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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