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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
Submission information
Preprint Link: scipost_202312_00015v1  (pdf)
Date submitted: 2023-12-06 10:54
Submitted by: Hock, Alexander
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


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, the topological vertex curve which computes Gromov-Witten invariants of $\mathbb{C}^3$, equivalently triple Hodge integrals on the moduli space of complex curves, orbifold Hurwitz numbers, or 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.

Current status:
In refereeing

Reports on this Submission

Anonymous Report 1 on 2024-3-27 (Invited 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.


No weaknesses.


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.

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No changes are requested.

  • validity: top
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  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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