$x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm

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 $e^x=F(y)e^{ay}$ 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.

arXiv comment30 pages, 3 figures

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