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For a given spectral curve, the theory of topological recursion generates twodifferent families $\omega_{g,n}$ and $\omega_{g,n}^\vee$ ofmulti-differentials, which are for algebraic spectral curves related via theuniversal $x-y$ duality formula. We propose a formalism to extend the validityof the $x-y$ duality formula of topological recursion from algebraic curves tospectral 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 inprinciple already observed in \cite{Dunin-Barkowski:2017zsd,Bychkov:2020yzy}.From topological recursion perspective the family $\omega_{g,n}^\vee$ would betrivial 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 whichwill 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 aconsequence, the $x-y$ symplectic transformation formula extends further toimportant 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 ofcomplex curves, orbifold Hurwitz numbers, or stationary Gromov-Witteninvariants of $\mathbb{P}^1$. The proposed formalism is related to the issuetopological recursion encounters for specific choices of framings for thetopological vertex curve.

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