x-y duality in topological recursion for exponential variables via quantum dilogarithm

For a given spectral curve, the theory of topological recursion generates two different families ωg,n ωg,n and ωg,n^ ωg,n∨ of multi-differentials, which are for algebraic spectral curves related via the universal x-y x−y duality formula. We propose a formalism to extend the validity of the x-y x−y duality formula of topological recursion from algebraic curves to spectral curves with exponential variables of the form e^x=F(e^y) ex=F(ey) or e^x=F(y)ea y ex=F(y)eay with F F rational and a a some complex number, which was in principle already observed in [Commun. Number Theory Phys. 13, 763 (2019); J. Lond. Math. Soc. 109, e12946 (2024)]. From topological recursion perspective the family ωg,n^ ωg,n∨ would be trivial for these curves. However, we propose changing the n=1 n=1 sector of ωg,n^ ωg,n∨ via a version of the Faddeev’s quantum dilogarithm which will lead to the correct two families ωg,n ωg,n and ωg,n^ ωg,n∨ related by the same x-y x−y duality formula as for algebraic curves. As a consequence, the x-y x−y symplectic transformation formula extends further to important examples governed by topological recursion including, for instance, Gromov-Witten invariants of C^3 ℂ3 (or, equivalently, triple Hodge integrals), orbifold Hurwitz numbers, and stationary Gromov-Witten invariants of P^1 ℙ1 . The proposed formalism is related to the issue topological recursion encounters for specific choices of framings for the topological vertex curve.