Dimers on Riemann surfaces and compactified free field

We consider the dimer model on a bipartite graph embedded into a locally flat Riemann surface with conical singularities and satisfying certain geometric conditions in the spirit of the work of [Chelkak, Laslier and Russkikh, Proceedings of the London Mathematical Society 126.5 (2023), pp. 1656-1739]. Following the approach developed by Dubédat in his work [J. Amer. Math. Soc. 28 (2015), pp. 1063-1167] we establish the convergence of dimer height fluctuations to the compactified free field in the small mesh size limit. This work is inspired by the series of works of [Nathanaël Berestycki, Benoît Laslier, and Gourab Ray, Annales de l’Institut Henri Poincaré D 12.2 (2024), pp. 363-444.] and [Nathanaël Berestycki, Benoît Laslier, and Gourab Ray, Probability and Mathematical Physics 5.4 (2024), pp. 961-1037], where a similar problem is addressed, and the convergence to a conformally invariant limit is established in the Temperlian setup, but the identification of the limit as the compactified free field is missing. This identification is the main result of our paper.