Real Riemann Surfaces: Smooth and Discrete

This paper develops a discrete theory of real Riemann surfaces based on quadrilateral cellular decompositions (quad-graphs) and a linear discretization of the Cauchy-Riemann equations. We construct a discrete analogue of an antiholomorphic involution and classify the topological types of discrete real Riemann surfaces, recovering the classical results on the number of real ovals and the separation of the surface. Central to our approach is the construction of a symplectic homology basis adapted to the discrete involution. Using this basis, we prove that the discrete period matrix admits the same canonical decomposition $\mathrm{\Pi }=\frac{1}{2}H+iT$ as in the smooth setting, where $H$ encodes the topological type and $T$ is purely imaginary. This structural result bridges the gap between combinatorial models and the classical theory of real algebraic curves.