Discrete Theory of Real Riemann Surfaces
Analysis
This paper presents a discrete approach to studying real Riemann surfaces, using quad-graphs and a discrete Cauchy-Riemann equation. The significance lies in bridging the gap between combinatorial models and the classical theory of real algebraic curves. The authors develop a discrete analogue of an antiholomorphic involution and classify topological types, mirroring classical results. The construction of a symplectic homology basis adapted to the discrete involution is central to their approach, leading to a canonical decomposition of the period matrix, similar to the smooth setting. This allows for a deeper understanding of the relationship between discrete and continuous models.
Key Takeaways
- •Develops a discrete theory of real Riemann surfaces.
- •Uses quad-graphs and a discrete Cauchy-Riemann equation.
- •Classifies topological types of discrete real Riemann surfaces.
- •Constructs a symplectic homology basis adapted to the discrete involution.
- •Proves a canonical decomposition of the discrete period matrix.
“The discrete period matrix admits the same canonical decomposition $Π= rac{1}{2} H + i T$ as in the smooth setting, where $H$ encodes the topological type and $T$ is purely imaginary.”