Klein foams as families of real forms of Riemann surfaces

TL;DR

Klein foams, which are structures with singularities inspired by physics, are shown to correspond to families of real forms of Riemann surfaces, simplifying their study by leveraging known properties of these surfaces.

Contribution

The paper establishes an equivalence between Klein foams and families of real forms of Riemann surfaces, enabling new approaches to their analysis.

Findings

Klein foams are equivalent to families of real forms of complex algebraic curves.

This correspondence allows the use of existing properties of real forms to analyze Klein foams.

The study provides insights into the topological and analytic properties of Klein foams.

Abstract

Klein foams are analogues of Riemann surfaces for surfaces with one-dimensional singularities. They first appeared in mathematical physics (string theory etc.). By definition a Klein foam is constructed from Klein surfaces by gluing segments on their boundaries. We show that, a Klein foam is equivalent to a family of real forms of a complex algebraic curve with some structures. This correspondence reduces investigations of Klein foams to investigations of real forms of Riemann surfaces. We use known properties of real forms of Riemann surfaces to describe some topological and analytic properties of Klein foams.