Real-normalized differentials with a single order $2$ pole

TL;DR

This paper develops a combinatorial model for real-normalized differentials with a single order 2 pole on Riemann surfaces, enabling analysis of their associated period foliation and topological properties of related moduli spaces.

Contribution

It introduces a novel combinatorial framework for studying real-normalized differentials with a specific pole order, advancing understanding of their topology and period foliation.

Findings

Constructed a combinatorial model for the differentials.

Analyzed the absolute period foliation using the model.

Linked the topology of the orbifolds to moduli spaces.

Abstract

A meromorphic differential on a Riemann surface is said to be {\it real-normalized} if all its periods are real. Real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles form real orbifolds whose topology is closely related to that of moduli spaces of Riemann surfaces with marked points. Our goal is to develop tools to study this topology. We propose a combinatorial model for the real normalized differentials with a single order $2$ pole and use it to analyze the corresponding absolute period foliation.