# Horizontal strips and spaces of quadratic differentials

## TL;DR

This paper provides a comprehensive, self-contained account of the theory of quadratic differentials, focusing on their families and parametrizations, which are crucial for understanding their role in moduli spaces and stability conditions.

## Contribution

It offers a complete, accessible framework for defining and parametrizing families of quadratic differentials, facilitating their application in moduli space and stability condition studies.

## Key findings

- Defines families of quadratic differentials and their parametrizations
- Clarifies the role of quadratic differentials in moduli space analysis
- Provides foundational tools for further research in related fields

## Abstract

A (meromorphic) quadratic differential is a (meromorphic) section of the tensor square of the canonical bundle of a Riemann surface. They arose in the study of quasiconformal mappings in the works of Oswald Teichm\"uller, and have played a mayor role in the study of the Riemann moduli, where they can be identified with cotangent vectors to the moduli space. In more recent years, inspired by several observations by Kontsevich, Soibelman and Seidel, in arXiv:1302.7030 Bridgeland and Smith identified certain spaces of meromorphic quadratic differentials with the space of stability conditions on a triangulated category related to the underlying topological surface. This work is an attempt to give a complete (and as self-contained as possible) account of the theory needed to define some families of quadratic differentials (and a natural parametrization of such families). This families are the basic building blocks used by Bridgeland and Smith in their study of the spaces of quadratic differentials.