Stability conditions on $\text{CY}_N$ categories associated to $A_n$-quivers and period maps

TL;DR

This paper extends the understanding of stability conditions on $N$-Calabi-Yau categories linked to $A_n$-quivers, describing their space via polynomial periods and generalizing previous $ ext{CY}_3$ results to higher dimensions.

Contribution

It generalizes the construction of stability conditions from $ ext{CY}_3$ categories to higher-dimensional Calabi-Yau categories, relating them to polynomial periods and the universal cover of polynomial spaces.

Findings

Describes the space of stability conditions as a universal cover of polynomial spaces.

Constructs central charges as periods of quadratic differentials with zeros of order $N-2$.

Generalizes Bridgeland and Smith's results to higher Calabi-Yau dimensions.

Abstract

In this paper, we study the space of stability conditions on a certain $N$-Calabi-Yau ($\text{CY}_N$) category associated to an $A_n$-quiver. Recently, Bridgeland and Smith constructed stability conditions on some $\text{CY}_3$ categories from meromorphic quadratic differentials with simple zeros. Generalizing their results to higher dimensional Calabi-Yau categories, we describe the space of stability conditions as the universal cover of the space of polynomials of degree $n+1$ with simple zeros. In particular, central charges of stability conditions on $\text{CY}_N$ categories are constructed as the periods of quadratic differentials with zeros of order $N-2$ which are associated to polynomials.