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

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.