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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 simplezeros. Generalizing their results to higher dimensional Calabi-Yau categories,we describe the space of stability conditions as the universal cover of thespace of polynomials of degree $n+1$ with simple zeros. In particular, centralcharges of stability conditions on $\text{CY}_N$ categories are constructed asthe periods of quadratic differentials with zeros of order $N-2$ which areassociated to polynomials.

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