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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.