  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.
