Inspired by mirror symmetry, I will investigate some differential geometric aspects of the space of Bridgeland stability conditions on a CY triangulated category. The aim is to develop theory of Weil-Petersson geometry on the K\"ahler moduli space. A few basic examples will be discussed. In particular, I will identify our Weil-Petersson metric with the Bergman metric on a Siegel modular variety in the case of the self-product of an elliptic curve. This is a joint work with Yu-Wei Fan and Shing-Tung Yau.