  Inspired by mirror symmetry, we investigate some differential geometric
aspects of the space of Bridgeland stability conditions on a Calabi-Yau
triangulated category. The aim is to develop theory of Weil-Petersson geometry
on the stringy K\"ahler moduli space. A few basic examples are studied. In
particular, we 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.
