Stability conditions and Teichm\"uller space

We consider a 3-Calabi-Yau triangulated category associated to an ideal triangulation of a marked bordered surface. Using the theory of harmonic maps between Riemann surfaces, we construct a natural map from a component of the space of Bridgeland stability conditions on this category to the enhanced Teichm"uller space of the surface. We describe a relationship between the central charges of objects in the category and shear coordinates on the Teichm"uller space.