The elementary excitations in the XY model are spin waves and vortices. The work of Vitelli and Ari Turner reveals that the vortices on a curved surface are coupled to the curvature in a way that reflects the conformal anomaly . In this talk, I will first review the basics of the XY model and some essential concepts of differential geometry. Then, I will construct the XY model on an arbitrarily curved surface [2,3]. Finally, I will analyze the effects of curvature on vortices and discuss the underlying physics.
 Vincenzo Vitelli and Ari M. Turner. Anomalous Coupling Between Topological Defects and Curvature. (PRL 93, 215301 (2004))
 Vincenzo Vitelli and David R. Nelson. Defect generation and deconfinement on corrugated topographies. (PRE 70, 051105(2004))
 D. R. Nelson and L. Peliti. Fluctuations in membranes with crystalline and hexatic order. (Journal de Physique, 48, 7(1987))