The rational elliptic genus has been refined to an elliptic genus with values in a ring of level-2 modular forms over the coefficient ring of real K-theory. The refined genus retains at least some of the properties of the rational genus (e.g., modularity, integrality). Other properties, such as rigidity, can be proven in special cases (e.g., for circle actions). The project suggests a systematic study of the refined genus, its connections with Lie group theory, Riemannian geometry, and index theory on the free loop space. The rational elliptic genus was introduced by the principal investigator in connection with a physics inspired question of E. Witten. Both the rigidity and modularity have an interpretation in Quantum Field Theory, which probably provides the best "explanation" for them. There are indications that the torsion invariants derived from the refined elliptic genus also have a physics interpretation. The interplay between theoretical physics and topology is one of the main attractions of the subject.