Circle actions and higher elliptic genera

For a manifold with an S-1-action, we define generalized elliptic genera by using the orbit map and generalize Hirzebruch-Slodowy's formula in [9]. As a result, we have vanishing theorems of higher elliptic genera and higher twisted (A) over cap-genera. We also generalize elliptic genera of level N for stable almost complex manifolds and have a similar vanishing theorem.