Circle Actions and Higher Elliptic Genera

Abstract

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 Â-genera. We also generalize elliptic genera of level N for stable almost complex manifolds and have a similar vanishing theorem.