η-INVARIANT AND MODULAR FORMS

We show that the Atiyah–Patodi–Singer reduced η-invariant of the twisted Dirac operator on a closed 4m−1-dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a meromorphic modular form of weight 2m up to an integral q-series. We prove this result by combining our construction of certain modular characteristic forms associated to a generalized Witten bundle on spin^c-manifolds with a deep topological theorem due to Hopkins.