Modular invariance and anomaly cancellation formulas in odd dimension

By studying modular invariance properties of some characteristic forms, we get some new anomaly cancellation formulas on (4r−1)(4r−1) dimensional manifolds. As an application, we derive some results on divisibilities of the index of Toeplitz operators on (4r−1)(4r−1) dimensional spin manifolds and some congruent formulas on characteristic number for (4r−1)(4r−1) dimensional spinc manifolds.