Cubic forms, anomaly cancellation and modularity

Recently Freed and Hopkins [11] proved that there is no parity anomaly in M-theory on pin+ manifolds in the low-energy field theory approximation, and they also developed an algebraic theory of cubic forms. Earlier Witten [33] proved the anomaly cancellation for spin manifolds by introducing the E8-bundle technique. Motivated by the cubic forms and the anomaly cancellation formulas of Witten-Freed-Hopkins, we give some new cubic forms on spin, spinc, spinω2 and orientable 12-manifolds respectively. We relate them to η-invariants when the manifolds are with boundary, and mod 2 indices on 10 dimensional characteristic submanifolds when the manifolds are spinc or spinω2. Our method of producing these cubic forms is a combination of (generalized) Witten classes and the character of the basic representation of affine E8.