  Motivated by the cubic forms and anomaly cancellation formulas of
Witten-Freed-Hopkins, we give some new cubic forms on spin, spin$^c$,
spin$^{w_2}$ and orientable 12-manifolds respectively. We relate them to
$\eta$-invariants when the manifolds are with boundary, and mod 2 indices on 10
dimensional characteristic submanifolds when the manifolds are spin$^c$ or
spin$^{w_2}$. Our method of producing these cubic forms is a combination of
(generalized) Witten classes and the character of the basic representation of
affine $E_8$.
