Cubic forms, anomaly cancellation and modularity

Abstract

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 $\hat{I}\cdot$-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 $E_8$.

final version