Several new $SL(2,Z)$ modular forms and anomaly cancellation formulas

In \cite{HLZ2} and \cite{HHLZ}, using $E_8$ bundles, some modular forms over $SL(2,\mathbf{Z})$ were constructed on $12$-dimensional manifolds and the Witten-Freed-Hopkins anomaly cancellation formula was derived by these $SL(2,Z)$ modular forms. In this paper, we construct several similar $SL(2,Z)$ modular forms on any dimensional manifolds and some new anomaly cancellation formulas and applications are given.