Several new two-variable elliptic genera

Abstract

In this paper, we define several new two-variable elliptic genera for even dimensional spin manifolds which are the index for some Dirac operators and holomorphic $SL(2,Z)$, $\hat{I}{“}_0(2)$, $\hat{I}{“}^0(2)$, $\hat{I}{“}_{\hat{I}}¸$-Jacobi forms. By these Jacobi forms, we can get some $SL(2,\mathbf{Z})$ and $\hat{I}{“}^0(2)$ modular forms. By these $SL(2,\mathbf{Z})$ and $\hat{I}{“}^0(2)$ modular forms, we get some interesting anomaly cancellation formulas for spin manifolds.