Modular forms and generalized anomaly cancellation formulas

In this paper, we generalize the anomaly cancellation formulas given by Alvarez-Gaumé and Witten (1983), Liu (1995) and Han and Zhang (2004) [1], [2] and [7] to the cases where an auxiliary bundle WW and a complex line bundle ξξ are involved with no conditions on the first Pontryagin forms being assumed.