Spin structure and its higher analogies play important roles in index theory and mathematical physics. In particular, Witten genera for String manifolds have very nice geometric implications. As a generalization of the work of Chen-Han-Zhang (2011), we introduce the general Stringc structures based on the algebraic topology of Spinc groups. It turns out that there are infinitely many distinct universal Stringc structures indexed by the infinite cyclic group. Furthermore, we can also construct a family of the so-called generalized Witten genera for Spinc manifolds, the geometric implications of which can be exploited in the presence of Stringc structures. As in the un-twisted case studied by Witten, Liu, etc, in our context there are also integrality, modularity, and vanishing theorems for effective non-abelian group actions. We will give some examples to illustrate our new structures and theorems. For instance, we will show that some homotopy complex projective spaces with prescribed stable almost complex structures do not admit non-abelian Lie group actions which preserve the almost complex structures.
This a joint work with Haibao Duan and Fei Han.