The Witten Genus , after Kevin Costello

Definition 1.1. Given a ring R, a genus with values in R is a ring homomorphism, Ω ⊗Q→ R, where Ω is the G-bordism ring. For example, the Â-genus and L-genus are ring maps from Ω⊗Q→ Q. The Atiyah-Singer theorem shows that  can be refined to a genus Ω⊗Q→ Z. The L-genus (or signature) is defined on Ω⊗Q, and the Todd genus is defined on the complex cobordism category. We can define genera via multiplicative sequences, which we won’t discuss here, but see e.g. [HBJ92]. Essentially, they are certain power series in characteristic numbers, and so may be evaluated on a manifold. An old theorem of Thom guarantees this will be a cobordism invariant, and certain properties of the power series force the invariant to give a ring homomorphism. One way to define the Witten genus is by some multiplicative sequence: