On mod 2 and higher elliptic genera
Elliptic genera have usually come in two ways: either by topological methods or by invoking properties of some differential operator, frequently an index formula. Although cobordism plays a crucial role, this paper essentially belongs to the latter category in that it uses the Atiyah-Singer mod 2 index formula for Dirac operators to construct mod 2 elliptic genera on compact smooth \(8k+1\)- and \(8k+2\)-dimensional spin manifolds. The basic idea is to lift the classical Dirac operator mod 2 index formula to a homomorphism from \(KO(X)\) to \(\mathbb{Z}_ 2\), where it is readily checked that it equals the standard cobordism \(KO\)-invariant. Cobordism is then invoked to arrive at the relationship with mod 2 elliptic genera as geometric objects. Higher elliptic genera are considered in the second part of the paper. It is interesting to note that in the course of the work an integral formula for the mod 2 index is obtained in the case of (compact) \(8k+2\)- dimensional spin manifolds.
scientific article; zbMATH DE number 95695
scientific article