Rigidity theorems for \(Spin^\mathbb{C}\)-manifolds
Assume that \(S^1\) acts on a compact manifold \(M\). Then the equivariant index of an \(S^1\)-invariant elliptic operator \(D\) is a virtual representation of \(S^1\). The operator \(D\) is called rigid, if the equivariant index is a trivial representation of \(S^1\), i.e. any \(z\in S^1\) acts on \(\text{ind}(D)\) via the identity. The following classical operators are rigid: the signature operator for oriented manifolds, the Dirac operator for spin manifolds and the Dolbeault operator for complex manifolds. \textit{E. Witten} [Lect. Notes Math. 1326, 161-181 (1988; Zbl 0679.58045)] considered analogues of the classical operators on the free loop space of a manifold. Although no rigorous definition of these operators is known until today, the invariants which correspond to the signature and the index of the Dirac operator are mathematically well-defined objects, called the elliptic genus and the Witten genus. Ochanine, Bott and Taubes proved that the elliptic genus is rigid for spin manifolds. In the article under consideration some rigidity results are derived for \(\text{Spin}^c\)-manifolds. We start with an \(S^1\)-equivariant complex vector bundle \(V\) and an \(S^1\)-equivariant Spin-vector bundle \(W\). Using \(V\) and \(W\), a \(q\)-series \(R_0\in K_{S^1}(M)q\) of virtual \(S^1\)-equivariant vector bundles is defined. One of the main results of the article is the rigidity of the Dirac operator twisted by \(R_0\) on a \(\text{Spin}^c\)-manifold under certain conditions. This generalizes the rigidity of the elliptic genus for Spin-manifolds with \(S^1\)-action. A similar statement is derived generalizing the elliptic genus of level \(N\). The author shows how to apply his results to \(\text{Spin}^c\)-manifolds with nice \(\text{Pin}(2)\)-actions.
scientific article; zbMATH DE number 1384246