\(\text{Spin}^c\)-manifolds with \(\text{Pin}(2)\)-action
It is a classical result of Atiyah and Hirzebruch that the \(\widehat A\)-genus (the index of the Dirac operator) on a Spin-manifold with a non-trivial \(S^1\)-action vanishes. The author proves here that the Witten genus (heuristically the index of the ``Dirac operator'' on the free loop space of the manifold) vanishes for \(\text{Spin}^c\)-manifolds with ``nice'' \(\text{Pin}(2)\)-action provided the first Chern class and first Pontrjagin class are torsion. The main tool is a vanishing theorem for certain indices of \(\text{Spin}^c\)-Dirac operators which the author proved in [\textit{A. Dessai}, ``Rigidity theorems for \(\text{Spin}^c\)-manifolds'', Topology 39, 239-258 (2000; Zbl 0944.58019)]. Applications are given in the cases when the manifold is a homotopy complex projective space and when it is of dimension four.
scientific article; zbMATH DE number 1395776