Twisted \(K\)-theory constructions in the case of a decomposable Dixmier-Douady class (Q6486649)

The purpose of this paper is to give some explicit examples related to twisted complex \(K\)-theory on \(X=S^1\times M\), where \(M\) is a compact manifold, and the twisting class \(\sigma\) is the cup product of the canonical generator of \(H^1(S^1)\) with a class \(\beta_M\in H^2(M, \mathbb Z)\). The authors start with the complex line bundle over \(M\) classified by \(\beta_M\), and a connection \(\nabla_M\) over it having a de Rham representative for \(\beta_M\) as its curvature. From this data they construct a projective Fock bundle over \(X\) and a gerbe with Dixmier-Douady class \(\sigma\). They realize twisted \(K\)-theory classes from Fredholm sections \(Q\) of a bundle over \(X\) naturally coming from this data. Attached to \(Q\) they construct a superconnection on the covering space \(\widetilde X=\mathbb R\times M\), giving an explicit realization of the Chern character of the twisted \(K\)-theory class.