LAUSR

We introduce the notion of a {\theta}-summable Fredholm module over a locally convex dg algebra {\Omega} and construct its Chern character as an entire cyclic cocyle in the entire cyclic complex of {\Omega}, leading to a cohomology class in the entire cyclic cohomology of {\Omega}. This extends the cocycle of Jaffe, Lesniewski and Osterwalder to the differential graded case. Using this Chern character, we prove an index theorem involving an abstract version of a Bismut-Chern character constructed by Getzler, Jones and Petrack. Our theory leads to a rigorous construction of the path integral for N=1/2 supersymmetry and provides a framework that allows to conduct a proof of the Atiyah-Singer theorem by means of a well-defined Duistermaat-Heckman type localization formula on loop space.