  We introduce the notion of a {\vartheta}-summable Fredholm module over a
locally convex dg algebra {\Omega} and construct its Chern character as a
cocycle on the entire cyclic complex of {\Omega}, extending the construction of
Jaffe, Lesniewski and Osterwalder to a differential graded setting. Using this
Chern character, we prove an index theorem involving an abstract version of a
Bismut-Chern character constructed by Getzler, Jones and Petrack in the context
of loop spaces. Our theory leads to a rigorous construction of the path
integral for N=1/2 supersymmetry which satisfies a Duistermaat-Heckman type
localization formula on loop space.
