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

The Chern Character of {\theta}-summable Fredholm Modules over dg Algebras and Localization on Loop Space