Twistor spaces and spinors over loop spaces

In this paper two natural twistor spaces over the loop space of a Riemannian manifold are constructed and their equivalence is shown in the Kählerian case. This relies on a detailed study of frame bundles of loop spaces on the one hand and, on the other hand, on an explicit local trivialization of the Atiyah operator family [defined in Atiyah (SMF 131:43–59, 1985)] associated to a loop space.We relate these constructions to the Dixmier-Douady obstruction class against the existence of a string structure, as well as to pseudo- line bundle gerbes in the sense of Brylinski (Loop spaces, characteristic classes and geometric quantization. Birkhäuser, Basel, 1993).