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\r\nin the Kählerian case. This relies on a detailed study of frame bundles of loop spaces\r\non the one hand and, on the other hand, on an explicit local trivialization of the\r\nAtiyah operator family [defined in Atiyah (SMF 131:43–59, 1985)] associated\r\nto a loop space.We relate these constructions to the Dixmier-Douady obstruction\r\nclass against the existence of a string structure, as well as to pseudo- line bundle gerbes in the sense of Brylinski (Loop spaces, characteristic\r\nclasses and geometric quantization. Birkhäuser, Basel, 1993).