In this talk I describe a method by which one can construct over a suitable loop space an operator which is the analogue of the Dirac operator on a finite dimensional manifold.

The key step is to adapt an idea due to Jack Morava to construct an inner product on the cotangent bundle of the loop space. There is then a Hilbert bundle which is the fibrewise completion of the cotangent bundle. This bundle is used to construct the spin bundle so that the Clifford multiplication map extends to the domain of a connection allowing one to define the Dirac operator.

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This talk was given at the Fields Institute as part of the Workshop on Forms of Homotopy Theory: Elliptic Cohomology and Loop Spaces.

• Abstract on the Atlas of Confrerences
• Audio and Slides courtesy of the Fields Institute