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A LOOP SPACE FORMULATION FOR GEOMETRIC LIFTING PROBLEMS

Part of: Global differential geometry

Published online by Cambridge University Press:  09 June 2011

KONRAD WALDORF
KONRAD WALDORF*
Affiliation:
Department of Mathematics, University of California, 970 Evans Hall #3840, Berkeley, CA 94720, USA
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Abstract

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We review and then combine two aspects of the theory of bundle gerbes. The first concerns lifting bundle gerbes and connections on those, developed by Murray and by Gomi. Lifting gerbes represent obstructions against extending the structure group of a principal bundle. The second is the transgression of gerbes to loop spaces, initiated by Brylinski and McLaughlin and with recent contributions of the author. Combining these two aspects, we obtain a new formulation of lifting problems in terms of geometry on the loop space. Most prominently, our formulation explains the relation between (complex) spin structures on a Riemannian manifold and orientations of its loop space.

Keywords

lifting problembundle gerbetransgressionloop space

MSC classification

Secondary: 53C08: Gerbes, differential characters 53C27: Spin and Spin$^c$ geometry
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 1 , February 2011 , pp. 129 - 144
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

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