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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 characters53C27: Spin and Spin$^c$ geometry

Information

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

References

[1]Atiyah, M. F., ‘Circular symmetry and stationary phase approximation’, in: Proceedings of the Conference in Honor of L. Schwartz, Astérisque 131 (1985), 43–60.Google Scholar
[2]Brylinski, J.-L., Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics, 107 (Birkhäuser, Basel, 1993).Google Scholar
[3]Brylinski, J.-L. and McLaughlin, D. A., ‘The geometry of degree four characteristic classes and of line bundles on loop spaces I’, Duke Math. J. 75(3) (1994), 603–638.CrossRefGoogle Scholar
[4]Brylinski, J.-L. and McLaughlin, D. A., ‘The geometry of degree four characteristic classes and of line bundles on loop spaces II’, Duke Math. J. 83(1) (1996), 105–139.Google Scholar
[5]Carey, A. L., Johnson, S. and Murray, M. K., ‘Holonomy on D-branes’, J. Geom. Phys. 52(2) (2002), 186–216.Google Scholar
[6]Carey, A. L. and Wang, B.-L., ‘Fusion of symmetric D-branes and Verlinde rings’, Comm. Math. Phys. 277(3) (2008), 577–625.CrossRefGoogle Scholar
[7]Gomi, K., ‘Connections and curvings on lifting bundle gerbes’, J. Lond. Math. Soc. 67(2) (2003), 510–526.Google Scholar
[8]McLaughlin, D. A., ‘Orientation and string structures on loop space’, Pacific J. Math. 155(1) (1992), 143–156.CrossRefGoogle Scholar
[9]Murray, M. K., ‘Bundle gerbes’, J. Lond. Math. Soc. 54 (1996), 403–416.CrossRefGoogle Scholar
[10]Murray, M. K., ‘An introduction to bundle gerbes’, in: The Many Facets of Geometry. A Tribute to Nigel Hitchin, (eds. Garcia-Prada, O., Bourguignon, J. P. and Salamon, S.) (Oxford University Press, Oxford, 2010).Google Scholar
[11]Schreiber, U. and Waldorf, K., ‘Parallel transport and functors’, J. Homotopy Relat. Struct. 4 (2009), 187–244.Google Scholar
[12]Schweigert, C. and Waldorf, K., ‘Gerbes and Lie groups’, in: Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics, 600 (eds. Neeb, K.-H. and Pianzola, A.) (Birkhäuser, Basel, 2010).Google Scholar
[13]Spera, M. and Wurzbacher, T., ‘Twistor spaces and spinors over loop spaces’, Math. Ann. 338 (2007), 801–843.Google Scholar
[14]Stevenson, D., ‘The geometry of bundle gerbes’, PhD Thesis, University of Adelaide, 2000.Google Scholar
[15]Stolz, S. and Teichner, P., ‘The spinor bundle on loop spaces’, unpublished draft.Google Scholar
[16]Waldorf, K., ‘Transgression to loop spaces and its inverse, I: diffeological bundles and fusion maps’, Preprint.Google Scholar
[17]Waldorf, K., ‘Transgression to loop spaces and its inverse, II: gerbes and fusion bundles with connection’, Preprint.Google Scholar
[18]Waldorf, K., ‘More morphisms between bundle gerbes’, Theory Appl. Categ. 18(9) (2007), 240–273.Google Scholar
[19]Waldorf, K., ‘Multiplicative bundle gerbes with connection’, Differential Geom. Appl. 28(3) (2010), 313–340.CrossRefGoogle Scholar