Summary
Compact Lie groups do not only carry the structure of a Riemannian manifold, but also canonical families of bundle gerbes. We discuss the construction of these bundle gerbes and their relation to loop groups. We present several algebraic structures for bundle gerbes with connection, such as Jandl structures, gerbe modules and gerbe bimodules, and indicate their applications to Wess–Zumino terms in two-dimensional field theories.
Key words
2000 Mathematics Subject Classifications: 22E67, 55R65, 81T40.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray and D. Stevenson, Twisted K-Theory and K-Theory of Bundle Gerbes, Commun. Math. Phys. 228(1), 17–49 (2002), hep-th/0106194.
C. Bachas, M. Douglas and C. Schweigert, Flux Stabilization of D-Branes, JHEP 0005(048) (2000), hep-th/0003037v2.
N. Bourbaki, ´El´ements de Math´ematique. Fasc. XXXIV. Groupes et alg`ebres de Lie. Chapitre IV–VI, Hermann, Paris, 1968.
P. Bordalo, S. Ribault and C. Schweigert, Flux Stabilization in Compact Groups, JHEP 0110(036) (2001).
J.-L. Brylinski, Gerbes on Complex Reductive Lie Groups, math/0002158.
J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, volume 107 of Progress in Mathematics, Birkh¨auser, 1993.
D. S. Chatterjee, On the Construction of Abelian Gerbes, Ph.D. thesis, Cambridge Univ., Cambridge, UK, 1998.
A. L. Carey, S. Johnson and M. K. Murray, Holonomy on D-Branes, J. Geom. Phys. 52(2), 186–216 (2002), hep-th/0204199.
J. Fuchs, I. Runkel and C. Schweigert, TFT Construction of RCFT Correlators III: Simple Currents, Nucl. Phys. B 694, 277–353 (2004), hep-th/0403157.
J. Fuchs, C. Schweigert and K. Waldorf, Bi-Branes: Target Space Geometry for World Sheet Topological Defects, hep-th/0703145, J. Geom. Phys. 58, 576–598 (2008).
K. Gawedzki, Topological Actions in Two-Dimensional Quantum Field Theories, in Non-perturbative Quantum Field Theory, edited by G. Hooft, A. Jaffe, G. Mack, K. Mitter and R. Stora, pages 101–142, Plenum Press, New York, 1988.
K. Gawedzki, Abelian and Non-Abelian Branes in WZW Models and Gerbes, Commun. Math. Phys. 258, 23–73 (2005), hep-th/0406072.
K. Gawedzki and N. Reis, WZW Branes and Gerbes, Rev. Math. Phys. 14(12), 1281–1334 (2002), hep-th/0205233.
K. Gawedzki and N. Reis, Basic Gerbe over Non Simply Connected Compact Groups, J. Geom. Phys. 50(1–4), 28–55 (2003), math.dg/0307010.
K. Gawedzki, R. R. Suszek and K. Waldorf, WZW Orientifolds and Finite Group Cohomology, hep-th/0701071, Commun. Math. Phys. 284 1–49 (2008).
K. Gomi and Y. Terashima, Higher-Dimensional Parallel Transports, Math. Research Letters 8, 25–33 (2001).
E. Meinrenken, The Basic Gerbe over a Compact Simple Lie Group, Enseign. Math., II. Sr. 49(3–4), 307–333 (2002), math/0209194.
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Annals of Mathematical Studies, Princeton University Press, Princeton, NJ, 1976.
M. K. Murray and D. Stevenson, Bundle Gerbes: Stable Isomorphism and Local Theory, J. Lond. Math. Soc. 62, 925–937 (2000), math/9908135.
M. K. Murray, Bundle Gerbes, J. Lond. Math. Soc. 54, 403–416 (1996), dg-ga/9407015.
A. Pressley and G. Segal, Loop Groups, Oxford Univ. Press, Oxford, 1986.
U. Schreiber, C. Schweigert and K. Waldorf, Unoriented WZW Models and Holonomy of Bundle Gerbes, Commun. Math. Phys. 274(1), 31–64 (2007), hep-th/0512283.
D. Stevenson, The Geometry of Bundle Gerbes, Ph.D. thesis, University of Adelaide, Australia, 2000, math.DG/0004117.
K. Waldorf, More Morphisms Between Bundle Gerbes, Theory Appl. Categories 18(9), 240–273 (2007), math.CT/0702652.
E. Witten, Nonabelian Bosonization in Two Dimensions, Commun. Math. Phys. 92, 455–472 (1984).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Schweigert, C., Waldorf, K. (2011). Gerbes and Lie Groups. In: Neeb, KH., Pianzola, A. (eds) Developments and Trends in Infinite-Dimensional Lie Theory. Progress in Mathematics, vol 288. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4741-4_10
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4741-4_10
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4740-7
Online ISBN: 978-0-8176-4741-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)