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{{Expert-subject|Mathematics|date=November 2008}} | |||
In [[mathematics]], a '''bundle gerbe''' is a [[geometry|geometrical]] model of certain 1-[[gerbe]]s with [[connection (mathematics)|connection]], or equivalently of a 2-class in [[Deligne cohomology]]. | |||
==Topology== | |||
<math>U(1)</math>-[[principal bundles]] over a space <math>M</math> (see [[circle bundle]]) are geometrical realizations of 1-classes in Deligne cohomology which consist of 1-form [[connection (mathematics)|connections)]] and 2-form curvatures. The topology of a <math>U(1)</math> bundle is classified by its [[Chern class]], which is an element of <math>H^2(M, \mathbb{Z})</math>, the second integral cohomology of <math>M</math>. | |||
[[Gerbe]]s, or more precisely 1-gerbes, are abstract descriptions of Deligne 2-classes, which each define an element of <math>H^3(M, \mathbb{Z})</math>, the third integral cohomology of ''M''. | |||
== History == | |||
Historically the most popular construction of a gerbe is a [[category theory|category-theoretic]] model featured in Giraud's theory of gerbes, which are roughly [[sheaf (mathematics)|sheaves]] of [[groupoid]]s over ''M''. | |||
In 1994 Murray introduced bundle gerbes, which are geometric realizations of 1-gerbes. | |||
For many purposes these are more suitable for calculations than Giraud's realization, because their construction is entirely within the framework of classical geometry. In fact, as their name suggests, they are [[fiber bundle]]s. This notion was extended to higher gerbes the following year.<ref>in [http://arxiv.org/abs/hep-th/9511169 Higher Bundle Gerbes and Cohomology Classes In Gauge Theories] by [[Alan Carey (mathematician)|Alan Carey]], [[Michael Murray (mathematician)|Michael Murray]] and [[Bai-Ling Wang]]</ref> | |||
==Relationship with twisted ''K''-theory== | |||
In [http://xxx.lanl.gov/abs/hep-th/0106194 Twisted K-theory and the K-theory of Bundle Gerbes] <ref>by [[Peter Bouwknegt]], [[Alan Carey (mathematician)|Alan Carey]], [[Mathai Varghese|Varghese Mathai]], [[Michael Murray (mathematician)|Michael Murray]] and [[Danny Stevenson]]</ref> the authors defined modules of bundle gerbes and used this to define a [[K-theory]] for bundle gerbes. They then showed that this K-theory is isomorphic to Rosenberg's [[twisted K-theory]], and provides an [[mathematical analysis|analysis]]-free construction. | |||
In addition they defined a notion of [[twisted Chern character]] which is a [[characteristic class]] for an element of twisted K-theory. The twisted Chern character is a [[differential form]] that represents a class in the [[twisted cohomology]] with respect to the [[nilpotent]] operator | |||
:<math> d + H </math> | |||
where <math>d</math> is the ordinary [[exterior derivative]] and the ''twist'' <math>H</math> is a 3-form. This construction was extended to [[equivariant K-theory]] and to [[holomorphic K-theory]] by Mathai and Stevenson.<ref>in [http://arxiv.org/abs/hep-th/0201010 Chern Character in Twisted K-theory: Equivariant and Holomorphic Cases]</ref> | |||
==Relationship with field theory== | |||
Bundle gerbes have also appeared in the context of [[conformal field theory|conformal field theories]]. [[Gawedzki]] and [[Reis]] have interpreted the Wess-Zumino term in the [[Wess-Zumino-Witten model]] (WZW) of [[string theory|string]] propagation on a [[Lie group|group manifold]] as the [[connection (mathematics)|connection]] of a bundle gerbe. [[Urs Schreiber]], [[Christoph Schweigert]] and [[Konrad Waldorf]] have used this construction to extend WZW models to unoriented surfaces and, more generally, the global [[Kalb-Ramond field|Kalb-Ramond coupling]] to unoriented strings. | |||
More details can be found at the [http://golem.ph.utexas.edu/category/ n-Category Café]: | |||
*''[http://golem.ph.utexas.edu/category/2006/10/bundle_gerbes.html Bundle Gerbes: General Idea and Definition] | |||
*''[http://golem.ph.utexas.edu/category/2006/10/bundle_gerbes_connections_and.html Bundle Gerbes: Connections and Surface Transport] | |||
== References == | |||
*''[http://arxiv.org/abs/dg-ga/9407015 Bundle gerbes]'', by Michael Murray. | |||
*''[http://arxiv.org/abs/0712.1651 Introduction to bundle gerbes]'', by Michael Murray. | |||
*''[http://arxiv.org/abs/hep-th/0312154 Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory]'', by Paolo Aschieri, Luigi Cantini and Branislav Jurco. | |||
*[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=311272 Bundle gerbes on arxiv.org] | |||
==Notes== | |||
{{Reflist}} | |||
[[Category:Differential geometry]] | |||
Revision as of 03:47, 10 December 2013
In mathematics, a bundle gerbe is a geometrical model of certain 1-gerbes with connection, or equivalently of a 2-class in Deligne cohomology.
Topology
-principal bundles over a space (see circle bundle) are geometrical realizations of 1-classes in Deligne cohomology which consist of 1-form connections) and 2-form curvatures. The topology of a bundle is classified by its Chern class, which is an element of , the second integral cohomology of .
Gerbes, or more precisely 1-gerbes, are abstract descriptions of Deligne 2-classes, which each define an element of , the third integral cohomology of M.
History
Historically the most popular construction of a gerbe is a category-theoretic model featured in Giraud's theory of gerbes, which are roughly sheaves of groupoids over M.
In 1994 Murray introduced bundle gerbes, which are geometric realizations of 1-gerbes. For many purposes these are more suitable for calculations than Giraud's realization, because their construction is entirely within the framework of classical geometry. In fact, as their name suggests, they are fiber bundles. This notion was extended to higher gerbes the following year.[1]
Relationship with twisted K-theory
In Twisted K-theory and the K-theory of Bundle Gerbes [2] the authors defined modules of bundle gerbes and used this to define a K-theory for bundle gerbes. They then showed that this K-theory is isomorphic to Rosenberg's twisted K-theory, and provides an analysis-free construction.
In addition they defined a notion of twisted Chern character which is a characteristic class for an element of twisted K-theory. The twisted Chern character is a differential form that represents a class in the twisted cohomology with respect to the nilpotent operator
where is the ordinary exterior derivative and the twist is a 3-form. This construction was extended to equivariant K-theory and to holomorphic K-theory by Mathai and Stevenson.[3]
Relationship with field theory
Bundle gerbes have also appeared in the context of conformal field theories. Gawedzki and Reis have interpreted the Wess-Zumino term in the Wess-Zumino-Witten model (WZW) of string propagation on a group manifold as the connection of a bundle gerbe. Urs Schreiber, Christoph Schweigert and Konrad Waldorf have used this construction to extend WZW models to unoriented surfaces and, more generally, the global Kalb-Ramond coupling to unoriented strings.
More details can be found at the n-Category Café:
References
- Bundle gerbes, by Michael Murray.
- Introduction to bundle gerbes, by Michael Murray.
- Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory, by Paolo Aschieri, Luigi Cantini and Branislav Jurco.
- Bundle gerbes on arxiv.org
Notes
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