Bundle gerbe

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

"U"(1)-principal bundles over a space "M" (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 U(1) bundle is classified by its Chern class, which is an arbitrary element of H²(M), the second integral cohomology of "M".

Gerbes, or more precisely 1-gerbes, are abstract descriptions of Deligne 2-classes, which each define an element of H³(M), 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. [in [http://arxiv.org/abs/hep-th/9511169 Higher Bundle Gerbes and Cohomology Classes In Gauge Theories] by Alan Carey, Michael Murray and Bai-Ling Wang]

Relationship with twisted "K"-theory

In [http://xxx.lanl.gov/abs/hep-th/0106194 Twisted K-theory and the K-theory of Bundle Gerbes] [by Peter Bouwknegt, Alan Carey, Varghese Mathai, Michael Murray and Danny Stevenson] 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

:"d" + "H",

where "d" is the ordinary exterior derivative and the "twist" "H" is a 3-form. This construction was extended to equivariant K-theory and to holomorphic K-theory by Mathai and Stevenson. [in [http://arxiv.org/abs/hep-th/0201010 Chern Character in Twisted K-theory: Equivariant and Holomorphic Cases] ]

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 [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/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