- Bundle gerbe
In

mathematics , a**bundle gerbe**is a geometrical model of certain 1-gerbe s with connection, or equivalently of a 2-class inDeligne cohomology .**Topology**"U"(1)-

principal bundles over a space "M" (seecircle 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 itsChern class , which is an arbitrary element of H^{2}(M), the second integral cohomology of "M".Gerbe s, or more precisely 1-gerbes, are abstract descriptions of Deligne 2-classes, which each define an element of H^{3}(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

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. [*in [*]*http://arxiv.org/abs/hep-th/9511169 Higher Bundle Gerbes and Cohomology Classes In Gauge Theories*] byAlan Carey , Michael Murray andBai-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*] the authors defined modules of bundle gerbes and used this to define aPeter Bouwknegt ,Alan Carey , Varghese Mathai, Michael Murray andDanny Stevenson K-theory for bundle gerbes. They then showed that this K-theory is isomorphic to Rosenberg'stwisted K-theory , and provides an analysis-free construction.In addition they defined a notion of

twisted Chern character which is acharacteristic class for an element of twisted K-theory. The twisted Chern character is adifferential form that represents a class in thetwisted cohomology with respect to thenilpotent operator:"d" + "H",

where "d" is the ordinary

exterior derivative and the "twist" "H" is a 3-form. This construction was extended toequivariant K-theory and toholomorphic 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 andReis have interpreted the Wess-Zumino term in theWess-Zumino-Witten model (WZW) of string propagation on a group manifold as the connection of a bundle gerbe.Urs Schreiber ,Christoph Schweigert andKonrad Waldorf have used this construction to extend WZW models to unoriented surfaces and, more generally, the globalKalb-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**

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Gerbe**— In mathematics, a gerbe is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud harv|Giraud|1971 following ideas of Alexandre Grothendieck as a tool for non commutative cohomology in degree 2. They can be seen as … Wikipedia**Twisted K-theory**— In mathematics, twisted K theory (also called K theory with local coefficients ) is a variation on K theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory. More specifically, twisted K… … Wikipedia**List of mathematics articles (B)**— NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… … Wikipedia**Sheaf (mathematics)**— This article is about sheaves on topological spaces. For sheaves on a site see Grothendieck topology and Topos. In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.… … Wikipedia**List of algebraic geometry topics**— This is a list of algebraic geometry topics, by Wikipedia page. Contents 1 Classical topics in projective geometry 2 Algebraic curves 3 Algebraic surfaces 4 … Wikipedia**Jean-Luc Brylinski**— (born in 1951) is a French mathematician. Educated at the Ecole Normale Supérieure in Paris, after an appointment as researcher with the C. N. R. S., he became a Professor of Mathematics at Pennsylvania State University. He proved the Kazhdan… … Wikipedia**Bikini variants**— Since the bikini was introduced in 1946, it has generated a number of variations, primarily smaller and more revealing than the original. The bikini has spawned many stylistic variations. A regular bikini is defined as a two pieces of garments… … Wikipedia**апробационный сноп (образец)**— Сноп (образец) из растений или их частей, отбираемый в соответствии с установленной методикой по проведению апробации сортовых посевов. [ГОСТ 20081 74] Тематики семеноводство Обобщающие термины апробация EN bundle for certification DE… … Справочник технического переводчика