- Bundle gerbe
principal bundlesover 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".
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-theoryfor 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 characterwhich is a characteristic classfor an element of twisted K-theory. The twisted Chern character is a differential formthat represents a class in the twisted cohomologywith respect to the nilpotentoperator
:"d" + "H",
where "d" is the ordinary
exterior derivativeand the "twist" "H" is a 3-form. This construction was extended to equivariant K-theoryand to holomorphic K-theoryby 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.
Gawedzkiand Reishave 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 Schweigertand Konrad Waldorfhave used this construction to extend WZW models to unoriented surfaces and, more generally, the global Kalb-Ramond couplingto 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]
*" [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]
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