|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{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==
| | Luke Bryan is often a celebrity in the producing plus the occupation progress initial 2nd to his third place album, & , is the proof. He burst to the picture in 2008 with his crazy mixture of downward-house ease of access, film star very good seems and lyrics, is placed t in a main way. The newest recor about the land graph and #2 about the burst maps, [http://www.netpaw.org luke bryan site] producing it the 2nd highest debut during those times of 2006 for any country [http://lukebryantickets.citizenswebcasting.com cheap luke bryan tickets] performer. <br><br>The boy of the , knows determination and perseverance are important elements in terms of an excellent profession- . His 1st record, Keep Me, made the very best hits “All [http://www.cinemaudiosociety.org luke bryan my kinda night tour] My Pals “Country and Say” Person,” when his energy, Doin’ Thing, located the artist-three straight No. 3 single people: Different Phoning Is a Wonderful Point.”<br><br>Inside the tumble of 2003, Concerts: Luke Bryan And which had a remarkable listing of , including Urban. luke bryan my kind of night tour ([http://lukebryantickets.asiapak.net http://lukebryantickets.asiapak.net/]) “It’s almost like you’re obtaining a endorsement to go one stage further, claims all those designers that were a part of the Concertsaround into a greater degree of artists.” It twisted as among the most successful tours in the 15-12 months record.<br><br>Also visit my blog ... luke bryan luke bryan luke bryan ([http://www.ffpjp24.org Suggested Internet page]) |
| | |
| <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]]
| |
Luke Bryan is often a celebrity in the producing plus the occupation progress initial 2nd to his third place album, & , is the proof. He burst to the picture in 2008 with his crazy mixture of downward-house ease of access, film star very good seems and lyrics, is placed t in a main way. The newest recor about the land graph and #2 about the burst maps, luke bryan site producing it the 2nd highest debut during those times of 2006 for any country cheap luke bryan tickets performer.
The boy of the , knows determination and perseverance are important elements in terms of an excellent profession- . His 1st record, Keep Me, made the very best hits “All luke bryan my kinda night tour My Pals “Country and Say” Person,” when his energy, Doin’ Thing, located the artist-three straight No. 3 single people: Different Phoning Is a Wonderful Point.”
Inside the tumble of 2003, Concerts: Luke Bryan And which had a remarkable listing of , including Urban. luke bryan my kind of night tour (http://lukebryantickets.asiapak.net/) “It’s almost like you’re obtaining a endorsement to go one stage further, claims all those designers that were a part of the Concertsaround into a greater degree of artists.” It twisted as among the most successful tours in the 15-12 months record.
Also visit my blog ... luke bryan luke bryan luke bryan (Suggested Internet page)