Formation matrix - Revision history

en>NHCLS: Adding may be too technical tag

2014-05-17T22:44:29Z

Adding may be too technical tag

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	The ~~title of the author~~ is ~~Luther~~. ~~Interviewing is what~~ she ~~does. Arizona has usually~~ been my residing ~~place~~ but ~~my spouse~~ desires us to ~~transfer~~. ~~Climbing is~~ what ~~adore~~ performing.<br><br>~~Here is~~ my ~~web page: extended auto warranty (~~[http://~~Www~~.~~True-Life~~.~~org~~/~~Activity~~-~~Feed/My~~-~~Profile/UserId/2850 mouse click the following web site~~])		The writer's name is Christy Brookins. For many years she's been residing in Kentucky but love [http://galab-work.cs.pusan.ac.kr/Sol09B/?document_srl=1489804 telephone psychic] - [http://gcjcteam.org/index.php?mid=etc_video&document_srl=696611&sort_index=regdate&order_type=desc click through the next web page], her husband desires them to move. It's not a typical factor but what I like performing is to climb but I don't have the time lately. Office supervising is where her primary earnings comes from but she's currently applied for an additional 1.<br><br>my weblog; [http://www.zavodpm.ru/blogs/glennmusserrvji/14565-great-hobby-advice-assist-allow-you-get-going free tarot readings]

en>BD2412: Fixing links to disambiguation pages using AWB

2014-02-24T15:56:19Z

Fixing links to disambiguation pages using AWB

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	In a field of [[mathematics]] known as [[differential geometry]], the '''Courant bracket''' is a generalization of the [[Lie derivative\|Lie bracket]] from an operation on the [[tangent bundle]] to an operation on the [[direct sum of vector bundles\|direct sum]] of the tangent bundle and the [[vector bundle]] of [[differential form\|''p''-forms]].		The title of the author is Luther. Interviewing is what she does. Arizona has usually been my residing place but my spouse desires us to transfer. Climbing is what adore performing.<br><br>Here is my web page: extended auto warranty ([http://Www.True-Life.org/Activity-Feed/My-Profile/UserId/2850 mouse click the following web site])

	The case ''p'' = 1 was introduced by [[Theodore James Courant]] in his 1990 doctoral dissertation as a structure that bridges [[Poisson manifold\|Poisson geometry]] and pre[[symplectic geometry]], based on work with his advisor [[Alan Weinstein]]. The ~~twisted version of the Courant bracket was introduced in 2001 by [[Pavol Severa]], and studied in collaboration with Weinstein.~~

	~~Today a [[complex number\|complex]] version~~ of the ~~''p''=1 Courant bracket plays a central role in the field of [[generalized complex geometry]], introduced by [[Nigel Hitchin]] in 2002. Closure under the Courant bracket~~ is ~~the [[integrability condition]] of a [[Almost_complex_manifold#Generalized_almost_complex_structure\|generalized almost complex structure]].~~

	~~==Definition==~~
	Let ''X'' and ''Y'' be [[vector field]]s on an N-dimensional real [[manifold (mathematics)\|manifold]] ''M'' and let ''ξ'' and ''η'' be ''p''-forms. Then ''X+ξ'' and ''Y+η'' are [[fiber_bundle#Sections\|sections]] of the direct sum of the tangent bundle and the bundle of ''p''-forms. ~~The Courant bracket of ''X+ξ'' and ''Y+η''~~ is ~~defined to be~~

	~~:<math>[X+\xi,Y+\eta]=[X,Y]~~
	~~+\mathcal{L}_X\eta-\mathcal{L}_Y\xi~~
	~~-\frac{1}{2}d(i(X)\eta-i(Y)\xi)</math>~~

	where <math>\mathcal{L}_X</math> is the [[Lie derivative]] along the vector field ''X'', ''d'' is the [[exterior derivative]] and ''i'' is the [[Exterior_algebra#The_interior_product_or_insertion_operator\|interior product]].

	~~==Properties==~~

	~~The Courant bracket is [[antisymmetric]]~~ but ~~it does not satisfy the [[Jacobi identity]] for ''p'' greater than zero.~~

	~~===The Jacobi identity===~~

	~~However, at least in the case ''p=1'', the [[Jacobiator]], which measures a bracket's failure~~ to ~~satisfy the Jacobi identity, is an [[exact form]]. It is the exterior derivative of a form which plays the role of the [[Nijenhuis tensor]] in generalized complex geometry.~~

	~~The Courant bracket is the antisymmetrization of the [[Courant_bracket#Dorfman_bracket\|Dorfman bracket]], which does satisfy a kind of Jacobi identity~~.

	~~===Symmetries===~~

	~~Like the Lie bracket, the Courant bracket~~ is ~~invariant under diffeomorphisms of the manifold ''M''~~. ~~It also enjoys an additional symmetry under the vector bundle [[automorphism]]~~

	~~:::~~<~~math~~>~~X+\xi\mapsto X+\xi+i(X)\alpha~~<~~/math~~>

	~~where ''α''~~ is ~~a closed ''p+1''-form. In the ''p=1'' case, which is the relevant case for the geometry of [[Compactification~~ (~~physics)#Flux compactification\|flux compactification]]s in~~ [~~[string theory]], this transformation is known in the physics literature as a shift in the [[Kalb-Ramond field\|B field]].~~

	~~==Dirac and generalized complex structures==~~

	The [[cotangent bundle]], <math>{\mathbf T}^</math> of M is the bundle of differential one-forms. In the case ''p''=1 the Courant bracket maps two sections of <math>{\mathbf T}\oplus{\mathbf{T}}^</math>, the direct sum of the tangent and cotangent bundles, to another section of <math>{\mathbf T}\oplus{\mathbf{T}}^</math>. The fibers of <math>{\mathbf T}\oplus{\mathbf{T}}^</math> admit [[inner product]]s with [[signature of a quadratic form\|signature]] (N,N) given by
	:::~~<math>\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X)).<~~/~~math>~~

	~~A [[linear subspace]] of <math>{\mathbf T}\oplus{\mathbf{T}}^<~~/~~math> in which all pairs of vectors have zero inner product is said to be an [[isotropic subspace]]~~. ~~The fibers of <math>{\mathbf T}\oplus{\mathbf{T}}^</math> are ''2N''-dimensional and the maximal dimension of an isotropic subspace is ''N''. An ''N''~~-~~dimensional isotropic subspace is called a maximal isotropic subspace.~~

	A [[Dirac structure]] is a maximally isotropic subbundle of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math> whose sections are closed under the Courant bracket. Dirac structures include as special cases [[symplectic geometry\|symplectic structures]], [[Poisson manifold\|Poisson structures]] and [[foliation\|foliated geometries]].

	A [[generalized complex structure]] is defined identically, but one [[tensor product\|tensors]] <math>{\mathbf T}\oplus{\mathbf{T}}^*</math> by the complex numbers and uses the [[complex dimension]] in the above definitions and one imposes that the direct sum of the subbundle and its [[complex conjugate]] be the entire original bundle ('''T'''<math>\oplus</math>
	~~'''T'''<sup>*</sup>)<math>\otimes<~~/~~math>'''C'''. Special cases of generalized complex structures include [[complex structure]]{{dn\|date=September 2012}} and a version of [[Kähler manifold\|Kähler structure]] which includes the B~~-~~field.~~

	~~==Dorfman bracket==~~

	~~In 1987 [[Irene Dorfman]] introduced the Dorfman bracket [,]<sub>D</sub>, which like the Courant bracket provides an integrability condition for Dirac structures. It is defined by~~

	~~::<math>[A,B]_D=[A,B]+d\langle A,B\rangle</math>.~~

	~~The Dorfman bracket is not antisymmetric, but it is often easier to calculate with than the Courant bracket because it satisfies a [[Leibniz rule]] which resembles the Jacobi identity~~

	~~::<math>[A,[B,C]_D]_D=[[A,B]_D,C]_D+[B,[A,C]_D]_D.<~~/~~math>~~

	~~==Courant algebroid==~~

	The Courant bracket does not satisfy the [[Jacobi identity]] and so it does not define a [[Lie algebroid]], in addition it fails to satisfy the Lie algebroid condition on the [[anchor map]]. Instead it defines a more general structure introduced by [[Zhang-~~Ju Liu]], [[Alan Weinstein]] and [[Ping Xu]] known as a [[Courant algebroid]].~~

	~~==Twisted Courant bracket==~~
	~~===Definition and properties===~~
	The Courant bracket may be twisted by a ''(p+2)''-form ''H'', by adding the interior product of the vector fields ''X'' and ''Y'' of ''H''. It remains antisymmetric and invariant under the addition of the interior product with a ''(p+1)''-form ''B''. When ''B'' is not closed then this invariance is still preserved if one adds ''dB'' to the final ''H''.

	If ''H'' is closed then the Jacobiator is exact and so the twisted Courant bracket still defines a Courant algebroid. In [[string theory]], ''H'' is interpreted as the [[Kalb-Ramond field\|Neveu-Schwarz 3-form]].

	~~===''p=0'': Circle-invariant vector fields===~~

	When ''p''=0 the Courant bracket reduces to the Lie bracket on a [[principal bundle\|principal]] [[circle bundle]] over ''M'' with [[Riemann curvature tensor\|curvature]] given by the 2-form twist ''H''. The bundle of 0-forms is the trivial bundle, and a section of the direct sum of the tangent bundle and the trivial bundle defines a circle [[invariant vector field]] on this circle bundle.

	~~Concretely, a section of the sum of the tangent and trivial bundles is given by a vector field ''X'' and a function ''f'' and the Courant bracket is~~
	~~::<math>[X+f,Y+g]=[X,Y]+Xg-Yf<~~/~~math>~~
	~~which is just the Lie bracket of the vector fields~~
	~~:::<math>[X+f,Y+g]=[X+f\frac{\partial}{\partial\theta},Y+g\frac{\partial}{\partial\theta}]_{Lie}<~~/~~math>~~
	~~where ''θ'' is a coordinate on the circle fiber. Note in particular that the Courant bracket satisfies the Jacobi identity in the case ''p=0''.~~

	~~===Integral twists and gerbes===~~

	~~The curvature of a circle bundle always represents an integral [[cohomology]] class,~~ the ~~[[Chern class~~]] of the circle bundle. Thus the above geometric interpretation of the twisted ''p=0'' Courant bracket only exists when ''H'' represents an integral class. Similarly at higher values of ''p'' the twisted Courant brackets can be geometrically realized as untwisted Courant brackets twisted by [[gerbe]]s when ''H'' is an integral cohomology class.

	~~==References==~~

	*Courant, Theodore, ''Dirac manifolds'', Trans. Amer. Math. Soc., 319:631-661, (1990).

	*Gualtieri, Marco, [http://xxx.lanl.gov/abs/math.DG/0401221 Generalized complex geometry], PhD Thesis (2004).

	~~[[Category:Differential geometry]]~~
	~~[[Category:Binary operations]]~~

212.105.160.164: Added a notational convention

2012-08-06T01:09:32Z

Added a notational convention

New page

In a field of [[mathematics]] known as [[differential geometry]], the '''Courant bracket''' is a generalization of the [[Lie derivative|Lie bracket]] from an operation on the [[tangent bundle]] to an operation on the [[direct sum of vector bundles|direct sum]] of the tangent bundle and the [[vector bundle]] of [[differential form|''p''-forms]].

The case ''p'' = 1 was introduced by [[Theodore James Courant]] in his 1990 doctoral dissertation as a structure that bridges [[Poisson manifold|Poisson geometry]] and pre[[symplectic geometry]], based on work with his advisor [[Alan Weinstein]]. The twisted version of the Courant bracket was introduced in 2001 by [[Pavol Severa]], and studied in collaboration with Weinstein.

Today a [[complex number|complex]] version of the ''p''=1 Courant bracket plays a central role in the field of [[generalized complex geometry]], introduced by [[Nigel Hitchin]] in 2002. Closure under the Courant bracket is the [[integrability condition]] of a [[Almost_complex_manifold#Generalized_almost_complex_structure|generalized almost complex structure]].

==Definition==
Let ''X'' and ''Y'' be [[vector field]]s on an N-dimensional real [[manifold (mathematics)|manifold]] ''M'' and let ''ξ'' and ''η'' be ''p''-forms. Then ''X+ξ'' and ''Y+η'' are [[fiber_bundle#Sections|sections]] of the direct sum of the tangent bundle and the bundle of ''p''-forms. The Courant bracket of ''X+ξ'' and ''Y+η'' is defined to be

:<math>[X+\xi,Y+\eta]=[X,Y]
+\mathcal{L}_X\eta-\mathcal{L}_Y\xi
-\frac{1}{2}d(i(X)\eta-i(Y)\xi)</math>

where <math>\mathcal{L}_X</math> is the [[Lie derivative]] along the vector field ''X'', ''d'' is the [[exterior derivative]] and ''i'' is the [[Exterior_algebra#The_interior_product_or_insertion_operator|interior product]].

==Properties==

The Courant bracket is [[antisymmetric]] but it does not satisfy the [[Jacobi identity]] for ''p'' greater than zero.

===The Jacobi identity===

However, at least in the case ''p=1'', the [[Jacobiator]], which measures a bracket's failure to satisfy the Jacobi identity, is an [[exact form]]. It is the exterior derivative of a form which plays the role of the [[Nijenhuis tensor]] in generalized complex geometry.

The Courant bracket is the antisymmetrization of the [[Courant_bracket#Dorfman_bracket|Dorfman bracket]], which does satisfy a kind of Jacobi identity.

===Symmetries===

Like the Lie bracket, the Courant bracket is invariant under diffeomorphisms of the manifold ''M''. It also enjoys an additional symmetry under the vector bundle [[automorphism]]

:::<math>X+\xi\mapsto X+\xi+i(X)\alpha</math>

where ''α'' is a closed ''p+1''-form. In the ''p=1'' case, which is the relevant case for the geometry of [[Compactification (physics)#Flux compactification|flux compactification]]s in [[string theory]], this transformation is known in the physics literature as a shift in the [[Kalb-Ramond field|B field]].

==Dirac and generalized complex structures==

The [[cotangent bundle]], <math>{\mathbf T}^*</math> of M is the bundle of differential one-forms. In the case ''p''=1 the Courant bracket maps two sections of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math>, the direct sum of the tangent and cotangent bundles, to another section of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math>. The fibers of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math> admit [[inner product]]s with [[signature of a quadratic form|signature]] (N,N) given by
:::<math>\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X)).</math>

A [[linear subspace]] of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math> in which all pairs of vectors have zero inner product is said to be an [[isotropic subspace]]. The fibers of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math> are ''2N''-dimensional and the maximal dimension of an isotropic subspace is ''N''. An ''N''-dimensional isotropic subspace is called a maximal isotropic subspace.

A [[Dirac structure]] is a maximally isotropic subbundle of <math>{\mathbf T}\oplus{\mathbf{T}}^*</math> whose sections are closed under the Courant bracket. Dirac structures include as special cases [[symplectic geometry|symplectic structures]], [[Poisson manifold|Poisson structures]] and [[foliation|foliated geometries]].

A [[generalized complex structure]] is defined identically, but one [[tensor product|tensors]] <math>{\mathbf T}\oplus{\mathbf{T}}^*</math> by the complex numbers and uses the [[complex dimension]] in the above definitions and one imposes that the direct sum of the subbundle and its [[complex conjugate]] be the entire original bundle ('''T'''<math>\oplus</math>
'''T'''<sup>*</sup>)<math>\otimes</math>'''C'''. Special cases of generalized complex structures include [[complex structure]]{{dn|date=September 2012}} and a version of [[Kähler manifold|Kähler structure]] which includes the B-field.

==Dorfman bracket==

In 1987 [[Irene Dorfman]] introduced the Dorfman bracket [,]<sub>D</sub>, which like the Courant bracket provides an integrability condition for Dirac structures. It is defined by

::<math>[A,B]_D=[A,B]+d\langle A,B\rangle</math>.

The Dorfman bracket is not antisymmetric, but it is often easier to calculate with than the Courant bracket because it satisfies a [[Leibniz rule]] which resembles the Jacobi identity

::<math>[A,[B,C]_D]_D=[[A,B]_D,C]_D+[B,[A,C]_D]_D.</math>

==Courant algebroid==

The Courant bracket does not satisfy the [[Jacobi identity]] and so it does not define a [[Lie algebroid]], in addition it fails to satisfy the Lie algebroid condition on the [[anchor map]]. Instead it defines a more general structure introduced by [[Zhang-Ju Liu]], [[Alan Weinstein]] and [[Ping Xu]] known as a [[Courant algebroid]].

==Twisted Courant bracket==
===Definition and properties===
The Courant bracket may be twisted by a ''(p+2)''-form ''H'', by adding the interior product of the vector fields ''X'' and ''Y'' of ''H''. It remains antisymmetric and invariant under the addition of the interior product with a ''(p+1)''-form ''B''. When ''B'' is not closed then this invariance is still preserved if one adds ''dB'' to the final ''H''.

If ''H'' is closed then the Jacobiator is exact and so the twisted Courant bracket still defines a Courant algebroid. In [[string theory]], ''H'' is interpreted as the [[Kalb-Ramond field|Neveu-Schwarz 3-form]].

===''p=0'': Circle-invariant vector fields===

When ''p''=0 the Courant bracket reduces to the Lie bracket on a [[principal bundle|principal]] [[circle bundle]] over ''M'' with [[Riemann curvature tensor|curvature]] given by the 2-form twist ''H''. The bundle of 0-forms is the trivial bundle, and a section of the direct sum of the tangent bundle and the trivial bundle defines a circle [[invariant vector field]] on this circle bundle.

Concretely, a section of the sum of the tangent and trivial bundles is given by a vector field ''X'' and a function ''f'' and the Courant bracket is
::<math>[X+f,Y+g]=[X,Y]+Xg-Yf</math>
which is just the Lie bracket of the vector fields
:::<math>[X+f,Y+g]=[X+f\frac{\partial}{\partial\theta},Y+g\frac{\partial}{\partial\theta}]_{Lie}</math>
where ''θ'' is a coordinate on the circle fiber. Note in particular that the Courant bracket satisfies the Jacobi identity in the case ''p=0''.

===Integral twists and gerbes===

The curvature of a circle bundle always represents an integral [[cohomology]] class, the [[Chern class]] of the circle bundle. Thus the above geometric interpretation of the twisted ''p=0'' Courant bracket only exists when ''H'' represents an integral class. Similarly at higher values of ''p'' the twisted Courant brackets can be geometrically realized as untwisted Courant brackets twisted by [[gerbe]]s when ''H'' is an integral cohomology class.

==References==

*Courant, Theodore, ''Dirac manifolds'', Trans. Amer. Math. Soc., 319:631-661, (1990).

*Gualtieri, Marco, [http://xxx.lanl.gov/abs/math.DG/0401221 Generalized complex geometry], PhD Thesis (2004).

[[Category:Differential geometry]]
[[Category:Binary operations]]