Courant bracket
Encyclopedia
In a field of mathematics
known as differential geometry, the Courant bracket is a generalization of the Lie bracket
from an operation on the tangent bundle
to an operation on the direct sum of the tangent bundle and the vector bundle
of p-forms
.
The case p = 1 was introduced by Theodore James Courant
in his 1990 doctoral dissertation as a structure that bridges Poisson geometry
and presymplectic 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
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 generalized almost complex structure.
s on an N-dimensional real manifold M and let ξ and η be p-forms. Then X+ξ and Y+η are 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
where
is the Lie derivative
along the vector field X, d is the exterior derivative
and i is the interior product.
but it does not satisfy the Jacobi identity
for p greater than zero.
The Courant bracket is the antisymmetrization of the Dorfman bracket, which does satisfy a kind of Jacobi identity.
where α is a closed p+1-form. In the p=1 case, which is the relevant case for the geometry of flux compactifications in string theory
, this transformation is known in the physics literature as a shift in the B field
.
,
of M is the bundle of differential one-forms. In the case p=1 the Courant bracket maps two sections of 
, the direct sum of the tangent and cotangent bundles, to another section of 
. The fibers of 
admit inner products with signature (N,N) given by
A linear subspace
of
in which all pairs of vectors have zero inner product is said to be an isotropic subspace. The fibers of 
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
whose sections are closed under the Courant bracket. Dirac structures include as special cases symplectic structures, Poisson structures
and foliated geometries
.
A generalized complex structure
is defined identically, but one tensors
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
T*)
C. Special cases of generalized complex structures include complex structure and a version of Kähler structure
which includes the B-field.
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
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
.
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 Neveu-Schwarz 3-form
.
circle bundle over M with 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
which is just the Lie bracket of the vector fields
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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
known as differential geometry, the Courant bracket is a generalization of the Lie bracket
Lie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
from an operation on the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
to an operation on the direct sum of the tangent bundle and the vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
of p-forms
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
.
The case p = 1 was introduced by Theodore James Courant
Theodore James Courant
Theodore James Courant is a mathematician who has conducted research in the fields of differential geometry and classical mechanics. In particular, he made seminal contributions to the study of Dirac manifolds, which generalize both symplectic manifolds and Poisson manifolds, and are related to...
in his 1990 doctoral dissertation as a structure that bridges Poisson geometry
Poisson manifold
In mathematics, a Poisson manifold is a differentiable manifold M such that the algebra C^\infty\, of smooth functions over M is equipped with a bilinear map called the Poisson bracket, turning it into a Poisson algebra...
and presymplectic geometry, based on work with his advisor Alan Weinstein
Alan Weinstein
Alan David Weinstein is a professor of mathematics at the University of California, Berkeley who works in symplectic geometry, Poisson Geometry and Mathematical Physics....
. The twisted version of the Courant bracket was introduced in 2001 by Pavol Severa, and studied in collaboration with Weinstein.
Today a complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
version of the p=1 Courant bracket plays a central role in the field of generalized complex geometry, introduced by Nigel Hitchin
Nigel Hitchin
Nigel Hitchin is a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.-Academic career:...
in 2002. Closure under the Courant bracket is the integrability condition of a generalized almost complex structure.
Definition
Let X and Y be vector fieldVector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s on an N-dimensional real manifold M and let ξ and η be p-forms. Then X+ξ and Y+η are 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
where
is the Lie derivativeLie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
along the vector field X, d is the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
and i is the interior product.
Properties
The Courant bracket is antisymmetricAntisymmetric
The word antisymmetric refers to a change to an opposite quantity when another quantity is symmetrically changed. This concept is related to that of Symmetry and Asymmetry. The difference between these three concepts can be simply illustrated with Latin letters. The character "A" is symmetric about...
but it does not satisfy the Jacobi identity
Jacobi identity
In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying 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 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 automorphismAutomorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
where α is a closed p+1-form. In the p=1 case, which is the relevant case for the geometry of flux compactifications in string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, this transformation is known in the physics literature as a shift in the B field
Kalb-Ramond field
In theoretical physics in general and string theory in particular, the Kalb–Ramond field, also known as the NS-NS B-field, is a quantum field that transforms as a two-form i.e. an antisymmetric tensor field with two indices....
.
Dirac and generalized complex structures
The cotangent bundleCotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
,
of M is the bundle of differential one-forms. In the case p=1 the Courant bracket maps two sections of , the direct sum of the tangent and cotangent bundles, to another section of . The fibers of admit inner products with signature (N,N) given by
A linear subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
of
in which all pairs of vectors have zero inner product is said to be an isotropic subspace. The fibers of 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
whose sections are closed under the Courant bracket. Dirac structures include as special cases symplectic structures, Poisson structuresPoisson manifold
In mathematics, a Poisson manifold is a differentiable manifold M such that the algebra C^\infty\, of smooth functions over M is equipped with a bilinear map called the Poisson bracket, turning it into a Poisson algebra...
and foliated geometries
Foliation
In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
.
A generalized complex structure
Generalized complex structure
In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure...
is defined identically, but one tensors
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
by the complex numbers and uses the complex dimensionComplex dimension
In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d...
in the above definitions and one imposes that the direct sum of the subbundle and its complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
be the entire original bundle (T
T*)
C. Special cases of generalized complex structures include complex structure and a version of Kähler structureKähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
which includes the B-field.
Dorfman bracket
In 1987 Irene Dorfman introduced the Dorfman bracket [,]D, which like the Courant bracket provides an integrability condition for Dirac structures. It is defined by-
.
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
Courant algebroid
The Courant bracket does not satisfy the Jacobi identityJacobi identity
In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...
and so it does not define a Lie algebroid
Lie algebroid
In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones...
, 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
Alan Weinstein
Alan David Weinstein is a professor of mathematics at the University of California, Berkeley who works in symplectic geometry, Poisson Geometry and Mathematical Physics....
and Ping Xu known as a Courant algebroid
Courant algebroid
In a field of mathematics known as differential geometry, a Courant algebroid is a combination of a Lie algebroid and a quadratic Lie algebra. It was originally introduced in 1990 by Theodore James Courant in his dissertation at UC Berkeley where he first called them Dirac Manifolds, and then were...
.
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
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, H is interpreted as the Neveu-Schwarz 3-form
Kalb-Ramond field
In theoretical physics in general and string theory in particular, the Kalb–Ramond field, also known as the NS-NS B-field, is a quantum field that transforms as a two-form i.e. an antisymmetric tensor field with two indices....
.
p=0: Circle-invariant vector fields
When p=0 the Courant bracket reduces to the Lie bracket on a principalPrincipal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
circle bundle over M with curvature
Curvature tensor
In differential geometry, the term curvature tensor may refer to:* the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds;* the curvature of an affine connection or covariant derivative ;...
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
which is just the Lie bracket of the vector fields
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 cohomologyCohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
class, the Chern class
Chern class
In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...
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
Gerbe
In mathematics, a gerbe is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as a generalization of principal bundles to the setting of 2-categories...
s when H is an integral cohomology class.