Courant bracket explained
In differential geometry, a field of mathematics, 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
was introduced by
Theodore James Courant in his 1990 doctoral dissertation as a structure that bridges
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 version of the
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.
Definition
Let
and
be vector fields on an
-dimensional real manifold
and let
and
be
-forms. Then
and
are sections of the direct sum
of the tangent bundle and the bundle of
-forms. The Courant bracket of
and
is defined to be[X+\xi,Y+η]=[X,Y]
+l{L}Xη-l{L}
d(i(X)η-i(Y)\xi)
where
is the
Lie derivative along the vector field
,
is the
exterior derivative and
is the
interior product.
Properties
The Courant bracket is antisymmetric but it does not satisfy the Jacobi identity for
greater than zero.The Jacobi identity
However, at least in the case
, 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 automorphism
X+\xi\mapstoX+\xi+i(X)\alpha
where
is a closed
-form. In the
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.Dirac and generalized complex structures
The cotangent bundle,
of
is the bundle of differential one-forms. In the case
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
given by
\langleX+\xi,Y+η\rangle=
(\xi(Y)+η(X)).
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
-dimensional and the maximal dimension of an isotropic subspace is
. An
-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.
}^* 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
}^*)\otimes \mathbf C. Special cases of generalized complex structures include
complex structure and a version of
Kähler structure which includes the B-field.
Dorfman bracket
In 1987 Irene Dorfman introduced the Dorfman bracket
, which like the Courant bracket provides an integrability condition for Dirac structures. It is defined by
[A,B]D=[A,B]+d\langleA,B\rangle
.
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
[A,[B,C]D]D=[[A,B]D,C]D+[B,[A,C]D]D.
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
-form
, by adding the interior product of the vector fields
and
of
. It remains antisymmetric and invariant under the addition of the interior product with a
-form
. When
is not closed then this invariance is still preserved if one adds
to the final
.If
is closed then the Jacobiator is exact and so the twisted Courant bracket still defines a Courant algebroid. In string theory,
is interpreted as the Neveu–Schwarz 3-form.p=0: Circle-invariant vector fields
When
the Courant bracket reduces to the Lie bracket on a
principal circle bundle over
with
curvature given by the 2-form twist
. 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
and a function
and the Courant bracket is
which is just the Lie bracket of the vector fields
| [X+f,Y+g]=[X+f | \partial | ,Y+g |
| \partial\theta |
]Lie
where
is a coordinate on the circle fiber. Note in particular that the Courant bracket satisfies the Jacobi identity in the case
.
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
Courant bracket only exists when
H represents an integral class. Similarly at higher values of
the twisted Courant brackets can be geometrically realized as untwisted Courant brackets twisted by
gerbes when
H is an integral cohomology class.
References
- Courant . Theodore . Dirac manifolds . Trans. Amer. Math. Soc. . 319 . 631–661 . 1990 .
- Gualtieri . Marco . math.DG/0401221 . Generalized complex geometry . PhD Thesis . 2004 .
