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. Generalized complex structures were introduced by Nigel Hitchin in 2002 and further developed by his students Marco Gualtieri and Gil Cavalcanti.
Contents
- The generalized tangent bundle
- Courant bracket
- The definition
- Classification
- Type
- Real index
- Canonical bundle
- Generalized almost complex structures
- Integrability and other structures
- Regular point
- Darbouxs theorem
- Local holomorphicity
- Complex manifolds
- Symplectic manifolds
- Relation to G structures
- Calabi versus Calabi Yau metric
- References
These structures first arose in Hitchin's program of characterizing geometrical structures via functionals of differential forms, a connection which formed the basis of Robbert Dijkgraaf, Sergei Gukov, Andrew Nietzke and Cumrun Vafa's 2004 proposal that topological string theories are special cases of a topological M-theory. Today generalized complex structures also play a leading role in physical string theory, as supersymmetric flux compactifications, which relate 10-dimensional physics to 4-dimensional worlds like ours, require (possibly twisted) generalized complex structures.
The generalized tangent bundle
Consider an N-manifold M. The tangent bundle of M, which will be denoted T, is the vector bundle over M whose fibers consist of all tangent vectors to M. A section of T is a vector field on M. The cotangent bundle of M, denoted T*, is the vector bundle over M whose sections are one-forms on M.
In complex geometry one considers structures on the tangent bundles of manifolds. In symplectic geometry one is instead interested in exterior powers of the cotangent bundle. Generalized geometry unites these two fields by treating sections of the generalized tangent bundle, which is the direct sum T
The fibers are endowed with a natural inner product with signature (N, N). If X and Y are vector fields and ξ and η are one-forms then the inner product of X+ξ and Y+η is defined as
A generalized almost complex structure is just an almost complex structure of the generalized tangent bundle which preserves the natural inner product:
such that
Like in the case of an ordinary almost complex structure, a generalized almost complex structure is uniquely determined by its
Such subbundle L satisfies the following properties:
(i) the intersection with its complex conjugate is the zero section:
(ii) L is maximal isotropic, i.e. its complex rank equals N and
Vice versa, any subbundle L satisfying (i), (ii) is the
Courant bracket
In ordinary complex geometry, an almost complex structure is integrable to a complex structure if and only if the Lie bracket of two sections of the holomorphic subbundle is another section of the holomorphic subbundle.
In generalized complex geometry one is not interested in vector fields, but rather in the formal sums of vector fields and one-forms. A kind of Lie bracket for such formal sums was introduced in 1990 and is called the Courant bracket which is defined by
where
The definition
A generalized complex structure is a generalized almost complex structure such that the space of smooth sections of L is closed under the Courant bracket.
Classification
There is a one-to-one correspondence between maximal isotropic subbundle of T
Given a pair (E,ε) one can construct a maximally isotropic subbundle L(E,ε) of T
To see that L(E, ε) is isotropic, notice that if Y is a section of E and ξ restricted to E* is ε(X) then ξ(Y) = ε(X, Y), as the part of ξ orthogonal to E* annihilates Y. Thesefore if X + ξ and Y + η are sections of T
and so L(E, ε) is isotropic. Furthermore, L(E, ε) is maximal because there are dim(E) (complex) dimensions of choices for E, and ε is unrestricted on the complement of E*, which is of (complex) dimension n − dim(E). Thus the total (complex) dimension in n. Gualtieri has proven that all maximal isotropic subbundles are of the form L(E,ε) for some E and ε.
Type
The type of a maximal isotropic subbundle L(E,ε) is the real dimension of the subbundle that annihilates E. Equivalently it is 2N minus the real dimension of the projection of L(E,ε) onto the tangent bundle T. In other words, the type of a maximal isotropic subbundle is the codimension of its projection onto the tangent bundle. In the complex case one uses the complex dimension and the type is sometimes referred to as the complex type. While the type of a subbundle can in principle be any integer between 0 and 2N, generalized almost complex structures cannot have a type greater than N because the sum of the subbundle and its complex conjugate must be all of (T
The type of a maximal isotropic subbundle is invariant under diffeomorphisms and also under shifts of the B-field, which are isometries of T
where B is an arbitrary closed 2-form called the B-field in the string theory literature.
The type of a generalized almost complex structure is in general not constant, it can jump by any even integer. However it is upper semi-continuous, which means that each point has an open neighborhood in which the type does not increase. In practice this means that subsets of greater type than the ambient type occur on submanifolds with positive codimension.
Real index
The real index r of a maximal isotropic subspace L is the complex dimension of the intersection of L with its complex conjugate. A maximal isotropic subspace of (T
Canonical bundle
As in the case of ordinary complex geometry, there is a correspondence between generalized almost complex structures and complex line bundles. The complex line bundle corresponding to a particular generalized almost complex structure is often referred to as the canonical bundle, as it generalizes the canonical bundle in the ordinary case. It is sometimes also called the pure spinor bundle, as its sections are pure spinors.
Generalized almost complex structures
The canonical bundle is a one complex dimensional subbundle of the bundle Λ*T
A spinor is said to be a pure spinor if it is annihilated by half of a set of a set of generators of the Clifford algebra. Spinors are sections of our bundle Λ*T, and generators of the Clifford algebra are the fibers of our other bundle (T
Given a generalized almost complex structure, one can also determine a pure spinor up to multiplication by an arbitrary complex function. These choices of pure spinors are defined to be the sections of the canonical bundle.
Integrability and other structures
If a pure spinor that determines a particular complex structure is closed, or more generally if its exterior derivative is equal to the action of a gamma matrix on itself, then the almost complex structure is integrable and so such pure spinors correspond to generalized complex structures.
If one further imposes that the canonical bundle is holomorphically trivial, meaning that it is global sections which are closed forms, then it defines a generalized Calabi-Yau structure and M is said to be a generalized Calabi-Yau manifold.
Canonical bundle
Locally all pure spinors can be written in the same form, depending on an integer k, the B-field 2-form B, a nondegenerate symplectic form ω and a k-form Ω. In a local neighborhood of any point a pure spinor Φ which generates the canonical bundle may always be put in the form
where Ω is decomposable as the wedge product of one-forms.
Regular point
Define the subbundle E of the complexified tangent bundle T
for some subbundle Δ. A point which has an open neighborhood in which the dimension of the fibers of Δ is constant is said to be a regular point.
Darboux's theorem
Every regular point in a generalized complex manifold has an open neighborhood which, after a diffeomorphism and shift of the B-field, has the same generalized complex structure as the Cartesian product of the complex vector space Ck and the standard symplectic space R2n-2k with the standard symplectic form, which is the direct sum of the two by two off-diagonal matrices with entries 1 and -1.
Local holomorphicity
Near non-regular points, the above classification theorem does not apply. However, about any point, a generalized complex manifoldis is, up to diffeomorphism and B-field, a product of a symplectic manifold with a generalized complex manifold which is of complex type at the point, much like Weinstein's theorem for the local structure of Poisson manifolds. The remaining question of the local structure is: what does a generalized complex structure look like near a point of complex type? In fact, it will be induced by a holomorphic Poisson structure.
Complex manifolds
The space of complex differential forms Λ*T
(n,0)-forms are pure spinors, as they are annihilated by antiholomorphic tangent vectors and by holomorphic one-forms. Thus this line bundle can be used as a canonical bundle to define a generalized complex structure. Restricting the annihilator from (T
As only half of a basis of vector fields are holomorphic, these complex structures are of type N. In fact complex manifolds, and the manifolds obtained by multiplying the pure spinor bundle defining a complex manifold by a complex,
Symplectic manifolds
The pure spinor bundle generated by
for a nondegenerate two-form ω defines a symplectic structure on the tangent space. Thus symplectic manifolds are also generalized complex manifolds.
The above pure spinor is globally defined, and so the canonical bundle is trivial. This means that symplectic manifolds are not only generalized complex manifolds but in fact are generalized Calabi-Yau manifolds.
The pure spinor
Up to a shift of the B-field, which corresponds to multiplying the pure spinor by the exponential of a closed, real 2-form, symplectic manifolds are the only type 0 generalized complex manifolds. Manifolds which are symplectic up to a shift of the B-field are sometimes called B-symplectic.
Relation to G-structures
Some of the almost structures in generalized complex geometry may be rephrased in the language of G-structures. The word "almost" is removed if the structure is integrable.
The bundle (T
A generalized almost Kähler structure is a pair of commuting generalized complex structures such that minus the product of the corresponding tensors is a positive definite metric on (T
Finally, a generalized almost Calabi-Yau metric structure is a further reduction of the structure group to SU(n)
Calabi versus Calabi-Yau metric
Notice that a generalized Calabi metric structure, which was introduced by Gualtieri, is a stronger condition than a generalized Calabi-Yau structure, which was introduced by Hitchin. In particular a generalized Calabi-Yau metric structure implies the existence of two commuting generalized almost complex structures.