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. Generalized complex structures were introduced by Nigel Hitchin in 2002 and further developed by his students Marco Gualtieri and Gil Cavalcanti.
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 Neitzke 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 generalized complex structures.
Definition
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 of the tangent and cotangent bundles, which are formal sums of a vector field and a one-form.
The fibers are endowed with a natural inner product with signature. 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 and
Like in the case of an ordinary almost complex structure, a generalized almost complex structure is uniquely determined by its -eigenbundle, i.e. a subbundle of the complexified generalized tangent bundle
given by
Such subbundle L satisfies the following properties:
Vice versa, any subbundle L satisfying, is the -eigenbundle of a unique generalized almost complex structure, so that the properties, can be considered as an alternative definition of generalized almost complex structure.
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 is the Lie derivative along the vector field X, d is the exterior derivative and i is the interior product.
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.Maximal isotropic subbundles
Classification
There is a one-to-one correspondence between maximal isotropic subbundle of and pairs where E is a subbundle of T and is a 2-form. This correspondence extends straightforwardly to the complex case.Given a pair one can construct a maximally isotropic subbundle of as follows. The elements of the subbundle are the formal sums where the vector field X is a section of E and the one-form ξ restricted to the dual space is equal to the one-form
To see that is isotropic, notice that if Y is a section of E and restricted to is then as the part of orthogonal to annihilates Y. Therefore if and are sections of then
and so is isotropic. Furthermore, is maximal because there are dimensions of choices for and is unrestricted on the complement of which is of dimension Thus the total dimension is n. Gualtieri has proven that all maximal isotropic subbundles are of the form for some and
Type
The type of a maximal isotropic subbundle is the real dimension of the subbundle that annihilates E. Equivalently it is 2N minus the real dimension of the projection of 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 ofThe type of a maximal isotropic subbundle is invariant under diffeomorphisms and also under shifts of the B-field, which are isometries of of the form
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 is a generalized almost complex structure if and only if r = 0.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 of complex differential forms on M. Recall that the gamma matrices define an isomorphism between differential forms and spinors. In particular even and odd forms map to the two chiralities of Weyl spinors. Vectors have an action on differential forms given by the interior product. One-forms have an action on forms given by the wedge product. Thus sections of the bundle act on differential forms. This action is a representation of the action of the Clifford algebra on spinors.A spinor is said to be a pure spinor if it is annihilated by half of a set of generators of the Clifford algebra. Spinors are sections of our bundle and generators of the Clifford algebra are the fibers of our other bundle Therefore, a given pure spinor is annihilated by a half-dimensional subbundle E of Such subbundles are always isotropic, so to define an almost complex structure one must only impose that the sum of E and its complex conjugate is all of This is true whenever the wedge product of the pure spinor and its complex conjugate contains a top-dimensional component. Such pure spinors determine generalized almost complex structures.
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.
Local classification
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 formwhere Ω is decomposable as the wedge product of one-forms.
Regular point
Define the subbundle E of the complexified tangent bundle to be the projection of the holomorphic subbundle L of to In the definition of a generalized almost complex structure we have imposed that the intersection of L and its conjugate contains only the origin, otherwise they would be unable to span the entirety of However the intersection of their projections need not be trivial. In general this intersection is of the formfor 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.