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 (possibly twisted) generalized complex structures.
In symplectic geometry one is instead interested in exterior powers of the cotangent bundle.
of the tangent and cotangent bundles, which are formal sums of a vector field and a one-form.
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
where the vector field X is a section of E and the one-form ξ restricted to the dual space
Thus the total (complex) dimension in n. Gualtieri has proven that all maximal isotropic subbundles are of the form
The 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.
However it is upper semi-continuous, which means that each point has an open neighborhood in which the type does not increase.
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.
This is true whenever the wedge product of the pure spinor and its complex conjugate contains a top-dimensional component.
These choices of pure spinors are defined to be the sections of the canonical bundle.
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.
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.
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
with the standard symplectic form, which is the direct sum of the two by two off-diagonal matrices with entries 1 and −1.
(n, 0)-forms are pure spinors, as they are annihilated by antiholomorphic tangent vectors and by holomorphic one-forms.
to the complexified tangent bundle one gets the subspace of antiholomorphic vector fields.
The pure spinor bundle generated by for a nondegenerate two-form ω defines a symplectic structure on the tangent space.
The above pure spinor is globally defined, and so the canonical bundle is trivial.
Therefore, these generalized complex structures are of the same type as those corresponding to a scalar pure spinor.
A scalar is annihilated by the entire tangent space, and so these structures are of type 0.
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.
Some of the almost structures in generalized complex geometry may be rephrased in the language of G-structures.
Generalized Kähler manifolds, and their twisted counterparts, are equivalent to the bihermitian manifolds discovered by Sylvester James Gates, Chris Hull and Martin Roček in the context of 2-dimensional supersymmetric quantum field theories in 1984.
Notice that a generalized Calabi metric structure, which was introduced by Marco Gualtieri, is a stronger condition than a generalized Calabi–Yau structure, which was introduced by Nigel Hitchin.