Generalized complex structure
Encyclopedia
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 functional
s 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.
of M, which will be denoted T, is the vector bundle
over M whose fibers consist of all tangent vector
s 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 complex geometry unites these two fields by treating sections of the direct sum (T
T*)
C of the tangent and cotangent bundles tensored with the complex number
s, which are formal sums of a complex vector field and a complex one-form.
The fibers are endowed with an inner product with complex 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 linear subspace
of (T
T*) 
C in which all pairs of vectors have zero inner product is said to be an isotropic subspace. A generalized almost complex structure is an isotropic subbundle
E of (T
T*) 
C
whose fibers are the maximal complex dimension
, N, and such that the direct sum of E and its complex conjugate
is all of (T
T*) 
C.
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.
of T
T* and pairs (E,ε) where E is a subbundle of T and ε is a 2-form. This correspondence extends straightforwardly to the complex case.
Given a pair (E,ε) one can construct a maximally isotropic subbundle L(E,ε) of T
T* as follows. The elements of the subbundle are the formal sums X + ξ where the vector field
X is a section of E and the one-form ξ restricted to the dual space
E* is equal to the one-form ε(X).
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
T* then
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.
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
T*)
C.
The type of a maximal isotropic subbundle is invariant
under diffeomorphisms and also under shifts of the B-field
, which are isometries
of T
T* 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
.
of L with its complex conjugate. A maximal isotropic subspace of (T
T*) 
C is a generalized almost complex structure if and only if r = 0.
in the ordinary case. It is sometimes also called the pure spinor bundle, as its sections are pure spinor
s.
C 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 (T
T*) 
C 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 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
T*) 
C.
Therefore a given pure spinor is annihilated by a half-dimensional subbundle E of (T
T*) 
C.
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 (T
T*) 
C. 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.
, 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.
Φ which generates the canonical bundle may always be put in the form
where Ω is decomposable as the wedge product of one-forms.
C to be the projection of the holomorphic subbundle L of (T
T*)
C to T
C. 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 (T
T*)
C. However the intersection of their projections need not be trivial. In general this intersection is of the form
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.
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.
Near non-regular points, however, the above classification theorem does not apply, and little is known for certain.
C has a complex conjugation operation given by complex conjugation in C. This allows one to define holomorphic and antiholomorphic one-forms and (m, n)-forms, which are homogeneous polynomials in these one-forms with m holomorphic factors and n antiholomorphic factors. In particular, all (n,0)-forms are related locally by multiplication by a complex function and so they form a complex line bundle.
(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
T*)
C to the complexified tangent bundle one gets the subspace of antiholomorphic vector fields. Therefore this generalized complex structure on (T
T*)
C defines an ordinary complex structure on the tangent bundle.
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,
-closed (2,0)-form, are the only type N generalized complex manifolds.
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
is related to a pure spinor which is just a number by an imaginary shift of the B-field, which is a shift of the Kahler form. 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. Manifolds which are symplectic up to a shift of the B-field are sometimes called B-symplectic.
s. The word "almost" is removed if the structure is integrable.
The bundle (T
T*) 
C with the above inner product is a O(2n, 2n) structure. A generalized almost complex structure is a reduction of this structure to a U(n, n) structure. Therefore the space of generalized complex structures is the coset
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
T*)
C.
Generalized Kahler structures are reductions of the structure group to U(n)
U(n). Generalized Kahler 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.
Finally, a generalized almost Calabi-Yau metric structure is a further reduction of the structure group to SU(n)
SU(n).
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, 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
Nigel Hitchin
Nigel Hitchin is a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.-Academic career:...
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 functional
Functional
Generally, functional refers to something able to fulfill its purpose or function.*Functionalism and Functional form, movements in architectural design*Functional group, certain atomic combinations that occur in various molecules, e.g...
s of differential forms, a connection which formed the basis of Robbert Dijkgraaf
Robbert Dijkgraaf
Robertus Henricus "Robbert" Dijkgraaf is a Dutch mathematical physicist and string theorist.Robertus Henricus Dijkgraaf was born on 24 January 1960 in Ridderkerk, Netherlands. He currently lives in Amsterdam, Netherlands...
, Sergei Gukov
Sergei Gukov
Sergei Gukov is a string theorist who is a professor of physics at Caltech and at UCSB .Gukov graduated from Moscow Institute of Physics and Technology in Moscow, Russia before obtaining a doctorate in physics from Princeton University under the supervision of Edward Witten.-External links:* * *...
, Andrew Nietzke and Cumrun Vafa
Cumrun Vafa
Cumrun Vafa is an Iranian-American leading string theorist from Harvard University where he started as a Harvard Junior Fellow. He is a recipient of the 2008 Dirac Medal.-Birth and education:...
's 2004 proposal that topological string theories
Topological string theory
In theoretical physics, topological string theory is a simplified version of string theory. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount of supersymmetry...
are special cases of a topological M-theory. Today generalized complex structures also play a leading role in physical 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...
, as supersymmetric
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
flux compactifications, which relate 10 dimensional physics to 4-dimensional worlds like ours, require (possibly twisted) generalized complex structures.
The tangent plus cotangent bundle
Consider an N-manifold M. The tangent bundleTangent 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...
of M, which will be denoted T, is 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...
over M whose fibers consist of all tangent vector
Tangent vector
A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....
s to M. A section of T is a vector field
Vector 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...
on M. The cotangent bundle
Cotangent 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, denoted T*, is the vector bundle over M whose sections are one-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...
on M.
In complex geometry
Complex geometry
In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....
one considers structures on the tangent bundles of manifolds. In symplectic geometry one is instead interested in exterior powers of the cotangent bundle. Generalized complex geometry unites these two fields by treating sections of the direct sum (T
T*)
C of the tangent and cotangent bundles tensored with the complex numberComplex 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...
s, which are formal sums of a complex vector field and a complex one-form.
The fibers are endowed with an inner product with complex 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 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 (T
T*) C in which all pairs of vectors have zero inner product is said to be an isotropic subspace. A generalized almost complex structure is an isotropic subbundleSubbundle
In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right....
E of (T
T*) Cwhose fibers are the maximal complex dimension
Complex 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...
, N, and such that the direct sum of E 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...
is all of (T
T*) C.Courant bracket
In ordinary complex geometry, an almost complex structure is integrable to a complex structure if and only if the Lie bracketLie 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...
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
Courant bracket
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....
which is defined by
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.
The definition
A generalized complex structure is a generalized almost complex structure whose sections are closed under the Courant bracket.Classification
There is a one-to-one correspondence between maximal isotropic subbundleSubbundle
In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right....
of T
T* and pairs (E,ε) where E is a subbundle of T and ε is a 2-form. This correspondence extends straightforwardly to the complex case.Given a pair (E,ε) one can construct a maximally isotropic subbundle L(E,ε) of T
T* as follows. The elements of the subbundle are the formal sums X + ξ where the 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...
X is a section of E and the one-form ξ restricted to the dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
E* is equal to the one-form ε(X).
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
T* thenand 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 projectionProjection (mathematics)
Generally speaking, in mathematics, a projection is a mapping of a set which is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a left inverse. Bot notions are strongly related, as follows...
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
T*)
C.The type of a maximal isotropic subbundle is invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
under diffeomorphisms and also under shifts of 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....
, which are isometries
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
of T
T* of the form
where B is an arbitrary closed 2-form called the B-field in the 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...
literature.
The type of a generalized almost complex structure is in general not constant, it can jump by any even integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
. 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
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
.
Real index
The real index r of a maximal isotropic subspace L is the complex dimension of the intersectionIntersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of L with its complex conjugate. A maximal isotropic subspace of (T
T*) C 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 bundleCanonical bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle\,\!\Omega^n = \omegawhich is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising...
in the ordinary case. It is sometimes also called the pure spinor bundle, as its sections are pure spinor
Pure spinor
In a field of mathematics known as representation theory pure spinors are spinor representations of the special orthogonal group that are annihilated by the largest possible subspace of the Clifford algebra. They were introduced by Élie Cartan in the 1930s to classify complex structures...
s.
Generalized almost complex structures
The canonical bundle is a one complex dimensional subbundle of the bundle Λ*TC of complex differential forms on M. Recall that the gamma matrices define an isomorphismIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
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 (T
T*) C act on differential forms. This action is a representationGroup representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of the action of the Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
on spinors.
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
T*) C.Therefore a given pure spinor is annihilated by a half-dimensional subbundle E of (T
T*) C.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 (T
T*) C. 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 closedClosed and exact differential forms
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero , and an exact form is a differential form that is the exterior derivative of another differential form β...
, 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 spinorPure spinor
In a field of mathematics known as representation theory pure spinors are spinor representations of the special orthogonal group that are annihilated by the largest possible subspace of the Clifford algebra. They were introduced by Élie Cartan in the 1930s to classify complex structures...
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 TC to be the projection of the holomorphic subbundle L of (TT*)C to TC. 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 (TT*)C. However the intersection of their projections need not be trivial. In general this intersection is of the form
for some subbundle Δ. A point which has an open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
neighborhood in which the dimension of the fibers of Δ is constant is said to be a regular point.
Darboux 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 productCartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of the complex vector space
Complex vector space
A complex vector space is a vector space over the complex numbers. It can also refer to:* a vector space over the real numbers with a linear complex structure...
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.
Near non-regular points, however, the above classification theorem does not apply, and little is known for certain.
Complex manifolds
The space of complex differential forms Λ*TC has a complex conjugation operation given by complex conjugation in C. This allows one to define holomorphic and antiholomorphic one-forms and (m, n)-forms, which are homogeneous polynomials in these one-forms with m holomorphic factors and n antiholomorphic factors. In particular, all (n,0)-forms are related locally by multiplication by a complex function and so they form a complex line bundle.(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
T*)
C to the complexified tangent bundle one gets the subspace of antiholomorphic vector fields. Therefore this generalized complex structure on (TT*)
C defines an ordinary complex structure on the tangent bundle.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,
-closed (2,0)-form, are the only type N generalized complex manifolds.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
is related to a pure spinor which is just a number by an imaginary shift of the B-field, which is a shift of the Kahler form. Therefore these generalized complex structures are of the same type as those corresponding to a scalarScalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
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. 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-structureG-structure
In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM of M....
s. The word "almost" is removed if the structure is integrable.
The bundle (T
T*) C with the above inner product is a O(2n, 2n) structure. A generalized almost complex structure is a reduction of this structure to a U(n, n) structure. Therefore the space of generalized complex structures is the coset
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
T*)
C.Generalized Kahler structures are reductions of the structure group to U(n)
U(n). Generalized Kahler manifolds, and their twisted counterparts, are equivalent to the bihermitian manifolds discovered by Sylvester James GatesSylvester James Gates
Sylvester James Gates, Jr. is a noted American theoretical physicist. He received BS and PhD degrees from Massachusetts Institute of Technology, the latter in 1977. His doctoral thesis was the first thesis at MIT to deal with supersymmetry. Gates is currently the John S...
, Chris Hull
Chris Hull (physicist)
Chris Hull is a professor of Theoretical Physics at Imperial College London. Hull is known for his work on string theory, M-theory, and generalized complex structures...
and Martin Roček
Martin Rocek
Martin Rocek is a professor of theoretical physics at the State University of New York at Stony Brook and a member of the C. N. Yang Institute for Theoretical Physics. He received A.B. and Ph.D. degrees from Harvard University in 1975 and 1979...
in the context of 2-dimensional supersymmetric
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
quantum field theories
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
in 1984.
Finally, a generalized almost Calabi-Yau metric structure is a further reduction of the structure group to SU(n)
SU(n).