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 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 physical string theory, as supersymmetric flux compactifications, which relate 10 dimensional physics to 4-dimensional worlds like ours, require (possibly twisted) generalized complex structures.

Definition

The tangent plus cotangent 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 complex geometry unites these two fields by treating sections of the direct sum (T${\displaystyle \oplus }$  T^*)${\displaystyle \otimes }$ C of the tangent and cotangent bundles tensored with the complex numbers, which are formal sums of a complex vector field and a complex one-form.

The fibers are endowed with the 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

${\displaystyle \langle X+\xi ,Y+\eta \rangle ={\frac {1}{2}}(\xi (Y)+\eta (X)).}$ 

A linear subspace of (T${\displaystyle \oplus }$  T^*)${\displaystyle \otimes }$ C in which all pairs have vectors have zero inner product is said to be an isotropic subspace. A generalized almost complex structure is an isotropic subbundle E of (T${\displaystyle \oplus }$  T^*)${\displaystyle \otimes }$ 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${\displaystyle \oplus }$  T^*)${\displaystyle \otimes }$ C.

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

${\displaystyle [X+\xi ,Y+\eta ]=[X,Y]+{\mathcal {L}}_{X}\eta -{\mathcal {L}}_{Y}\xi -{\frac {1}{2}}d(i(X)\eta -i(Y)\xi )}$ 

where ${\displaystyle {\mathcal {L}}_{X}}$  is the Lie derivative along the vector field X, d is the exterior derivative 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.

Maximal isotropic subbundles

Classification

There is a one-to-one correspondence between maximal isotropic subbundle of T${\displaystyle \oplus }$ T^* and pairs (E,ε) where E is a subbundle of 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${\displaystyle \oplus }$ 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${\displaystyle \oplus }$ T^* then

${\displaystyle \langle X+\xi ,Y+\eta \rangle ={\frac {1}{2}}(\xi (Y)+\eta (X))={\frac {1}{2}}(\epsilon (Y,X)+\epsilon (X,Y))=0}$ 

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 (complex) dimension of the subbundle that annihilates E. Equivalently it is n minus the (complex) dimension of the projection of L(E,ε) onto the tangent bundle T.

Canonical bundle

As in the case of ordinary complex geometry, there is a correspondence between generalized almost complex stuctures and complex line bundles. The complex line bundle corresponding to a particular generalized almost complex structure is often refered 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${\displaystyle \otimes }$ 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${\displaystyle \oplus }$  T^*)${\displaystyle \otimes }$ 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${\displaystyle \oplus }$  T^*)${\displaystyle \otimes }$ C. Therefore a given pure spinor is annihilated by a half-dimensional subbundle E of (T${\displaystyle \oplus }$  T^*)${\displaystyle \otimes }$ 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${\displaystyle \oplus }$  T^*)${\displaystyle \otimes }$ 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.

Other structures

If a pure spinor that determines a particular complex structure is closed, or more generally if its exterior derivative is in the image of the action of a gamma matrix, 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 trivial then it defines a generalized Calabi-Yau structure and M is said to be a generalized Calabi-Yau manifold.

Examples

Complex manifolds

The space of complex differential forms Λ^*T${\displaystyle \otimes }$ 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${\displaystyle \oplus }$  T^*)${\displaystyle \otimes }$ C to the complexified tangent bundle one gets the subspace of antiholomorphic vector fields. Therefore this generalized complex structure on (T${\displaystyle \oplus }$  T^*)${\displaystyle \otimes }$ C defines an ordinary complex structure on the tangent bundle.

Symplectic manifolds

The pure spinor bundle generated by

${\displaystyle \phi =e^{i\omega }}$ 

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 sympectic manifolds are not only generalized complex manifolds but in fact are generalized Calabi-Yau manifolds.

Relation to G-structures

Some of the structures in generalized complex geometry may be rephrased in the language of G-structures. The bundle (T${\displaystyle \oplus }$  T^*)${\displaystyle \otimes }$ C. with the above inner product is a O(2n,2n) structure. A generalized complex structure is a reduction of this structure to a U(n,n) structure.

A generalized Kahler 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${\displaystyle \oplus }$  T^*)${\displaystyle \otimes }$ C. Generalized Kahler structures are reductions of the structure group to U(n)${\displaystyle \times }$ U(n). Finally, a generalized Calabi-Yau structure is a further reduction of the structure group to SU(n)${\displaystyle \times }$ SU(n).

References

• Hitchin, Nigel Generalized Calabi-Yau manifolds, Quart.J.Math.Oxford Ser. 54 (2003) 281-308.

• Gualtieri, Marco, Generalized complex geometry, PhD Thesis (2004).

• Graña, Mariana Flux compactifications in string theory: a comprehensive review, Phys.Rept. 423 (2006) 91-158.