Utilisateur:Girianshiido/Brouillon

Dans la branche des mathématiques appelée géométrie différentielle, une structure complexe généralisée est une notion associée à celle de variété différentielle qui généralise à la fois la notion de structure complexe et celle de structure symplectique. Les structures complexes généralisées ont été introduites par Nigel Hitchin en 2002 et étudiées plus en détails par ses étudiants Marco Gualtieri et 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.

Définition

Le fibré tangent généralisé

Soit $M$ une variété réelle de dimension $n$. Le fibré tangent de $M$, noté $TM$, est le fibré vectoriel de base $M$ dont les fibres sont les espaces tangents en chaque point de $M$. Une section de $TM$ est un champ de vecteurs sur $M$. Le fibré cotangent de $M$, noté $T^{*}M$, est le fibré vectoriel de base $M$ dont les sections sont 1-formes sur $M$.

En géometrie complexe, on étudie des structures particulières sur le fibré tangent alors qu'en géométrie symplectique, on s'intéresse plutôt aux structures définies sur l'algèbre extérieure du fibré cotangent. La géométrie (complexe) généralisée réunit ces deux champs d'études en s'intéressant aux sections du fibré tangent généralisé, qui est la somme directe $TM\oplus T^{*}M$ des fibrés tangent et cotangent, dont les sections sont des sommes formelles de champs de vecteurs et de 1-formes.

Les fibres sont naturellement munies d'une structure d'espace pseudo-euclidien de signature $(n,n)$. Si $X$ et $Y$ sont des champs de vecteurs et $\xi$ et $\eta$ sont des 1-formes, alors le produit de $X+\xi$ par $Y+\eta$ est défini par : $\langle X+\xi ,Y+\eta \rangle ={\dfrac {1}{2}}(\xi (Y)+\eta (X)).$

Une structure presque complexe généralisée est une structure presque complexe sur le fibré tangent généralisé qui préserve le produit :

${\mathcal {J}}:\ TM\oplus T^{*}M\to TM\oplus T^{*}M$ doit vérifier ${\mathcal {J}}^{2}=-{\rm {Id}},$ (structure complexe) et

$\langle {\mathcal {J}}(X+\xi ),{\mathcal {J}}(Y+\eta )\rangle =\langle X+\xi ,Y+\eta \rangle .$ (préserve le produit)

Comme dans le cas des structure presque complexe, une structure presque complexe généralisée est uniquement déterminée par son $i$-espace propre, i.e. le sous-fibré $L$ de $(TM\oplus T^{*}M)\otimes \mathbb {C}$défini par

$L=\{X+\xi \in (TM\oplus T^{*}M)\otimes \mathbb {C} \ :\ {\mathcal {J}}(X+\xi )=i(X+\xi )\}$

Ce sous-fibré $L$ vérifie les deux propriétés :

(i) Son intersection avec son conjugué est la section nulle : $L\cap {\overline {L}}=0$;

(ii) $L$ est isotrope maximal, i.e. il est de dimension $n$ et $\langle \ell ,\ell '\rangle =0$ pour tout $\ell ,\ell '\in L.$

Réciproquement, tout sous-fibré $L$ qui satisfait ces propriétés est le $i$-espace propre d'une unique structure presque complexe généralisée. Elles pourraient donc servir de définition alternative à la notion de structure presque complexe généralisée.

Crochet de Courant

En géométrie complexe ordinaire, une structure presque complexe est intégrable en une structure structure complexe si et seulement si le crochet de Lie de deux sections du sous-fibré holomorphe est encore une telle section.

En géométrie complexe généralisée, on ne s'intéresse pas uniquement aux champs de vecteurs, mais plutôt aux sommes formelles d'un champ de vecteurs avec une 1-forme. Une sorte de crochet de Lie pour de telles sommes formelles a été introduit en 1990, appelé crochet de Courant, qui est défini par :

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

où ${\mathcal {L}}_{X}$ est la dérivée de Lie selon le champ de vecteur $X$, $d$ est la dérivée extérieure et $i$ est le produit intérieur.

Définition

Une structure complexe généralisée est une structure presque complexe généralisée telle que le fibré des sections lisses de L soit stable par le crochet de Courant.

Sous-fibrés isotropes maximaux

Classification

Il existe une correspondance biunivoque entre les sous-fibrés isotropes maximaux de $TM\oplus T^{*}M$ et les couples $(E,\varepsilon )$ où $E$ est un sous-fibré de $TM$ et $\varepsilon$ est une 2-forme. Cette correspondance s'étend de façon évidente au complexifié.

Étant donné un tel couple $(E,\varepsilon )$, on peut construire un sous-fibré isotrope maximal $L(E,\varepsilon )$ de $TM\oplus T^{*}M$ comme suit. Les éléments de ce sous-fibré sont les sommes formelles $X+\xi$ où le champ de vecteurs $X$ est une section de $E$ et la 1-forme $\xi$ restreinte au fibré dual $E^{*}$ est égale à la 1-forme $i_{X}\varepsilon$.

To see that $L(\mathbf {E} ,\varepsilon )$ is isotropic, notice that if Y is a section of E and $\xi$ restricted to $\mathbf {E} ^{*}$ is $\varepsilon (X)$ then $\xi (Y)=\varepsilon (X,Y),$ as the part of $\xi$ orthogonal to $\mathbf {E} ^{*}$ annihilates Y. Thesefore if $X+\xi$ and $Y+\eta$ are sections of $\mathbf {T} \oplus \mathbf {T} ^{*}$ then

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

and so $L(\mathbf {E} ,\varepsilon )$ is isotropic. Furthermore, $L(\mathbf {E} ,\varepsilon )$ is maximal because there are $\dim(\mathbf {E} )$ (complex) dimensions of choices for $\mathbf {E} ,$ and $\varepsilon$ is unrestricted on the complement of $\mathbf {E} ^{*},$ which is of (complex) dimension $n-\dim(\mathbf {E} ).$ Thus the total (complex) dimension in n. Gualtieri has proven that all maximal isotropic subbundles are of the form $L(\mathbf {E} ,\varepsilon )$ for some $\mathbf {E}$ and $\varepsilon .$

Type

The type of a maximal isotropic subbundle $L(\mathbf {E} ,\varepsilon )$ is the real dimension of the subbundle that annihilates E. Equivalently it is 2N minus the real dimension of the projection of $L(\mathbf {E} ,\varepsilon )$ 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 $(\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .$

The type of a maximal isotropic subbundle is invariant under diffeomorphisms and also under shifts of the B-field, which are isometries of $\mathbf {T} \oplus \mathbf {T} ^{*}$ of the form

$X+\xi \longrightarrow X+\xi +i_{X}B$

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 $(\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C}$ 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 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 $\mathbf {\Lambda } ^{*}\mathbf {T} \otimes \mathbb {C}$ 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 $(\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C}$ 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 $\mathbf {\Lambda } ^{*}\mathbf {T} ,$ and generators of the Clifford algebra are the fibers of our other bundle $(\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .$ Therefore, a given pure spinor is annihilated by a half-dimensional subbundle E of $(\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .$ 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 $(\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .$ 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 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 form

$\Phi =e^{B+i\omega }\Omega$

where Ω is decomposable as the wedge product of one-forms.

Regular point

Define the subbundle E of the complexified tangent bundle $\mathbf {T} \otimes \mathbb {C}$ to be the projection of the holomorphic subbundle L of $(\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C}$ to $\mathbf {T} \otimes \mathbb {C} .$ 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 $(\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .$ However the intersection of their projections need not be trivial. In general this intersection is of the form

$E\cap {\overline {E}}=\Delta \otimes \mathbb {C}$

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 $\mathbb {C} ^{k}$ and the standard symplectic space $\mathbb {R} ^{2n-2k}$ 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 manifold 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.

Examples

Complex manifolds

The space of complex differential forms $\mathbf {\Lambda } ^{*}\mathbf {T} \otimes \mathbb {C}$ has a complex conjugation operation given by complex conjugation in $\mathbb {C} .$ 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 $(\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C}$ to the complexified tangent bundle one gets the subspace of antiholomorphic vector fields. Therefore, this generalized complex structure on $(\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C}$ 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, $\partial$-closed (2,0)-form, are the only type N generalized complex manifolds.

Symplectic manifolds

The pure spinor bundle generated by

$\phi =e^{i\omega }$

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

The pure spinor $\phi$ 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 Kähler 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.

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 $(\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C}$ with the above inner product is an 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

${\frac {O(2n,2n)}{U(n,n)}}.$

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 $(\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .$ Generalized Kähler structures are reductions of the structure group to $U(n)\times U(n).$ 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.

Finally, a generalized almost Calabi-Yau metric structure is a further reduction of the structure group to $SU(n)\times SU(n).$

Calabi versus Calabi–Yau metric

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. In particular a generalized Calabi–Yau metric structure implies the existence of two commuting generalized almost complex structures.

References

• Nigel Hitchin, « Generalized Calabi-Yau manifolds », Quarterly Journal of Mathematics, vol. 54, n^o 3,‎ 2003, p. 281–308 (DOI 10.1093/qmath/hag025)
• Marco Gualtieri, Generalized complex geometry (thèse), 2004 (arXiv math.DG/0401221)
• Marco Gualtieri, « Generalized complex geometry », Annals of Mathematics, vol. 174, n^o 1,‎ 2011, p. 75–123 (DOI 10.4007/annals.2011.174.1.3)
• Mariana Graña, « Flux compactifications in string theory: a comprehensive review », Phys. Rep., vol. 423,‎ 2006, p. 91–158 (arXiv hep-th/0509003)
• Robbert Dijkgraaf, Sergei Gukov, Andrew Neitzke et Cumrun Vafa, « Topological M-theory as unification of form theories of gravity », Advances in Theoretical and Mathematical Physics, vol. 9, n^o 4,‎ 2005, p. 603–665 (DOI 10.4310/ATMP.2005.v9.n4.a5)

Category:Géométrie différentielle Category:Structures on manifolds