Differential Geometry and its Applications??(????)pp.99 North-Holland PROOFS 1
Natural quotients on split supercotangent
bundles and their canonical supersymplectic
structures∗
J.Mu˜n oz-Masqu´e1
C.S.I.C.,I.F.A.,C/Serrano144,28006–Madrid,Spain
O.A.S´a nchez-Valenzuela2
CIMAT,Apdo.Post402,C.P.36000Guanajuato,Gto.,M´e xico
Communicated by P.Michor
Received8September1998
Revid20January1999
Abstract:The Batchelor model of the supercotangent bundle of a given ba supermanifold is studied.Under the assumption that the supercotangent bundle splits,two differentfibrations over the given ba can be globally defined.The total spaces of thefibrations are in turn quotient supermanifolds of the supercotangent bundle,and each of them is equipped with a supersymplectic structure.Their corresponding supersymplectic 2-forms are actually exact,and homogeneous of different degrees.The homogeneous1-forms from which they come from are natural with respect to Batchelor trivializations.Each of the1-forms can be pulled back to the supercotangent bundle via the quotient maps,and can be added together in the supercotangent bundle to produce a nonhomogeneous1-form there.Such a1-form in the supercotangent bundle is canonical;
it is characterized by the fact that the pullback of it under any1-form on the ba supermanifold yields the same1-form on the ba.The exterior derivative of this canonical1-form is degenerate.Its radical produces an example of an involutive subsheaf,which is not integrable.This phenomenon is explained at the light of Frobenius for supermanifolds.The radicals of its homogeneous components,on the other hand,taken parately,do produce two globally defined foliations on the supercotangent bundle,and the corresponding spaces of leaves are precily the two quotients of the supercotangent bundle we started with.
Keywords:.
MS classification:.
1.Introduction and description of results
高家庄生态园The general framework of this paper is that of Z2-graded manifolds as defined in[13,18,20]. We refer the reader to[2,7,27,28]for alternative generalizations and further examples.Fol-lowing[20]a Z2-graded manifold is a pair(M,A),where M is a smooth,finite-dimensional manifold,and A is a sheaf of Z2-graded algebras over M locally isomorphic to the sheaf of c-tions of the exterior algebra bundle E→M of a vector bundle E→M—called the Batchelor *Partially supported by DGES grant PB95-0124;FRN.96and CONACyT grant28491-E.
1jaime@iec.csic.es.
0926-2245/??/$??.??c ????Elvier Science B.V.All rights rerved
2J.Mu˜n oz-Masqu´e,O.A.S´a nchez-Valenzuela
伯牙绝铉
bundle—canonically attached to A.More specifically,there is always defined a sheaf morphism A→Ŵ( E)which restricts to an isomorphism of the corresponding Z2-graded algebras of local ctions over a suitable neighborhood of each point.When A is globally isomorphic toŴ( E)—i.e.,when A→Ŵ( E)can be inverted—the supermanifold is called split(or “Batchelor-trivial”).Split supermanifolds exhaust all the examples of smooth supermanifolds (e[3,9,15,20]).In the holomorphic category,this is far from being the ca(e[10,29]). This does not mean,however,that the category of smooth supermanifolds gets subsummed into the category of smooth vector bundles:Supermanifold morphisms are a lot more general than the induced vector bundle morphisms at the level of the Batchelor trivializations involved.In fact,the class of split graded manifolds is interesting,and wide enough so as to derve special consideration.
小水母A fundamental relationship between splittings and connections has been given by Koszul in[15]:A supermanifold is split if and only if it admits a Z2-graded connection.When such a connection exists,it induces a connection on E→M.Since,connections always exist on smooth vector bundles,but may not exist in the holomorphic category(e[1],and[22]),the Koszul approach sheds new light into the earlier results on non-split holomorphic supermanifolds given in[10]and[29].For the purpos of our work,the relevant corollary of[15]is that if a connection fails to exist on either T M or
E,the supertangent and the supercotangent bundles of a split supermanifold do not split(this was also pointed out by M.Rothstein in a beautiful paper[30]where the general theory for even symplectic structures on split supermanifolds was studied,and canonical forms for such structures were given).
A vector bundle over(M,A)is a locally free sheaf F of A-modules over M with a given ,F=F0⊕F1.Given such an F,a geometrical and functorial construction of a Z2-graded manifold(V F(M),V F(A))equipped with a Z2-graded submersion
π: V F(M),V F(A) →(M,A),(1) can be performed in such a way that the abstract ctions of F correspond in a one-to-one fashion with the geometric ctions of the submersionπ:
{β|β∈F(M)}←→{β:(M,A)→(V F(M),V F(A))|π◦β=id}.(2) It was one of the main issues of the early supermanifold theory to prove(or to guarantee by means of suitable axioms),that F=Der A,and F∗=Hom A(Der A,A)were actually locally free sheaves of A-modules.In particular,the construction of(1)above can be applied to them, and respectively the tangent and cotangent Z2-graded manifolds associated to(M,A)can be obtained.They shall be hereby denoted by ST(M,A),and ST∗(M,A),respectively,and we are specifically interested in the ca when A=Ŵ( E).
When a connection∇exists on the Batchelor bundle E→M,it can be ud to establish an isomorphism
between the derivation sheaf DerŴ( E)and F=Ŵ(( E)⊗(T M⊕E∗))which directly exhibits itsŴ( E)-module structure(cf.[21,30]).Similarly,a connection on E→M makes HomŴ( E)(DerŴ( E),Ŵ( E))isomorphic to F∗=Ŵ( E⊗(T∗M⊕E)).If further-more a connection∇′exists on T M,the supermanifolds ST(M,Ŵ( E))and ST∗(M,Ŵ( E)) are split themlves,and the functorial construction referred to above is particularly simple to describe globally.In particular,the supercotangent bundle ST∗(M,Ŵ( E))is the super-manifold(T∗M⊕E,Ŵ( (T M⊕E∗⊕E))),and thefibration(1)becomes(e Section3
Natural quotients on split supercotangent bundles3 below)
π: T∗M⊕E,Ŵ( (T M⊕E∗⊕E)) →(M,Ŵ( E)),(3) which is induced by the inclusionŴ( E)֒→Ŵ( (T M⊕E∗⊕E)).(Note:the bundle projection T∗M⊕E→M is ud to pull the sheaves of ctions of the bundles E,T M,and E∗back to T∗M⊕E.Nevertheless,we have writtenŴ( (E⊕T M⊕E∗))instead of showing explicitly the map to keep the notation simple.In what follows the bundles who ctions we refer to, must be understood as pullbacks under the obvious submersion.)
The main objective of this work is to address the question of whether or not a canonical 1-form can be defined on a split supercotangent bundle ST∗(M,Ŵ( E))as in nongraded differential geometry,and
to e whether or not its exterior derivative can define a canonical supersymplectic structure on it.It will be thus assumed throughout this work that the following “splitting hypothesis”is satisfied:Connections exist on E and T M.
Now,a Z2-graded1-formβin(M,Ŵ( E))is,according to the basic definitions in[13],a ction of HomŴ( E)(DerŴ( E),Ŵ( E))which in turn gets uniquely identified with a Z2-graded manifold morphism
β:(M,Ŵ( E))→ST∗(M,Ŵ( E))(4) such thatπ◦β=id;πbeing the supercotangent bundle projection(3).Since Z2-graded1-forms can be pulled back under Z2-graded manifold morphisms,an equality likeβ∗ =βmakes n for a given1-form in ST∗(M,Ŵ( E)).In fact,by requiring it to hold true for anyβ(regardless of whether it might be Z2-homogenous or not)one picks up the canonical1-form of the graded cotangent bundle ST∗(M,Ŵ( E))(cf.[32]and Section4below).
It turns out that is nonhomogeneous in the Z2-grading of HomŴ( E)(DerŴ( E),Ŵ( E)), i.e., = 0+ 1,and µ=0(µ=0,1).Whence,ω=−d =ω0+ω1(with d as in[13]) is nonhomogeneous,too.Thefirst difference obrved with respect to nongraded differential geometry is that the2-formωdegenerates in the graded tting(cf.Proposition5.1below).We then look at the distribution defined by its radical,Radω={D|i Dω=0},where D stands for any graded derivation of the supercotangent bundle str
ucture sheafŴ( (T M⊕E∗⊕E)).It is immediate to verify that Radωis an involutive subsheaf of the locally free module of graded derivations.But,even though Radωisfinitely generated and locally free,its behavior as an involutive distribution is rather pathological.The reason is that Radωis not generated by a t of homogeneous derivations;in other words,it is not a direct subsheaf in the n of[26].At this point the expert reader may take a look at Example5.2below where we explicitly exhibit the pathology referred to above;plainly,there is no way to define a reasonable subsupermanifold structure on what would be the leaves of the foliation generated by Radω.In fact,if such a structure were possible,there would also be—locally,at least—a supermanifold structure on the quotient modulo the foliation,but thefibers of any quotient projection always have a homogeneously generated tangent sheaf structure(e[26]).Put in different words:Radωgives an example of an involutive distribution which is not integrable.
李清照有几个老公One may look,however,at the homogeneous components ofω,and try to determine parately whether or not Radω0,and Radω1are homogeneously generated involutive distributions on the supercotangent bundle.The result is(cf.Section6below)that both are involutive direct subsheaves;but more remarkably:Their corresponding foliations F(Radωµ)are regular in the
4J.Mu˜n oz-Masqu´e,O.A.S´a nchez-Valenzuela
n of[26].That is,there are supermanifold structures which are globally defined on the space of leaves of each distribution,equipped with graded submersions
qµ:ST∗(M,Ŵ( E))→ST∗(M,Ŵ( E))/F(Radωµ),µ=0,1.(5) Furthermore,each quotient ST∗(M,Ŵ( E))/F(Radωµ),µ=0,1has a globally defined Z2-graded symplectic structureˆωµwhich is homogeneous of degreeµ,and
qµ∗ˆωµ=ωµ=−d µ,µ=0,1.(6) More specifically,the quotients ST∗(M,Ŵ( E))/F(Radωµ),are
q0: T∗M⊕E,Ŵ( (E⊕T M⊕E∗)) → T∗M,Ŵ( (E⊕E∗)) (6a) and
q1: T∗M⊕E,Ŵ( (E⊕T M⊕E∗)) → E,Ŵ( (E⊕T M)) (6b) which are defined by the canonical inclusionsŴ( (E⊕E∗))֒→Ŵ( (E⊕T M⊕E∗))and Ŵ( (E⊕T M))֒→Ŵ( (E⊕T M⊕E∗)),respectively.
We furthermorefind in Section6coordinate-free expressions for ,ωand the symplectic formsˆω0on(T∗M,Ŵ( (E⊕E∗))),andˆω1on(E,Ŵ( (E⊕T M))),respectively:First of all,using the fact(Lemma3.3)that DerŴ( (E⊕E∗))is generated as aŴ( (E⊕E∗))-module over T∗M by the derivations∇X(with X∈Der C∞(T∗M)),iϕ(withϕ∈Ŵ(E)),and iξ(withξ∈Ŵ(E∗)),the canonical1-form 0of(T∗M,Ŵ( (E⊕E∗)))is defined as the Ŵ( (E⊕E∗))-linear extension of the mapping given on generators by
∇X+iϕ+iξ| 0 = X|θ0 +(ϕ|ξ)(7a) whereθ0is the canonical1-form on the cotangent bundle,(X,θ0)→ X|θ0 is just the evaluation of the1-formθ0on the tangent vector X,and(ϕ,ξ)→(ϕ|ξ)stands for the duality bilinear pairing of ctions from E and E∗.Its Z2-graded exterior derivative d 0=−ω0is computed in Proposition6.2.
Similarly,since DerŴ( ˜p∗
1(E⊕T M))is generated as aŴ( (E⊕T M))-module over
the manifold E by∇X,∇ξ,iη,and iϕ(with X∈Der C∞(M),ξ∈Ŵ(E),η∈ 1(M), andϕ∈Ŵ(E∗)—e Lemma3.3below),the canonical1-form 1of the graded manifold (E,Ŵ( (E⊕T M)))is defined as theŴ( (E⊕T M))-linear extension of the mapping given on generators by
∇X+∇ξ+iη+iϕ| 1 = X|η +(ϕ|ξ)(7b) and its exterior derivative d 1=−ω1is computed in Proposition6.3.
2.Geometric supervector bundles
Let(M,Ŵ( E))be a split graded manifold.We shall adhere ourlves to the following nota-
tion:Write E and E instead ofŴ(E)andŴ( E),respectively,and write X M and 1
M instead of
Ŵ(T M)andŴ(T∗M),respectively.We recall in this ction two basic results(Theorem2.1,and Proposition2.2,resp.),and provide a sketch of their proofs for the sake of lf-containedness.
Natural quotients on split supercotangent bundles 5
2.1.Theorem.Let (M ,A M )be a supermanifold.There exists a functorial correspondence between the isomorphism class of locally free sheaves of A -modules over M ,and the isomor-phism class of fibrations π:(H ,A H )→(M ,A M )over (M ,A M )given in such a way that the abstract ctions of a given locally free sheaf of A M -modules correspond in a one-to-one fashion with the geometric ctions (M ,A M )→(H ,A H )of its corresponding fibration (Z 2-graded vector bundle ).
Sketch of proof.Basically,one must perform local constructions over a given open cover {U i }of M ,in such a way that on the overlaps the sheaf maps involved satisfy the appropriate compatibility conditions in terms of transition functions.One then invokes the sheaf axioms to conclude the existence of the globally defined structures.We shall only indicate here how the local construction works,and will leave the details to the reader;therefore,we may assume that (M , E )is a split graded manifold of dimension (m ,n )and that F is a locally free sheaf of E -modules of rank (p ,q )having th
e following structure:
F =( E )⊗(F 0⊕F 1),
where F 0(resp.,F 1)is the sheaf of ctions of some vector bundle F 0→M (resp.,F 1→M ),of rank p (resp.q ).In other words,the Z 2-grading is such that the ctions from F 0are even and tho from F 1are odd.Morphisms of the fibrations obtained are defined locally by the condition of being E -linear on ctions in order to produce the desired correspondence on isomorphism class.But then,the central point is to prove the following (e [24,32]):
2.2.Proposition.The total space F =F 0⊕F 1of the Whitney sum bundle admits the structure of an (m +p +q ,n +p +q )-dimensional graded manifold :Its structure sheaf A F is ˜π∗(E ⊕F ∗0⊕F ∗1),and the bundle projection ˜π:F 0⊕F 1→M is ud to pull the bundle E ⊕F ∗0⊕F ∗1back to F 0⊕F 1.Moreover ,there is a graded-manifold submersion
π:(F 0⊕F 1,A F )→(M , E )
uniquely defined by the following conditions :
(1)M can be covered by open subts U such that there is a graded diffeomorphism (defined through
an isomorphism of E |U -modules )
ϕU : ˜π−1(U ),A F |˜π−1(U ) →(U , E |U )×(V 0⊕V 1,A V 0⊕V 1)
satisfying p 2◦ϕU =π,where p 2is the projection of the product onto the fixed graded manifold
(V 0⊕V 1,A V 0⊕V 1)who underlying manifold is a Z 2-graded vector space of dimension (p +q ,p +q ).
(2)The abstract ctions from F (U )correspond in a one-to-one fashion with graded manifold morphisms
σU :(U , E |U )→(V 0⊕V 1,A V 0⊕V 1)
over a trivializing neighborhood U of the form described in (1).
Sketch of proof.Actually,one first establishes property (2)of the statement,and then proceeds to (1).Now,(2)is in turn a conquence of the following fact (e [6,26,34]for its proof):
6J.Mu˜n oz-Masqu´e ,O.A.S´a nchez-Valenzuela
积极乐观的句子
(*)There is a universal object in the category of (say ,real )supermanifolds ,R 1|1,such that E ↔Maps (M , E ),R 1|1 .
That is ,the abstract ctions from E correspond in a one-to-one fashion with actual super-manifold morphisms.In fact ,R 1|1=(R ,Ŵ( T )),where T =R ×R →R is the rank 1trivial vector bundle.
年终总结会议通知Taking this result into account,the correspondence in (2)can be made evident by first thinking of it algebraically:the ctions from
F = E ⊗(F 0⊕F 1)≃Hom (F 0⊕F 1)∗, E can be extended to maps of Z -graded algebras
Hom (F 0⊕F 1)∗, E ≃ (E ⊕F 0⊕F 1).
Thus,if A F is to have property (2),then it must be the sheaf of ctions of the vector bundle
˜π∗(E ⊕F ∗0⊕F ∗1)→F 0⊕F 1.One then proves that this works analytically (with its Z 2-
grading)as well.Finally,the graded manifold morphism πis given by the composition E →
˜π∗˜π∗ E ֒→˜π∗˜π∗ (E ⊕F ∗0⊕F ∗1)(cf.[36]).
什么言什么语四字成语3.Supertangent and supercotangent bundles
For example,the structure of the graded tangent manifold associated to the locally free sheaf Der E ,is completely determined by the fact that,locally F 0≃X M and F 1≃E ∗.If a connection exists,the statement becomes global:
3.1.Proposition.Let (M , E )be a split supermanifold ,and let ∇be a connection on E .Then ,
Der E = E ⊗(X M ⊕E ∗).
The proof of this proposition can be found in veral references:cf.[13,21,24,30];it is bad on the proofs given in [8]and [12]for derivations on the space of differential forms.Now,Proposition 3.1implies that Hom (Der E , E )≃ E ⊗(X M ∗⊕E ),and one may immediately apply the construction indicated in the previous ction to obtain the following:
3.2.Proposition.The Z 2-graded fibrations corresponding to the rank (m ,n )locally free sheaves of E -modules over M ,Der E ,and Hom (Der E , E ),are respectively given under the bijection of Theorem 2.1by
Der E ←→π: T M ⊕E ∗, ˜π∗(E ⊕X M ∗⊕E ) →(M , E ),
and
Hom (Der E , E )←→π: T ∗M ⊕E , ˜π∗(E ⊕X M ⊕E ∗) →(M , E ).Moreover ,(T ∗M ⊕E , ˜π∗(E ⊕X M ⊕E ∗))has two natural quotients induced by the inclusions (E ⊕E ∗)֒→ (E ⊕X M ⊕E ∗),and (E ⊕X M )֒→ (E ⊕X M ⊕E ∗):语篇分析
q 0: T ∗M ⊕E , ˜π∗(E ⊕X M ⊕E ∗) → T ∗M , ˜p ∗0(E ⊕E ∗) ,
