Induced differential forms on manifolds of functions
aa r X i v : . [ m a t h . DG ] N ov Induced differential forms onmanifolds of functions
Cornelia Vizman
West University of Timi¸soara, Department of Mathematicse-mail: [email protected]
Abstract
Differential forms on the Fr´echet manifold F ( S, M ) of smooth functionson a compact k –dimensional manifold S can be obtained in a natural wayfrom pairs of differential forms on M and S by the hat pairing. Specialcases are the transgression map Ω p ( M ) → Ω p − k ( F ( S, M )) (hat pairingwith a constant function) and the bar map Ω p ( M ) → Ω p ( F ( S, M )) (hatpairing with a volume form). We develop a hat calculus similar to the tildacalculus for non-linear Grassmannians [HV04].
Pairs of differential forms on the finite dimensional manifolds M and S inducedifferential forms on the Fr´echet manifold F ( S, M ) of smooth functions. Moreprecisely, if S is a compact oriented k –dimensional manifold, the hat pairing is:Ω p ( M ) × Ω q ( S ) → Ω p + q − k ( F ( S, M )) [ ω · α = − Z S ev ∗ ω ∧ pr ∗ α, where ev : S × F ( S, M ) → M denotes the evaluation map, pr : S × F ( S, M ) → S the projection and − R S fiber integration. We show that the hat pairing is compatiblewith the canonical Diff(M) and Diff( S ) actions on F ( S, M ), and with the exteriorderivative. As a consequence we obtain a hat pairing in cohomology.The hat (transgression) map is the hat pairing with the constant function1, so it associates to any form ω ∈ Ω p ( M ) the form d ω · b ω = − R S ev ∗ ω ∈ Ω p − k ( F ( S, M )). Since X ( M ) acts infinitesimally transitive on the open subsetEmb( S, M ) ⊂ F ( S, M ) of embeddings of the k –dimensional oriented manifold S into M [H76], the expression of b ω at f ∈ Emb(
S, M ) is b ω ( X ◦ f, . . . , X p − k ◦ f ) = Z S f ∗ ( i X p − k . . . i X ω ) , X , . . . , X p − k ∈ X ( M ) . S is the circle, then one obtains the usual transgression map with values inthe space of ( p − M .Let Gr k ( M ) be the non-linear Grassmannian of k –dimensional oriented sub-manifolds of M . The tilda map associates to every ω ∈ Ω p ( M ) a differential( p − k )-form on Gr k ( M ) given by [HV04]˜ ω ( ˜ Y N , . . . , ˜ Y p − kN ) = Z N i Y p − kN · · · i Y N ω, ∀ ˜ Y N , . . . , ˜ Y p − kN ∈ Γ( T N ⊥ ) = T N Gr k ( M ) , for ˜ Y N section of the orthogonal bundle T N ⊥ represented by the section Y N of T M | N . The natural map π : Emb( S, M ) → Gr k ( M ) , π ( f ) = f ( S )provides a principal bundle with the group Diff + ( S ) of orientation preservingdiffeomorphisms of S as structure group.The hat map on Emb( S, M ) and the tilda map on Gr k ( M ) are related by b ω = π ∗ ˜ ω . This is the reason why for the hat calculus one has similar properties tothose for the tilda calculus. The tilda calculus was used to study the non-linearGrassmannian of co-dimension two submanifolds as symplectic manifold [HV04].We apply the hat calculus to the hamiltonian formalism for p -branes and open p -branes [AS05] [BZ05].The bar map ¯ ω = d ω · µ is the hat pairing with a fixed volume form µ on S , so¯ ω ( Y f , . . . , Y pf ) = Z S ω ( Y f , . . . , Y pf ) µ, ∀ Y f , . . . , Y pf ∈ Γ( f ∗ T M ) = T f F ( S, M ) . We use the bar calculus to study F ( S, M ) with symplectic form ¯ ω induced by asymplectic form ω on M . The natural actions of Diff ham ( M, ω ) and Diff ex ( S, µ ),the group of hamiltonian diffeomorphisms of M and the group of exact vol-ume preserving diffeomorphisms of S , are two commuting hamiltonian actionson F ( S, M ). Their momentum maps form the dual pair for ideal incompressiblefluid flow [MW83] [GBV09].We are grateful to Stefan Haller for extremely helpful suggestions.
We denote by F ( S, M ) the set of smooth functions from a compact oriented k –dimensional manifold S to a manifold M . It is a Fr´echet manifold in a naturalway [KM97]. Tangent vectors at f ∈ F ( S, M ) are identified with vector fields on M along f , i.e. sections of the pull-back vector bundle f ∗ T M .Let ev : S × F ( S, M ) → M be the evaluation map ev( x, f ) = f ( x ) andpr : S × F ( S, M ) → S the projection pr( x, f ) = x . A pair of differential forms ω ∈ Ω p ( M ) and α ∈ Ω q ( S ) determines a differential form [ ω · α on F ( S, M ) by the2ber integral over S (whose definition and properties are listed in the appendix)of the ( p + q )-form ev ∗ ω ∧ pr ∗ α on S × F ( S, M ): [ ω · α = − Z S ev ∗ ω ∧ pr ∗ α (1)In this way we obtain a bilinear map called the hat pairing :Ω p ( M ) × Ω q ( S ) → Ω p + q − k ( F ( S, M )) . An explicit expression of the hat pairing avoiding fiber integration is:( [ ω · α ) f ( Y f , . . . , Y p + q − kf ) = Z S f ∗ ( i Y p + q − kf . . . i Y f ( ω ◦ f )) ∧ α, (2)for Y f , . . . Y p + q − kf vector fields on M along f ∈ F ( S, M ). Here we denote by f ∗ β f the ”restricted pull-back” by f of a section β f of f ∗ (Λ m T ∗ M ), which is adifferential m –form on S given by f ∗ β f : x ∈ S (Λ m T ∗ x f )( β f ( x )) ∈ Λ m T ∗ x S ,where T ∗ x f : T ∗ f ( x ) M → T ∗ x S denotes the dual of T x f .The fact that (1) and (2) provide the same differential form on F ( S, M ) canbe deduced from the identity(ev ∗ ω ) ( x,f ) ( Y f , . . . , Y p − kf , X x , . . . , X kx ) = f ∗ ( i Y p − kf . . . i Y f ( ω ◦ f ))( X x , . . . , X kx )for Y f , . . . , Y p − kf ∈ T f F ( S, M ) and X x , . . . , X kx ∈ T x S .Since X ( M ) acts infinitesimally transitive on the open subset Emb( S, M ) ⊂F ( S, M ) of embeddings of the k –dimensional oriented manifold S into M , weexpress b ω at f ∈ Emb(
S, M ) as:( [ ω · α ) f ( X ◦ f, . . . , X p + q − k ◦ f ) = Z S f ∗ ( i X p + q − k . . . i X ω ) ∧ α. (3)One uses the fact that the ”restricted pull-back” by f of i X p + q − k ◦ f . . . i X ◦ f ( ω ◦ f )is f ∗ ( i X p + q − k . . . i X ω ).Next we show that the hat pairing is compatible with the exterior derivativeof differential forms. Theorem 1.
The exterior derivative d is a derivation for the hat pairing, i.e. d ( d ω · α ) = \ ( d ω ) · α + ( − p \ ω · d α, (4) where ω ∈ Ω p ( M ) and α ∈ Ω q ( S ) .Proof. Differentiation and fiber integration along the boundary free manifold S commute, so d ( [ ω · α ) = d − Z S ev ∗ ω ∧ pr ∗ α = − Z S d (ev ∗ ω ∧ pr ∗ α )= − Z S ev ∗ d ω ∧ pr ∗ α + ( − p − Z S ev ∗ ω ∧ pr ∗ d α = \ ( d ω ) · α + ( − p \ ω · d α for all ω ∈ Ω p ( M ) and α ∈ Ω q ( S ). 3he differential form [ ω · α is exact if ω is closed and α exact (or if α is closedand ω exact). In the special case p + q = k these conditions imply that the function [ ω · α on F ( S, M ) vanishes.
Corollary 2.
The hat pairing induces a bilinear map on de Rham cohomologyspaces H p ( M ) × H q ( S ) → H p + q − k ( F ( S, M )) . (5) In particular there is a bilinear map H p ( M ) × H q ( M ) → H p + q − k (Diff( M )) . Remark . The cohomology group H q ( S ) is isomorphic to the homology group H k − q ( S ) by Poincar´e duality. With the notation n = k − q , the hat pairing (5)becomes H p ( M ) × H n ( S ) → H p − n ( F ( S, M )) , and it is induced by the map ( ω, σ )
7→ − R σ ev ∗ ω , for differential p -forms ω on M and n -chains σ on S .If S is a manifold with boundary, then formula (4) receives an extra termcoming from integration over the boundary. Let i ∂ : ∂S → S be the inclusion and r ∂ : F ( S, M ) → F ( ∂S, M ) the restriction map. Proposition 4.
The identity d ( d ω · α ) = \ ( d ω ) · α + ( − p \ ω · d α + ( − p + q − k r ∗ ∂ ( \ ω · i ∗ ∂ α ∂ ) (6) holds for ω ∈ Ω p ( M ) and α ∈ Ω q ( S ) , where the upper index ∂ assigned to the hatmeans the pairing Ω p ( M ) × Ω q ( ∂S ) → Ω p + q − k +1 ( F ( ∂S, M )) . Proof.
For any differential n –form β on S × F ( S, M ), the identity d − Z S β − − Z S d β = ( − n − k − Z ∂S ( i ∂ × F ( S,M ) ) ∗ β holds because of the identity (19) from the appendix. The obvious formulaspr ◦ ( i ∂ × F ( S,M ) ) = i ∂ ◦ pr ∂ , ev ◦ ( i ∂ × F ( S,M ) ) = ev ∂ , for ev ∂ : ∂S × F ( S, M ) → M and pr ∂ : ∂S × F ( S, M ) → ∂S , are used to compute d ( [ ω · α ) = d − Z S ev ∗ ω ∧ pr ∗ α = − Z S d (ev ∗ ω ∧ pr ∗ α ) + ( − p + q − k − Z ∂S ( i ∂ × F ( S,M ) ) ∗ (ev ∗ ω ∧ pr ∗ α )= − Z S ev ∗ d ω ∧ pr ∗ α + ( − p − Z S ev ∗ ω ∧ pr ∗ d α + ( − p + q − k − Z ∂S ev ∗ ∂ ω ∧ pr ∗ ∂ i ∗ ∂ α = \ ( d ω ) · α + ( − p \ ω · d α + ( − p + q − k r ∂ ∗ ( \ ω · i ∗ ∂ α ∂ ) , thus obtaining the requested identity. 4 eft Diff( M ) action. The natural left action of the group of diffeomorphismsDiff( M ) on F ( S, M ) is ϕ · f = ϕ ◦ f . The infinitesimal action of X ∈ X ( M ) is thevector field ¯ X on F ( S, M ):¯ X ( f ) = X ◦ f, ∀ f ∈ F ( S, M ) . We denote by ¯ ϕ the diffeomorphism of F ( S, M ) induced by the action of ϕ ∈ Diff( M ), so ¯ ϕ ( f ) = ϕ ◦ f is the push-forward by ϕ . Proposition 5.
Given ω ∈ Ω p ( M ) and α ∈ Ω q ( S ) , the identity ¯ ϕ ∗ d ω · α = \ ( ϕ ∗ ω ) · α (7) and its infinitesimal version L ¯ X d ω · α = \ ( L X ω ) · α (8) hold for all ϕ ∈ Diff( M ) and X ∈ X ( M ) .Proof. Using the expression (1) of the hat pairing and identity (15) from theappendix, we have:¯ ϕ ∗ [ ω · α = ¯ ϕ ∗ − Z S ev ∗ ω ∧ pr ∗ α = − Z S (1 S × ¯ ϕ ) ∗ (ev ∗ ω ∧ pr ∗ α )= − Z S ev ∗ ϕ ∗ ω ∧ pr ∗ α = \ ( ϕ ∗ ω ) · α, since pr ◦ (1 S × ¯ ϕ ) = pr and ev ◦ (1 S × ¯ ϕ ) = ϕ ◦ ev.A similar result is obtained for any smooth map η ∈ F ( M , M ) and its push-forward ¯ η : F ( S, M ) → F ( S, M ), ¯ η ( f ) = η ◦ f :¯ η ∗ [ ω · α = \ η ∗ ω · α, for all ω ∈ Ω p ( M ) and α ∈ Ω q ( S ). Lemma 6.
For all vector fields X ∈ X ( M ) , the identity i ¯ X d ω · α = \ ( i X ω ) · α holds.Proof. The vector field 0 S × ¯ X on S × F ( S, M ) is ev-related to the vector field X on M , so i ¯ X [ ω · α = i ¯ X − Z S ev ∗ ω ∧ pr ∗ α = − Z S i S × ¯ X (ev ∗ ω ∧ pr ∗ α )= − Z S ev ∗ ( i X ω ) ∧ pr ∗ α = \ ( i X ω ) · α. At step two we use formula (18) from the appendix.5 ight
Diff( S ) action. The natural right action of the diffeomorphism groupDiff( S ) on F ( S, M ) can be transformed into a left action by ψ · f = f ◦ ψ − . Theinfinitesimal action of Z ∈ X ( S ) is the vector field ˆ Z on F ( S, M ): b Z ( f ) = − T f ◦ Z, ∀ f ∈ F ( S, M ) . We denote by b ψ the diffeomorphism of F ( S, M ) induced by the action of ψ , so b ψ ( f ) = f ◦ ψ − is the pull-back by ψ − . Proposition 7.
Given ω ∈ Ω p ( M ) and α ∈ Ω q ( S ) , the identity b ψ ∗ d ω · α = \ ω · ψ ∗ α and its infinitesimal version L b Z d ω · α = \ ω · L Z α hold for all orientation preserving ψ ∈ Diff( S ) and Z ∈ X ( S ) .Proof. The obvious identities ev ◦ (1 S × b ψ ) = ev ◦ ( ψ − × F ), pr ◦ (1 S × b ψ ) = prand pr ◦ ( ψ × F ) = ψ ◦ pr are used in the computation b ψ ∗ [ ω · α = b ψ ∗ − Z S ev ∗ ω ∧ pr ∗ α = − Z S (1 S × b ψ ) ∗ (ev ∗ ω ∧ pr ∗ α )= − Z S (cid:0) ( ψ − × F ) ∗ ev ∗ ω (cid:1) ∧ pr ∗ α = − Z S ev ∗ ω ∧ ( ψ × F ) ∗ pr ∗ α = − Z S ev ∗ ω ∧ pr ∗ ψ ∗ α = \ ω · ψ ∗ α, together with formula (17) from the appendix at step four. Lemma 8.
The identity i b Z d ω · α = ( − p \ ω · i Z α holds for all vector fields Z ∈ X ( S ) , if ω ∈ Ω p ( M ) .Proof. The infinitesimal version of the first identity in the proof of proposition 7is T ev . (0 S × b Z ) = T ev . ( − Z × F ( S,M ) ), so we compute: i b Z [ ω · α = i b Z − Z S ev ∗ ω ∧ pr ∗ α = − Z S i S × b Z (ev ∗ ω ∧ pr ∗ α )= − Z S ( i S × b Z ev ∗ ω ) ∧ pr ∗ α = − Z S ( i − Z × F ( S,M ) ev ∗ ω ) ∧ pr ∗ α = − Z S i − Z × F ( S,M ) (ev ∗ ω ∧ pr ∗ α ) − − Z S ( − p ev ∗ ω ∧ i − Z × F ( S,M ) pr ∗ α = ( − p − Z S ev ∗ ω ∧ pr ∗ ( i Z α ) = ( − p \ ω · i Z α. At step two we use formula (18) from the appendix.6
Tilda map and hat map
Let Gr k ( M ) be the non-linear Grassmannian (or differentiable Chow variety) ofcompact oriented k –dimensional submanifolds of M . It is a Fr´echet manifold[KM97] and the tangent space at N ∈ Gr k ( M ) can be identified with the space ofsmooth sections of the normal bundle T N ⊥ = ( T M | N ) /T N . The tangent vectorat N determined by the section Y N ∈ Γ( T M | N ) is denoted by ˜ Y N ∈ T N Gr k ( M ).The tilda map [HV04] associates to any p –form ω on M a ( p − k )–form ˜ ω onGr k ( M ) by: ˜ ω N ( ˜ Y N , . . . , ˜ Y p − kN ) = Z N i Y p − kN · · · i Y N ω. (9)Here all ˜ Y jN are tangent vectors at N ∈ Gr k ( M ), i.e. sections of T N ⊥ representedby sections Y jN of T M | N . Then i Y p − kN · · · i Y N ω ∈ Ω k ( N ) does not depend on rep-resentatives Y jN of ˜ Y jN , and integration is well defined since N ∈ Gr k ( M ) comeswith an orientation.Let S be a compact oriented k –dimensional manifold. The hat map is thehat pairing with the constant function 1 ∈ Ω ( S ). It associates to any form ω ∈ Ω p ( M ) the form b ω ∈ Ω p − k ( F ( S, M )): b ω = d ω · − Z S ev ∗ ω. (10)On the open subset Emb( S, M ) ⊂ F ( S, M ) of embeddings, formula (2) gives b ω ( X ◦ f, . . . , X p − k ◦ f ) = Z S f ∗ ( i X p − k . . . i X ω ) . (11) Remark . The hat map induces a transgression on cohomology spaces H p ( M ) → H p − k ( F ( S, M )) . When S is the circle, then one obtains the usual transgression map with values inthe ( p − M .Let π denote the natural map π : Emb( S, M ) → Gr k ( M ) , π ( f ) = f ( S ) . where the orientation on f ( S ) is chosen such that the diffeomorphism f : S → f ( S ) is orientation preserving. The image π (Emb( S, M )) is the manifold Gr Sk ( M )of k –dimensional submanifolds of M of type S . Then π : Emb( S, M ) → Gr Sk ( M )is a principal bundle over Gr Sk ( M ) with structure group Diff + ( S ), the group oforientation preserving diffeomorphisms of S .Note that there is a natural action of the group Diff( M ) on the non-linearGrassmannian Gr k ( M ) given by ϕ · N = ϕ ( N ). Let ˜ ϕ be the diffeomorphism7f Gr k ( M ) induced by the action of ϕ ∈ Diff( M ). Then ˜ ϕ ◦ π = π ◦ ¯ ϕ for therestriction of ¯ ϕ ( f ) = ϕ ◦ f to a diffeomorphism of Emb( S, M ) ⊂ F ( S, M ). As aconsequence, the infinitesimal generators for the Diff( M ) actions on Gr k ( M ) andon Emb( S, M ) are π –related. This means that for all X ∈ X ( M ), the vector fields˜ X on Gr k ( M ) given by ˜ X ( N ) = X | N and ¯ X on Emb( S, M ) given by ¯ X ( f ) = X ◦ f are π –related. Proposition 10.
The hat map on
Emb(
S, M ) and the tilda map on Gr k ( M ) arerelated by b ω = π ∗ ˜ ω , for any k –dimensional oriented manifold S .Proof. For the proof we use the fact that X ( M ) acts infinitesimally transitive onEmb( S, M ), so T f Emb(
S, M ) = { X ◦ f : X ∈ X ( M ) } . With (9) and (11) wecompute:( π ∗ ˜ ω ) f ( X ◦ f, . . . , X p − k ◦ f ) = ˜ ω f ( S ) ( X | f ( S ) , . . . , X p − k | f ( S ) )= Z f ( S ) i X p − k . . . i X ω = Z S f ∗ ( i X p − k . . . i X ω ) = b ω f ( X ◦ f, . . . , X p − k ◦ f ) , since ¯ X and ˜ X are π –related.From the properties of the hat pairing presented in proposition 5, lemma 6and theorem 1, a hat calculus follows easily: Proposition 11.
For any ω ∈ Ω p ( M ) , ϕ ∈ Diff( M ) , X ∈ X ( M ) , and η ∈F ( M ′ , M ) with push-forward ¯ η : F ( S, M ′ ) → F ( S, M ) , the following identitieshold:1. ¯ ϕ ∗ b ω = d ϕ ∗ ω and ¯ η ∗ b ω = d η ∗ ω L ¯ X b ω = d L X ω i ¯ X b ω = d i X ω d b ω = c d ω .Remark . If S is a manifold with boundary, then the formula 4. above receivesan extra term coming from integration over the boundary ∂S as in proposition 4: d b ω = c d ω + ( − p − k r ∗ ∂ b ω ∂ (12)for ω ∈ Ω p ( M ). As before, r ∂ : F ( S, M ) → F ( ∂S, M ) denotes the restriction mapon functions and ω ∈ Ω p ( M ) b ω ∂ ∈ Ω p − k +1 ( F ( ∂S, M )).Now the properties of the tilda calculus follow imediately from proposition 11. Proposition 13. [HV04] For any ω ∈ Ω p ( M ) , ϕ ∈ Diff( M ) and X ∈ X ( M ) , thefollowing identities hold: . ˜ ϕ ∗ ˜ ω = g ϕ ∗ ω L ˜ X ˜ ω = g L X ω i ˜ X ˜ ω = g i X ω d ˜ ω = f d ω .Proof. We verify the identities 1. and 4. From relation 1. from proposition 11 weget that π ∗ ˜ ϕ ∗ ˜ ω = ¯ ϕ ∗ π ∗ ˜ ω = ¯ ϕ ∗ b ω = d ϕ ∗ ω = π ∗ g ϕ ∗ ω, and this implies the first identity. Using identity 4. from proposition 11 wecompute π ∗ d ˜ ω = d π ∗ ˜ ω = d b ω = c d ω = π ∗ f d ω, which shows the last identity. Hamiltonian formalism for p -branes In this section we show how the hat calculus appears in the hamiltonian formalismfor p -branes and open p -branes [AS05] [BZ05].Let S be a compact oriented p -dimensional manifold. The phase space for the p -brane world volume S × R is the cotangent bundle T ∗ F ( S, M ), where the canon-ical symplectic form is twisted. The twisting consists in adding a magnetic term,namely the pull-back of a closed 2-form on the base manifold, to the canonicalsymplectic form on a cotangent bundle [MR99]. These twisted symplectic formsappear also in cotangent bundle reduction.We consider a closed differential form H ∈ Ω p +2 ( M ). Since dim S = p , the hatmap (10) provides a closed 2-form b H on F ( S, M ). If π F : T ∗ F ( S, M ) → F ( S, M )denotes the canonical projection, the twisted symplectic form on T ∗ F ( S, M ) isΩ H = − d Θ F + 12 π ∗F b H, where Θ F is the canonical 1-form on T ∗ F ( S, M ).For the description of open branes one considers a compact oriented p -dimensionalmanifold S with boundary ∂S and a submanifold D of M . The phase space is inthis case the cotangent bundle T ∗ F D ( S, M ) over the manifold [M80] F D ( S, M ) = { f : S → M | f ( ∂S ) ⊂ D } . The twisting of the canonical symplectic form is done with a closed differentialform H ∈ Ω p +2 ( M ) with i ∗ H = d B for some B ∈ Ω p +1 ( D ), where i : D → M denotes the inclusion. The twisted symplectic form on T ∗ F D ( S, M ) isΩ ( H,B ) = − d Θ F D + 12 π ∗F D ( b H − ∂ ∗ b B ∂ )9ith ∂ : F D ( S, M ) → F ( ∂S, D ) the restriction map and π F D : T ∗ F D ( S, M ) →F D ( S, M ). To distinguish between the hat calculus for F ( S, M ) and the hatcalculus for F ( ∂S, M ), we denote b ∂ : Ω n ( M ) → Ω n − p +1 ( F ( ∂S, M )).The only thing we have to verify is the closedness of b H − ∂ ∗ b B ∂ . We first noticethat (12) implies d b H = d d H + r ∗ ∂ b H ∂ , where r ∂ : F ( S, M ) → F ( ∂S, M ) denotesthe restriction map, and identity 4 from proposition 11 implies c d B ∂ = d b B ∂ . Onthe other hand identity 1 from proposition 11 ensures that d i ∗ H ∂ = ¯ i ∗ b H ∂ , with¯ i : F ( ∂S, D ) → F ( ∂S, M ) denoting the push-forward by i : D → M . Knowingthat r ∂ = ¯ i ◦ ∂ , we compute: d b H = d d H + r ∗ ∂ b H ∂ = ∂ ∗ ¯ i ∗ b H ∂ = ∂ ∗ d i ∗ H ∂ = ∂ ∗ c d B ∂ = d ∂ ∗ b B ∂ , so the closed 2–form b H − ∂ ∗ b B ∂ provides a twist for the canonical symplectic formon the cotangent bundle T ∗ F D ( S, M ). Non-linear Grassmannians as symplectic manifolds
In this subsection we recall properties of the co-dimension two non-linear Grass-mannian as a symplectic manifold.
Proposition 14. [I96] Let M be a closed m –dimensional manifold with volumeform ν . The tilda map provides a symplectic form ˜ ν on Gr m − ( M )˜ ν N ( ˜ X N , ˜ Y N ) = Z N i Y N i X N ν, for ˜ X N and ˜ Y N sections of T N ⊥ determined by sections X N and Y N of T M | N .Proof. The 2–form ˜ ν is closed since d ˜ ν = f d ν by the tilda calculus. To verify thatit is also (weakly) non-degenerate, let X N be an arbitrary vector field along N such that R N i Y N i X N ν = 0 for all vector fields Y N along N . Then X N must betangent to N , so ˜ X N = 0.In dimension m = 3 the symplectic form ˜ ν is known as the Marsden–Weinsteinsymplectic from on the space of unparameterized oriented links, see [MW83] [B93]. Hamiltonian
Diff ex ( M, ν ) action. The action of the group Diff(
M, ν ) of vol-ume preserving diffeomorphisms of M on Gr m − ( M ) preserves the symplectic form˜ ν : ˜ ϕ ∗ ˜ ν = g ϕ ∗ ν = ˜ ν, ∀ ϕ ∈ Diff(
M, ν ) . The subgroup Diff ex ( M, ν ) of exact volume preserving diffeomorphisms acts ina hamiltonian way on the symplectic manifold (Gr m − ( M ) , ˜ ν ). Its Lie algebra is X ex ( M, ν ), the Lie algebra of exact divergence free vector fields, i.e. vector fields10 α such that i X α ν = d α for a potential form α ∈ Ω m − ( M ). The infinitesimalaction of X α is the vector field ˜ X α . By the tilda calculus ˜ α ∈ F (Gr m − ( M )) is ahamiltonian function for the hamiltonian vector field ˜ X α : i ˜ X α ˜ ν = g i X α ν = f d α = d ˜ α. It depends on the particular choice of the potential α of X α . A fixed continuousright inverse b : d Ω m − ( M ) → Ω m − ( M ) to the differential d picks up a potential b ( d α ) of X α . The corresponding momentum map is: J : M → X ex ( M, ν ) ∗ , h J ( N ) , X α i = ^ b ( d α )( N ) = Z N b ( d α ) . On the connected component M of N ∈ Gr m − ( M ), the non-equivariance of J is measured by the Lie algebra 2–cocycle on X ex ( M, ν ) σ N ( X, Y ) = h J ( N ) , [ X, Y ] op i − ˜ ν ( ˜ X, ˜ Y )( N ) = ^ ( b d i Y i X ν )( N ) − ^ ( i Y i X ν )( N )= ^ ( P i X i Y ν )( N ) = Z N P i X i Y ν. Here P = 1 Ω m − ( M ) − b ◦ d is a continuous linear projection on the subspace ofclosed ( m − X, Y ) [ P i Y i X ν ] ∈ H m − ( M ) is the universal Liealgebra 2–cocycle on X ex ( M, ν ) [R95]. The cocycle σ N is cohomologous to theLichnerowicz cocycle σ η ( X, Y ) = Z M η ( X, Y ) ν, (13)where η is a closed 2-form Poincar´e dual to N [V09].If ν is an integral volume form, then σ N is integrable [I96]. The connectedcomponent M of Gr m − ( M ) is a coadjoint orbit of a 1–dimensional central Liegroup extension of Diff ex ( M, ν ) integrating σ N , and ˜ ν is the Kostant-Kirillov-Souriau symplectic form. [HV04]. When a volume form µ on the compact k –dimensional manifold S is given, onecan associate to each differential p -form on M a differential p -form on F ( S, M )¯ ω ( Y f , . . . , Y pf ) = Z S ω ( Y f , . . . , Y pf ) µ, ∀ Y if ∈ T f F ( S, M ) , where ω ( Y f , . . . , Y pf ) : x ω f ( x ) ( Y f ( x ) , . . . , Y pf ( x )) defines a smooth function on S . In this way a bar map is defined. Formula (2) assures that this bar map is justthe hat pairing of differential forms on M with the volume form µ ¯ ω = d ω · µ = − Z S ev ∗ ω ∧ pr ∗ µ. (14)11rom the properties of the hat pairing presented in proposition 5, lemma 6and theorem 1, one can develop a bar calculus. Proposition 15.
For any ω ∈ Ω p ( M ) , ϕ ∈ Diff( M ) and X ∈ X ( M ) , the followingidentities hold:1. ¯ ϕ ∗ ¯ ω = ϕ ∗ ω L ¯ X ¯ ω = L X ω i ¯ X ¯ ω = i X ω d ¯ ω = d ω . F ( S, M ) as symplectic manifold Let (
M, ω ) be a connected symplectic manifold and S a compact k –dimensionalmanifold with a fixed volume form µ , normalized such that R S µ = 1. The followingfact is well known: Proposition 16.
The bar map provides a symplectic form ¯ ω on F ( S, M ) : ¯ ω f ( X f , Y f ) = Z S ω ( X f , Y f ) µ. Proof.
That ¯ ω is closed follows from the bar calculus: d ¯ ω = d ω = 0. The (weakly)non-degeneracy of ¯ ω can be verified as follows. If the vector field X f on M along S is non-zero, then X f ( x ) = 0 for some x ∈ S . Because ω is non-degenerate, onecan find another vector field Y f along f such that ω ( X f , Y f ) is a bump functionon S . Then ¯ ω ( X f , Y f ) = R S ω ( X f , Y f ) µ = 0, so X f does not belong to the kernelof ¯ ω , thus showing that the kernel of ¯ ω is trivial. Hamiltonian action on M . Let G be a Lie group acting in a hamiltonian wayon M with momentum map J : M → g ∗ . Then F ( S, M ) inherits a G -action:( g · f )( x ) = g · ( f ( x )) for any x ∈ S . The infinitesimal generator is ξ F = ¯ ξ M for any ξ ∈ g , where ξ M denotes the infinitesimal generator for the G -action on M . The bar calculus shows quickly that G acts in a hamiltonian way on F ( S, M )with momentum map J = ¯ J : F ( S, M ) → g ∗ , ¯ J ( f ) = Z S ( J ◦ f ) µ, ∀ f ∈ F ( S, M ) . Indeed, for all ξ ∈ g i ξ F ¯ ω = i ¯ ξ M ¯ ω = i ξ M ω = d h J, ξ i = d h ¯ J , ξ i . M be connected and let σ be the R -valued Lie algebra 2–cocycle on g measuring the non-equivariance of J , i.e. σ ( ξ, η ) = h J ( x ) , [ ξ, η ] i − ω ( ξ M , η M )( x ) , x ∈ M, (both terms are hamiltonian function for the vector field [ ξ, η ] M = − [ ξ M , η M ]).Then the non-equivariance of J = ¯ J is also measured by σ : for all f ∈ F ( S, M ) h ¯ J ( f ) , [ ξ, η ] i − ¯ ω ( ξ F , η F )( f ) = h J, [ ξ, η ] i ( f ) − ω ( ξ M , η M )( f ) = σ ( ξ, η ) . Hamiltonian
Diff ham ( M, ω ) action. The action of the group Diff(
M, ω ) of sym-plectic diffeomorphisms preserves the symplectic form ¯ ω :¯ ϕ ∗ ¯ ω = ϕ ∗ ω = ¯ ω, ∀ ϕ ∈ Diff(
M, ω ) . The subgroup Diff ham ( M, ω ) of hamiltonian diffeomorphisms of M acts in ahamiltonian way on the symplectic manifold F ( S, M ). The infinitesimal actionof X h ∈ X ham ( M, ω ), h ∈ F ( M ), is the hamiltonian vector field ¯ X h on F ( S, M )with hamiltonian function ¯ h . This follows by the bar calculus: d ¯ h = d h = i X h ω = i ¯ X h ¯ ω. The hamiltonian function ¯ h of ¯ X h depends on the particular choice of thehamiltonian function h . To solve this problem we fix a point x ∈ M and wechoose the unique hamiltonian function h with h ( x ) = 0, since M is connected.The corresponding momentum map is J : F ( S, M ) → X ham ( M, ω ) ∗ , h J ( f ) , X h i = ¯ h ( f ) = Z S ( h ◦ f ) µ. The Lie algebra 2–cocycle on X ham ( M, ω ) measuring the non-equivariance ofthe momentum map is σ ( X, Y ) = − ω ( X, Y )( x ) , by the bar calculus σ ( X, Y )( f ) = h J ( f ) , [ X, Y ] op i − ¯ ω ( X F , Y F )( f )= ω ( X, Y ) − ω ( X, Y )( x )( f ) − ¯ ω ( ¯ X, ¯ Y )( f ) = − ω ( X, Y )( x ) . This is a Lie algebra cocycle describing the central extension0 → R → F ( M ) → X ham ( M, ω ) → F ( M ) is enowed with the canonical Poisson bracket. A group cocycleon Diff ham ( M, ω ) integrating the Lie algebra cocycle σ if ω exact is studied in[ILM06]. 13 amiltonian Diff ex ( S, µ ) action. The (left) action of the group Diff(
S, µ ) ofvolume preserving diffeomorphisms preserves the symplectic form ¯ ω : b ψ ∗ ¯ ω = b ψ ∗ d ω · µ = \ ω · ψ ∗ µ = d ω · µ = ¯ ω, ∀ ψ ∈ Diff(
S, µ ) . The subgroup Diff ex ( S, µ ) of exact volume preserving diffeomorphisms actsin a hamiltonian way on the symplectic manifold F ( S, M ). The infinitesimalaction of the exact divergence free vector field X α ∈ X ex ( S, µ ) with potential form α ∈ Ω k − ( S ) is the hamiltonian vector field b X α on F ( S, M ) with hamiltonianfunction [ ω · α . Indeed, from i X α µ = d α follows by the hat calculus that d ( [ ω · α ) = \ d ω · α + \ ω · d α = \ ω · i X α µ = i b X α d ω · µ = i b X α ¯ ω. If the symplectic form ω is exact, then the corresponding momentum map is J : F ( S, M ) → X ex ( S, µ ) ∗ , h J ( f ) , X α i = \ ( ω · α )( f ) = Z S f ∗ ω ∧ α. It takes values in the regular part of X ex ( S, µ ) ∗ , which can be identified with d Ω ( S ), so we can write J ( f ) = f ∗ ω under this identification.In general the hamiltonian function [ ω · α of b X α depends on the particularchoice of the potential form α of X α . To fix this problem we consider as inSection 3 a continuous right inverse b : d Ω m − ( M ) → Ω m − ( M ) to the differential d , so b ( d α ) is a potential for X α . The corresponding momentum map is J : F ( S, M ) → X ex ( S, µ ) ∗ , h J ( f ) , X α i = \ ( ω · b d α )( f ) = Z S f ∗ ω ∧ b ( d α ) . On a connected component F of F ( S, M ), the non-equivariance of J is mea-sured by the Lie algebra 2–cocycle σ F ( X, Y ) = h J ( f ) , [ X, Y ] i − ¯ ω ( ˆ X, ˆ Y )( f ) = ( ω · b d i Y i X µ )ˆ( f ) − ( ω · i Y i X µ )ˆ( f )= ( ω · P i X i Y µ )ˆ( f ) = Z S f ∗ ω ∧ P i X i Y µ on the Lie algebra of exact divergence free vector fields, for P = 1 − b d theprojection on the subspace of closed ( m − f ∈ F ,because the cohomology class [ f ∗ ω ] ∈ H ( S ) does not depend on the choice of f .The cocycle σ F is cohomologous to the Lichnerowicz cocycle σ f ∗ ω defined in (13)[V09]. Since R S µ = 1, the cocycle σ F is integrable if and only if the cohomologyclass of f ∗ ω is integral [I96]. Remark . The two equivariant momentum maps on the symplectic manifold F ( S, M ), for suitable central extensions of the hamiltonian group Diff ham ( M, ω )and of the group Diff ex ( S, µ ) of exact volume preserving diffeomorphisms, formthe dual pair for ideal incompressible fluid flow [MW83] [GBV09].14
Appendix: Fiber integration
Chapter VII in [GHV72] is devoted to the concept of integration over the fiber inlocally trivial bundles. We particularize this fiber integration to the case of trivialbundles S × M → M , listing its main properties without proofs.Let S be a compact k –dimensional manifold. Fiber integration over S assignsto ω ∈ Ω n ( S × M ) the differential form − R S ω ∈ Ω n − k ( M ) defined by( − Z S ω )( x ) = Z S ω x ∈ Λ n − k T ∗ x M, ∀ x ∈ M, where ω x ∈ Ω k ( S, Λ n − k T ∗ x M ) is the retrenchment of ω to the fiber over x : h ω x ( Z s , . . . , Z n − ks ) , X x ∧ · · · ∧ X kx i = ω ( s,x ) ( X x , . . . , X kx , Z s , . . . , Z n − ks )for all X ix ∈ T x M and Z js ∈ T s S .The properties of the fiber integration used in the text are special cases of thepropositions (VIII) and (X) in [GHV72]:1. Pull-back of fiber integrals: f ∗ − Z S ω = − Z S (1 S × f ) ∗ ω, ∀ f ∈ F ( M ′ , M ) , (15)with infinitesimal version L X − Z S ω = − Z S L S × X ω, ∀ X ∈ X ( M ) . (16)2. Invariance under pull-back by orientation preserving diffeomorphisms of S : − Z S ( ϕ × M ) ∗ ω = − Z S ω, ∀ ϕ ∈ Diff + ( S ) , (17)with infinitesimal version − R S L Z × M ω = 0 , ∀ Z ∈ X ( S ).3. Insertion of vector fields into fiber integrals: i X − Z S ω = − Z S i S × X ω, ∀ X ∈ X ( M ) . (18)4. Integration along boundary free manifolds commutes with differentiation.When ∂S denotes the boundary of the k –dimensional compact manifold S and i ∂ : ∂S → S the inclusion, d − Z S β − − Z S d β = ( − n − k − Z ∂S ( i ∂ × M ) ∗ β (19)holds for any differential n –form β on S × M .15 eferences [AS05] A. Alekseev and T. Strobl, Current algebras and differential geometry ,arXiv:hep-th/0410183v2, 2005.[BZ05] G. Bonelli and M. Zabzine,
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