Symmetries of the space of connections on a principal G-bundle and related symplectic structures
aa r X i v : . [ m a t h . DG ] D ec Symmetries of the space of connections on a principal G -bundle and related symplectic structures Grzegorz Jakimowicz, Anatol Odzijewicz, Aneta Sli˙zewskaOctober 9, 2018
Contents
Aut
T G
T P on the space of connections 74 Space of generalized canonical forms on T ∗ P
95 The Marsden-Weinstein reduction 15
Abstract
We investigate G -invariant symplectic structures on the cotangent bundle T ∗ P ofa principal G -bundle P ( M, G )which are canonically related to automorphisms of thetangent bundle
T P covering the identity map of P and commuting with the action of T G on T P . The symplectic structures corresponding to connections on P ( M, G ) arealso investigated. The Marsden-Weinstein reduction procedure for these symplecticstructures is discussed.
The phase space of a typical Hamiltonian system is the cotangent bundle T ∗ P of itsconfigurations space P equiped with the standard symplectic form. Usually one considersthe case when Hamiltonian of this system is invariant with respect to the cotangent liftof an action of some group G on P . Therefore, it is sometimes reasonable to replace thestandard symplectic form with another G -invariant symplectic form on T ∗ P which retainscertain properties. Assuming that the action of G on P is free and the quotient space P/G is a manifold M one can consider P as the total space of the principal G -bundle P ( M, G ).Motivated by the above we consider here the following structures related to P ( M, G ) in anatural way. 1i) The space
CanT ∗ P of fibre-wise linear non-singular differential one-forms γ on thecotangent bundle T ∗ P , which annihilate the vectors tangent to the fibres of T ∗ P .(ii) The space ConnP ( M, G ) of connections on the principal G -bundle P ( M, G ).(iii) The group
Aut
T G
T P of automorphisms of the tangent bundle
T P covering theidentity map of P which commute with the action of the tangent group T G on T P .Since the canonical one-form γ belongs to CanT ∗ P and other elements of CanT ∗ P posses some properties of γ , hence they are called generalized canonical forms here.There are important relations between the above structures. Namely the group Aut
T G
T P acts on the both subspaces mentioned above. The action on
ConnP ( M, G ) is transitiveand the action on
CanT ∗ P is free. The orbit Can
T G T ∗ P of Aut
T G
T P through the canon-ical form γ consists of G -invariant generalized canonical forms γ such that the momentummaps corresponding to symplectic forms ω = dγ coincide with the momentum map whichcorresponds to the standard symplectic form ω = dγ . In this way one obtains a familyof symplectic forms ω A on T ∗ P enumerated by the elements A of the group Aut
T G
T P .The group
Aut
T G
T P is investigated in Section 2, see Proposition 2.In Section 3 we describe the action of the group
Aut
T G
T P on ConnP ( M, G ) and showthat
Aut
T G
T P can be defined equivalently as the group of symmetries of
ConnP ( M, G ),see Proposition 4.The relations between the group
Aut
T G
T P and the space
Can
T G T ∗ P are investigatedin Section 4, see Proposition 5 and Proposition 6. Among other things it is shown thatfixing a reference connection α one embeds the space of all connections into the space Can
T G T ∗ P of generalized canonical forms, see Corrolary 7. The choice of the referenceconnection α also allows ones to define a G -equivariant diffeomorphism I α : T ∗ P → P × T ∗ e G , see [7, 12], where P is the total space of the pull back P ( T ∗ M, G ) of P ( M, G )on T ∗ M and T ∗ e G is the dual of the Lie algebra T e G . The generalized canonical formsand related symplectic forms ω A written on P × T ∗ e G obtain the form consistent with thestructure of the bundle P × T ∗ e G → T ∗ M , see (47) and (48).In Section 5 we discuss the G -Hamiltonian system on ( T ∗ P, ω A , J ) which could beconsidered as a natural generalization of the ones investigated in [7, 11, 12]. The Marsden-Weinstein reduction procedure is applied to these systems. Let P ( M, G ) be a principal bundle over a manifold M . Throughout the paper we willdenote the right action of the Lie group G on P by κ : P × G → P and write µ : P → M for the bundle projection. We will use also the shorter notation pg := κ ( p, g ). For a fixed p ∈ P and g ∈ G one has the corresponding maps κ p : G → P and κ g : P → P defined by κ p := κ ( p, · ) and κ g := κ ( · , g ) . T G of G is a Lie group itself with the product and theinverse defined as follows X g • Y h := T L g ( h ) Y h + T R h ( g ) X g , X − g := − T L g − ( e ) ◦ T R g − ( g ) X g , where X g ∈ T g G , Y h ∈ T h G and L g ( h ) := gh , R g ( h ) := hg . Let e ∈ G be the unit elementof G and : G → T G be the zero section of the tangent bundle
T G . Then one has X e • Y e = X e + Y e , g • h = gh ,X g • Y e • X − g = ( T R g − ( g ) ◦ T L g ( e )) Y e =: Ad g Y e So, the corresponding Lie algebra T e G can be considered as an abelian normal subgroupof T G and the zero section : G → T G is a group monomorphism.The diffeomorphism I : G × T e G ∋ ( g, X e ) T R g ( e ) X e =: X g ∈ T G (1)allows us to consider
T G as the semidirect product G ⋉ Ad G T e G of G by the T e G , wherethe group product of ( g, X e ) , ( h, Y e ) ∈ G ⋉ Ad G T e G is given by( g, X e ) • ( h, Y e ) = I − ( I ( g, X e ) · I ( h, Y e )) == ( gh, X e + T ( R g − ◦ L g )( e ) Y e ) = ( gh, X e + Ad g Y e ) . Using the Lie group isomorphism (1) and the equality κ g ◦ κ p = κ p ◦ R g , we obtain the action Φ ( g,X e ) ( v p ) = T κ g ( p )( v p + T κ p ( e ) X e ) (2)of G ⋉ Ad G T e G on the tangent bundle T P .Applying the above action one obtains the following isomorphisms
T P/T v P ∼ = T P/T e G, (3) T P/T G ∼ = ( T P/T e G ) /G ∼ = ( T P/G ) /T e G, (4) T M = T ( P/G ) ∼ = T P/T G, (5)of vector bundles, where we write T v P := KerT µ for the vertical subbundle of
T P . Theseisomorphisms will be useful in subsequent considerations.Another group important here which acts on
T P is the group
Aut T P of smoothautomorphisms A : T P → T P of the tangent bundle covering the identity map of P , i.e.for any p ∈ P one has the map A ( p ) : T p P → T p P which is an isomorphism of the tangentspace T p P and A ( p ) depends smoothly on p . Note here that Aut T P is a normal subgroupof the group
AutT P of all automorphisms of
T P .3y
Aut
T G
T P ⊂ Aut T P we denote the subgroup consisting of those elements of
Aut T P whose action on
T P commutes with the action (2) of
T G ∼ = G ⋉ Ad g T e G on T P ,i.e. A ( pg ) ◦ Φ ( g,X e ) = Φ ( g,X e ) ◦ A ( p ) . (6)From the isomorphisms (4) and (5) it follows that the group Aut
T G
T P acts also on vectorbundles
T P/G → M and T M → M . Proposition 1. A ∈ Aut
T G
T P if and only if A ( p ) ◦ T κ p ( e ) = T κ p ( e ) (7) A ( pg ) ◦ T κ g ( p ) = T κ g ( p ) ◦ A ( p ) (8) for any g ∈ G and p ∈ P .Proof. Substituting (2) into (6) we obtain the equality A ( pg )[ T κ g ( p ) v p + ( T κ g ( p ) ◦ T κ p ( e )) X e ] = T κ g ( p ) A ( p ) v p + ( T κ g ( p ) ◦ T κ p ( e )) X e , (9)which is valid for all v p ∈ T p P and X e ∈ T e G . From the relation κ g ◦ κ p = κ pg ◦ I − g , where I g := L g ◦ R − g and Ad g = T I g , setting v p = 0 in (9) we obtain the equality( A ( pg ) ◦ T κ pg ( e ) ◦ Ad g − ( e )) X e = ( T κ pg ( e ) ◦ Ad g − ( e )) X e , valid for any X e ∈ T e G . The above gives (7). In order to show (8) we substitute X e = 0into (9) and apply the same arguments as for the previous case.Now we define the subgroup Aut N T P ⊂ Aut
T G
T P consisting of A ∈ Aut
T G
T P suchthat A ( p ) = id p + B ( p ), where B ( p ) : T p P → T vp P . Conditions (7) and (8) imposed on A ( p ) written in terms of B ( p ) take the form B ( p ) ◦ T κ p ( e ) = 0 B ( pg ) ◦ T κ g ( p ) = T κ g ( p ) ◦ B ( p ) . From the definition of B ( p ) and (2) one has ImB ( p ) ⊂ T vp P ⊂ KerB ( p ) . (10)Thus it follows that B ( p ) B ( p ) = 0 for any id + B , id + B ∈ Aut N P . So, one has A ( p ) ◦ A ( p ) = (id p + B ( p ))(id p + B ( p )) = id p + B ( p ) + B ( p ) , for A ( p ) , A ( p ) ∈ Aut N T P . This shows that
Aut N T P is a commutative subgroup of
Aut
T G
T P . Therefore, we may identify
Aut N T P with the vector subspace
End N T P of4 ndT P which consists of such endomorphisms B ( p ) : T p P → T p P that the property (10)is valid for any p ∈ P .Now, let us recall that by the definition a connection form on P is a T e G -valueddifferential one-form α satisfying the conditions α p ◦ T κ p ( e ) = id T e G , (11) α pg ◦ T κ g ( p ) = Ad g − ◦ α p (12)for the value α p of α at p ∈ P and g ∈ G . Using α one defines the decomposition T p P = T vp P ⊕ T α,hp P (13)of T p P on the vertical T vp P and the horizontal T α,hp P := Kerα p subspaces. Using thedecomposition (13) one defines the vector spaces isomorphismΓ α ( p ) : T µ ( p ) M ∼ → T α,hp P (14)such that Γ α ( pg ) = T κ g ( p ) ◦ Γ α ( p ) and id T µ ( p ) M = T µ ( p ) ◦ Γ α ( p ) . Let us take the decomposition id T p P = Π vα ( p ) + Π hα ( p ) (15)of the identity map of T p P into the sum of projections corresponding to (13). Then wehave Π hα ( p ) = Γ α ( p ) ◦ T µ ( p ) and Π vp ( p ) = T κ p ( e ) ◦ α p . (16) Proposition 2. (i) One has the short exact sequence { id T P } →
Aut N T P ι → Aut
T G
T P λ → Aut T M → { id T M } (17) of the group morphisms, where ι is the inclusion map and λ is an epimorphism of Aut
T G
T P on the group
Aut T M of the automorphisms of the tangent space
T M covering the identity map of M defined by ( λ ( A )( µ ( p ))( T µ ( p )) v p ) := ( T µ ( p ) ◦ A ( p )) v p , (18) where v p ∈ T p P .(ii) Fixing a connection α one defines the injection σ α : Aut T M → Aut
T G
T P by σ α ( ˜ A )( p ) := Π vα ( p ) + Γ α ( p ) ◦ ˜ A ( µ ( p )) ◦ T µ ( p ) , (19) where ˜ A ∈ Aut T M , and the surjection β α : Aut
T G
T P → Aut N T P by β α ( A ) := Aσ α ( λ ( A )) − , (20)5 here A ∈ Aut
T G
T P , which are arranged into the short exact sequence { id T M } →
Aut T M σ α −→ Aut
T G
T P β α −→ Aut N T P → { id T P } , (21) inverse to the sequence (17), i.e. Imσ α = β − α (id T P ) , σ α is a right inverse π ◦ σ α = id T M of π and β α is the left inverse β α ◦ ι = id T P of ι . The map σ α is amonomorphism σ α ( ˜ A ˜ A ) = σ α ( ˜ A ) σ α ( ˜ A ) of the groups and β α satisfies β α ( A A ) = β α ( A ) σ α ( λ ( A )) β α ( A ) σ α ( λ ( A )) − . (iii) The decomposition A ( p ) = (id p + B ( p )) σ α ( ˜ A )( p ) (22) of A ∈ Aut
T G
T P , where id p + B ( p ) ∈ Aut N T P and ˜ A ∈ Aut T M , defines anisomorphism of
Aut
T G
T P with the semidirect product group
Aut T M ⋉ α End N T P ,where the product of ( ˜ A , B ) , ( ˜ A , B ) ∈ Aut T M ⋉ α End N T P is given by [( ˜ A , B ) · ( ˜ A , B )]( p ) := ( ˜ A ( µ ( p )) ˜ A ( µ ( p )) , B ( p )+ B ( p ) ◦ Γ α ( p ) ◦ ˜ A − ( µ ( p )) ◦ T µ ( p )) . (23) Proof.
From the definition (18) of π , for all A , A ∈ Aut
T G
T P , we obtain that λ ( A A )( µ ( p ))( T µ ( p ) v p ) = ( T µ ( p ) ◦ A ( p ))( A ( p ) v p ) == λ ( A )( µ ( p ))( T µ ( p ) ◦ A ( p )) v p = ( λ ( A )( µ ( p )) ◦ λ ( A ( µ ( p )))( T µ ( p ) v p )for T µ ( p ) v p ∈ T µ ( p ) M . Also, for ˜ A , ˜ A ∈ Aut T M , from (19) we have σ α ( ˜ A )( p ) ◦ σ α ( ˜ A )( p ) == (Π vα ( p ) + Γ α ( p ) ◦ ˜ A ( µ ( p )) ◦ T µ ( p ))(Π vα ( p ) + Γ α ( p ) ◦ ˜ A ( µ ( p )) ◦ T µ ( p )) == Π vα ( p ) + Γ α ( p ) ◦ ˜ A ( µ ( p )) ◦ T µ ( p ) ◦ Γ α ( p ) ◦ ˜ A ( µ ( p )) ◦ T µ ( p ) == Π vα ( p ) + Γ α ( p ) ◦ ˜ A ( µ ( p )) ◦ ˜ A ( µ ( p )) ◦ T µ ( p ) = σ α ( ˜ A ˜ A )( p ) . In order to obtain these equalities we used Π vα = Π vα and Π vα ◦ Γ α ( p ) = 0 and T µ ( p ) ◦ Π vα =0. Additionally we have the following equalities( β α ◦ ι )(id T P + B ) = (id T P + B ) σ α (id T P ) = id
T P + B, ( λ ◦ β α )( A ) = λ ( Aσ α ( λ ( A )) − ) = λ ( A ) λ ( A ) − = id T M and ( λ ◦ σ α )( ˜ A )( µ ( p ))( T µ ( p ) v p ) = ( λ ( σ α ( ˜ A ))( µ ( p ))( T µ ( p ) v p ) = T µ ( p )( σ α ( ˜ A )( p ) v p ) =6 T µ ( p )(Π vα ( p ) + Γ α ( p ) ◦ ˜ A ( µ ( p )) ◦ T µ ( p )) v p = ˜ A ( µ ( p ))( T µ ( p ) v p ) . Let us also note that β α ( A ) = id T P if and only if A = σ α ( λ ( A )). This implies that β − α (id T P ) =
Imσ α . Summing up the above statements we prove the points (i) and (ii) ofthe proposition.In order to prove (iii) we first note that the decomposition (22) of A ∈ Aut
T G
T P intothe product of id
T P + B ∈ Aut N T P and σ α ( ˜ A ) ∈ σ α ( Aut T M ) follows from λ ◦ σ α = id T M and from
Aut N T P = kerλ . Taking this fact into account one finds σ α ( ˜ A )( p )(id p + B ( p )) σ α ( ˜ A − )( p ) = id p + B ( p ) ◦ Γ α ( p ) ◦ ˜ A − ( µ ( p )) ◦ T µ ( p )which yields (23).The structural properties of Aut
T G
T P described in the above proposition will be usefulfor the subsequent considerations.
Aut
T G
T P on the space of connections
In this section we describe the relationship between the space
ConnP ( M, G ) of allconnections on P ( M, G ) and the groups from the short exact sequence (17). To this endwe define by φ A ( α ) p := α p ◦ A ( p ) − (24)the left action φ A : ConnP ( M, G ) → ConnP ( M, G ) of
Aut
T G
T P on ConnP ( M, G ), i.e. φ satisfies φ A A = φ A ◦ φ A for A , A ∈ Aut
T G
T P . Proposition 3.
For the groups
Aut
T G
T P , Aut N T P and
Aut T M one has:(i) The action of
Aut
T G
T P defined in (24) is transitive.(ii) The horizontal lift Γ α defined by α ∈ ConnP ( M, G ) , see (14), satisfies the relation A ( p ) ◦ Γ α ( p ) = Γ φ A ( α ) ( p ) ◦ λ ( A )( µ ( p )) (25) for all A ∈ Aut
T G
T P .(iii) The action (24) restricted to the subgroup
Aut N T P is free and transitive.(iv) The subgroup σ α ( Aut T M ) is the stabilizer of α with respect to the action (24).Proof. (i) In order to show that φ A ( α ) ∈ ConnP ( M, G ) we note that it satisfies conditions(11) and (12) if A satisfies (7) and (8). Next let us take the decompositions v p = v vp + v hp v p = v v ′ p + v h ′ p of v p ∈ T p P on the vertical and horizontal parts with respect to the connections α and α ′ ,respectively. One easily sees that A ( p ) : T p P → T p P , defined by A ( p ) v p := v vp + v h ′ p , (26)7atisfies conditions (7) and (8) and α ′ = φ A ( α ). Thus A ∈ Aut
T G ( T P ) and the action(24) is transitive.(ii) The equivariance property (25) follows from
Kerφ A ( α p ) = A ( p ) Kerα p and from T α,hp P = Kerα p . (iii) For any two connections α , α ′ their difference α − α ′ is a T e G -valued tensorialone-form, i.e. Ker ( α − α ′ ) = T vp P and ( α pg − α ′ pg ) ◦ T κ g ( p ) = Ad g − ( α p − α ′ p ) for any p ∈ P . Since, for any p ∈ P , α define the vector space isomorphism α p : T vp P → T e G which is the inverse to T κ p ( e ) : T e G → T vp P , see (11), it follows that B ( p ) := T κ p ( e ) ◦ ( α p − α ′ p ) (27)satisfies (10). From (27) one obtains α ′ = φ A ( α ) where A ( p ) := id p + B ( p ). The aboveproves point (iii).(iv) Straightforward verification.The next proposition shows that one can define the subgroup Aut
T G
T P ⊂ Aut T P interms of the connection space
ConnP ( M, G ). Proposition 4. If A ∈ Aut ( T P ) and φ A ( ConnP ( M, G )) ⊂ ConnP ( M, G ) then A ∈ Aut
T G ( T P ) .Proof. Let A ∈ Aut ( T P ) be such that φ A ( ConnP ( M, G )) ⊂ ConnP ( M, G ) then for any α ∈ ConnP ( M, G ) one has α p ◦ A ( p ) ◦ T κ p ( e ) = α p ◦ T κ p ( e )and α pg ◦ A ( pg ) − ◦ T κ g ( e ) = α pg ◦ T κ g ( p ) ◦ A ( p ) − . The above equalities imply (7) and (8) if \ α ∈ ConnP ( M,G ) Ker α p = { } for any p ∈ P . In order to prove (3) we observe that for any vector subspace H p ⊂ T p P transversal to T vp P and an open subset Ω ⊂ M such that µ − (Ω) ∼ = G × Ω p and µ ( p ) ∈ Ωthere exists a local connection form α on µ − (Ω) for which Ker α p = H p . Assumingparacompactness of M and using the decomposition of the unity for the properly chosencovering of M by Ω p one can construct a connection form α on M such that H p = Ker α p .Let us note that, given an arbitrary c ∈ R \ { } , any connection α ∈ ConnP ( M, G )defines a multiplicative one-parameter subgroup of
Aut
T G
T P , i.e. A c α ◦ A c α = A c c α , for c , c ∈ R \ { } by A cα ( p ) := Π vα ( p ) + c Π hα ( p ) = σ α ( c id T M ) . Space of generalized canonical forms on T ∗ P We recall for further considerations that the standard symplectic form on T ∗ P is ω = dγ , where γ ∈ C ∞ T ∗ ( T ∗ P ) is the canonical one-form on T ∗ P defined at ϕ ∈ T ∗ P by h γ ϕ , ξ ϕ i := h ϕ, T π ∗ ( ϕ ) ξ ϕ i , where π ∗ : T ∗ P → P is the projection of T ∗ P on the base and ξ ϕ ∈ T ϕ ( T ∗ P ).Let us mention also that by definition a linear vector field on T ∗ P is a pair ( ξ, χ ) ofvector fields ξ ∈ C ∞ T ( T ∗ P ) and χ ∈ C ∞ T P such that T ∗ P T ( T ∗ P ) P T P ❄ ❄✲✲ π ∗ T π ∗ ξχ defines a morphism of vector bundles. Note here that T π ∗ ( ϕ ) ξ ϕ = χ π ∗ ( ϕ ) . Regarding thetheory of linear vector fields over vector bundles see e.g. Section 3.4 of [5], where theirvarious properties are discussed.In the sequel we will denote by LinC ∞ T ( T ∗ P ) the Lie algebra of linear vector fieldsover the vector bundle π ∗ : T ∗ P → P . The Lie bracket of ( ξ , χ ), ( ξ , χ ) ∈ LinC ∞ T ( T ∗ P )is defined by [( ξ , χ ) , ( ξ , χ )] := ([ ξ , ξ ] , [ χ , χ ])and the vector space structure on LinC ∞ T ( T ∗ P ) by c ( ξ , χ ) + c ( ξ , χ ) := ( c ξ + c ξ , c χ + c χ ) . Let
LinC ∞ ( T ∗ P ) denote the vector space of smooth fibre-wise linear functions on T ∗ P .Notice that spaces LinC ∞ ( T ( T ∗ P )) and LinC ∞ ( T ∗ P ) have structures of C ∞ ( P )-modulesdefined by f ( ξ, χ ) := (( f ◦ π ∗ ) ξ, f χ ) and by f l := ( f ◦ π ∗ ) l , respectively, where f ∈ C ∞ ( P )and l ∈ LinC ∞ ( T ∗ P ). Definition 1.
A differential one-form γ ∈ C ∞ T ∗ ( T ∗ P ) is called a generalized canonicalform on T ∗ P if:(i) γ ϕ = 0 for any ϕ ∈ T ∗ P ,(ii) kerT π ∗ ( ϕ ) ⊂ ker γ ϕ := { ξ ϕ ∈ T ϕ ( T ∗ P ) : h γ ϕ , ξ ϕ i = 0 } ,(iii) h γ, ξ i ∈ LinC ∞ ( T ∗ P ) for any ξ ∈ LinC ∞ T ( T ∗ P ).The space of generalized canonical forms on T ∗ P will be denoted by CanT ∗ P . Let usnote here that γ ∈ CanT ∗ P . 9 roposition 5. (i) The map Θ :
Aut T P → CanT ∗ P defined by h Θ( A ) ϕ , ξ ϕ i := h ϕ, A ( π ∗ ( ϕ )) T π ∗ ( ϕ )) ξ ϕ i , (28) where ξ ϕ ∈ T ϕ ( T ∗ P ) , is bijective.(ii) The natural left action L ∗ : Aut T P × CanT ∗ P → CanT ∗ P of Aut T P on CanT ∗ P defined by h ( L ∗ A ( γ )) ϕ , ξ ϕ i := h γ A ∗ ( ϕ ) , T A ∗ ( ϕ ) ξ ϕ i , (29) where A ∗ : T ∗ P → T ∗ P is the dual of A ∈ Aut T P , is a transitive and free action.Furthermore, L ∗ A ◦ Θ = Θ ◦ L A , (30) where L A A ′ := AA ′ , i.e. L ∗ A Θ( A ′ ) = Θ( AA ′ ) . Proof. (i) If Θ( A ) = Θ( A ), then using (29), for any T π ∗ ( ϕ ) ξ ϕ ∈ T π ∗ ( ϕ ) P we obtain h ( A ( π ∗ ( ϕ )) ∗ − A ( π ∗ ( ϕ )) ∗ ) ϕ, T π ∗ ( ϕ ) ξ ϕ i = 0 . This gives A = A . So, Θ is an injection.Let us take γ ∈ CanT ∗ P . Then h γ, ξ i ∈ LinC ∞ ( T ∗ P ) if h ξ, χ i ∈ LinC ∞ ( T ( T ∗ P )).By virtue of the point (ii) of Definition 1 the fibre-wise linear functions h γ, ξ i depend onlyon vector fields χ = T π ∗ ξ ∈ C ∞ T P and this dependence defines a morphism of C ∞ ( P )-modules. On the other hand one can consider h γ, ξ i as a section of T ∗∗ P ∼ = T P . Thus wecan represent it as h γ, ξ i ( ϕ ) = h ϕ, χ ′ ( π ∗ ( ϕ ) i (31)by some vector field χ ′ ∈ C ∞ T P . The dependence between h γ, ξ i and χ ′ given by (31) isalso a morphism of C ∞ ( P )-modules. Therefore, there exists A ∈ End T P such that χ ′ ( p ) = A ( p ) χ ( p ) (32)and we have γ = Θ( A ). Substituting (32 ) into (31 ) we obtain h γ ϕ , ξ ϕ i = h ϕ, A ( π ∗ ( ϕ )) T π ∗ ( ϕ ) ξ ϕ i = h A ( π ∗ ( ϕ )) ∗ ϕ, T π ∗ ( ϕ ) ξ ϕ i and, thus γ = Θ( A ).Let us assume that A / ∈ Aut T P . Then there exists ϕ such that A ( π ∗ ( ϕ )) ∗ ϕ = 0. From(34) we see that for this ϕ we have γ ϕ = 0, which contradicts the point (i) of Definition 1.So, A ∈ Aut T P and thus Θ is a surjection. The above proves (i).(ii) Since any element of
CanT ∗ P can be written as Θ( A ′ ) for some A ′ ∈ Aut T P weobtain from the definition (29) that h L ∗ A (Θ( A ′ )) ϕ , ξ ϕ i = h Θ( A ′ ) A ∗ ( ϕ ) , T A ∗ ( ϕ ) ξ ϕ i = h A ∗ ( ϕ ) , A ′ ( π ∗ ( ϕ )) ◦ T π ∗ ( ϕ ) ◦ T A ∗ ( ϕ ) ξ ϕ i == h ϕ, A ( π ∗ ( ϕ )) ◦ A ′ ( π ∗ ( ϕ )) ◦ T ( π ∗ ◦ A ∗ )( ϕ ) ξ ϕ i = h ϕ, A ( π ∗ ( ϕ )) ◦ A ′ ( π ∗ ( ϕ )) ◦ T ( π ∗ )( ϕ ) ξ ϕ i == h Θ( AA ′ ) ϕ , ξ ϕ i , which proves (30). From (30) and from the point (i) of the proposition it follows that L ∗ is a transitive and free action. 10rom the above proposition we conclude that γ ∈ CanT ∗ P is the pull-back γ = Θ( A ) = L ∗ A γ of the canonical form γ . So, ω A := d Θ( A ) is a symplectic form.The lift Φ ∗ g : T ∗ P → T ∗ P of the action κ g : P → P to the cotangent bundle T ∗ P isdefined by Φ ∗ g ( ϕ )( pg ) = ( T κ g ( p ) − ) ∗ ϕ (33)where p = π ∗ ( ϕ ).If γ ∈ CanT ∗ P is G -invariant with respect to (33), then L ξ X γ = 0 for X ∈ T e G , where ξ X ∈ C ∞ T ( T ∗ P ) is the fundamental vector field, i.e. the vector field tangent to the flow t → Φ ∗ exp tx . So, for a G -invariant symplectic form ω A = dγ = d Θ( A ) one has ξ X x ω = − d h J A , X i where the G -equivariant momentum map J A : T ∗ P → T ∗ e G is given by J A = J ◦ A ∗ . Wenote here that for the standard symplectic form ω = dγ the momentum map is J ( ϕ ) = ϕ ◦ T κ π ∗ ( ϕ ) ( e )It is reasonable to define the space Can
T G T ∗ P := Θ( Aut
T G
T P )which is an
Aut
T G
T P -invariant subspace of the space
CanT ∗ P . Proposition 6. (i) The generalized canonical form Θ( A ) belongs to Can
T G T ∗ P if andonly if ( φ ∗ g ) ∗ Θ( A ) = Θ( A ) and J A = J .(ii) One can consider Can
T G T ∗ P as the orbit of the subgroup Aut
T G
T P ⊂ Aut T P taken through γ with respect to the free action L ∗ defined in (29).(iii) If A ∈ Aut T P and L ∗ A ( Can
T G T ∗ P ) ⊂ Can
T G T ∗ P then A ∈ Aut
T G
T P .Proof. (i) The canonical form Θ( A ) is G -invariant if and only if h Θ( A ) Φ ∗ g ( ϕ ) , T Φ ∗ g ( ϕ ) ξ ϕ i = h Θ( A ) ϕ , ξ ϕ i (34)for any g ∈ G . For the left hand side of (34) we have h ( φ ∗ g ) ∗ Θ( A ) ϕ , ξ ϕ i = h Θ( A ) Φ ∗ g ( ϕ ) , T Φ ∗ g ( ϕ ) ξ ϕ i = (35)= h Φ ∗ g ( ϕ ) , A ( π ∗ (Φ ∗ g ( ϕ )) T π ∗ (Φ ∗ g ( ϕ )) T Φ ∗ g ( ϕ ) ξ ϕ i == h ϕ, T κ g ( p ) − A ( π ∗ ◦ Φ ∗ g )( ϕ ) T ( π ∗ ◦ Φ ∗ g )( ϕ ) ξ ϕ i == h ϕ, T κ g ( p ) − A ( π ∗ ( ϕ ) g ) T κ g ( π ∗ ( ϕ )) T π ∗ ( ϕ ) ξ ϕ i = h Θ( A ) ϕ , ξ ϕ i for any ϕ ∈ T ∗ P and ξ ϕ ∈ T ϕ ( T ∗ P ). Note that the second equality in (35) follows from π ∗ ◦ Φ ∗ g = κ g ◦ π ∗ . From (35) and from the definition (28) we obtain the condition (8).11rom J A = J we have h J ( ϕ ) , X i = h Θ( A ) ϕ , ξ Xϕ i = h ϕ, A ( π ∗ ( ϕ )) T κ π ∗ ( ϕ ) ( e ) X i == h ϕ, T κ π ∗ ( ϕ ) ( e ) X i for all ϕ ∈ T ∗ P and X ∈ T e G . This shows that an element A ∈ Aut T P satisfies (7).(ii) This statement follows from (i) and from L ∗ A γ = Θ( A ).(iii) If L ∗ A Can
T G T ∗ P ⊂ Can
T G T ∗ P then L ∗ A Θ( A ) = Θ( AA ) ∈ Can
T G T ∗ P forany A ∈ Aut
T G
T P . So, due to the property (i) we have AA ∈ Aut
T G
T P and, thus A ∈ Aut
T G
T P.
In Section 2 we fixed a reference connection α in order to investigate the structure ofgroup Aut
T G
T P , see Proposition 2. Now, taking into consideration Proposition 6 we studythe structure of generalized canonical forms Θ( A ) ∈ Can
T G T ∗ P using decompositions (22)and (15), and (16). We obtain A ( p ) = Π vα ( p ) + Π vα ( p ) A ( p )Π hα ( p ) + Π hα ( p ) A ( p )Π hα ( p ) = (36)= T κ p ( e ) ◦ α p + (id T P + B )( p ) ◦ Γ α ( p ) ◦ ˜ A ( µ ( p )) ◦ T µ ( p ) , where id T P + B ∈ Aut N T P and ˜ A ∈ Aut T M . Substituting A given by (36) into thedefinition (28) we find the corresponding formula for Θ( A )Θ( A )( ϕ ) = ϕ ◦ T κ π ∗ ( ϕ ) ◦ α π ∗ ( ϕ ) + (37)+ ϕ ◦ (id T P + B )( π ∗ ( ϕ )) ◦ Γ α ( π ∗ ( ϕ )) ◦ ˜ A (( µ ◦ π ∗ )( ϕ )) ◦ T ( µ ◦ π ∗ )( ϕ ) . In particular cases when ˜ A = id T M and B = 0 we haveΘ(id + B )( ϕ ) = ϕ ◦ T π ∗ ( ϕ ) + ϕ ◦ B ( π ∗ ( ϕ )) ◦ T π ∗ ( ϕ ) = (38)= ϕ ◦ T π ∗ ( ϕ ) + ϕ ◦ T κ π ∗ ( ϕ ) ( e ) ◦ ( α ′ π ∗ ( ϕ ) − α π ∗ ( ϕ ) ) ◦ T π ∗ ( ϕ )andΘ( σ α ( ˜ A ))( ϕ ) = J ( ϕ ) ◦ α π ∗ ( ϕ ) ◦ T π ∗ ( ϕ ) + ϕ ◦ Γ α ( π ∗ ( ϕ )) ◦ ˜ A (( µ ◦ π ∗ )( ϕ )) ◦ T ( µ ◦ π ∗ )( ϕ ) , respectively. Let us note that Θ( σ α (id T M )) = Θ(id
T P ) = ϕ ◦ T π ∗ ( ϕ ), i.e. Θ( σ α (id T M )) isthe canonical one-form γ . Corrolary 7.
Fixing a connection α one obtains from (38) an embedding ι α : ConnP ( M, G ) ֒ → Can
T G T ∗ P of the connection space into the space of generalized canonical forms definedas follows ι α ( α ′ ) := ϕ ◦ T π ∗ ( ϕ ) + ϕ ◦ T κ π ∗ ( ϕ ) ( e ) ◦ ( α ′ π ∗ ( ϕ ) − α π ∗ ( ϕ ) ) ◦ T π ∗ ( ϕ ) . (39) The symplectic form dι α ( α ′ ) is the pullback L ∗ id TP + B ω of the standard symplectic form ω by the bundle morphism (id T P + B ) ∗ : T ∗ P → T ∗ P , where id T P + B ∈ Aut N T P is definedin (27). G -bundle in general, suchconnections exist in some particular cases. For example, if the principal bundle P ( M, G )is trivial, P = M × G , or when P is a Lie group and G is its subgroup. In the lastcase there exists the connection which is invariant with respect to the left action of P on itself and this connection is defined in a unique way, see Theorem 11.1 in [3]. Let usmention that in the case when the reference connection α is determined by some additionalconditions, for example by the symmetry properties, then the connection α ′ , see (39), canbe naturally interpreted as an external field which interacts with a particle localized inthe configuration space P .In [7, 11] a G -equivariant diffeomorphism I α : T ∗ P ∼ → P × T ∗ e G dependent on a fixedconnection α was considered, where P := { ( ˜ ϕ, p ) ∈ T ∗ M × P : ˜ π ∗ ( ˜ ϕ ) = µ ( p ) } is the total space of the principal bundle P ( T ∗ M, G ) being the pullback of the principal-bundle P ( M, G ) to T ∗ M by the projection ˜ π ∗ : T ∗ M → M of T ∗ M on the base M . Thisdiffeomorphism is defined as follows I α ( ϕ ) := (Γ ∗ α ( π ∗ ( ϕ ))( ϕ ) , π ∗ ( ϕ ) , J ( ϕ )) . The correctness of the above definition follows from ˜ π ∗ ◦ Γ ∗ α = µ ◦ π ∗ . The map I − α : P × T ∗ e G → T ∗ P given by I − α ( ˜ ϕ, p, χ ) = ˜ ϕ ◦ T µ ( p ) + χ ◦ α p (40)is the inverse to I α .The natural right action of Aut
T G
T P on T ∗ P , defined for A ∈ Aut
T G
T P by ( A ∗ ϕ )( π ∗ ( ϕ )) := ϕ ◦ A ( π ∗ ( ϕ )), and the action of G on T ∗ P defined in (33) transported by I α to P × T ∗ e G are given by Λ α ( A )( ˜ ϕ, p, χ ) := ( I α ◦ A ∗ ◦ I − α )( ˜ ϕ, p, χ ) = (41)= (( ˜ ϕ ◦ T µ ( p ) + χ ◦ α p ) ◦ A ( p ) ◦ Γ α ( p ) , p, χ )and by ψ ∗ g ( ˜ ϕ, p, χ ) := ( I α ◦ φ ∗ g ◦ I − α )( ˜ ϕ, p, χ ) = ( ˜ ϕ, pg, Ad ∗ g − χ ) , (42)respectively. Setting A = id T P + B or A = σ α ( ˜ A ) in (41) we obtainΛ α (id T P + B )( ˜ ϕ, p, χ ) = ( ˜ ϕ + χ ◦ α p ◦ B ( p ) ◦ Γ α ( p ) , p, χ ) (43)or Λ α ( σ α ( ˜ A ))( ˜ ϕ, p, χ ) = ( ˜ ϕ ◦ ˜ A, p, χ ) , (44)respectively. Summarizing, let us mention some properties of the above two actions:(i) The action Λ α of Aut
T G
T P on P × T ∗ e G is reduced to an action of Aut
T G
T P on T ∗ M which preserves the cotangent spaces T ∗ m M , m ∈ M , and is realized on themby affine maps, see (41), (43) and (44).13ii) The action (42) of G does not change ˜ ϕ and commute with the action (41) of thegroup Aut
T G
T P .Using I − α : P × T ∗ e G → T ∗ P we pull the generalized canonical form Θ( A ) back to P × T ∗ e G . For this reason note that a vector ξ ( ˜ ϕ,p,χ ) ∈ T ( ˜ ϕ,p,χ ) ( T ∗ M × P × T ∗ e G ) is tangentto P × T ∗ e G ⊂ T ∗ M × P × T ∗ e G if and only if T (˜ π ∗ ◦ pr )( ˜ ϕ, p, χ ) ξ ( ˜ ϕ,p,χ ) = T ( µ ◦ pr )( ˜ ϕ, p, χ ) ξ ( ˜ ϕ,p,χ ) , (45)where pr ( ˜ ϕ, p, χ ) := ˜ ϕ and pr ( ˜ ϕ, p, χ ) := p . The equality (45) follows from ˜ π ∗ ◦ pr = µ ◦ pr . For A = (id T P + B ) σ α ( ˜ A ) we have h ( I − α ) ∗ Θ( A )( ˜ ϕ, p, χ ) , ξ ( ˜ ϕ,p,χ ) i = (46)= h I − α ( ˜ ϕ, p, χ ) , A ( π ∗ ( I − α ( ˜ ϕ, p, χ ))) ◦ T π ∗ ( I − α ( ˜ ϕ, p, χ )) ◦ T I − α ( ˜ ϕ, p, χ ) ξ ( ˜ ϕ,p,χ ) i == h I − α ( ˜ ϕ, p, χ ) , A ( π ∗ ◦ I − α )( ˜ ϕ, p, χ ) ◦ T ( π ∗ ◦ I − α )( ˜ ϕ, p, χ ) ξ ( ˜ ϕ,p,χ ) i == h ˜ ϕ ◦ T µ ( p ) + χ ◦ α p , A ( p ) T pr ( ˜ ϕ, p, χ ) ξ ( ˜ ϕ,p,χ ) i == h ˜ ϕ ◦ ˜ A ( µ ( p )) , T µ ( p ) ◦ T pr ( ˜ ϕ, p, χ ) ξ ( ˜ ϕ,p,χ ) i + h χ ◦ α p , A ( p ) T pr ( ˜ ϕ, p, χ ) ξ ( ˜ ϕ,p,χ ) i == h ˜ ϕ ◦ ˜ A ( µ ( p )) , T (˜ π ∗ ◦ pr )( ˜ ϕ, p, χ ) ξ ( ˜ ϕ,p,χ ) i + h χ ◦ α p , A ( p ) ◦ T pr ( ˜ ϕ, p, χ ) ξ ( ˜ ϕ,p,χ ) i , where we have used (40), (45) and π ∗ ◦ I − α = pr . Omitting ξ ( ˜ ϕ,p,χ ) in (46) we obtain( I − α ) ∗ Θ( A )( ˜ ϕ, p, χ ) = (47)= ˜ ϕ ◦ ˜ A ( µ ( p )) ◦ T (˜ π ∗ ◦ pr )( ˜ ϕ, p, χ ) + χ ◦ α p ◦ A ( p ) ◦ T pr ( ˜ ϕ, p, χ ) == pr ∗ ( ˜Θ( ˜ A )( ˜ ϕ, p, χ ) + h pr ( ˜ ϕ, p, χ ) , pr ∗ (Φ A − ( α ))( ˜ ϕ, p, χ ) i , where pr ( ˜ ϕ, p, χ ) := χ . The symplectic form corresponding to (47) is given by d (( I − α ) ∗ Θ( A )) = (48)= pr ∗ ( d ˜Θ( ˜ A )) + h d pr ∧ , pr ∗ (Φ A − ( α )) i + h pr , pr ∗ ( d Φ A − ( α )) i . Let us note that ( I − α ) ∗ Θ( A ) consists of the pull back on P × T ∗ e G of the generalizedcanonical form ˜Θ( ˜ A ) ∈ CanT ∗ M by pr : P × T ∗ e G → T ∗ M and the part defined by theconnection form Φ A − ( α ). 14 The Marsden-Weinstein reduction
Considering P as the configuration space of a physical system which has a symmetrydescribed by G one consequently assumes that its Hamiltonian H ∈ C ∞ ( T ∗ P ) is a G -invariant function on T ∗ P , i.e. H ◦ φ ∗ g = H for g ∈ G . Hence it is natural to consider theclass of Hamiltonian systems on G -symplectic manifold ( T ∗ P, ω A , J ) with a G -invariantHamiltonians H .Using the isomorphism ( T ∗ P, ω A , J ) ∼ = ( P × T ∗ e G, ( I − α ) ∗ ω A , pr ) of G -symplectic man-ifolds, where the symplectic form ( I − α ) ∗ ω A is presented in (48) and the momentum mapis J ◦ I α = pr , one defines (see [7, 11]) the G -invariant Hamiltonian H ∈ C ∞ ( P × T ∗ e G )as follows H ( ˜ ϕ, p, χ ) := ( ˜ H ◦ µ )( ˜ ϕ, p, χ ) + ( C ◦ pr )( ˜ ϕ, p, χ ) , where µ : P → T ∗ M is the projection of the total space P of the principal G -bundle P ( T ∗ M, G ) on the base T ∗ M and ˜ H ∈ C ∞ ( T ∗ M ). Coming back to the phase space( T ∗ P, ω A , J ) one obtains the G -Hamiltonian system with the Hamiltonian H α ( ϕ ) := ( H ◦ I α )( ϕ ) = ( ˜ H ◦ Γ ∗ α )( ϕ ) + ( C ◦ J )( ϕ ) . Let us stress that in the case of ( T ∗ P, ω A , J , H α ) only the Hamiltonian H α of the systemdepends on α ∈ ConnP ( M, G ) and in the case of ( P × T ∗ e G, ( I − α ) ∗ ω A , pr , H ) the onlysymplectic form ( I − α ) ∗ ω A .In [2, 4, 7, 8, 9, 10, 11, 12, 13] there were presented various models of the descriptionof motion of a classical particle in the external Yang-Mills field given by the connection α and by the Hamiltonian H . In these models the basic symplectic structure on T ∗ P isgiven by the standard symplectic form ω . Here allowing the generalized symplectic form ω A we extend the class of models under investigations.The G -invariance of the Hamiltonian system ( T ∗ P, ω A , J , H α ) allows ones to applythe Marsden-Weinstein reduction procedure [6]. For this reason we consider the dual pairof Poisson manifolds ( T ∗ P, ω A )( T ∗ P/G, {· , ·} A/ G ) ( T ∗ e G, {· , ·} L − P ) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✠ ❅❅❅❅❅❅❘ π ∗ G J (49)in the sense of Subsection 9.3 in [1]. Recall that the symplectic form ω A is a G -invarianttwo-form. The Poisson bracket { f, g } A/G of f, g ∈ C ∞ ( T ∗ P/G ) is defined by { f ◦ π ∗ G , g ◦ π ∗ G } A , where we identify C ∞ ( T ∗ P/G ) with the Poisson subalgebra C ∞ G ( T ∗ P ) ⊂ C ∞ ( T ∗ P )of G -invariant functions and {· , ·} A is the Poisson bracket on C ∞ ( T ∗ P ) defined by ω A . By {· , ·} L − P we denoted Lie-Poisson bracket on the dual T ∗ e G of the Lie algebra T e G .Note that surjective submersions in (49) are Poisson maps and the Poisson subalgebras( π ∗ G ) ∗ ( C ∞ ( T ∗ P/G )) and J ∗ ( C ∞ ( T ∗ e G )) are mutually polar. As a consequence of the aboveone obtains the one-to-one correspondence between the coadjoint orbits O ⊂ T ∗ e G of G S ⊂ T ∗ P/G of the Poisson manifold ( T ∗ P/G, {· , ·} A/G ) whichis defined as follows S = π ∗ G ( J − ( O )) and O = J ( π ∗ G − ( S )) . Let us stress that the manifold structure of a symplectic leaf S does not depend on thechoise of A ∈ Aut
T G
T P , but its symplectic structure ω S A does. The action of Aut
T G
T P on T ∗ P commutes with the action (33) of G on T ∗ P , so, it defines an action of Aut
T G
T P on the quotient manifold T ∗ P/G . By the definition of
Can
T G T ∗ P , the momentum map J A for ω A coincides with J = J A . Thus we conclude that the action of A ′ ∈ Aut
T G
T P on T ∗ P/G preserves the symplectic leaves S and transforms their symplectic forms in thefollowing way ω S A → ω S A ′ A .Since I α : T ∗ P → P × T ∗ e G is a G -equivariant map it defines a diffeomorphism[ I α ] : T ∗ P/G → P × Ad ∗ G T ∗ e G of the quotient manifolds which transports the Poisson structure {· , ·} A/G of T ∗ P/G onthe total space P × Ad ∗ G T ∗ e G of the vector bundle P × Ad ∗ G T ∗ e G → T ∗ M over the symplecticmanifold ( T ∗ M, d ˜Θ( ˜ A )). Using (5) one obtains the isomorphisms [ I α, O ] = π ∗ G ( J − ( O )) ∼ → P × Ad ∗ G O of symplectic leaves. If A = σ α ( ˜id T M ) = id
T P one obtains the diffeomorphismsof symplectic leaves constructed in [7, 11] where the coadjoint orbit O is the phase spacefor inner degrees of freedom. In this case the symplectic manifold ( T ∗ M, d ˜ γ ) is the phasespace for external degrees of freedom and P × Ad ∗ G O is the total phase space of a classicalparticle interacted with Yang-Mills field described by α which was constructed in [11].If ρ ∈ T ∗ e G is such that Ad ∗ G ρ = ρ then O = { ρ } . Hence the symplectic leaf S = P × Ad ∗ G O is isomorphic as a manifold with T ∗ M but the reduced symplectic form ω SA of S depends on the choice of A ∈ Aut
T G
T P . For example the above situation happens forall ρ ∈ T ∗ e G if G is a commutative group or if ρ = 0.Ending let us mention that all constructions presented above have an equivarianceproperties with respect to the group Aut
T G
T P . References [1] Coste, A., Dazord, P., Weinstein, A.: Groupo¨ıdes symplectiques, Publications duD´epartement de Math´ematiques de l’Universit´e Lyon, Dept. Math. Univ. Claude-Bernard Lyon I, 1–67 (1987)[2] B. Kerner..
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Canonical formulations of a classical particle in a Yang-Mills fieldand Wong’s equations . Lett. Math.Phys. 8 (1984) 59-67.[8] J.´Sniatycki Geometric Quantization and Quantum Mechanics
Springer-Verlag , 1977.[9] J.´Sniatycki On Hamiltonian dynamics of particles with gauge degrees of freedom
J.Hadronic , 2, 642-656 (1979).[10] J.M.Souriau Structure des Systemes Dynamiques
Dunod, Paris , 1970.[11] S.Sternberg Minimal coupling and the symplectic mechanics of a classical particle inthe presence of a Yang-Mills field.
Proc.Natl.Acad.Sci.USA. ,74(12):5253-5254, 1977.[12] A. Weinstein. A universal phase space for particles in Yang-Mills fields.
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