Symplectic Diﬀeology - The Hebrew...

CHAPTER 9

Symplectic Di!eology

The generalization of classical symplectic geometry to di!eology needs first anappropriate extension of the classical notion of moment map.1 We know alreadythat di!eology is suitable for describing, in a unique and satisfactory way, manifoldsor infinite dimensional spaces, as well as singular quotients. But, if di!eology excelswith covariant objects, like di!erential forms, it is more subtle when it is a questionof contravariant objects like vector fields, Lie algebra,2 kernels, etc. Thus, in orderto build a good di!eological theory of the moment map and to avoid useless debates,we need to get rid from everything related to contravariant geometrical objects.

Actually, the notion of moment map is not really an object of the symplecticworld, but relates more generally to the category of spaces equipped with closed2-forms. The nondegeneracy condition is secondary and can be first skipped fromthe data. This has been underlined explicitly by Souriau in his symplectic formula-tion of Noether’s theorem, which involves presymplectic manifolds. On symplecticmanifolds Noether’s theorem is silent.3 The moment map is just an object of theworld of di!erential closed forms, and there is no reason a priori that it couldnot be extended to di!eology which o!ers a pretty well developed framework forCartan-De Rham calculus.

In order to generalize the moment map in di!eology, we need to understand itsmeaning, and this meaning lies in the following simplest possible case. Let M be amanifold equipped with a closed 2-form!. Let G be a Lie group acting smoothly onM and preserving !, that is, g!

M(!) = ! for all elements g of G, where gM denotesthe action of g on M. Let us assume that ! is exact, ! = d", and moreover that" also is invariant by the action of G. Then, for every point m of M, the pullbackof " by the orbit map m : g !! gM(m) is a left-invariant 1-form of G, that is, anelement of the dual of the Lie algebra G!. The map µ : m !! m!(") is exactly themoment map of the action of G on the pair (M,!)—at least one of the momentmaps, since they are defined up to constants. As we can see, this construction doesnot involve really the Lie algebra of G but the space G! of left-invariant 1-forms onG. Since this space is well defined in di!eology, we just have to replace “manifold”by “di!eological space”, and “Lie group” by “di!eological group”, and everythingworks the same way. Thus, let us change the manifold M for a di!eological space4

1The notion of moment map in the framework of classical symplectic geometry was originallyintroduced in the early 1970s by Souriau; see [Sou70].

2Several authors, beginning with Souriau, proposed some generalizations of Lie algebra indi!eology. But it does not seem to exist a unique good choice. Such generalizations rely actuallyon the kind of problem treated.

3Noether’s theorem states that the moment map is constant on the characteristics of the2-form; if the form is nondegenerate, then the characteristics are trivial.

4The space X will be assumed to be connected, as many results need this hypothesis.

299

300 9. SYMPLECTIC DIFFEOLOGY

X, and let G be some di!eological group. Let us continue to denote the space of left-invariant 1-forms on G by G!, even if the star does not refer a priori to some duality,and let us simply call it the space of momenta of the group G. Note that the groupG continues to act on G! by pullback of its adjoint action Ad : (g, k) !! gkg!1, sowe do not lose the notions of coadjoint action and coadjoint orbits.

Next, if we got the good space of momenta, which is the space where themoment maps are assumed to take their values, the problem remains that not everyG-invariant closed 2-form is exact. And moreover, even if such form is exact, thereis no reason for some of its primitives to be G-invariant. We shall pass over thisdi"culty by introducing an intermediary, on which we can realize the simple casedescribed above. This intermediary is the space Paths(X), of all the smooth pathsin X, where the group G acts naturally by composition. And since Paths(X) carriesa natural functional di!eology, it is legitimate to consider its di!erential forms, andthis is what we do. By integrating ! along the paths, we get a di!erential 1-formdefined on Paths(X), and invariant by the action of G. The exact tool used hereis the Chain-Homotopy Operator K. The 1-form " = K!, defined on Paths(X), isa G-invariant primitive of the 2-form # = (1! ! 0!)(!), where 1 and 0 map everypath in X to its ends. Thus, thanks to the construction described above, we get amoment map $ for the 2-form # = d" and the action of G on Paths(X). But thispaths moment map $ is not the one we are waiting for. We need to push it downon X, or rather on X" X. Now, if we get this way a 2-points moment map % welldefined on X" X, it no longer takes its value in G!, as $ does, but in the quotientG!/& , where & is the image by $ of all the loops in X. Fortunately, & = $(Loops(X))is a subgroup of (G!,+) and depends on the loops only through their free homotopyclasses. In other words, & is a homomorphic image of the Abelianized fundamentalgroup 'Ab

1 (X) of X. Well, it is not a big deal to have the moment map taking itsvalues in some quotient of the space of momenta, we can live with that, especiallyif the group & is invariant under the coadjoint action of G, which is actually thecase.5 But we are not completely done: the usual moment map is not a 2-pointsfunction, but a 1-point function. Hence, we have to extract our usual moment mapsfrom this 2-points function %. This is fairly easy, thanks to its very definition, themoment map $ satisfies an additive property for the concatenation of paths, andthe moment map % inherits this property as a cocycle condition: for any threepoints x, x " and x "" of X we have %(x, x ") +%(x ", x "") = %(x, x ""). Therefore, for Xconnected, there exists always a map µ such that %(x, x ") = µ(x ")! µ(x), and anytwo such maps di!er just by a constant. We get finally our wanted moment mapsµ, defined in the di!eological framework. The only di!erence, with the simplestcase described above, is that a moment map takes its values in some quotient of thespace of momenta, instead the space of momenta itself. But this is in fact alreadythe case in the classical theory. It does not appear explicitly because people focusmore on Hamiltonian actions than just on symplectic actions. Actually, the group& represents the very obstruction, for the action of G on (X,!), to be Hamiltonian.We shall call & the holonomy of the action of G.

Now, let us come back to some properties of the various moment maps intro-duced above. The paths moment map $ and its projection % are equivariant withrespect to the action of G on X and the coadjoint action of G on G!, or the projection

5More precisely, the elements of & are not just elements of G! but are moreover closed, andtherefore invariant, each of them, by the coadjoint action of G.

9. SYMPLECTIC DIFFEOLOGY 301

of the coadjoint action on G!/! . But this is no longer the case for the moment mapsµ. The variance of the maps µ reveals a family of cocycles " from G to G!/! dif-fering just by coboundaries, and generalizing the Souriau cocycles [Sou70]. Theircommon class # belongs to the cohomology group H1(G,G!/!), and will be calledthe Souriau class of the action of G of (X,$). The Souriau class # is precisely theobstruction for the 2-points moment map % to be exact, that is, for some momentmap µ to be equivariant. Actually, # is just the pullback, at the group level, ofthe class of % regarded as a cocycle. In parallel with the classical situation, everySouriau cocycle " defines a new action of G on G!/! , which we still call the a!necoadjoint action (associated with "). And the image of a moment map µ is a col-lection of coadjoint orbits for this action. We call these orbits the (!, ")-coadjointorbits of G. Two di"erent cocycles give two families of orbits translated by thesame constant.

Let us remark that the holonomy group ! and the Souriau class # appear clearlyon a di"erent level of meaning: the first one is responsible for the non-Hamiltoniancharacter of the action of G and the second characterizes the lack of equivarianceof the moment maps.

Well, until now we did not use all the facilities o"ered by the di"eologicalframework. Since we do not restrict ourselves to the category of Lie groups, nothingprevents us from considering the group of all the automorphisms of the pair (X,$),that is, the group Di"(X,$) of all the di"eomorphisms of X preserving $. Thisgroup is a natural di"eological group, acting smoothly on X. Thus, everything builtabove applies to Di"(X,$), and every other action preserving$, of any di"eologicalgroup, passes through Di"(X,$), and through the associated object of the theorydeveloped here. Therefore, considering the whole group of automorphisms of theclosed 2-form $ of X, we get a natural notion of universal moment maps &!,%! and µ!, universal holonomy !!, universal Souriau cocycles "!, and universalSouriau class #!. By the way, this universal construction suggests a simple andnew characterization, for any di"eological space X equipped with a closed 2-form $,of the group of Hamiltonian di!eomorphisms Ham(X,$), as the largest connectedsubgroup of Di"(X,$) whose holonomy vanishes.

It is interesting to notice that, contrary to the original constructions [Sou70]and most of their generalizations, the theory described above is essentially global,more or less algebraic, does not refer to any di"erential or partial di"erential, equa-tion, and does not involve any notion of vector field or functional analysis tech-niques.

Considering the classical case of a closed 2-form $ defined on a manifold M,we show in particular that $ is nondegenerate if and only if the group Di"(M,$)is transitive on M and if a universal moment map µ! is injective. In other words,symplectic manifolds are identified, by the universal moment maps, with somecoadjoint orbits—in our general sense—of their group of symplectomorphisms. Thisidea that every symplectic manifold is a coadjoint orbit is not new, it is actuallyvery natural and suggested by a well known classification theorem for symplectichomogeneous Lie group actions [Kir76], [Kos70], [Sou70]. This has been statedalready in a di"erent context, for example in [Omo86]. But the real question is,How can we make this statement rigorous at a good price, without involving theheavy functional analysis apparatus? This is what brings di"eology.


The examples and exercises at the end of this chapter show several situationsinvolving di!eological groups which are not Lie groups, or involving di!eologicalspaces which are not manifolds. We can see, for example, the general theory ap-plying meaningfully to the singular symplectic irrational tori for which topologyis irrelevant. These general constructions of moment maps are also applied to afew examples in infinite dimension, and also when finite and infinite dimensionsare mixed. Finally, two exercises on orbifolds exhibit a strong di!erence betweenclassical symplectic geometry and what we expect from its di!eological counter-part. These examples show without any doubt the ability of this theory to treatcorrectly, in a unique framework, avoiding heuristic arguments, the large variety ofsituations we can find in the mathematical literature today. Infinite dimensionalheuristic examples can be found for instance in [Dnl99]. The solutions of someof the exercises need tedious computations, which just shows di!eology at work inthis particular field.

In conclusion, besides the point that the construction developed in this chapteris a first step in the elaboration of the symplectic di!eology program, I would em-phasize the fact that, since {Manifolds} is a full subcategory of {Di!eology}, all theconstructions developed here apply to manifolds and give a faithful description ofthe classical theory of moment maps. As we have seen, there is no mention, and nouse, of Lie algebra or vector fields in this expose. This reveals the fact that theseobjects also are useless in the traditional approach, and can be avoided. And, Iwould add, they should be avoided. Not just because they can then be extended tolarger categories, but because the use of contravariant objects hides the deep factthat the theory of moment maps is a pure covariant theory. Let us take an exam-ple. We know that since coadjoint orbits of Lie groups are symplectic they are evendimensional. This is often regarded as a miracle, since it is not necessarily the casefor adjoint orbits. But if we think that the Lie algebra has little to do with the spaceof momenta of a Lie group, there is no more miracle, just di!erent behaviors fordi!erent objects, which is unsurprising. Moreover I would add, but this can appearas more or less subjective, that avoiding all this va-et-vient between Lie algebra anddual of Lie algebra, the di!eological approach of the moment maps is much simpler,and even deeper, than the classical approach. Compare for example the Souriaucocycle constructions in the original “Structure des systemes dynamiques” [Sou70]and in di!eology. The only crucial property used here is connectedness, that is,the existence of enough smooth paths connecting points in spaces. Finally, it goeswithout saying that the di!eological approach respects scrupulously the principleof minimality required by mathematics.

The Paths Moment Map

We shall introduce the various flavors of moment map in di!eology step by step.The first step consists, in this section, in defining the paths moment map.

9.1. Definition of the paths moment map. Let X be a di!eological space,and let ! be a closed 2-form defined on X. Let G be a di!eological group, and let" : G ! Di!(X) be a smooth action. Let us denote by the same letter the naturalaction of G on Paths(X), induced by the action " of G on X, that is, for all g ! G,for all p ! Paths(X),

"(g)(p) = "(g) " p = [t #! "(g)(p(t))].

THE PATHS MOMENT MAP 303

Let us assume now that the action ! of G on X preserves ", that is, for all g ! G,

!(g)!(") = " or ! ! Hom!(G,Di!(X,")).

Let K be the Chain-Homotopy Operator (art. 6.83), so K" is a 1-form on Paths(X),and the action of G on Paths(X) preservesK". This is a consequence of the varianceof the Chain-Homotopy Operator (art. 6.84). Thus, for all g ! G,

!(g)!(K") = K".

Now, let p be a path in X, and let p : G ! Paths(X) be the orbit map, p(g) =!(g) " p. Then, the pullback p!(K") is a left-invariant 1-form of G, that is, anelement of G!. The map

# : Paths(X) ! G!, defined by #(p) = p!(K"),

is smooth with respect to the functional di!eology, # ! C!(Paths(X),G!). Themap # will be called the paths moment map.

9.2. Evaluation of the paths moment map. Let X be a di!eological spaceand " be a closed 2-form defined on X. Let G be a di!eological group, and let! be a smooth action of G on X, preserving ". Let p be a path in X. Thanksto the explicit expression of the Chain-Homotopy Operator (art. 6.83), we get theevaluation of the momentum #(p) on any n-plot P of G,

#(p)(P)r($r) =

"1

0

"

!"su

##! (! " P)(u)(p(s+ t))

$

(s=0u=r)

"10

#"0$r

#dt, ($)

for all r in def(P) and all $r in Rn. Now, as a di!erential 1-form, #(p) is charac-terized by its values on the 1-plots (art. 6.37). Then, let f : t #! ft be a 1-plot of Gcentered at the identity 1G, that is, f ! Paths(G) and f(0) = 1G. For every t ! R,let Ft be the path in Di!(X,")—centered at the identity 1X—defined by

Ft : s #! !(f!1t " ft+s).

We have then

#(p)(f)t(1) = !

"

p

iFt(") = !

"1

0

iFt(")(p)s(1)ds, (%)

where iFt(") is the contraction of " by Ft (art. 6.56). But, as an invariant 1-formon G, the moment #(p) is characterized by its value at the identity, for t = 0,

#(p)(f)0(1) = !

"

p

iF(") = !

"1

0

iF(")(p)t(1)dt with F = ! " f. (&)

Note. Let f ! Hom!(R, G), then #(p)(f) is an invariant 1-form on R whosecoe"cient is just

#p iF("), that is,

#(p)(f) = hf(p)' dt where hf(p) = !

"

p

iF(").

The smooth map hf : Paths(X) ! R is the Hamiltonian of f, or the Hamilton-ian of the 1-parameter group f(R). Also note that the map h : Hom!(R, G) !C!(Paths(X),R), defined above, is smooth.


Proof. Let us prove (!). Let us recall that for every p " Paths(X) and everyg " G, p(g) = !(g) # p = [t $! !(g)(p(t))]. Thus, by definition,

"(p)(P)r(#r) = p!(K$)(P)r(#r)

= K$(p # P)r(#r)

=

"1

0

$

!"sr

#$! p # P(r)(s+ t)

$

(0r)

"10

#"0#r

#dt

=

"1

0

$

!"sr

#$! (! # P)(r)(p(s+ t))

$

(0r)

"10

#"0#r

#dt .

Let us prove (%). Let us apply the general formula (!) for P = f. Introducingu " = u!t and s "" = s+s ", using the compatibility property of $(P#Q) = Q!($(P))and the !(ft) invariance of $, we get

"(p)(f)t(1) =

"1

0

$

!"su

#$! !(fu)(p(s+ s "))

$

(s=0u=t)

"10

#"01

#ds "

=

"1

0

$

!"s ""

u "

#$! !(ft+u !)(p(s ""))

$

(s!!=s !u !=0 )

"10

#"01

#ds "

=

"1

0

$

!"s ""

u "

#$! !(ft # f!1

t # ft+u !)(p(s ""))

$

(s!!=s !u !=0 )

"10

#"01

#ds "

=

"1

0

$

!"s ""

u "

#$! !(ft)

"Ft(u

")(p(s ""))

#$

(s!!=s !u !=0 )

"10

#"01

#ds "

=

"1

0

$

!"s ""

u "

#$! Ft(u

")(p(s ""))

$

(s!!=s !u !=0 )

"10

#"01

#ds "

=

"1

0

$

!"u "

s ""

#$! Ft(u

")(p(s ""))

$

( u !=0s !!=s !)

"01

#"10

#ds "

= !

"1

0

$

!"u "

s ""

#$! Ft(u

")(p(s ""))

$

( u !=0s !!=s !)

"10

#"01

#ds "

= !

"1

0

iFt($)(p)s !(1)ds "

= !

"

p

iFt($).

Let us prove the Note. Let f " Hom!(R, G). By definition of di!erential forms andpullbacks, "(p)(f) = f!("(p)), but since f is a homomorphism from R to Di!(X,$)and "(p) is a left-invariant 1-form on Di!(X,$), f!("(p)) is an invariant 1-formof R, then "(p)(f) = f!("(p)) = a & dt, for some real a. Thus, "(p)(f)r ="(p)(f)0(1) & dt = hf(p) & dt, with hf(p) = "(p)(f)0(1) = !

#p iF($), where dt

is the canonical 1-form on R. !9.3. Variance of the paths moment map. Let X be a di!eological space, andlet $ be a closed 2-form defined on X. Let G be a di!eological group and ! bea smooth action of G on X, preserving $. The paths moment map ", defined in

THE PATHS MOMENT MAP 305

(art. 9.1), is equivariant under the action of G, that is, for all g ! G,

! " "(g) = Ad(g)! " !.

Proof. Let us denote here the orbit map p of p ! Paths(X) by R(p), that is,R(p)(g) = "(g) " p and !(p) = R(p)!(K#). Then, !("(g)(p)) = !("(g) " p) =R("(g) " p)!(K#). But R("(g) " p)(g ") = "(g ")("(g) " p) = "(g ") " "(g) " p ="(g "g) " p = R(p)(g "g) = R(p) " R(g)(g "), thus R("(g) " p) = R(p) " R(g), and!("(g)(p)) = (R(p) "R(g))!(K#) = R(g)!(R(p)!(K#)) = R(g)!(!(p)). But since!(p) is left-invariant, R(g)!(!(p)) = Ad(g)!(!(p)). !9.4. Additivity of the paths moment map. Let X be a di!eological spaceand # be a closed 2-form defined on X. Let G be a di!eological group and " bea smooth action of G on X, preserving #. The paths moment map !, defined in(art. 9.1), satisfies the following additive property,

!(p! p ") = !(p) + !(p ") and !(p) = !!(p), with p(t) = p(1! t),

for any two juxtaposable paths p and p " in X.

Proof. This is a direct application of the expression given in (art. 9.2, (#)), andof the additivity of the integral of di!erential forms on paths. !9.5. Di!erential of the paths moment map. Let X be a di!eological space,and let # be a closed 2-form defined on X. Let G be a di!eological group, and let" be a smooth action of G on X, preserving #. Let p be a path in X. Then, theexterior derivative of the paths momentum !(p) is given by

d(!(p)) = x!1(#)! x!0(#),

where x0 = p(0) and x1 = p(1), and the xi denote the orbit maps.

Proof. This is a direct application of the main property of the Chain-HomotopyOperator, d " K + K " d = 1! ! 0!. Since d# = 0, d(K#) = 1!(#) ! 0!(#),composed with p! we get p!"d(K#) = p!"1!(#)!p!"0!(#), that is, d(p!(K#)) =(1 " p)!(#)! (0 " p)!(#). Thus, d(!(p)) = x!1(#)! x!0(#). !9.6. Homotopic invariance of the paths moment map. Let X be a di!eolo-gical space and # be a closed 2-form defined on X. Let G be a di!eological group,and let " be a smooth action of G on X, preserving #. Let p0 and p1 be any twopaths in X. If p0 and p1 are fixed-ends homotopic, then !(p0) = !(p1). In otherwords, ! passes on the Poincare groupoid of X (art. 5.15).

Proof. Let s $! ps be a fixed-ends homotopy connecting p0 to p1, for examplelet ps(0) = x0 and ps(1) = x1, for all s. Let f be a 1-plot of G centered at theidentity 1G, that is, f(0) = 1G, and let F = " " f. We use the fact that the momentof paths is characterized by its value at the identity, !(ps)(f)0(1) = !

"ps

iF(#);see (art. 9.2, (#)). Let us di!erentiate this equality with respect to s,

$

$s

!!(ps)(f)0(1)

"= !%

#

ps

iF(#), with % =$

$s.

The variation of the integral of di!erential forms on cubes (art. 6.70) gives


!

!

ps

iF(") =

!1

0

d [iF(")](!ps) +

!1

0

d[iF(")(!ps)]

=

!1

0

d [iF(")](!ps) +

!iF(")(!ps)

"1

0.

The second summand of the right term vanishes because !ps vanishes at the ends:ps(0) = cst and ps(1) = cst. Now, thanks to the Cartan formula (art. 6.72),d[iF(")] = £F(")!iF(d"). But" is invariant under the action of G, thus £F(") =0, and d" = 0, so d[iF(")] = 0. Thus, !

"ps

iF(") = 0 and therefore #(p0) =

#(ps) = #(p1), for all s. !9.7. The holonomy group. Let X be a connected di!eological space, and let" be a closed 2-form defined on X. Let G be a di!eological group, and let $ be asmooth action of G on X, preserving ". Let # be the paths moment map (art. 9.1).We define the holonomy % of the action $ by

% = {#(&) | & ! Loops(X)}.

1. The holonomy % is an additive subgroup of the subspace of closed mo-menta, % " Z (art. 7.17), that is,

d' = 0 and '! ' ! ! %,

for any two elements ' and ' ! in % .2. The paths moment map #, restricted to Loops(X), factorizes through a

homomorphism from (1(X) to G". Thus, % is a homomorphic image of theAbelianized fundamental group (Ab

1 (X).3. In particular, every element ' of % is invariant by the coadjoint action of

G on G". For all g in G,

Ad"(g)(') = '.

The holonomy % is the obstruction for the action $ to be Hamiltonian. Precisely, theaction of G on X will be said to be Hamiltonian if % = {0}. Note that if (Ab

1 (X) = {0},or if the group G has no Ad"-invariant 1-form except 0, the action $ is necessarilyHamiltonian.

Proof. We get immediately that ' ! % is closed, by application of (art. 9.5).Indeed, for all paths p ! Paths(X), d(#(p)) = x"1(")! x"0("), where x0 = p(0) andx1 = p(1). Thus, for a loop &, since &(0) = &(1), d(#(&)) = 0. Now, let us choose abasepoint x0 ! X. For every loop & ! Loops(X, x0), the momentum #(&) dependson & only through its homotopy class (art. 9.6), so % is the image of (1(X, x0).And, thanks to the additive property of # (art. 9.4), the map class(&) ## #(&)is a homomorphism. Now, since X is connected, for every other point x1 of X,there exists a path c connecting x0 to x1, c = t ## c(1 ! t) connects x1 to x0.Thanks again to the additive property of #, #(c ! & ! c) = #(c) + #(&) + #(c) =!#(c) + #(&) + #(c) = #(&). Then, since the map class(&) ## class(c ! & ! c) isa conjugation from (1(X, x0) to (1(X, x1), % is the same homomorphic image of(1(X, x), for every point x ! X. Hence, we proved points 1 and 2. Point 3 is adirect consequence of (art. 7.17). !

THE 2-POINTS MOMENT MAP 307

Exercise

! Exercise 142 (Compact supported real functions I). Let us denote by X thespace of compact supported real functions defined on R, equipped with the di!eo-logy defined in Exercise 25, p. 23. Precisely, P : U ! X is a plot if and only if a)(r, t) !! P(r)(t) is a real smooth map defined on U " R, and b) for every r0 # Uthere exist an open neighborhood V $ U of r0 and a compact K $ R such that forevery r # V, P(r) and P(r0) coincide out of K. We want to consider the followingbilinear form ! as a di!erential 2-form on X,

!(f, g) =

"+!

!!f(t)g(t)dt,

where the dot denotes the derivative with respect to the parameter t of f. For alln # N, for every n-plot P : U ! X, for all r # U and "r, " !r # Rn, we define

!(P)r("r, "!r) =

"+!

!!

#

#r

!#P(r)(t)

#t

"("r)

#P(r)(t)

#r(" !r)dt .

1) Show that ! is a 2-form on X.

2) Check that ! realizes !, that is, for any f, g # X,

!(f, g) = !

!!ss !

"!! sf+ s !g

"

( ss !)

!10

"!01

".

Let us consider now the group G = (X,+) acting on itself by translation, that is,for all u # X,

Tu(f) = f+ u for all f # X.

3) Show that ! is invariant by G.

4) Say why the holonomy group $ , associated with the action of G on (X,!), mustvanish.

5) Show that, for any path p connecting f = p(0) to g = p(1), the paths momentmap is given, for any plot F of G, that is, a plot of X, by

%(p)(F)r("r) =

"+!

!!

!g(t)! f(t)

"#F(r)(t)

#r"r dt .

The 2-points Moment Map

The definition of the paths moment map leads immediately to the 2-points mo-ment map. The 2-points moment map satisfies a cocycle condition inherited fromthe additive property of the paths moment map. This is the second step in thegeneral construction.

9.8. Definition of the 2-points moment map. Let X be a connected di!eo-logical space, and let ! be a closed 2-form defined on X. Let G be a di!eologicalgroup and & be a smooth action of G on X, preserving !. Let % be the pathsmoment map (art. 9.1), and let $ be the holonomy of the action & (art. 9.7). Then,there exists a smooth map ' : X " X ! G"/$ such that the following diagramcommutes,


Paths(X) G!

X! X G!/!

"

ends pr

#

where pr is the canonical projection from G! onto its quotient, and ends maps pto (p(0), p(1)). The map # " C!(X! X,G!/!) will be called the 2-points momentmap.

1. The 2-points moment map # satisfies the Chasles cocycle relation, for anythree points x, x ", x "" of X,

#(x, x "") = #(x, x ") +#(x ", x ""). (#)

2. The 2-points moment map # is equivariant under the action of G. Pre-cisely, for any g " G, and any pair of points x and x " of X,

#($(g)(x), $(g)(x ")) = Ad!!(g)(#(x, x ")).

Proof. By construction # is defined by #(x, x ") = class! ("(p)), where p "Paths(X), x = p(0), x " = p(1), and class! (%) denotes the class of % " G! inG!/! . The map # is smooth simply by the general properties of subductions in dif-feology. Next, the first point is a consequence of the additivity of the paths momentmap (art. 9.4). The second point is a consequence of the equivariance of the pathsmoment map of the Ad! invariance of ! (art. 9.3), and of the definition of the Ad!!action (art. 7.16). !

The Moment Maps

From the construction of the paths moment map given in (art. 9.1) and the 2-points moment map given in (art. 9.8) we get the notion of 1-point moment maps,or simply moment maps. This is the third step of the general construction, and thegeneralization of the classical notion of moment maps.

9.9. Definition of the moment maps. Let X be a connected di!eologicalspace, and let & be a closed 2-form defined on X. Let G be a di!eological groupand $ be a smooth action of G on X, preserving &. Let # be the 2-points momentmap defined in (art. 9.8). There exists always a smooth map µ : X ! G!/! , calleda primitive of #, such that, for any two points x and x " of X,

#(x, x ") = µ(x ")! µ(x).

For every point x0 " X, for every constant c " G!/! , the map µ defined by

µ(x) = #(x0, x) + c

is a primitive of #. Every primitive µ of # is of this kind, and any two primitivesµ and µ " of # di!er only by a constant. The 2-points moment map # will be saidto be exact if there exists a primitive µ, equivariant by the action of G, that is, ifthere exists a primitive µ such that, for all g " G,

µ $ $(g) = Ad!!(g) $ µ.

THE MOMENT MAPS 309

The primitives µ of !, equivariant or not, will be called the moment maps.6

Note. By the identity (!) of (art. 9.8), ! is a 1-cocycle of the G-equivariantcohomology of X with coe!cients in G!/" , twisted by the coadjoint action. Twococycles ! and ! " are cohomologous if and only if there exists a smooth equivariantmap µ : X ! G!/" such that ! "(x, x ") = !(x, x ") + #µ(x, x "), where #µ(x, x ") =µ(x ") ! µ(x), and #µ is a coboundary. So, the 2-points moment map ! defines aclass belonging to H1

G(X,G!/") which depends only on the form $ and on the action

% of G on X. If the moment map ! is exact, that is, if class(!) = 0, we shall saythat the action % of G on X is exact, with respect to $. In this case, there existsa point x0 of X and a constant c such that µ : x "! !(x0, x) + c is an equivariantprimitive for !.

Proof. Let us choose a basepoint x0 # X. Since X is connected, for any x # Xthere exists always a path p # X such that p(0) = x0 and p(1) = x. Thus, definingµ(x) = !(x0, x) = class(&(p)), and thanks to the cocycle properties of !, we have!(x, x ") = !(x, x0) + !(x0, x ") = !(x0, x ") ! !(x0, x) = µ(x ") ! µ(x). Now, since! is smooth, µ is smooth. Therefore, the equation !(x ", x) = µ(x ") ! µ(x) alwayshas a solution in µ. Now, let µ and µ " be two primitives of !. For each pair x,x " of points of X we have µ "(x ") ! µ "(x) = µ(x ") ! µ(x), that is, µ "(x ") ! µ(x ") =µ "(x)!µ(x). So, the map x "! µ "(x)!µ(x) is constant. There exists c # G!/" suchthat µ "(x) ! µ(x) = c, that is, µ "(x) = µ(x) + c. Since the map x "! !(x0, x) is aspecial solution of the equation in µ : !(x ", x) = µ(x ") ! µ(x), any solution writesµ(x) = !(x0, x) + c for some point x0 # X and some constant c # G!/" . !9.10. The Souriau cocycle. Let X be a connected di"eological space, and let $be a closed 2-form defined on X. Let G be a di"eological group and % be a smoothaction of G on X, preserving $. Let ! be the 2-points moment map defined in(art. 9.8), and let µ be a primitive of ! as defined in (art. 9.9). Then, there existsa map ' # C!(G,G!/") such that

µ(%(g)(x)) = Ad!!(g)(µ(x)) + '(g).

This map ' is a (G!/")-cocycle, as defined in (art. 7.16). For all g, g " # G,

'(gg ") = Ad!!(g)('(g")) + '(g).

We shall call the cocycle ' the Souriau cocycle of the moment µ.

1. Two Souriau cocycles ' and ' ", associated with two moment maps µ andµ " are cohomologous , they di"er by a coboundary

#c : g "! Ad!!(g)(c)! c, where c # G!/".

2. For the a!ne coadjoint action of G on G!/" defined by ' (art. 7.16), themoment map µ is equivariant. For all g # G,

µ $ %(g) = Ad!,"! (g) $ µ.

3. For every cocycle ', associated with some moment map µ, there alwaysexist a point x0 # X and a constant c # G!/" such that,

'(g) = !(x0, %(g)(x0)) + #c(g) for all g # G.

6These maps should have been called “1-point moment maps”, but to conform to the usualdenomination, we chose to call them simply “moment maps”.


4. The cohomology class ! of " belongs to a cohomology group denoted byH1(G,G!/#). It depends only on the cohomology class of the 2-pointsmoment map $. This class ! will be called the Souriau cohomology class.

Note 1. Let x0 be some point of X. The 2-points moment map $ (which canalso be regarded as a 1-cocycle) defines a 1-cocycle f from G to G!/# by f(g, g ") =$(%(g)(x0), %(g ")(x0)). The cocycle f " associated with another point x "

0 will di!eronly by a coboundary. So, the Souriau cocycle ! represents just the class of thepullback f = x!0($) by the orbit map x0, where x!0 : H1

!(X,G!/#) ! H1(G,G!/#).

And, by the way, it depends only on the restriction of & to any one orbit of G onX. Hence, a good choice of the point x0 can simplify the computation of !.

Note 2. The nature of the action % has strong consequences on the Souriau class.For example, thanks to the third item, if the group G has a fixed point x0, thatis, %(g)(x0) = x0 for all g in G, then the Souriau class is zero and the cocycle $ isexact, i.e., there exists an equivariant primitive µ of $.

Proof. Thanks to (art. 9.9), every moment map µ writes µ(x) = $(x0, x) + c,where x0 is some fixed point of X and c ! G!/# . Thus, µ(%(g)(x))!Ad"!(g)(µ(x)) =$(x0, %(g)(x))+c!Ad"!(g)($(x0, x)+c) = $(x0, %(g)(x))+c!Ad"!(g)($(x0, x))!Ad"!(g)(c) = $(x0, %(g)(x)) ! $(%(g)(x0), %(g)(x)) ! 'c(g) = $(x0, %(g)(x)) +$(%(g)(x), %(g)(x0)) ! 'c(g) = $(x0, %(g)(x0)) ! 'c(g). Therefore, µ(%(g)(x)) !Ad"!(g)(µ(x)) is constant with respect to x. That proves points 1 and 4. Now,the variance of ", with respect to the multiplication of G, is a classical result ofcohomology (see for example [Kir76]). It is then obvious that, since two momentmaps µ and µ " di!er only by a constant, the associated cocycles " and " " di!erby a coboundary. The remaining items are just the results of elementary algebraiccomputations. !

Exercise

! Exercise 143 (Compact supported real functions, II). Let us consider the dataand notations of Exercise 142, p. 307.

1) Show that the moment maps of the action of G on (X,&) are given, for any plotF of G (that is, X) by

µ(f)(F)r((r) =

"+!

!!f(t)

)F(r)(t)

)r(r dt+cst.

2) Compute the associated Souriau cocycle ". Is it trivial?

The Moment Maps for Exact 2-Forms

The special case where a closed 2-form is the exterior derivative of an invariant1-form deserves special care, since it justifies a posteriori the constructions above,by analogy with the classical moment maps [Sou70].

9.11. The exact case. Let X be a connected di!eological space, and let & be aclosed 2-form defined on X. Let G be a di!eological group, and let % be a smoothaction of G on X, preserving &. Let us assume that & = d* and that * also isinvariant under the action of G, that is, %(g)!(*) = * for all g in G. Let + bethe paths moment map defined in (art. 9.1), and $ be the 2-points moment map

FUNCTORIALITY OF THE MOMENT MAPS 311

defined in (art. 9.8). Then, for every p ! Paths(X)

!(p) = "(x, x !) = x"1(#)! x"0(#),

where x1 = p(1) and x0 = p(0). Moreover, the 2-points moment map " is exact,and every equivariant moment map is cohomologous to

µ : x "! x"(#).

The action of G is Hamiltonian, $ = {0} and exact % = 0; see (art. 9.7) and(art. 9.10). This shows, in particular, the coherence of the general constructionsdeveloped until now.

Proof. By definition of the paths moment map !(p) = p"(K&), that is,!(p) = p"(K(d#)). But K(d#) + d(K#) = 1"(#) ! 0"(#), hence K(d#) =p"[1"(#)! 0"(#)!d(K#)], and !(p) = (1# p)"(#)! (0# p)"(#)!d[p"(K(#))]. But1 # p = x1 and 0 # p = x0, then !(p) = x"1(#)! x"0(#)!d[p"(K#)]. Now, K# is thereal function

K# : p "!"

p

#,

since p"(K#) = K# # p, for all g ! G,

K#(p(g)) =

"

!(g)#p# =

"

p

'(g)"(#) =

"

p

#.

Thus, the function p"(K#) :G!R is constant and equal to#p #. Then, d[p"(K#)]=

0, and !(p) = x"1(#)! x"0(#). Hence, !(p) = "(x0, x1) and $ = {0}.Next, the function µ : x "! x"(#) is clearly a primitive of ", that is, "(x0, x1) =

µ(x1) ! µ(x0). But R('(g)(x)) = x # R(g), where R('(g)(x)) is the orbit map of'(g)(x), g ! G. Thus, µ('(g)(x)) = (x#R(g))"(#) = R(g)"(x"(#)) = R(g)"(µ(x)) =Ad"(g)(µ(x)). Hence, µ is an equivariant primitive of " and % = 0. !

Functoriality of the Moment Maps

In this section we focus on the behavior of the moment maps, and the variousassociated objects, under natural transformations.

9.12. Images of the moment maps by morphisms. Let X be a connecteddi!eological space, and let & be a closed 2-form defined on X. Let G be a di!eologi-cal group, and let ' be a smooth action of G on X, preserving &. Let G ! be anotherdi!eological group, and let h : G ! ! G be a smooth homomorphism. Let ' ! = ' #hbe the induced action of G ! on X. Let us recall that the pullback h" : G" ! G !" isa linear smooth map.

1. Let ! : Paths(X) ! G, and ! ! : Paths(X) ! G ! be the paths momentmaps with respect to the actions of G and G ! on X. Then, ! ! = h" # !.

2. Let $ and $ ! be the holonomy groups with respect to the actions of G andG ! on X. Then, $ ! = h"($).

3. The linear map h" projects on a smooth homomorphism h"" : G/$ !

G !"/$ !, such that the following diagram commutes.


G! G "!

G!/! G "!/! "

h!

pr pr "

h!!

4. Let " and " " be the 2-points moment maps with respect to the actionsof G and G ". Then, " " = h!

! !".5. Let µ be a moment map relative to the action # of G. Then, µ " = h!

! ! µis a moment map relative to the action # " of G ".

6. Let µ be a moment map relative to the action # of G, and let µ " = µ !h!!

be the associated moment map relative to the action # " of G ". Then, theassociated Souriau cocycles satisfy $ " = h!

! ! $ ! h, which is summarizedby the following commutative diagram.

G G "

G!/! G "!/! "

h

$ $ "

h!!

Said di!erently, if $ is the Souriau cocycle associated with a moment µ ofthe action # of G, and µ " is a moment of the action # " of G ", then $ " andh!! ! $ ! h are cohomologous.

Note. Thanks to the identification between the space of momenta of a di!eologicalgroup and any of its extensions by a discrete group (art. 7.13), the moment mapsof the action of a group or the moment map of the restriction of this action toits identity component coincide. Said di!erently, the moment maps do not sayanything about actions of discrete groups.

Proof. To avoid confusion, let us denote by R(p) and R "(p) the orbit maps ofG and G " of p " Paths(X), that is, R(p)(g) = #(g) ! p and R "(p)(g) = # "(g) ! p.We have then, R "(p)(g) = # "(g) ! p = #(h(g)) ! p = (R(p) ! h)(g). Thus, R "(p) =R(p) ! h.1. By definition of the paths moment map, % "(p)=R "(p)!(K&)=(R(p)!h)!(K&) =h!(R(p)!(K&)) = h!(%(p)), that is, % " = h! ! %.2. Since ! " = % "(Loops(X)), and thanks to item 1, ! " = h!(!).

3. The map h!! is defined by class! (') #! class! !(h!(')), for all ' " G!. If ( = '+),

with ) " ! , then h!(() = h!(') + ) ", with ) " = h!()) " ! " (item 2). Thus,class! !(h!(()) = class! !(h!(')) and h!

! is well defined. Thanks to the linearity ofh!, h!

! is clearly a homomorphism. For G!/! and G "!/! " equipped with the quotientdi!eologies, h!

! is naturally smooth.

4. With the notations above, " and " " are defined by pr ! % = " ! ends andpr " ! % " = " " ! ends, where ends(p) = (p(0), p(1)), with p " Paths(X). Thus, byitem 1 and 3, pr " ! h! ! % = h!

! ! " ! pr, that is, pr " ! % " = (h!! ! ") ! pr. Hence,

h!! !" = " ".

FUNCTORIALITY OF THE MOMENT MAPS 313

5. Let µ ! = h"! ! µ and x, y " X. Then, µ !(y) ! µ !(x) = h"

! ! µ(y) ! h"! ! µ(y) =

h"! (µ(y)!µ(x)) = h"

! !!(y, x) = ! !(y, x). Thus, µ ! is a moment map for the action" ! of G.

6. According to (art. 9.10), there exists a point x0 " X such that, for allg ! " G !, # !(g !) = ! !(x0, " !(g !)(x0)). Thus, thanks to the previous items, # !(g !) =(h"

! ! !)(x0, "(h(g !))(x0)) = h"! (!(x0, "(h(g !))(x0))) = h"

! (#(h(g!))) = (h"

! ! # !h)(g !). Hence, # ! = h"

! ! # ! h. !9.13. Pushing forward moment maps. Let X and X ! be two connecteddi!eological spaces. Let $ and $ ! be two closed 2-forms, defined respectively on Xand X !. Let G be a di!eological group, let " be a smooth action of G on X preserving$, and let " ! be a smooth action, of G on X !, preserving $ !. Let f : X ! X ! be asmooth map such that $ = f"($ !) and f ! "(g) = " !(g) ! f, for all g " G.

1. Let f" : Paths(X) ! Paths(X !) defined by f"(p) = f ! p. Then, the pathsmoment maps % and % ! relative to the actions " and " ! are related by

% = % ! ! f",

and the associated holonomy groups & and & ! satisfy

& = {% !(f ! ') | ' " Loops(X)} # & !.

2. Let ( : G"/& ! G"/& ! be the projection induced by the inclusion & # & !.Let ! and ! ! be the 2-points moment maps relative to the actions " and" !. Then, for any two points of X, x1 and x2,

! !(f(x1), f(x2)) = ((!(x1, x2)).

3. For every moment map µ relative to the action ", there exists a momentmap µ ! relative to the action " !, such that

µ ! ! f = ( ! µ.

4. Let # and # ! be two Souriau cocycles relative to the actions " and " !.Then, the map (!# is a Souriau cocycle, cohomologous to # ! and the twoSouriau classes ) and ) ! satisfy ) ! = ("()), where (" denotes the actionof ( on cohomology, ("(class(#)) = class(( ! #).

Proof. 1. By definition %(p) = p"(K$), that is, %(p) = p"(K(f"($ !))). Thanksto the variance of the Chain-Homotopy Operator K ! f" = (f")" ! K ! (art. 6.84),%(p) = p"!(f")"(K !$ !) = (f"!p)"(K !$ !). But for all g " G, f"!p(g) = f!"(g)!p =" !(g) ! f ! p = p !(g), where p ! = f ! p. Then, %(p) = p !"(K !$ !) = % !(p !) =% !(f"(p)). Therefore, % = % ! ! f". Now, by definition of the holonomy groups,& = %(Loops(X)) = % !(f"(Loops(X))), and since f"(Loops(X)) # Loops(X !), & # & !.

2. Since & # & !, the map ( : class! (*) $! class! !(*), from G"/& to G"/& !, is welldefined. Now, let x !

1 = f(x1) and x !2 = f(x2). There exists then p " Paths(X)

connecting x1 to x2, and the path f"(p) connects x !1 to x !

2. By definition of ! !,! !(x !

1, x!2) = class! !(% !(p !)) = class! !(% ! ! f"(p)), and thanks to the first item,

class! !(% !(p !)) = class! !(%(p)) = ((class! (%(p))). But class! (%(p)) = !(x1, x2).Hence, ! !(x !

1, x!2) = ((!(x1, x2)), that is, ! !(f(x1), f(x2)) = !(x1, x2).

3. According to (art. 9.9), for every moment map µ there exist a point x0 " Xand a constant c " G"/& such that µ(x) = !(x0, x) + c. Let us define µ ! byµ !(x !) = ! !(x !

0, x!) + c !, where x !

0 = f(x0) and c ! = ((c). Then, thanks to item 2,


! !(f(x0), f(x))="(!(x0, x)), and µ !(f(x))="(!(x0, x))+"(c)="(!(x0, x)+c)="(µ(x)). Hence, µ ! satisfies µ ! ! f = " ! µ.

4. Let # be a Souriau cocycle for the action $. According to (art. 9.10), # iscohomologous to % : g "! !(x0, $(g)(x)), where x0 is some point of X. Then, letx !0 = f(x0), and % ! : g "! ! !(x !

0, $!(g)(x !

0)). Thus, %!(g) = ! !(f(x0), $ !(g)(f(x0))) =

! !(f(x0), f($(g)(x0))) = "(!(x0, $(g)(x0))) = " ! %(g). Now, since all Souriaucocycles, with respect to a given action of G, are cohomologous, the cocycle # ! iscohomologous to % !, and then cohomologous to "!%, that is, cohomologous to "!#.Hence, & ! = class(# !) = class(" ! #) = ""(class(#)) = ""(&). !

The Universal Moment Maps

In this section we build the universal moment maps, and related objects, as-sociated with the whole group of automorphisms Di!(X,'), where ' is a closed2-form on a di!eological space X. We show then how these objects, associated witha smooth automorphic action of some arbitrary di!eological group G, relate to theuniversal construction.

9.14. Universal moment maps. Let X be a connected di!eological space,and let ' be a closed 2-form defined on X. We consider the group Di!(X,'), ofall the automorphisms of (X,'), equipped with the functional di!eology of groupof di!eomorphisms. We shall also denote this group by G!. Every constructionintroduced in the previous sections—the space of momenta, the paths moment map,the holonomy group, the 2-points moment map, the moment maps, the Souriaucocycles, and the Souriau class—apply for G!. We shall distinguish these objectsby the index '. So, we shall denote by G"

! the space of momenta of G!, by(! : Paths(X) ! G"

! the paths moment map, by )! = (!(Loops(X)) the holonomygroup, by !! the 2-points moment map, by µ! the moment maps, by #! theSouriau cocycles, and by &! the Souriau class. Since G! and its action on X areuniquely defined by ', these objects depend only on the 2-form '.

Now, let G be a di!eological group, and let $ be a smooth action of G on X,preserving ', that is, a smooth homomorphism $ from G to G!. The values of thevarious objects (, ) , !, µ, #, with respect to the action $ of G on X, depend only onthe pullback $" and on (!, )!, !!, µ!, and #!, as it is described in (art. 9.12),

"#

$

( = $" ! (!

) = $"()!)! = $""! !!!

and

%µ # $""! ! µ!

# # $""! ! #! ! $.

In this sense the objects G!, )!, (!, )!, !!, µ!, #!, and &! are universal.That is why we shall call (! the universal paths moment map, )! the universalholonomy, !! the universal 2-points moment map, µ! the universal moment maps,#! the universal Souriau cocycles, and &! the universal Souriau class of '.

Note. The universal holonomy leads naturally to the notion of Hamiltonian spaces,the ones for which, for one reason or another, )! = {0}.

9.15. The group of Hamiltonian di!eomorphisms. Let X be a connecteddi!eological space, equipped with a closed 2-form '. There exists a largest con-nected subgroup Ham(X,') $ Di!(X,') whose action is Hamiltonian, that is,

THE UNIVERSAL MOMENT MAPS 315

whose holonomy is trivial. The elements of Ham(X,!) are called Hamiltonian dif-feomorphisms . An action " of a di!eological group G on X is Hamiltonian if andonly if, restricted to the identity component of G, " takes its values in Ham(X,!).

The group Ham(X,!) is precisely built as follows. Let us denote by G! thegroup Di!(X,!) and by G!

! its identity component. Let # : G!! ! G!

! be theuniversal covering. Since the universal holonomy $! is made of closed momenta,every % ! $! defines a unique homomorphism k(%) from G!

! to R such that #"(%) =d[k(%)] (art. 7.17). Let

!H! ="

"##!

ker(k(%)),

and let !H!! be its identity component. Then,

Ham(X,!) = #(!H!!).

Note 1. The map f : G!! ! Hom(#1(X),R), defined by f(g) = [& "! k(%)(g)],

with & = class(') and % = (('), is a homomorphism, and !H! = ker(f). In classicalsymplectic geometry, the image F = val(f) is called, by some authors, the group offlux of !.

Note 2. Since the Hamiltonian nature of a group of automorphisms depends onlyon its identity component (see (art. 7.13) and (art. 7.14)) every extension H #Di!(X,!) of Ham(X,!), such that H/Ham(X,!) is discrete,7 is Hamiltonian. In

particular #(!H!) is Hamiltonian, or if $! = {0}, then Di!(X,!) is Hamiltonian,and Ham(X,!) is the identity component of Di!(X,!).

Note 3. Let us choose a point x0 in X, and let µ be the moment map with respectto the group Ham(X,!), defined by µ(x0) = 0. Let f be a 1-parameter subgroup ofHam(X,!). Applying (art. 9.2, Note), we get the expression of µ(x), for all x ! X,evaluated on f

µ(x)(f) = hf(x)$ dt with hf(x) = !

"x

x0

if(!).

The smooth function hf : X ! R is the Hamiltonian (vanishing at x0) of the1-parameter subgroup f.

Proof. Let us remark first of all that for every % ! $!, #"(%) ! !H! = 0. Indeed,

#"(%) ! !H! = d[k(%)] ! !H! = d[k(%) ! !H!]. But, by the very definition of !H!,

k(%) ! !H! = 0, thus #"(%) ! !H! = 0.

a) Let us prove that the holonomy of Ham(X,!) is trivial. Let H! = #(!H!), and

let us denote by jH! the inclusion H! # G!, by j!H!the inclusion !H! # G!

!,

and by #H! : !H! ! H! the projection, so that jH! % #H! = # % j!H!. Let $H!

be the holonomy of H!, then, according to (art. 9.12), $H! = j"H!($!). Thus,

for every % ! $H! there exists % ! $! such that % = % ! H! = j"H!(%). Hence,

for all % ! $H! , #"H!

(%) = #"H!

(j"H!(%)) = (jH! % #H!)"(%) = (# % j!H!

)"(%) =

j"!H!(#"(%)) = #"(%) ! !H!. But #"(%) ! !H! = 0, thus #"

H!(%) = 0, and since #H!

is a subduction, % = 0. Therefore, the holonomy of H! trivial, $H! = {0}.

7where H and Ham(X,!) are equipped with the subset di!eology of the functional di!eologyof Di!(X,!).


b) Let us prove now that every connected subgroup H ! G! whose action is

Hamiltonian is a subgroup of Ham(X,!). Let !H = "!1(H) and !H! be its identity

component. Let jH be the inclusion H ! G!, and j!H! be the inclusion !H! ! G!!.

Let "H = " ! !H!, so that jH " "H = " " j!H! . Let #H be the holonomy of H. Since#H = j"H(#!) and #H = {0}, for all $ # #!, j"H($) = 0. Thus, for all $ # #!,""H(j

"H($)) = 0. But ""

H(j"H($)) = (jH " "H)"($) = (" " j!H!)"($) = j"!H!(""($)) =

""($) ! !H!, thus, for all $ # #!, ""($) ! !H! = 0. Now, ""($) = d[k($)], hence

d[k($) ! !H!] = 0. Then, since H! is connected, k($) is constant on !H!, and since

k($) is a homomorphism to R, this constant is necessarily 0. Thus, !H! ! ker(k($)),

for all $ # #!, that is, !H! ! !H!. But since H! is connected, !H! ! !H!! ! H!, and

thus H = "(!H!) ! Ham(X,!) = "(!H!!). "

9.16. Time-dependent Hamiltonian. Let X be a connected di!eological space,and let ! be a closed 2-form defined on X. A di!eomorphism f of X belongs toHam(X,!) if and only if the two following conditions are fulfilled.

1. There exists a smooth path t $! ft in Di!(X,!) connecting the identity1M = f0 to f = f1.

2. There exists a smooth path t $! %t in C!(X,R) such that

iFt(!) = !d%t with Ft : s $! f!1t " ft+s,

for all t. According to the tradition of classical symplectic geometry, the patht $! %t may be called a time-dependent Hamiltonian of the 1-parameter family ofHamiltonian di!eomorphisms t $! ft.

Proof. Let us assume first that f satisfies the condition above, that there existsa smooth path t $! ft in Di!(X,!) such that f0 = 1M, f1 = f, and there existsa smooth path t $! %t in C!(X,R) such that iFt(!) = !d%t, for all t, where

Ft : s $! f!1t " ft+s. Let us recall that Ham(X,!) = "(!H!

!), with !H!! the identity

component of !H! ="

"##!ker(k($)), and let f # G!

! be the homotopy class of thepath t $! ft, notations of (art. 9.15). Then, let $ # #!, that is, $ = &!('), where' is some loop in M. By definition,

k($)(f) =

"

[t $"ft]$ =

"

[t $"ft]&!(') =

"1

0

&!(')([t $! ft])t(1)dt .

Now, thanks to (art. 9.2, (%)),

&!(')([t $! ft])t(1) = !

"

$

iFt(!) =

"

$

d%t =

"

%$

%t = 0.

Thus, k($)(f) = 0 for all $ # #!, and f belongs to !H!, more precisely to the

identity component of !H!. Therefore, f # Ham(X,!).Conversely, let f # Ham(M,!). Since Ham(M,!) is connected, there exists

a path t $! ft in Ham(M,!) connecting 1M to f. And, since the projection

" ! !H!! : !H!

! ! Ham(M,!) is a covering, there exists a (unique) lift t $! ft of

t $! f in !H!!, along " ! !H!

!, such that f0 = 1!H!. This lift is actually given by

ft = class(pt), with pt : s $! fst. Thus, for all t, ft # !H!! ! !H! =

""##!

ker(k($)),

that is, for all $ # #!, k($)(ft) = 0. In other words, for all ' # Loops(M),

THE UNIVERSAL MOMENT MAPS 317

k(!!("))(ft) = 0. But

k(!!("))(ft) =

!

pt

!!(")

=

!1

0

!!(")(s !" fst)s(1)ds

=

!1

0

!!(")(s !" st !" fst)s(1)ds

=

!1

0

[!!(")(u !" fu)]u=st

!d(st)

ds

"ds

=

! t

0

!!(")(u !" fu)u(1)du .

Thus, k(!!("))(ft) = 0, which implies 1t

#t0 !!(")(u !" fu)u(1)du = 0, and

!!(")(t !" ft)t(1) = limt! 01t

#t0 !!(")(u !" fu)u(1)du = 0. Next, since

!!(")([t !" ft])t(1) = !#" iFt(#) (art. 9.2, (")), for all t and all " # Loops(X),#

" iFt(#) = 0. Then, since Ft is a path in Di!(X,#) centered at the identity, the Liederivative of # by Ft vanishes, and by application of the Cartan formula (art. 6.72),we get £Ft# = 0, which implies d[iFt(#)] + iFt(d#) = d[iFt(#)] = 0. Thus, the1-form iFt(#) is closed and its integral on any loop " in X vanishes, hence iFt(#)is exact (art. 6.89). Therefore, for all real number t, there exists a real function$t # C"(X,R) such that iFt(#) = !d$t. The fact that t !" $t is a smooth mapfrom R to C"(X,R), for the functional di!eology, is a consequence of the explicitconstruction of the function $t by integration along the paths. !9.17. The characteristics of moment maps. Let X be a connected di!eologi-cal space equipped with a closed 2-form #. Let G be a di!eological group, and let% be a smooth action of G on X, preserving #. Let & be the 2-points moment map(art. 9.8). Thanks to the additive property of &, the relation R, defined on X by

xR x ! if &(x, x !) = 0G!/# ,

is an equivalence relation. The classes of this equivalence relation are the preimagesof the values of a moment map µ, solution of &(x, x !) = µ(x !) ! µ(x) (art. 9.9).We shall define the characteristics of the moment map µ (or &) as the connectedcomponents of the equivalence classes of R, that is, the connected components ofthe preimages of µ.

Note. This definition applies obviously to the universal moment map µ!. Since,in this case, the characteristics depend only on #, it is tempting to call them thecharacteristics of the 2-form #, especially when we have in mind the particular caseof homogeneous manifolds, treated in (art. 9.26). We get then a general picture:by equivariance, the image of the universal moment map µ! is a union of coadjointorbits of Di!(X,#), images of its orbits in X. The moment map factorizes thenthrough the space of characteristics of #, denoted here by Chars(X,#), by a mapµ! : Chars(X,#) " G"

!/'!. The preimages by µ! are the connected componentsof the preimages by µ!, that is, µ!1

! (m) = (0(µ!1! (m)). The projection char!,

associating with each x # X the characteristic passing through x, is a kind ofsymplectic reduction. But we do not know if, in general, the 2-form # passes to


the quotient, especially when the group of automorphisms has more than one orbit.This is still an open question, but the framework is here.

X G!!/!!

Chars(X,")

µ!

char!µ!

This construction is related to the well-known Marsden-Weinstein symplectic re-duction [MaWe74], when it is applied to some subspaces W ! X for the restriction" ! W. There is, however, a small di!erence: we reduce first by the characteristicsof " and then by the moment map, which can be interpreted as a regularization ofthe reduction by the characteristics.8

The Homogeneous Case

Because of its simple character, the case of a homogeneous action of a di!eologi-cal group G on a space X, preserving a closed 2-form ", deserves a special attention.We shall see in (art. 9.23) how this applies to classical symplectic geometry.

9.18. The homogeneous case. Let X be a connected di!eological spaceequipped with a closed 2-form ". Let # be a smooth action of a di!eologicalgroup G on X, preserving ". Let us assume that X is homogeneous for this action(art. 7.8). Let ! be the holonomy of the action #, let µ be a moment map, and let$ be the cocycle associated to µ. Let x0 be any point of X, and let µ0 = µ(x0).Let StAd!,"

!(µ0) be the stabilizer of µ0 for the a"ne coadjoint action of G on G!/! .

Thanks to the equivariance of the moment map µ, with respect to the $-a"necoadjoint action of G on G!/! , µ " #(g) = Ad",#! (g) " µ, the image O = µ(X) isa (!, $)-orbit of G, and St$(x0) ! StAd!,"

!(µ0). Let us equip O with the pushfor-

ward of the di!eology of G by the orbit map µ0 : g #! Ad",#! (g)(µ0). Then, theorbit map x0 : G ! X is a principal fibration with structure group St$(x0), theorbit map µ0 : G ! O is a principal fibration with structure group StAd!,"

!(µ0),

and the moment map µ : X ! O is a fibration, with fiber the homogeneous spaceStAd!,"

!(µ0)/St$(x0).

Note 1. The moment maps µ are defined up to a constant. But the characteristicsof µ, that is, the connected components of the subspaces defined by µ(x) = cst, arenot (art. 9.17).

Note 2. Let us consider the simple case d%, where % $ G!. Its moment mapµ : G ! G! is the coadjoint orbit map µ : g #! Ad!(g)(%). A natural question isthen, is there a closed 2-form ", defined on the coadjoint orbit O% = µ(G), suchthat µ!(") = d% ? I do not yet have a definitive answer to this question; the bestI can say is contained in the following proposition. Let us consider O%, equippedwith the quotient di!eology of G.

Proposition. There exists a closed 2-form " on O% such that µ!(") = d%, if andonly if % ! StAd!(%) is closed.

8It is not impossible that in particular, for the two-bodies problem [Sou83], it be preciselythe universal moment map which regularizes the space of motions.

THE HOMOGENEOUS CASE 319

There are particular cases, or special group di!eologies, for which the restrictionof ! on its stabilizer StAd!(!) is closed. For example, if G is a Lie group, then wecan use the Cartan formula to check it. But even simpler, if the stabilizer StAd!(!)is discrete or 1-dimensional, then every 1-form is closed. Paul Donato has givenin [Don94] an interesting example of a discrete stabilizer. Let us give here thedi!eological version of his construction. We consider G = Di!(S1) and its universalcovering G, described in Exercise 28, p. 27, as the subgroup of di!eomorphisms "of R such that "(x + 2#) = "(x) + 2#. The kernel of the projection # : G ! Gis the subgroup 2#Z of the translations T2!k : x !! x + 2#k, commuting with all" " G. Let P : U ! G be an n-plot, and let r " U, $r " Rn, and ! be defined by

!(P)r($r) =!"

n=0

1

2n%

%s

#P(r)!1 # P(s)(n)

$

s=r

($r).

One can check that ! is a left invariant 1-form, and by the way an element of G!.The moment map µ is then given by

µ(")(P)r($r) =!"

n=0

1

2n%

%s

#"!1 # P(r)!1 # P(s) #"(n)

$

s=r

($r).

Because a momentum is characterized by its values on arcs centered at the identity,and because every arc & centered at the identity in G is tangential to some rayh " Hom!(R, G), the computation of the stabilizer of !, for the coadjoint action,is reduced to Donato’s computation in his paper, and it coincides with the orbits of2#Z. Thus, d! passes to the coadjoint orbit O", which is actually di!eomorphic toDi!(S1) itself, and symplectic according to the meaning we define below (art. 9.19).

Proof. The triple fibration is an application of (art. 8.15).

G

X O

x0 µ0

µ

G

X O

St#(x0) StAd!,"!

(µ0)

StAd!,"!

(µ0)/St#(x0)

Let us focus on Note 2. There exists a (closed) 2-form ' on O" such that µ!(') =d! if and only if, for two plots P and P " of G such that µ # P = µ # P ", thend!(P) = d!(P ") (art. 6.38). But µ # P = µ # P " means that there exists aplot ( of StG!(!) such that P "(r) = P(r) · ((r). Then, thanks to Exercise 127,p. 227, ![r !! P(r) · ((r)]r = [L(P(r))!(!)](()r + [R(((r))!(!)](P)r = !(()r+[Ad!(((r))(!)](P)r, but since ( is a plot of the stabilizer of ! for the coadjointaction, Ad!(((r))(!) = ! and !(P ") = !(P) + !((). Thus, d! passes to the quo-tient O" $ G/StG!(!) if and only if d!(() = 0 for all plots ( of StG!(!), i.e., if andonly if d[! ! StG!(!)] = 0. "9.19. Symplectic homogeneous di!eological spaces. Let X be a connecteddi!eological space and ' be a closed 2-form defined on X.


Definition. We say that (X,!) is a homogeneous symplectic space if it is homoge-neous under the action of Di!(X,!) and if the characteristics of a universal momentmap µ! are reduced to points (art. 9.17).

For example, the 2-form d" on Di!(S1) described in the previous article (art. 9.18)is symplectic in this sense. Let us recall that being homogeneous under the action ofDi!(X,!) means that the orbit map R(x) : Di!(X,!) ! X, defined by R(x)(#) =#(x), is a subduction, where x ! X and Di!(X,!) is equipped with the functionaldi!eology (art. 1.61).

Note 1. The characteristics of a universal moment map µ! are reduced to pointsif and only if it is a covering onto its image, equipped with the quotient di!eology.If it is the case for one universal moment map, then it is the case for every one.

Note 2. The homogeneous situation where a moment map µ! is not a coveringonto its image can be regarded as the presymplectic homogeneous case, as suggestedby (art. 9.26).

Note 3. Let G be a di!eological group, and let $ be a smooth action of G on X,preserving !. If the action $ of G on X is homogeneous, then X is a homogeneousspace of Di!(X,!). And, if a moment map µ : X ! G!/% is a covering onto itsimage, then any universal moment map µ! : X ! G!

!/%! is a covering onto itsimage. Thus, to check that a homogeneous pair (X,!) is symplectic it is su"cientto find a homogeneous smooth action, of some di!eological group G, for which amoment map is a covering onto its image.

Proof. Note 1 is obvious, by definition of the characteristics (art. 9.17) and byhomogeneity (art. 9.18). Let us prove Note 3. To be homogeneous under the actionof G means that, for some point (and thus for every point) x ! X, the orbit mapx : G ! X, defined by x(g) = $(g)(x), is a subduction. So, x is surjective and,for any plot P : U ! X, for any r0 ! U, there exist an open neighborhood V of r0and a plot Q : V ! G such that P ! V = x "Q, that is, P(r) = $(Q(r))(x) for allr ! V. Since $ is smooth, Q = $ " Q is a plot of Di!(X,!), and P ! V = x " Q.Since, x : Di!(X,!) ! X is surjective, it is a subduction and X is a homogeneousspace of Di!(X,!). Now, let us remark that, since the moment maps di!er onlyby a constant, if a moment map µ is a covering onto its image O equipped with thequotient di!eology of G, then every other moment map µ " = µ + cst is a coveringonto its image O " = O+ cst. Then, let x0 be a point of X, and let µ(x) = &(x0, x),where& is the 2-points moment map. Let µ! = &!(x0, x). According to (art. 9.14),µ = $!"! " µ!. Let O = µ(X) = $!"!(µ!(X)) = $!"!(O!), with O! = µ!(X). Letm! ! O! and m = $!"!(m!). If µ!(x) = m!, then $!"!(µ!(x)) = $!"!(m!), thatis, µ(x) = m. Thus, µ!1

! (m!) # µ!1(m). Therefore, if µ!1(m) is discrete, thenµ!1! (m!) is discrete, a fortiori , and if µ is injective, then µ! is injective. "

Exercise

! Exercise 144 (Compact supported real functions III). Let us consider thespace (X,!) as defined in Exercise 142, p. 307. Show that (X,!) is a homogeneoussymplectic space.

ABOUT SYMPLECTIC MANIFOLDS 321

About Symplectic Manifolds

The case of classical symplectic manifolds (M,!) deserves special care. Weshall see in this section that, in this case, any universal moment map µ! is injec-tive and therefore identifies M with a coadjoint orbit of Di!(M,!), in the generalmeaning that we gave in (art. 7.16).

9.20. Value of the moment maps for manifolds. Let M be a connectedmanifold equipped with a closed 2-form !. In this context, the paths moment map"! takes a special expression. Let p be a path in M and F : U ! Di!(M,!) bean n-plot, then

"!(p)(F)r(#r) =

"1

0

!p(t)(p(t), #p(t))dt , (!)

for all r " U and #r " Rn, where #p is the lift in the tangent space TM of the pathp, defined by

#p(t) = [D(F(r))(p(t))]!1$F(r)(p(t))

$r(#r). (#)

Proof. By definition, "(p)(F) = p!(K!)(F) = K!(p $ F). The expression of theoperator K (art. 6.83), applied to the plot p $ F : r %! F(r) $ p of Paths(X), gives

(K!)(p $ F)r(#r) ="1

0

!

!"su

#%! (p $ F)(u)(s+ t)

$

(s=0u=r)

"10

#"0#r

#dt .

But (p $ F)(u)(s + t) = F(u)(p(s + t)), let us denote temporarily by %t the plot(s, u) %! F(u)(p(s+ t)), so F(u)(p(s+ t)) writes %t(s, u). Now, let us denote by Ithe integrand of the right term of this expression. We have

I = !

!"su

#%! %t(s, u)

$

(s=0u=r)

"10

#"0#r

#

= %!t(!)(0r)

"10

#"0#r

#

= !"t(0r)

"D(%t)(0r)

"10

#, D(%t)(0r)

"0#v

##

= !F(r)(p(t))

"$

$s

#F(r)(p(s+ t))

$

s=0

,$

$r

#F(r)(p(t))

$(#r)

#.

But,

$

$s

#F(r)(p(s+ t))

$

s=0

= D(F(r))(p(t))

"$p(s+ t)

$s

%%%%s=0

#

= D(F(r))(p(t))(p(t)).

Then, using this last expression and the fact that F is a plot of Di!(M,!), that is,for all r in U, F(r)!! = !, we have

I = !F(r)(p(t))

"D(F(r))(p(t))(p(t)),

$F(r)(p(t))

$r(#r)

#

= !p(t)

"p(t), [D(F(r))(p(t))]!1$F(r)(p(t))

$r(#r)

#

= !p(t)(p(t), #p(t)).


Therefore, !!(p)(F)r("r) = K#(p ! F)r("r) =!1

0

#p(t)(p(t), "p(t))dt. !

9.21. The paths moment map for symplectic manifolds. Let M be aHausdor! manifold, and let # be a nondegenerate closed 2-form defined on M. Letm0 and m1 be two points of M connected by a path p. Let f " C!(M,R) withcompact support. Let F be the exponential of the symplectic gradient9 grad!(f), Fis a 1-plot of Di!(M,#), and precisely a 1-parameter homomorphism. Then, theuniversal paths moment map !!, computed at the path p, evaluated on the 1-plotF, is the constant 1-form of R

!!(p)(F) = [f(m1)! f(m0)]# dt with F : t $" et grad!(f),

where dt is the standard 1-form of R. Note that we are in the special case whereF is actually a 1-parameter homomorphism of Ham(M,#) % Di!(M,#), and thefunction f is one Hamiltonian of F.

Proof. Let us remark that, in our case, the lift "p defined by (art. 9.20, (&))writes simply

"p(t) = [D(er")(p(t))]!1$er"(p(t))

$r("r) = %(p(t))# "r,

with % = grad!(f), and where r and "r are real numbers. Thus, the expression(art. 9.20, (')) becomes

!!(p)(F)r("r) =

!1

0

#p(t)(p(t), %(p(t)))dt#"r

=

!1

0

#p(t)(p(t), grad!(f)(p(t)))dt#"r

=

!1

0

df

!dp(t)

dt

"dt#"r

= [f(p(1))! f(p(0))]# "r,

that is, !!(p)(F) = [f(m1)! f(m0)]# dt. !9.22. Hamiltonian di!eomorphisms of symplectic manifolds. Let (M,#)be a connected Hausdor! symplectic manifold. According to Banayaga [Ban78], adi!eomorphism f is said to be Hamiltonian if it can be connected to the identity1M by a smooth path t $" ft in Di!(M,#) such that

#(ft, ·) = d&t with ft(x) =d

ds

#fs ! f!1

t (x)

$

s=t

,

where (t, x) $" &t(x) is a smooth real function. If f is Hamiltonian according to thisdefinition, then it belongs to Ham(M,#), as defined in (art. 9.15). Conversely, anyelement f of Ham(M,#) satisfies the above Banyaga’s condition. Thus, the defini-tion of Hamiltonian di!eomorphisms given in (art. 9.15) is a faithful generalizationof the usual definition for symplectic manifolds.

Proof. This proposition is a direct consequence of the general statement givenin (art. 9.16) and the following comparison between the above 1-parameter family

9Let us recall that the symplectic gradient is defined by #(grad!(f), ·) = !df.


of vector fields ft and the family Ft of the (art. 9.16). Since ft ! ! f!1t = ft!

(f!1t ! ft !) ! f!1

t , the vector fields ft and Ft are conjugated by ft, precisely

ft = (ft)!(Ft) or ft(x) = D(ft)(f!1t (x))(Ft(f

!1t (x))).

This implies in particular that if the vector field ft satisfies Banyaga’s conditionfor the function !t, then the vector field Ft satisfies Banyaga’s condition for thefunction "t = !!t ! ft, and conversely, that is,

#(ft, ·) = d!t ! #(Ft, ·) = !d"t with "t = !!t ! ft.

Let us check it. Let x " M, x " = ft(x), $x " TxM, and $x " = D(ft)(x)($x), then

#x !(ft(x"), $x ") = [d!t]x !($x ")

#ft(x)(ft(ft(x)), D(ft)(x)($x)) = [d!t]ft(x)(D(ft)(x)($x))

#ft(x)(D(ft)(x)(Ft(x)), D(ft)(x)($x)) = [f!t(d!t)]x($x)

[f!t(#)]x(Ft(x), $x) = d[f!t(!t)]x($x)

#x(Ft(x), $x) = d[!t ! ft]x($x).Thus, "t = !!t ! ft. !9.23. Symplectic manifolds are coadjoint orbits. Let M be a manifold, andlet # be a closed 2-form defined on M. We assume M connected and Hausdor!.Then, the form # is nondegenerate, thus symplectic, if and only if the two followingconditions are fulfilled.

A. The manifold M is a homogeneous space of Di!(M,#).

B. The universal moment map µ! : M " G!!/%! is injective.

In that case, the image O! = µ!(M) " G!!/%! of the universal moment map10

(art. 9.14) is a (%!, &!)-coadjoint orbit of Di!(M,#) (art. 7.16), and µ! identifiesM with O!, where O! is equipped with the quotient di!eology of Di!(M,#). Inother words, every symplectic manifold is a coadjoint orbit.

Note 1. Let Ham(M,#) be the group of Hamiltonian di!eomorphisms, and letH!!

be the space of its momenta. Let µ!! : M " H!

! be a moment map associated withthe action of Ham(M,#), and let &!! be the associated Souriau cocycle. Then, µ!

!

is also injective, and identifies M to a &!!-coadjoint orbit O!! # H!

! of Ham(M,#).

Note 2. Let us consider the example M = R2 and # = (x2 + y2) dx! dy. SinceR2 is contractible, the holonomy %! is trivial (art. 9.7). Next, # is nondegenerateon R2! {0}, but degenerates at the point (0, 0). Thus, (0, 0) is an orbit of the groupDi!(R2,#), and actually R2 ! {0} is the other orbit. Hence, the universal momentmap µ! such that µ!(0, 0) = 0G"

!is equivariant (art. 9.10, Note 2). Moreover, µ!

is injective. The closed 2-form # not being symplectic, with an injective universalmoment map, shows that the hypothesis of transitivity of Di!(M,#) on M is notsuperfluous is this proposition.

Proof. Let us assume first that # is nondegenerate, that is, symplectic. Then,the group Di!(M,#) is transitive on M [Boo69]. Moreover, for every m " M,the orbit map m : ' $" '(m) is a subduction [Don84]. Thus, the image of themoment moment map µ! is one orbit O! of the a"ne coadjoint action of G! on

10The universal moment maps are defined up to a constant, but if one is injective, then theyare all injectives.


G!!/!!, associated with the cocycle "!. Hence, the orbit O! being equipped with

the quotient di!eology of G!, the moment map µ! is a subduction.Now, let m0 and m1 be two points of M such that µ!(m0) = µ!(m1), that

is, #!(m0,m1) = µ!(m1)!µ!(m0) = 0. Let p ! Paths(M) such that p(0) = m0

and p(1) = m1. Thus, #!(m0,m1) = 0 is equivalent to $!(p) = $!(%), where% is some loop in M, we can choose %(0) = %(1) = m0. Now, let us assume thatm0 "= m1. Since M is Hausdor! there exists a smooth real function f ! C!(M,R),with compact support, such that f(m0) = 0 and f(m1) = 1. Let us denote by & thesymplectic gradient field associated with f and by F the exponential of &. Thanks to(art. 9.21), on the one hand we have $(p)(F) = [f(m1)!f(m0)]dt = dt, and on theother hand $!(%)(F) = [f(m0)! f(m0)]dt = 0. But dt "= 0, thus #!(m0,m1) "= 0,and the moment map µ! is injective. Therefore, µ! is an injective subduction onO!, that is, a di!eomorphism.

Conversely, let us assume that M is a homogeneous space of Di!(M,') andthat µ! is injective. Let us notice first that since Di!(M,') is transitive, the rankof ' is constant. In other words, dim(ker(')) = cst. Now, let us assume ' isdegenerate, that is, dim(ker(')) # 1. Since m $! ker('m) is a smooth foliation,for every point m of M there exists a smooth path p in M such that p(0) = m andfor t belonging to a small interval around 0 ! R, p(t) "= 0 and p(t) ! ker('p(t))for all t in this interval. Then, we can reparametrize the path p and assume nowthat p is defined on the whole R and satisfies p(0) = m, p(1) = m " with m "= m ",and p(t) ! ker('p(t)) for all t. Next, since p(t) ! ker('p(t)) for all t, using theexpression (art. 9.20, (%)), we get $!(p) = 0G!

!, that is, µ!(m) = µ!(m "). But

since m "= m " and we assumed µ! injective, this is a contradiction. Thus, thekernel of ' is reduced to {0}, and ' is nondegenerate, that is, symplectic.

Let us prove Note 1. According to a Boothby’s theorem, the group Ham(M,')acts transitively on M [Boo69]. With respect to this group, and by construction,the holonomy is trivial: the associated paths moment map $!

! and the momentmaps µ!

! take their values in the space H!!. Let j : Ham(M,') ! Di!(M,')

be the inclusion, thus the universal holonomy !! is in the kernel of j!, and weget a natural mapping j!"! : G!

!/!! ! H!!. Now, the paths moment maps satisfy

$!! = j!"! &$!, and µ!

! = j!"! &µ! (art. 9.14). Then, since (art. 9.21) involves onlyplots of Ham(X,'), the proof above applies mutatis mutandis to the Hamiltoniancase, and we deduce that the moment maps µ!

! are injective. By transitivity, theyidentify M with a "!!-coadjoint orbit of Ham(M,').

Let us finish by proving the second note, that is, the universal moment mapµ! of ' = (x2 + y2) dx ! dy is injective. First of all µ!(0, 0) = 0G! . Now, ifz = (x, y) and z " = (x ", y ") are two di!erent points of R2 and di!erent from (0, 0),then there is a smooth function with compact support contained in a small ball, notcontaining (0, 0) nor z, such that f(z ") = 1. Then, the 1-parameter group generatedby grad!(f) belongs to Di!(R2,'), and a similar argument as the one of the proofabove shows that µ!(z) "= µ!(z "). We still need to prove that if z "= (0, 0), thenµ!(z) "= 0G! . Let us consider p(t) = tz and F(r) be the positive rotation of angle2(r, where r ! R. The application of (art. 9.20, (%)), computed at the pointr = 0 and applied to the vector )r = 1, gives (2(/3)(x2 + y2)2 which is not zero.Therefore, the moment map µ! is injective. !9.24. The classical homogeneous case. Let (M,') be a symplectic manifold.Let G be a Lie group together with a homogeneous Hamiltonian action on (M,'),


that is, the holonomy ! of G is trivial. For the sake of simplicity we assume Mconnected and G a Lie subgroup of Di!(M,"). By functoriality of the momentmaps (art. 9.12), we know that if a moment map µ of G is injective, then everyuniversal moment map µ! is injective (art. 9.23). But we are now in the oppositecase, since " is symplectic every universal moment map µ! is injective, but whatabout µ? This is actually the original case treated by Souriau in [Sou70]. Heshowed that the moment map µ is a covering onto its image, which is some coadjointorbit O ! G! (a"ne or not) of G. We give here the proof of this theorem, accordingto the present framework. This case is illustrated by Exercise 146, p. 328.

Proof. Let p be a path in M such that µ"p = cst, that is, #(p) = 0G! , where # isthe paths moment map ofG. Then, for any 1-parameter subgroup F # Hom!(R, G),#(p)(F)r($r) = 0, for all r and all $r belonging to R. Adapting to our case theexpression of # given in (art. 9.20), we get

!1

0

"p(t)(p(t), Z(p(t)))dt = 0 where Z(m) =%F(t)(m)

%t

!!!!t=0

.

Now, considering the 1-parameter family of paths ps : t $" p(st), the derivative ofthe above expression gives "p(0)(p(0), Z(p(0))) = 0. But since G is transitive, byrunning over all the 1-parameter subgroups F of G we describe the whole tangentspace TmM, where m = p(0). And since " is nondegenerate, p(0) = 0. The pathp is thus constant, p(t) = m for all t in R. Therefore the preimages of the valuesof the moment map µ are discrete. But, µ is a fibration (see (art. 9.18)), thus µ isa covering (art. 8.24) onto its image which is, by transitivity, a coadjoint orbit. !9.25. The Souriau-Nœther theorem. Let M be a manifold, and let " be aclosed 2-form on M. We say that two points m and m " are on the same character-istic11 of " if there exists a path p connecting m to m " such that p(t) # ker("p(t))for all t. Then, the universal moment maps µ! are constant on the characteristicsof ".

Note 1. In particular, for any smooth action of a di!eological group G, preserving", the moment maps µ are constant on the characteristics of ".

Note 2. This is an analogue to the first Nœther theorem, relating symmetries toconserved quantities for Lagrangian systems. For a comprehensive expose on thesubject, see the book of Y. Kosmann-Schwarzbach [YKS10].

Proof. We have µ!(m ")!µ!(m) = &!(m,m ") = class(#!(p)) # G!!/!!, where

p is any (smooth) path connecting m to m ". Then, thanks to the hypothesis, wecan use a path p contained in the characteristic of " containing m and m ", that is,p(t) # ker("p(t)) for all t. Now, thanks to the explicit formula of (art. 9.20), forevery n-plot F of Di!(M,"), n # N, for every r # def(F), for every $r # Rn,

#!(p)(F)r($r) =

!1

0

"p(t)(p(t), $p(t))dt =

!1

0

0% dt = 0.

Thus, #!(p) = 0 and therefore µ!(m ") = µ!(m). The note is a consequence of thefunctoriality of the moment map (art. 9.14). Indeed, let ' be the morphism from Gto Di!(M,"), then the paths moment map #, relative to G, satisfies # = '! "#!.

11This is the classical definition of the characteristics in the case of a closed 2-form " definedon a manifold M. They are the integral submanifolds of the distribution m !! ker("m).


Thus, !!(p) = 0 implies !(p) = 0 which implies µ(m) = µ(m !), for any momentmap µ relative to G. !9.26. Presymplectic homogeneous manifolds. Let M be a connected Haus-dor! manifold, and let " be a closed 2-form on M. Let G ! Di!(M,") be aconnected subgroup. If M is a homogeneous space of G, then the characteristics of" are the connected components of the preimages of the moment maps µ.

Note. In particular, if M is a homogeneous space of Di!(M,"), and thus ofits identity component (art. 7.9), then the characteristics of " are the connectedcomponents of the preimages of the values of a universal moment map µ!. Thisjustifies a posteriori the definition of the characteristics of moment maps, for thegeneral case of homogeneous di!eological spaces, in (art. 9.17).

Proof. The Souriau-Nœther theorem states that if m and m ! are on the samecharacteristic, then µ(m) = µ(m !) (art. 9.25). We shall prove the converse in a fewsteps.

a) Let us consider first the case when the holonomy of # is trivial, # = {0}. Let usassume m and m ! connected by a path p such that µ(p(t)) = µ(m) for all t. Then,let s "! ps be defined by ps(t) = p(st), for all s and t. We have µ(ps(1)) = µ(ps(0)),that is, !(ps) = 0"G, for all s. Thus, for all n-plots F of G, for all r # def(F) and all$r # Rn, !(ps)(F)r($r) = 0, and hence

0 =%!(ps)(F)r($r)

%s=

%

%s

"1

0

"ps(t)(ps(t), $ps(t))dt = "p(s)(p(s), $p(s)),

where $p(t) is given by (art. 9.20, ($)). Next, let v # Tp(t)(M), then there existsa path c of M such that c(0) = p(t) and dc(s)/ds|s=0 = v. Since M is assumedhomogeneous under the action of G, there exists a 1-plot s "! F(s) centered at theidentity, that is, F(0) = 1M, such that F(s)(p(t)) = c(s). Then, for s = 0 and$s = 1, we get from (art. 9.20, ($)),

$p(t) = 1Tp(t)MdF(s)(p(t))

ds

!!!!s=0

=dc(s)

ds

!!!!s=0

= v.

Hence, for every v # Tp(t)M, "(p(t), v) = 0, i.e., p(t) # ker("p(t)) for all t.Therefore, the connected components of the preimages of the values of the momentmap µ are the characteristics of ".

b) Let us consider the general case. Let "M be the universal covering of M, & : "M !M the projection, and let " = &"("). Let #G be the group defined by

#G = {g # Di!("M, ") | %g # G and & & g = g & &}.

Let ' : #G ! G be the morphism g "! g. The group #G is an extension of G by thehomotopy group &1(M)

1 ''! &1(M) ''! #G'

''! G ''! 1.

1. The morphism ' is surjective. Let g # G, let t "! gt be a smooth path in G

connecting 1G to g. Let m # "M and m = &(m), the path t "! gt(m) has a unique

lift t "! mt in "M starting at m (art. 8.25). We can check that, g : m "! m1 is

a di!eomorphism of "M, satisfying by construction & & g = g & &. Next, g"(") =


g!(!!(")) = g! ! !!(") = (! ! g)!(") = (g ! !)!(") = !!(g!(")) = !!(") = ".

Thus, g " !G.2. The group !G is transitive on "M. Let m and m " be two points of "M, let

m = !(m) and m " = !(m "). Since G is transitive on M there exists g " G suchthat g(m) = m ". The lift g defined in part 1 maps m to m1, and we have !(m1) =!(m ") = m ". So there exists an element k " !1(M) such that kM(m1) = m "

(art. 8.26). Let g = kM ! g, since ! ! g = g ! ! and g!(") = (kM ! g)!(") =

g!(k!M(")) = g!(") = ", g belongs to !G, and maps m to m ".

3. The kernel of # is reduced to !1(M). Let g " !G such that #(g) = 1M,that is, ! ! g = !. Thus, for every m " M there exists $(m) " !1(M) such that

g(m) = $(m)M(m). But the map $ : "M ! !1(M) is smooth, and !1(M) discrete,so $ is constant. Therefore, there exists k " !1(M) such that g(m) = kM(m), for

all m " "M.4. Since "M is simply connected, !G has no holonomy. Let µ be a moment

map of the action of !G on "M. Since !G is a discrete extension of G, their space ofmomenta coincide (art. 7.13), thus µ takes its values in G!, and since the action of

G on M is the image by the morphism # of the action of !G on "M, for every m " "M,µ(!(m)) = class(µ(m)) " G!/% (art. 9.13). Next, let c = µ(m) = class(µ(m)) "G!/% , m = !(m), and C = µ!1(c). The preimage !!1(C) = !!1(µ!1(c)) is equalto (µ ! !)!1(c) = (class !µ)!1(c) = µ!1(class!1(c)) = µ!1(µ(m) + %). Thus,

µ!1(c) = !

# $

!#"

µ!1(&+ ')

%with & = µ(m) and c = class(&).

Since !G is transitive on "M and since there is no holonomy, we can apply the resultof part a): for every ' " % , µ!1(&+') is a union of characteristics of ". Thus, theunion over all the ' " % is still a union of characteristics of ". Hence, µ!1(c) is the

!-projection of a union of characteristics of ". But, since ! : "M ! M is a coveringand since " = !!("), the !-projection of a characteristic of " is a characteristic of". Thus µ!1(c) is a union of characteristics of ", and the connected componentsof the preimages of the values of µ are the characteristics of ". !

Exercises

! Exercise 145 (The classical moment map). Let M be a connected Hausdor!manifold equipped with a closed 2-form ". Let G be a Lie group (that is, adi!eological group which is a manifold). Let # : g #! gM be a Hamiltonian actionof G on M. Let us recall that a 1-parameter subgroup F " Hom!(R, G) is uniquelydefined by

Z =dF(t)

dt

&&&&t=0

and Z " G = T1G(G).

We denote F(t) = etZ. The fundamental vector field ZM is defined on M by

ZM(m) =(etZM (m)

(t

&&&&t=0

" Tm(M).

Let µ be a moment map of the action of G on M, we shall denote

µZ(m) = µ(m)(t #! etZ)0(1).


1) Show that the moment map µ is defined, up to a constant, by the di!erentialequation

iZM(!) = !dµZ, that is, !m(ZM(m), "m) = !d[µZ]m("m),

for all m ! M and all "m ! Tm(M).

2. Show then, using a basis of G, that Z "! µZ is linear.

! Exercise 146 (The cylinder and SL(2, R)). We consider the space R2, equippedwith the standard symplectic form ! = dx!dy, with X = (x, y) ! R2. Check thatthe special linear group SL(2,R) preserves !, and that its action on R2 has twoorbits, the origin 0 ! R2 and the cylinder M = R2 ! {0}.

1) Justify, without computation, that the action of SL(2,R) on R2 is Hamiltonianand exact.

2) For every X ! R2, let #X = [t "! tX] ! Paths(R2) connecting 0 to X. Use thegeneral expression of the paths moment map given in (art. 9.20), for p = #X andF! = [s "! es!], with $ ! sl(2,R) — the Lie algebra of SL(2,R), that is, the spaceof 2# 2 traceless matrices — to show that

µ(X)(F!) =12!(X,$X)# dt.

3) Deduce that the moment map µM = µ ! M of SL(2,R) on M is a nontrivialdouble sheet covering onto its image.

Examples of Moment Maps in Di!eology

The few following examples want to illustrate how the theory of moment mapsin di!eology can be applied to the field of infinite dimensional situations, but alsoto the less familiar case of singular spaces. It is, at the same time, the opportunityto familiarize ourselves with the computational techniques in di!eology.

9.27. On the intersection 2-form of a surface, I. Let % be a closed surfaceoriented by a 2-form Surf, chosen once and for all. Let us consider &1(%), the infi-nite dimensional vector space of 1-forms of %, equipped with functional di!eology.Let us consider the antisymmetric bilinear map defined on &1(%) by

(',() "!"

"

'! (,

for all ', ( in &1(%). Since the wedge-product '!( is a 2-form of %, there exists areal smooth function ) ! C!(%,R) such that '!( = )#Surf. Thus, by definition,#" '! ( =

#" )# Surf.

1. A well defined di!erential 2-form ! of &1(X) is naturally associated with theabove bilinear form. For every n-plot P : U ! X, for all r ! U, "r and " !r in Rn,

!(P)r("r, "!r) =

"

"

*P(r)

*r("r)!

*P(r)

*r(" !r).

2. The 2-form ! is the di!erential of the 1-form + defined on &1(%) by

+(P)r("r) =12

"

"

P(r)!*P(r)

*r("r) and ! = d+.

EXAMPLES OF MOMENT MAPS IN DIFFEOLOGY 329

3. Consider now the additive group (C!(!,R),+) of smooth real functions on !,and let us define the following action of C!(!,R) on "1(!),

for all f ! C!(!,R), f "! f = [# "! #+ df].

Then, the additive group C!(!,R) acts by automorphisms on the pair ("1(!),$),

for all f in C!(!,R), f!($) = $.

Note that the kernel of the action f "! f is the subgroup of constant maps, and theimage of C!(!,R) is the group B1

dR(!) of exact 1-forms of !.

4. Let p ! Paths("1(!)) be a path connecting #0 to #1. The paths moment map%(p) is then given by

%(p) =

!#!1(&) + d

"f "! 1

2

"

!

f# d#1

#$!

!#!0(&) + d

"f "! 1

2

"

!

f# d#0

#$.

On this expression, we get immediately the 2-points moment map '(#0,#1) =%(p), for any path p connecting #0 to #1. Note that, since "1(!) is contractible,the holonomy of the action of C!(!,R) is trivial, ( = {0}, and the action of C!(!,R)is Hamiltonian.

5. The moment maps of this action of C!(!,R) on "1(!) are, up to a constant,equal to

µ : # "! d

"f "!

"

!

f# d#

#.

Moreover, the moment map µ is equivariant, that is, invariant, since the groupC!(!,R) is Abelian,

for all f ! C!(!,R), µ $ f = µ.

In summary, the action of C!(!,R) on "1(!) is exact and Hamiltonian.

Note. The moment map µ(#) is fully characterized by d#. This is why we find inthe mathematical literature on the subject that the moment map for this action isthe exterior derivative (or curvature, depending on the authors) # "! d#. But aswe see again on this example, the di!eological framework gives to this assertion aprecise meaning. Let us also remark that the moment map µ is linear, for all realnumbers t and s, and for all # and ) in "1(!), µ(t#+ s)) = t µ(#)+ s µ()). Thekernel of µ is the subspace of closed 1-forms,

ker(µ) = Z1dR(!) =

## ! "1(!) | d# = 0

$.

The orbit of the zero form 0 ! "1(!) by C!(!,R) is just the subspace B1dR(!) %

Z1dR(!), see (art. 9.29, Note 3) for a discussion about that.

Proof. 1. Let us check that $ defines a di!erential 1-form on "1(!). Note that,for any r ! U = def(P), P(r) is a section of the ordinary cotangent bundle T!!,P(r) = [x "! P(r)(x)] ! C!(!, T!!), where P(r)(x) ! T!

x(!). Thus,

*P(r)

*r(+r) = [x "! *P(r)(x)

*r(+r)] and

*P(r)(x)

*r(+r) ! T!

x(!),

where *P(r)(x)/*r denotes the tangent linear map D(r "! P(r)(x))(r). The formulagiving $ is then well defined. Now, $(P)r is clearly antisymmetric and dependssmoothly on r. Hence, $(P) is a smooth 2-form of U. Let us check that P "! $(P)defines a 2-form on "1(!), that is, satisfies the compatibility condition $(P $ F) =


F!(!(P)), for all F ! C!(V,U), where V is a real domain. Let s ! V, "s and " "stwo tangent vectors to V at s, let r = F(s), and compute

!(P " F)s("s, " "s) =

!

!

#P " F(s)#s

("s)!#P " F(s)

#s(" "s)

=

!

!

#P(r)

#r

#F(s)

#s("s)!

#P(r)

#r

#F(s)

#s(" "s)

= !(P)F(s)(DFs("s), DFs(""s))

= F!(!(P))s("s, ""s).

Thus, !(P " F) = F!(!(P)), and ! is a well defined 2-form on $1(%).

2. First of all, the proof that the map P #" &(P) is a well defined di!erential 1-formof $1(%) is analogous to the proof of the first item. Now, let us recall that ! = d&means d(&(P)) = !(P), for all plots P of $1(%). Let us apply the usual formula ofdi!erentiation of 1-forms on real domains,

d'r("r, ""r) = "('r("

"r))! " "('r("r)),

where " and " " are two independent variations. For the sake of simplicity let usdenote

( = P(r), "( =#P(r)

#r("r), " "( =

#P(r)

#r(" "r).

Then,

d(&(P))r("r, ""r) = 1

2

!"

!

!

(! " "( ! " "!

!

(! "(

"

= 12

! !

!

"(! " "( + (! "" "(!

!

!

" "(! "( + (! " ""(

",

but "" "( = " ""(, thus

d(&(P))r("r, ""r) = 1

2

! !

!

"(! " "(!

!

!

" "(! "(

"

= 12

! !

!

"(! " "(+

!

!

"(! " "(

"

=

!

!

"(! " "(

= !r("r, ""r).


3. Let us compute the pullback of ! by the action of f ! C!(",R). Let P : U !#1(") be an n-plot, and let r ! U and $r ! Rn,

f!(!)(P)r($r) = !(f " P)r($r)= !(r #! P(r) + df)r($r)

= 12

"

!

(P(r) + df)!%P(r)

%r($r)

= 12

"

!

P(r)!%P(r)

%r($r) + 1

2

"

!

df!%P(r)

%r($r)

= !(P)r($r) +%

%r

#12

"

!

df! P(r)

$($r)

= !(P)r($r)!%

%r

#12

"

!

f$ d(P(r))

$($r).

Next, we define, for every f ! C!(",R), &(f) : #1(") ! R by

&(f) : ' #! 12

"

!

f$ d'.

Then,

d(&(f))(P)r($r) =%

%r

#12

"

!

f$ d(P(r))

$($r).

Thus,f!(!)(P)r($r) = !(P)r($r)! (d&(f))(P)r($r),

that is,f!(!) = !! d(&(f)).

Hence, d[f!(!)] = d!, and ( = d! is invariant by the action of C!(",R).

4. Let p be a path in #1(") connecting '0 to '1. By definition )(p) = p!(K().Applying the property of the Chain-Homotopy Operator d "K +K " d = 1! ! 0!

to ( = d!, we get

)(p) = p!(Kd!)

= p!(1!(!)! 0!(!)! d(K!))

= (1 " p)!(!)! (0 " p)!(!)! d[(K!) " p]= '!

1(!)! '!0(!)! d[f #! K!(p(f))].

But K!(p(f)) = K!(f " p) =%f"p ! =

%p f!(!), and since f!(!) = ! ! d(&(f)),

K!(p(f)) =%p !!

%p d(&(f)) =

%p !!&(f)('1) +&(f)('0). Therefore,

)(p) = '!1(!)! '!

0(!)! d[f #! !&(f)('1) +&(f)('0)]

= '!1(!)! '!

0(!) + d

!f #! 1

2

"

!

f$ d'1 !12

"

!

f$ d'0

".

We get then the paths moment map ),

)(p) =

#'!1(!) + d

!f #! 1

2

"

!

f$ d'1

"$!

#'!0(!) + d

!f #! 1

2

"

!

f$ d'0

"$.

Concerning the 2-points moment map *, we clearly have *('0,'1) = )(p), for anypath connecting '0 to '1.


5. The 1-point moment maps are given by µ(!) = "(!0,!) for any origin !0. Letus choose !0 = 0. So,

µ(!) = !!(#) + d

!f !! 1

2

"

!

f" d!

"! 0!(#).

But 0!(!) is not necessarily zero. Let us compute generally !!(#). Let P : U !$1(%) be an n-plot, !!(#)(P) = #(! # P) = #(r !! !(P(r)) = #(r !! ! + d(P(r))),and

#(r !! !+ d(P(r))) = 12

"

!

(!+ P(r))!&

&r(!+ d(P(r)))

= 12

"

!

(!+ P(r))!&d(P(r))

&r

= 12

"

!

!!&d(P(r))

&r+ 1

2

"

!

P(r)!&d(P(r))

&r.

Then,

(!!(#)! 0!(#))(P) = 12

"

!

!!&d(P(r))

&r.

Therefore,

µ(!)(P)r = (!!(#)! 0!(#))(P)r + d

!f !! 1

2

"

!

f" d!

"(P)r

= 12

"

!

!!&d(P(r))

&r+

&

&r

#12

"

!

P(r)" d!

$

= 12&

&r

#"

!

!! d(P(r)) + P(r)" d!

$

=&

&r

#"

!

P(r)" d!

$,

which gives finally

µ(!) = d

!f !!

"

!

f" d!

".

Now, let us express the variance of µ. Let f $ C!(%,R) and F(!) be the real functionF(!) : f !!

%! f " d!, such that µ(!) = dF(!). We have µ(f(!)) = µ(! + df) =

dF(! + df) but, for every h $ C!(%,R), F(! + df)(h) =%! h " d(! + df) =%

! h" d! = F(!)(h). Then, for all f $ C!(%,R), µ # f = µ. The moment map µ isinvariant by the group C!(%,R). The Souriau class is zero, the action of C!(%,R)is then exact and Hamiltonian.

Let us end with the computation of the kernel of the moment map µ. Clearly,µ(!) = 0 if and only if dF(!) = 0. But since C!(%,R) is connected (actuallycontractible as a di!eological vector space), dF(!) = 0 if and only if F(!) = cst =F(!)(0) = 0. But F(!) = 0 if and only if, for all f $ C!(%,R),

%! f" d! = 0, that

is, if and only if d! = 0. !9.28. On the intersection 2-form of a surface, II. We continue with theexample of (art. 9.27) using the same notations. Let us introduce the group G ofpositive di!eomorphisms of (%, Surf), that is,

G =

#g $ Di!(%)

####g!(Surf)

Surf> 0

$.


The group G acts by pushforward on !1("). For all g ! G, for all # ! !1("),g!(#) ! !1("), and for all pairs g, g " of elements of G, (g " g ")! = g! " g "

!, andthis action is smooth.

1. The pushforward action of G on !1(") preserves the 1-form $, and thus the2-form %. For all g ! G, (g!)!($) = $, and (g!)!(%) = %. Thus, the action of Gis exact, & = 0, and Hamiltonian, ' = {0}.

2. The moment maps are, up to a constant, equal to the moment µ,

µ(#)(P)r((r) =12

!

!

#! P(r)!!)P(r)!(#)

)r((r)

",

for all # ! !1("), for all n-plots P, where r ! def(P) and (r ! Rn. In particular,applied to any 1-plot F centered at the identity 1G, that is, F(0) = 1G, we get thespecial expression

µ(#)(F)0(1) = !12

!

!

#! £F(#) = !

!

!

iF(#)# d#,

where £F(#) and iF(#) are the Lie derivative and the contraction of # by F. Notethat it is not surprising that the Lie derivative of # is closely associated with themoment map of the action of the group of di!eomorphisms.

Proof. 1. Let us compute the pullback of $ by the action of g ! G, that is,(g!)!($). Let P : U " !1(") be an n-plot, let r ! U, and (r ! Rn, then

(g!)!($)(P)r((r) = $(g! " P)r((r)

= 12

!

!

g!(P(r))!)g!(P(r))

)r((r)

= 12

!

!

g!(P(r))! g!

!)P(r)

)r((r)

"

= 12

!

!

g!

!P(r)!

)P(r)

)r((r)

"

= 12

!

g!(!)P(r)!

)P(r)

)r((r)

= 12

!

!

P(r)!)P(r)

)r((r)

= $(P)r((r).

Thus, $ is invariant by G, and so is % = d$.

2. Since the 1-form $ is invariant by the action of G, we can use directly the resultsof the exact case detailed in (art. 9.11). The moment maps are, up to a constant,equal to µ : # $" #!($). Then, let P : U " G be an n-plot, let r ! U, (r ! Rn and


let us compute,

µ(!)(P)r("r) = !!(#)(P)r("r)

= #(! ! P)r("r)= #(r "! P(r)!(!))r("r)

= 12

"

!

P(r)!(!)!$P(r)!(!)

$r("r)

= 12

"

!

!! P(r)!!$P(r)!(!)

$r("r)

".

Now, let P = F be a 1-plot centered at the identity, F(0) = 1G. Let us changethe variable r for the variable t. The previous expression, computed at t = 0 andapplied to the vector "t = 1 gives immediately

µ(!)(F)0(1) = 12

"

!

!!$F(t)!(!)

$t

####t=0

.

But, by definition of the Lie derivative,#$F(t)!(!)

$t

$

t=0

=

#$(F(t)!1)!(!)

$t

$

t=0

= !£F(!).

Thus, we get the first expression of the moment map µ applied to F, that is,

µ(!)(F)0(1) = !12

"

!

!! £F(!).

Now, on a surface ! ! d! = 0, and iF(! ! d!) = iF(!) # d! ! ! ! iF(d!),thus iF(!) # d! = ! ! iF(d!). Then, using the Cartan-Lie formula £F(!) =iF(d!) + d(iF(!)),

"

!

!! £F(!) =

"

!

!! [iF(d!) + d(iF(!))]

=

"

!

iF(!)d!+

"

!

!! d(iF(!))

=

"

!

iF(!)d!+

"

!

iF(!)d!!

"

!

d[!! iF(!)]

= 2

"

!

iF(!)d!.

And finally, we get the second expression for the moment map, that is,

µ(!)(F)0(1) = !

"

!

iF(!)# d!,

for any 1-plot of the group of positive di!eomorphisms of the surface %, centeredat the identity. !9.29. On the intersection 2-form of a surface, III. We continue again withthe example of (art. 9.27), using the same notations. Let us consider the space&1(%) as an additive group acting onto itself by translations. Let us denote by t"the translation t" : ! "! !+ ', where ! and ' belong to &1(%).

1. The 2-form ( is invariant by translation, that is, t!#(() = ( for all ! $ &1(%).This action of &1(%) onto itself is Hamiltonian but not exact.


2. The moment maps of the additive action of !1(") onto itself are equal, up to aconstant, to

µ : # !! d

!$ !!

"

!

#! $

".

In other words, µ(#) = d[%(#)], where % is regarded as the smooth linear function%(#) : $ !! %(#,$), defined on !1("). Moreover, the moment map µ is linear andinjective.

3. The moment map µ is its own Souriau cocycle, & = µ. The moment map µidentifies !1(") with the &-a!ne coadjoint orbit of 0 " !1(")!. Be aware that!1(")! denotes the space of invariant 1-forms of the Abelian group !1("), and notits algebraic dual.

Note 1. This example is analogous to finite dimension symplectic vector spaces.The 2-form % can be regarded as a real 2-cocycle of the additive group !1(").This cocycle builds up a central extension by R,

(#, t) · (# ", t ") =

##+ # ", t+ t " +

"

!

#! # "$

for all (#, t) and (# ", t ") in !1(") # R. This central extension acts on !1("),preserving %. This action is Hamiltonian, and now exact. The lack of equivariance,characterized by the Souriau class, has been absorbed in the extension. This groupcould be named, by analogy, the Heisenberg group of the oriented surface (", Surf).

Note 2. According to (art. 9.19), the space !1(") equipped with the 2-form % is ahomogeneous symplectic space. Thus, we have a first simple example of an infinitedimensional symplectic di!eological space, avoiding any consideration on the kernelof %.

Note 3. The preimage of zero by the moment map of the Abelian group C!(",R)on !1(") in (art. 9.27) is the subgroup of closed forms Z1

dR(") $ !1("). Thisgroup acts homogeneously on itself and its moment map for the restriction of the2-form % is the projection of the moment map of the action of C!(",R), that is,µ " : # !! d

%$ !!

#! #! $

&, but now # and $ are closed. The characteristics of

this moment map are the orbits of the subgroup µ "!1(0), that is, the subgroup of# " Z1

dR(") such that#! #!$ = 0 for all $ " Z1

dR("). This is the subgroup of exact1-forms B1

dR("). Then, the image of µ " identifies naturally with the quotient spaceH1

dR(") = Z1dR(")/B

1dR("). Moreover, the closed 2-form passes to this quotient, it

is the well known intersection form, denoted here by %int, % ! Z1dR(") = µ "!(%int).

This is an example of symplectic reduction in di"eology (see also (art. 9.17, Note)),the full general case will be addressed in a future work.


Proof. Let us compute the pullback of ! by a translation. Let P : U ! X be ann-plot, let r ! U, and let "r ! Rn,

t!!(!)(P)r("r) = !(t! " P)r("r)= ![r #! P(r) + #]r("r)

= 12

"

"

(P(r) + #)!$(P(r) + #)

$r("r)

= 12

"

"

P(r)!$P(r)

$r("r) + 1

2

"

"

#!$P(r)

$r("r)

= !(P)r("r) + d

!% #! 1

2

"

"

#! %

"(P)r("r).

Let us define next, for all # ! &1('), the smooth real function F(#) by

F(#) : % #! 12

"

"

#! %.

Then,t!!(!) = !+ d(F(#)) and t!!(() = (.

Next,&1('), as an additive group, acts on itself by automorphisms. Let us computethe moment maps. Let p be a path in &1('), connecting #0 to #1, then

)(p) = #!1(!)! #!

0(!)! d

!% #!

"

p

d(F(%))

"

= #!1(!)! #!

0(!)! d[% #! F(%)(#1)! F(%)(#0)]

= {#!1(!)! d[% #! F(%)(#1)]}! {#!

0(!)! d[% #! F(%)(#0)]}

= {#!1(!) + d(F(#1))}! {#!

0(!) + d(F(#0))}.

Thus, the 2-points moment map is clearly given by *(#0,#1) = )(p). Now, themoment maps are, up to a constant, equal to

µ(#) = *(0,#) = #!1(!) + d(F(#))! 0!(!).

But for any plot P : U ! &1('),

#!(!)(P)! 0!(!)(P) = !(# " P)! !(0 " P)= !(r #! P(r) + #)! !(r #! P(r))

= d

!% #! 1

2

"

"

#! %

"(P)

= d(F(#))(P).

Hence, #!(!)(P)! 0!(!) = d(F(#)) and µ is finally given by

µ(#) = 2d(F(#)) = d

!% #!

"

"

#! %

".

The moment map µ is not equivariant, the Souriau cocycle + is given by

µ(t!!(%)) = µ(#+ %) = µ(%) + +(#) with +(#) = µ(#).

Considering the Note, the moment map µ is clearly smooth and linear. Let # !ker(µ), µ(#) = 0 if and only if d(F(#)) = 0, that is, if and only if F(#) = cst =F(#)(0) = 0. But F(#)(%) = 0, for all % ! &1('), implies # = 0. Therefore, themoment map µ is injective. !


9.30. On symplectic irrational tori. Consider the smooth space Rn, for someinteger n. For all u ! Rn, let tu be the translation by u, that is, tu : x "! x + u.Let ! be a 2-form of Rn invariant by translation, that is, for all u ! Rn, t!u(!) =!. Thus, ! is a constant bilinear 2-form, thus closed, d! = 0. Let us considerthe moment maps associated with the translations (Rn,+). Since Rn is simplyconnected, the holonomy is trivial, " = {0}. Let p be a path in Rn connectingx = p(0) to y = p(1), the paths moment map #(p), and the 2-points moment map$(p) are given by

#(p) = $(x, y) = !(y! x),

where !(u) is regarded as the linear 1-form !(u) : v "! !(u, v). The momentmaps are, up to constant, equal to the linear map

µ : x "! !(x),

and the Souriau cocycle % associated with µ is equal to µ. For all u ! Rn,

%(u) = µ(u) = !(u).

Consider now a discrete di!eological subgroup K # Rn. Let us denote by Q thequotient Q = Rn/K and by & : Rn ! Q the projection. Let us continue to denoteby tu the translation on Q, by u ! Rn, that is, tu(q) = &(x + u), for any x suchthat q = &(x). Now, since ! is invariant by translation, ! is invariant by K, andsince K is discrete, ! projects on Q as a Rn-invariant closed 2-form denoted by!Q, that is,

!Q = &!(!) or ! = &!(!Q).

Note that the translation by any vector u of Rn on Q is still an automorphism of!Q, that is, t!u(!Q) = !Q.

1. The holonomy "Q of the action of (Rn,+) on (Q,!Q) is the image of the subgroupK by µ,

"Q = µ(K), "Q # Rn!.

Thus, if ! $= 0 and if K is not reduced to {0}, then the action of (Rn,+) on (Q,!Q)is not Hamiltonian and not exact.

2. The moment map µ : Rn ! Rn! projects on a moment µQ such that the followingdiagram commutes.

Rn Rn!

Q = Rn/K Rn!/µ(K)

µ

& pr

µQ

For all q ! Q, µQ(q) = pr(!(x)) for any x such that q = &(x). The Souriau cocycle%Q associated with µQ, for all u ! Rn, is given by

%Q(u) = µQ(&(u)).

Hence, if we consider the space Q as an additive group acting on itself by transla-tions, then the moment map µQ coincides with its Souriau cocycle %Q.

3. The map µ is a fibration onto its image whose fiber is the kernel of µ, thatis, val(µ) % Rn/E, E = ker(µ). And, the map µQ is a fibration onto its image


µ(Rn)/µ(K) whose fiber is ker(µQ) = E/(K ! E). If ! : Rn ! Rn! is injective(which implies that n is even), then the moment map µQ is a di!eomorphismwhich identifies Q with its image Rn!/µ(K).

Note 1. Regarded as a group, Q = Rn/K acts onto itself by projection of thetranslations of Rn. Since the pullback by " : Rn ! Q is an isomorphism from Q!

to Rn! (Rn is the universal covering of Q), the moment maps computed above givethe moment maps associated with this action.

Note 2. This construction applies to the torus T2 = R2/Z2. The action of (R2,+),is obviously not Hamiltonian, but the moment map µT2 is well defined. And, µT2

identifies T2 with the quotient of R2! — the (#Q, $Q)-coadjoint orbit of the point0 — by the holonomy #Q = !(Z2) " R2!, according to the meaning of a coadjointorbit we gave in (art. 7.16). The torus T2, equipped with the standard symplecticform !, is a coadjoint orbit of R2, or even a coadjoint orbit of itself. This is aspecial case of the (art. 9.23) discussion.

Note 3. All this construction above also applies to situations regarded as moresingular that the simple quotient of Rn by a lattice. It applies, for example, to theproduct of any irrational tori. An (n-dimensional) irrational torus TK is the quotientof Rn by any generating discrete strict subgroup K of Rn. See for example [IgLa90]for an analysis of 1-dimensional irrational tori. For example, we can consider theproduct of two 1-dimensional irrational tori Q = TH#TK, quotient of R2 = R#R bythe discrete subgroup %H(Zp)#%K(Zq), where %H : Rp ! R and %K : Rq ! R aretwo linear 1-forms with coe"cients independent over Q. In this case, the momentmap µQ will also identify TH # TK with the quotient of R2! — (#Q, $Q)-coadjointorbit of 0 — by #Q = !(%H(Zp)#%K(Zq)). This is the simplest example of totallyirrational symplectic space, and totally irrational coadjoint orbit. Note that thesecases escape completely the usual analysis, of course, but also the analysis in termsof Sikorski or Frolicher spaces; see Exercise 80, p. 99.

Proof. First of all, the fact that there exists a closed 2-form !Q on R/K suchthat "!(!Q) = ! is an application of the criterion of pushing forward forms, inthe special case of a covering (art. 8.27). Now, the computation of the momentmap of a linear antisymmetric form ! on Rn is well known, and independently ofthe method gives the same result µ(x) = !(x). The additive constant is set hereby the condition µ(0) = 0. But the value of the paths moment map &(p) can becomputed as well by the general method, applying the particular expression

K!p('p) =

"1

0

!p(t)(p(t), 'p(t))dt , with p(t) =dp(t)

dt,

of the Chain-Homotopy operator for manifold, where p is a path and 'p is a varia-tion of p. Then, since the result depends only on the ends of the path, choose, forany two points x and y in Rn, the connecting path p : t $! x+t(y!x). Let us recallthat &(p) = p!(K!), let u and 'u in Rn, note that p!(tu) = tu%p = [t $! p(t)+u].


Then,

!(p)u("u) = p!(K#)u("u)

= (K#)tu"p("(tu ! p)), with "p = 0

=

!1

0

#(p(t), "u) d t

= #(y! x, "u).

Thus, !(p) = $(x, y) = #(y! x) = #(y)!#(x), and µ : x "" #(x), x # Rn. Con-sider now #Q, since Rn is the universal covering of Q, every loop % # Loops(Q, 0)can be lifted into a path p in Rn starting at 0 and ending in K. In other words,

& = {!(%) | % # Loops(Q)} = {!(t "" tk) | k # K} = #(K).

The other propositions are then a direct application of the functoriality of themoment map described in (art. 9.13), and standard analysis on quotients and fi-brations. !9.31. The corner orbifold. Let us consider the quotient Q of R2 by the actionof the finite subgroup K $ {±1}2, embedded in GL(2,R) by

K =

#!' 00 ' #

" #### ', '# # {±1}

$.

The space Q = R2/K (Figure 9.1) is an orbifold, according to [IKZ05]. It is di!eo-morphic to the quarter space [0,%[ % [0,%[ & R2, equipped with the pushforwardof the standard di!eology of R2 by the map ( : R2 " [0,%[ % [0,%[, defined by,

((x, y) = (x2, y2) and Q $ (!(R2).

The letter Q will denote indi!erently the quotient R2/K or the quarter space (!(R2),and the meaning of the letter ( follows. Be aware that the corner orbifold is nota manifold with boundary, Q is not di!eomorphic to the corner equipped with theinduced di!eology of R2. That said, we remark that the decomposition of Q interms of point’s structure is given by

Str(0, 0) = {±1}2, Str(x, 0) = Str(0, y) = {±1}, and Str(x, y) = {1},

where x and y are positive real numbers. Then, since the structure group of apoint is preserved by di!eomorphisms [IKZ05], there are at least three orbits ofDi!(Q), the point 0Q = (0, 0), the regular stratum Q = ]0,%[2 and the union of thetwo axes, ox and oy. In particular, any di!eomorphism of Q preserves the origin0Q. Actually, these are exactly the orbits of Di!(Q). Let us remark that, sincedim(Q) = 2 (art. 1.78), every 2-form is closed.

1. Every 2-form of Q is proportional to the 2-form # defined on Q by

(!(#) :

!xy

""" 4xy% dx! dy,

that is, for any other 2-form # # there exists a smooth function ) # C!(Q,R) suchthat # # = )%#.

2. The space (Q,#) is Hamiltonian, &! = {0}, and the action of G! is exact, thatis, *! = 0. In particular, the universal moment map µ! defined by µ!(0Q) = 0, isequivariant.


0 ox

oy

Corner OrbifoldPlane

x! x

x!! x!!!

Figure 9.1. The corner orbifold Q.

3. The universal equivariant moment map µ! vanishes on the singular strata {0},ox and oy, and is injective on the regular stratum Q. Therefore, the image µ!(Q)is di!eomorphic to an open disc with a point attached somehow on the boundary.

Proof. 1. Let ! ! be a 2-form on Q, and let ! ! be its pullback by ", ! ! = ""(! !).Thus, there exists a smooth real function F such that ! ! = F! dx! dy. But since" " k = ", for all k # K, we get ## !F(#x, # !y) = F(x, y), for all (x, y) # R2 and all#, # ! in {±1}. Thus, F(!x, y) = !F(x, y) and F(x,!y) = !F(x, y). In particular,F(0, y) = 0 and F(x, 0) = 0. Therefore, since F is smooth, there exists f # C!(R2,R)such that F(x, y) = 4xyf(x, y), with f(#x, # !y) = f(x, y). Therefore, ! ! = f ! !,with ! = 4xy! dx! dy, that is, ! = d(x2)! d(y2), but x $! x2 and y $! y2 areinvariant by K so, they are the pullback by " of some smooth real functions on Q.Thus, d(x2) and d(y2) are the pullback of 1-forms on Q, let us say d(x2) = ""(ds)and d(y2) = ""(dt), so ! = ""(!), where ! = ds ! dt is a well defined 2-formon Q. Now, since f($x, $ !y) = f(x, y) means just that f is the pullback of a smoothreal function % on Q, it follows that any 2-form ! ! on Q is proportional to !, thatis, ! ! = %!!, with % # C!(Q,R).

2. The orbifold is contractible. The deformation retraction (s, x, y) $! (sx, sy) of R2

to {(0, 0)} projects on a smooth deformation retraction of Q. Thus, the holonomyis trivial, & = {0}. Now, since the origin 0Q is the only point with structure {±1},every di!eomorphism of Q preserves the origin 0Q. Then, the 2-point moment mapis exact, see (art. 9.10, Note 2), the Souriau class is zero, '! = 0. Let q be anypoint of Q and let µ!(q) = ((0Q, q). This is an equivariant moment map andµ!(0Q) = ((0Q, 0Q) = 0.

3. Let q # Q, thus µ!(q) = )(p) for any path p connecting 0Q to q. Let q belongto a semi-axis ox or oy, and let us choose p = t $! *(t)q, where * is a smashingfunction equal to 0 on ]!", 0] and equal to 1 on [1,+"[. Thus, for all t # R, p(t)belongs to the same semi-axis as q. Thanks to (art. 9.2, (%)), for any 1-plot % ofDi!(Q,!!) centered at the identity,

)(p)(%)0(1) =

#1

0

!

!"sr

#$! %(r)(*(s+ t)q)

$

(00)

"10

#"01

#dt .


x!

x

x!!

0

Cone OrbifoldPlane

x0

Figure 9.2. The cone orbifold Q3.

But (s, r) !! !(r)("(s+t)q) is a plot of the semi-axis, and thanks to item 1, the form# vanishes on the semi-axis. Thus, the integrand vanishes and $(p)(!)0(1) = 0.Then, since 1-forms are characterized by 1-plots and since momenta are character-ized by centered plots, µ!(q) = 0 for all q " Q belonging to any semi-axis.

On the other hand, let q and q ! be two points of the regular stratum Q. Since% ! {(x, y) | x > 0 and y > 0} is a di!eomorphism, and since # ! {(x, y) | x > 0 andy > 0} is a symplectic Hausdor! manifold, there exists always a symplectomorphism! with compact support S # {(x, y) | x > 0 and y > 0} which exchanges q and q !.Then, the image of this di!eomorphism on Q can be extended by the identity on thewhole Q. Therefore, the automorphisms of # are transitive on the regular stratum.The fact that the universal moment map is injective on the regular stratum comesfrom what we know already on symplectic manifolds (art. 9.23). "9.32. The cone orbifold. Let Qm be the quotient of the smooth complex planeC by the multiplicative action of the cyclic subgroup

Zm $ {& " C | &m = 1} with m > 1.

The space Qm (Figure 9.2) is an orbifold, according to [IKZ05]. We identify Qm

to the complex plane C, equipped with the pushforward of the standard di!eologyby the map %m : z !! zm. Precisely, the plots of Qm are the parametrizations Pof C which write locally P(r) = !(r)m, where ! is a smooth parametrization in C.Let us remark first that the decomposition of Qm, in terms of structure group, isgiven by

Str(0) = Zm, and Str(z) = {1} if z %= 0.

And secondly that there are two orbits of Di!(Qm), the point 0 and the regularstratum Qm = C ! {0}. In particular any di!eomorphism of Qm preserves theorigin 0. It is not di"cult to check that {%m} is a minimal generating family, thusdim(Qm) = 2 (art. 1.78), and every 2-form on Qm is closed.

1. Every 2-form of Qm is proportional to the 2-form # defined by

%"m(#) : z !! dx! dy with z = x+ iy.

For every other 2-form # ! there exists a smooth function f " C!(Qm,R) such that# ! = f&#.


2. The space (Q,!) is Hamiltonian, "! = {0}, and the action of G! is exact, thatis, #! = 0. In particular, the universal moment map µ!, defined by µ!(0) = 0, isequivariant.

3. The universal moment map µ! is injective. Its image is the reunion of twocoadjoint orbits, the point 0 ! G!

!, value of the origin of Qm, and the image of theregular stratum Qm.

Proof. Let us first prove that the usual surface form Surf = dx ! dy is thepullback of a 2-form ! defined on Qm. We shall apply the standard criterion andprove that for any two plots $1 and $2 of C such that %m " $1 = %m " $2, wehave Surf($1) = Surf($2), that is, $1(r)m = $2(r)m implies Surf($1) = Surf($2).First of all let us recall that, since we are dealing with 2-forms, is is su!cient toconsider 2-plots. So, let the $i be defined on some real domain U # R2. Letr0 ! U, we split the problem into two cases.

1. $1(r0) $= 0 — Thus $2(r0) $= 0, there exists an open disc B centered at r0 onwhich the $i do not vanish. Thus, the map r %! &(r) = $2(r)/$1(r) defined on Bis smooth with values in Zm. But, since Zm is discrete there exists & ! Zm suchthat $2(r) = & & $1(r) on B. Now, Surf is invariant by U(1) ' Zm. ThereforeSurf($1) = Surf($2) on B.

2. $1(r0) = 0 — Thus, $2(r0) = 0. Now, we have Surf($i) = det(D($i)) & Surf,where D($i) denotes the tangent map of $i. We split this case in two subcases.

2.a. D($1)r0 is not degenerate — Thus, thanks to the implicit function theorem,there exists a small open disc B around r0 where $1 is a local di"eomorphism ontoits image. Since $1(r)m = $2(r)m, the common zero r0 of both $1 and $2 isisolated. Thus, the map r %! &(r) = $2(r)/$1(r) defined on B! {r0} is smooth, andfor the same reason as in the first case, & is constant. Then, $2(r) = && $1(r) onB ! {r0}. But since $i(r0) = 0, this equality extends on B. Therefore Surf($1) =Surf($2) on B.

2.b. D($1)r0 is degenerate — Let u be in the kernel of D($1)r0 . We have$1(r0+su)m = $2(r0+su)m for s small enough. Then, di"erentiating this equalitym times with respect to s, we get at s = 0, D($1)r0(u)

m = D($2)r0(u)m = 0.

Therefore, D($2)r0 is also degenerate at r0 and thus Surf($1)r0 = Surf($2)r0 = 0.

Thus, we proved that for all r ! U, Surf($1)r = Surf($2)r. Therefore, thereexists a 2-form ! on Qm such that %!

m(!) = Surf, and this form ! is completelydefined by its pullback. Now, since the pullback by %m of any other 2-form ! "

on Qm is proportional to Surf, the form ! " is proportional to !. Now, for thesame reasons as in (art. 9.31), the universal holonomy "! is trivial, the Souriauclass #! is zero, and the universal moment map µ! defined by µ!(0) = 0G! isequivariant. Moreover, the regular stratum Q is just a symplectic manifold for therestriction of !. Every symplectomorphism with compact support which does notcontain 0 can be extended to an automorphism of (Q,!). Thus, since the compactlysupported symplectomorphisms of a connected symplectic manifold are transitive[Boo69], the regular stratum Q is an orbit of Di"(Q,!) and, the universal momentmap is injective on this stratum (art. 9.23). Therefore, the moment map µ! mapsinjectively Q onto the two orbits, {0G! } and µ!(Q). !


9.33. The infinite projective space. Let H be the Hilbert space of the squaresummable complex series

H =

!Z = (Zi)

!i=1

!!!!n"

i=1

Zi · Zi < #$,

where the dot denotes the Hermitian product. The spaceH is equipped with the finestructure of complex di!eological vector space (art. 3.15). Let P : U % H be a plot,then for every r0 ! U there exist an integer n, an open neighborhood V " U of r0and a finite family F = {(!a, Za)}a!A, where the Za ! H, and the !a ! C!(V,Cn),such that P ! V : r #%

&a!A !a(r)$Za. Such a family {(!a, Za)}a!A is called a local

family of P at the point r0. We introduced the symbol dZ in Exercise 99, p. 159,which associates every local family F = {(!a, Za)}a!A, defined on the domain V,with the complex valued 1-form of V

dZ(F) : r #%"

a!A

d!a(r)Za.

For every !a = xa + iya, where xa and ya are real smooth parametrizations,d!a = dxa + idya. There exists on H a 1-form " defined by

" = 12i[Z · dZ! dZ · Z].

1. As an additive group (H,+) acts on itself, preserving d". Let Z ! H, and lettZ be the translation by Z, then t"Z(d") = d". This action is Hamiltonian but notexact. Let µ be the moment map of the translations (H,+), defined by µ(0H) = 0,

µ(Z) = 2d[w(Z)] with w(#) : Z #% 12i[# · Z! Z · #] ! C!(H,R).

The moment map µ is injective and (H, d") is a homogeneous symplectic space.

2. Let U(H) be the group of unitary transformations of H, equipped with thefunctional di!eology. The group U(H) acts on H preserving ". The action ofU(H) on (H, d") is exact and Hamiltonian. Let P : U % U(H) be an n-plot. Thevalue of the moment map µ, for the action of U(H) on (H, d"), evaluated on P, isgiven by

µ(Z)(P)r($r) =12i

"P(r)(Z) · %P(r)(Z)

%r($r)!

%P(r)(Z)

%r($r) · P(r)(Z)

#,

where, r ! U , $r ! Rn and if

P(r)(Z) =loc

"

!!A

!!(r)Z!, then%P(r)(Z)

%r($r) =loc

"

!!A

%!!(r)

%r($r)Z!.

3. The unit sphere S " H is a homogeneous space of U(H); see Exercise 126,p. 221. The fibers of the equivariant moment map µ of the action of U(H) on(S, d" ! S) are the fibers of the infinite Hopf fibration & : S % P = S/S1, whereS1 ! C acts multiplicatively on S. There exists a symplectic form ' on P, suchthat &"(') = d" ! S; see Exercise 100, p. 160. The equivariant moment map of theinduced action of U(H) on P is injective. Therefore, the infinite projective spaceP, equipped with the Fubini-Study form, is a homogeneous symplectic space andcan be regarded as a coadjoint orbit of U(H).


Proof. Let us prove what is claimed here and has not been already proved in aprevious paragraph or exercise.

1. Since H is contractible, there is no holonomy. Now, let ! ! H, and let t! bethe translation t!(Z) = Z + !. A direct computation shows that t!!(") = " +d[w(!)]. Thus, d" is invariant by translation t!!(d") = d". Now, let p be a path

connecting 0H to Z, we have µ(Z) = #(p) = p!K(d") = Z!(")! 0!H(")!d[K"" p].But on the one hand we have Z = tZ, thus Z!(") ! 0!H(") = t!Z(") ! 1!H(") ="+ d[w(Z)]! " = d[w(Z)], and on the other hand, p(!) = t! " p. Thus, K" " p =!t!"p " =

!p t!!(") =

!p " +

!p d[w(!)] =

!p " + w(!)(Z), since w(!)(0H) = 0.

Hence, µ(Z) = d[w(Z)]! d[! #" w(!)(Z)]. But w(!)(Z) = !w(Z)(!), then µ(Z) =d[w(Z)] ! d[! #" !w(Z)(!)] = 2d[w(Z)]. Next, let Z be in the kernel of µ, thusw(Z) = cst = w(0H) = 0. But w(Z)(Z #) = 0 for all Z # ! H if and only if Z = 0H,we have just to decompose Z into real and imaginary parts and use the fact thatthe Hermitian norm on H is nondegenerate. Therefore, µ is injective.

2. Since the 1-form " is invariant by U(H), this statement is a direct applicationof (art. 9.11). !9.34. The Virasoro coadjoint orbits. Let Imm(S1,R2) be the space of allthe immersions of the circle S1 = R/2$Z into R2, equipped with the functionaldi!eology.

1. For all integers n and all n-plots P : U " Imm(S1,R2), let "(P) be the 1-formdefined on U by

"(P)r(%r) =

#2"

0

1

$P(r) #(t)$2

!P(r) ##(t)

""""&P(r) #(t)

&r(%r)

#dt ,

where r ! U and %r ! Rn. The prime denotes the derivative with respect to theparameter t, and the brackets %· | ·& denote the ordinary scalar product. Then, " isa 1-form on Imm(S1,R2).

2. We consider now the group Di!+(S1), of positive di!eomorphisms of the circle,and its action on Imm(S1,R2) by reparametrization. For all ' ! Di!+(S1), for allx ! Imm(S1,R1), we denote by '(x) the pushforward of x by ',

'(x) = '!(x) = x "'!1.

Let F : Di!+(S1) " C!(Imm(S1,R2),R) be the map defined, for all ' ! Di!+(S1),by

F(') : x #"#2"

0

log $x #(t)$ d log(' #(t)).

Then, the map F is smooth and for every ' ! Di!(S1),

'!(") = "! d[F(')].

It follows that the exact 2-form ( = d", defined on Imm(S1,R2), is invariant bythe action of Di!(S1). Moreover, the action of Di!(S1) is Hamiltonian.

3. Let x0 : class(t) #" (cos(t), sin(t)) be the standard immersion from S1 to R2. Themoment maps for ( restricted to the connected component of x0 ! Imm(S1,R2),relative to Di!+(S1), are given by

µ(x)(r #" ')r(%r) =

#2"

0

$$x ##(u)$2

$x #(u)$2 !d2

du2log $x #(u)$2

%%u du + cst,


where r !! ! is any plot of Di!+(S1) defined on some n-domain U, r is a point ofU, "r " Rn, u = !!1(t), and "u = D(r !! u)(r)("r).

4. With the same conventions as in item 3, the Souriau cocycles of the Di!+(S1)action on Imm(S1,R2) are cohomologous to

#(g)(r !! !)r("r) =

"2!

0

3$ !!(u)2 ! 2$ !!!(u)$ !(u)

$ !(u)2"u du ,

where g " Di!+(S1) and $ = g!1. We recognize in the integrand of the right handside the Schwartzian derivative of $.

5. Let % be the function defined, for all g and h in Di!+(S1), by

%(g, h) =

"2!

0

log(g # h) !(t) d log h !(t).

Then, for all g and h in Di!+(S1),

F(g # g !) = F(g) # g ! + F(g !)! %(g, g !).

This function % is known as the Bott cocycle [Bot78]. The central extension ofDi!+(S1) by % is the Virasoro group. Its action on Imm(S1,R2), through Di!+(S1),is still Hamiltonian, and now exact. This is a well known construction which willnot be more developed here.

Note. This example, built on purpose [Igl95], gathers the main ingredients foundin the literature on the construction of Virasoro’s group. It illustrates the wholetheory, connecting objects which appeared originally in disorder.

Proof. The proof is actually a long and tedious series of computations. To makeit as clear as possible, we shall split the computations into a few steps.

The 1-form &. We prove first that & is a well defined 1-form on Imm(S1,R2). LetF : U ! U be a smooth m-parametrization. Let s " V, "s " Rm. Denoting by rthe point F(s), we have

&(P # F)s("s) =

"2!

0

1

$(P # F)(s) !(t)$2

!(P # F)(s) !!(t)

""""'(P # F)(s) !(t)

's("s)

#dt

=

"2!

0

1

$P(F(s)) !(t)$2

!P(F(s)) !!(t)

""""'P(F(s)) !(t)

's("s)

#dt

=

"2!

0

1

$P(r) !(t)$2

!P(r) !!(t)

""""'P(r) !(t)

'r

$'F(s)

's("s)

%#dt

= &(P)r=F(s)

$'F(s)

's("s)

%

= F"(&(P))s("s).

Thus, &(P # F) = F"(&(P)), & satisfies the compatibility condition and is then adi!erential 1-form on Imm(S1,R2).

Let us consider now the action of Di!+(S1) on Imm(S1,R2). This action isobviously smooth from the very definition of the functional di!eology of Di!+(S1).Let us denote !!1 by ( such that

!"(&)(P) = &(! # P) = &[r !! P(r) #!!1] = &[r !! P(r) # (].


Note that Di!+(S1) acts on speed and acceleration of any immersion x, by

(x ! !) !(t) = x !(!(t)) · ! !(t),

(x ! !) !!(t) = x !!(!(t)) · ! !(t)2 + x !(!(t)) · ! !!(t).(")

Let us denote by Q the plot " ! P, that is, Q = [r #! P(r) ! !], such that

#(" ! P)r($r) ="2!

0

1

$Q(r) !(t)$2

!Q(r) !!(t)

""""%Q(r) !(t)

%r($r)

#dt

for all r % U and all $r % Rn. Now, from (") we have

Q(r) !(t) = (P(r) ! !) !(t) = P(r) !(!(t)) · ! !(t),

Q(r) !!(t) = (P(r) ! !) !!(t) = P(r) !!(!(t)) · ! !(t)2 + P(r) !(!(t)) · ! !!(t).

Let us write #(" ! P)r($r) according to this decomposition,

#(" ! P)r($r) ="2!

0

Adt+

"2!

0

Bdt.

One has first,

A =1

$P(r) !(!(t)) · ! !(t)$2

!P(r) !!(!(t)) · ! !(t)2

""""%P(r) !(!(t)) · ! !(t)

%r($r)

#,

that is,

A =1

$P(r) !(!(t))$2

!P(r) !!(!(t))

""""%P(r) !(!(t))

%r($r)

#! !(t).

Since ", and thus !, is a positive di!eomorphism, after the change of variablet #! !(t) under the integral, we get already

"2!

0

Adt = #(P)r($r).

Next,

B =1

$P(r) !(!(t)) · ! !(t)$2

!P(r) !(!(t)) · ! !!(t)

""""%P(r) !(!(t)) · ! !(t)

%r($r)

#,

then,

"2!

0

Bdt =

"2!

0

1

$P(r) !(!(t))$2

!P(r) !(!(t))

""""%P(r) !(!(t))

%r($r)

#! !!(t)

! !(t)dt.

Let us then denote for short,

x = P(r), x ! = P(r) !, and $x ! =

$t #! %P !(r)(t)

%r($r)

%.

Using that, for any variation $, we have the identities

$$v$ =1

$v$&v | $v' and $ log $v$ =1

$v$$$v$ =1

$v$2 &v | $v',


we get, with a change of variable s = !!1(t) in the middle,!2!

0

Bdt =

!2!

0

1

!x !("(t))!2 "x!("(t)) | #x !("(t))# " !!(t)

" !(t)dt

=

!2!

0

# log !x !("(t))!d log(" !(t))

= #

!2!

0

log !x !("(t))!d log(" !(t))

= #

!2!

0

log !x !(!!1(t))!d log((!!1) !(t))

= #

!2!

0

log !x !(s)!d log[(!!1) !(!(s))]

= ! #

!2!

0

log !x !(s)!d log(! !(s))

= !$

$r

"!2!

0

log !P(r) !(s)!d log(! !(s))

#(#r)

=

!2!

0

# log !x !("(t))!d log(" !(t))

= !$

$r

"F(!)(P(r))

#(#r)

= ! d[F(!)](P)r(#r).

Coming back to %(! $ P)r(#r), we get finally

%(! $ P)r(#r) = %(P)r(#r)! d[F(!)](P)r(#r),

that is,!"(%) = %! d[F(!)].

Hence, the exterior derivative & = d% is invariant by the action of Di!+(S1), andsince the di!erence !"(%)! % is exact, this action is Hamiltonian.

The 2-points moment map. Now, let us compute the 2-points moment map 'of the action of Di!+(S1) on (Imm(S1,R2),&). Let p be a path connecting twoimmersions x0 and x1. Thus, ((p) = p"(K&) = p"(Kd%) = p"(1"(%) ! 0"(%) !d(K%)) = x"1(%)! x"0(%)! d(K% $ p). But, for all ! % Di!+(S1),

K% $ p(!) =

!

"(p)% =

!

p

!"(%)

=

!

p

% !

!

p

dF(!) =

!

p

% ! F(!)(x1) + F(!)(x0).

We get from there

((p) = '(x0, x1)

= {x"1(%) + d[! &$ F(!)(x1)]}! {x"0(%) + d[! &$ F(!)(x0)]}.

But note that x"(%) + d[! &$ F(!)(x)] is not a momentum of Di!+(S1).

The 1-point moment maps. Let us compute the moment map '(x0, x). Let

m = {x"(%) + d[! &$ F(!)(x)]}(r &$ !)r(#r).


Let us denote for short

A = x!(!)(r !! ")r(#r),

B = d[" !! F(")(x)](r !! ")r(#r) =$F(")(x)

$r#r.

We shall use the notation m0, A0 and B0 for the immersion x0, thus

%(x0, x)(r !! ")r(#r) = m!m0 = A+ B!A0 ! B0.

But x!(!)(r !! ") = !(x " [r !! "]) = !(r !! x ""!1), then let & = "!1, we get

A =

"2!

0

1

#(x " &) "(t)#2

!(x " &) ""(t)

""""$(x " &) "(t)

$r(#r)

#.

Let us now introduce

u = &(t), u " = &(t), and u "" = & ""(t).

Then, the decomposition given by ($) writes

(x " &) "(t) = x "(u) · u " and (x " &) ""(t) = x ""(u) · u "2 + x "(u) · u "".

Next, let us use the prefix # for the various variations associated with #r, that is,#! = D(r !! !)(r)(#r). Then,

$(x " &) "(t)$r

(#r) = #[x "(u) · u "] = x ""(u) · #u · u " + x "(u) · #u ".

Thus,

A =

"2!

0

1

#x "(u)#2u "2 %x""(u)u "2 + x "(u)u "" | x ""(u)u "#u+ x "(u)#u "&dt

=

"2!

0

#x ""(u)#2

#x "(u)#2 #u u " dt+

"2!

0

%x "(u), x ""(u)&#x "(u)#2

$#u " +

u ""

u " #u

%dt

+

"2!

0

u ""

u " #u" dt .

Now,

B =$F(")(x)

$r#r = !

$F(&)(x)

$r#r = !#[F(&)(x)],

with

F(&)(x) =

"2!

0

log #x "(&(t))# d log& "(t) =

"2!

0

log #x "(u)# d log(u ").

Then, after the variation with respect to #r and an integration by parts, we get

B = !

"2!

0

%x "(u), x ""(u)&#x "(u)#2 #u

u ""

u " dt!

"2!

0

log #x "(u)# #d log(u ")

= !

"2!

0

%x "(u), x ""(u)&#x "(u)#2 #u

u ""

u " dt+

"2!

0

%x "(u), x ""(u)&#x "(u)#2 u " # log(u ")dt

= !

"2!

0

%x "(u), x ""(u)&#x "(u)#2 #u

u ""

u " dt+

"2!

0

%x "(u), x ""(u)&#x "(u)#2 #u " dt .


Therefore, grouping the terms and integrating again by parts, we get

A+ B =

!2!

0

!x !!(u)!2

!x !(u)!2 !udu+2

!2!

0

"x !(u), x !!(u)#!x !(u)!2 !u ! dt+

!2!

0

u !!

u ! !u! dt

=

!2!

0

!x !!(u)!2

!x !(u)!2 !udu!2

!2!

0

d2

du2log !x !(u)!!udu+

!2!

0

u !!

u ! !u ! dt

=

!2!

0

"!x !!(u)!2

!x !(u)!2 !d2

du2log !x !(u)!2

#!udu+

!2!

0

u !!

u ! !u ! dt .

Now, since !x !0(t)! = 1 we get the value of the 2-points moment map,

"(x0, x)(r $$ #)r(!r) =

!2!

0

"!x !!(u)!2

!x !(u)!2 !d2

du2log !x !(u)!2

#!udu

!

!2!

0

!udu .

The second term of the right hand side of this equality is a constant momentumof Di!+(S1), so it can be avoided. Thus, every moment map, up to a constant, isequal to this moment µ.

Souriau cocycles. The Souriau cocycle associated with the immersion x0 is definedby $(g) = "(x0, g(x0)); see (art. 9.10). We replace then, in the expression of" above, x by g(x0) = x0 % g!1, that is, x = x0 % %, and $(g)(r $$ #)r(!r) ="(x0, x0 % %). Let us note next that

(x0 % %) !(u) = x !0(%(u))%

!(u) and (x0 % %) !!(u) = x !!0 (%(u))%

!(u)2 + x !0(u)%

!!(u),

and let us recall that !x !0! = !x !!

0 ! = 1, and that "x !0 | x !!

0 # = 0. We get,

!x !(u)!2 = % !(u)2 and !x !!(u)!2 = % !(u)4 + % !!(u)2,

which gives

!x !!(u)!2

!x !(u)!2 = % !(u)2 +% !!(u)2

% !(u)2,

d2

du2log !x !(u)!2 = 2

% !!!(u)% !(u)! % !!(u)2

% !!(u)2.

Thus,

$(g)(r $$ #)r(!r) =

!2!

0

3% !!(u)2 ! 2% !!!(u)% !(u)

% !(u)2!udu

+

!2!

0

% !(u)2 !udu!

!2!

0

!udu .

But, after a change of variable u $$ v = %(u), we get!2!

0

% !(u)2 !udu =

!2!

0

(!u% !(u)) % !(u)du =

!2!

0

!v dv.

Then, the two last terms cancel each other, and we get the value claimed in theproposition for the Souriau cocycle $.

Bott’s cocycle. The real function F(g % h) ! F(g) % h ! F(h) is constant since X isconnected, and its di!erential is equal to (g % h)"(&)! h"(g"(&)), that is, 0. Now,to make '(g, g !) = F(g) % g ! + F(g !) ! '(g, g !) ! F(g % g !) explicit, it is su"cient


to compute the right hand member on the standard immersion x0, for which thespeed norm is equal to 1, and thus log !x !(t)! = 0 for all t. We get then

!(g, h) = F(g)(x0 " h!1)! F(h)(x0)! F(g " h)(x0)

= +

!2!

0

log !(x0 " h!1) !(t)! d log g !(t)

= +

!2!

0

log(h!1) !(t) d log g !(t)

= !

!2!

0

logh !(h!1(t)) d log g !(t)

= !

!2!

0

logh !(s) d log g !(h(s))

= +

!2!

0

log(g " h) !(t) d logh !(t).

And this is the standard expression of Bott’s cocycle. !

Exercise

! Exercise 147 (The moment of imprimitivity). Let X be a di!eological space.Let "1(X) be the vector space of 1-forms of X, equipped with the functional di!eo-logy (art. 6.45). Let Taut be the tautological 1-form defined on X#"1(X) and letLiouv be the Liouville 1-form defined on the cotangent bundle T"X; see (art. 6.48)and (art. 6.49). Let us consider then the additive di!eological group of smoothfunctions C!(X,R), acting smoothly on X#"1(X) by right action,

f : (x,#) $" (x,#! df)

for all f % C!(X,R) and (x,#) % X#"1(X). This action has a natural projectionon the cotangent T"X, and this action will be denoted the same way,

f : (x, a) $" (x, a! df(x))

for all f % C!(X,R) and (x, a) % T"X.

1) Show that, for all f % C!(X, R), the variance of the tautological form and theLiouville form are given by

f"(Taut) = Taut! pr"1(df) and f"(Liouv) = Liouv ! $"(df).

Observe that the exterior derivatives dTaut and % = dLiouv are invariant by theaction of C!(X,R).

2) Let p be a path in T"X, connecting (x0, a0) = p(0) to (x1, a1) = p(1). Showthat the paths moment map & and the 2-points moment map ', with respect tothe 2-form % = dLiouv, are given by

&(p) = '((x0, a0), (x1, a1)) = d[f $" f(x1)]! d[f $" f(x0)].

3) Check that, for all x % X, the real function [f $" f(x)] is smooth. We call it theDirac function of the point x, and we denote it by (x.

(x = [f $" f(x)] % C!(C!(X,R),R).

DISCUSSION ON SYMPLECTIC DIFFEOLOGY 351

Show that the di!erential d!x = d[f !! f(x)] is an invariant 1-form12 of the additivegroup C!(X,R). Show that every moment map of the action of C!(X, R) on T!Xis, up to a constant, equal to the invariant moment map

µ : (x, a) !! d!x = d[f !! f(x)].

Note that the moment µ is constant on the fibers T!xX = "!1(x), and if the real

smooth functions separate13 the points of X, then the image of the moment map µis the space X, identified with the space of Dirac’s functions.

4) Show that the action of C!(X,R) on (T!X,#) is Hamiltonian and exact, that is,$ = {0} and % = 0.

This example, in the case of di!erential manifolds, appears informally in Ziegler’sconstruction of a symplectic analogue for systems of imprimitivity in representa-tion theory [Zie96]. It is why the moment map µ may be called the moment ofimprimitivity. The di!eological framework then gives to it a full formal status andeven extends it.

Discussion on Symplectic Di!eology

Symplectic di!eology is more a program than a complete theory. The lastdecades of theoretical research in mechanics—in completely integrable systems,quantum mechanics, or quantum field theory etc.—have shown a special interest instructures which seem to be symplectic even if they do not live on manifolds but onspaces, generally infinite dimensional, where the formal constructions of symplecticgeometry do not apply as is. Building a formal framework for these symplectic-likestructures is not just a desire of formalism, but a need to embed these heuristicconstructions in a well delimited and workable mathematical construction. Thereare di!erent ways to approach these problems, such as functional analysis, infinitedimensional manifolds a la Banach, or maybe others. A di!eological approach is oneof them, but has the virtue of involving a very light apparatus of mathematical tools.Axiomatics, reduced to only three axioms, cannot be simpler and the great stabilityof the category, under set theoretic operations, is a gift for this kind of problem,where infinite dimensions and what is admitted to be considered as singularities area burden for classical geometry. On the other hand, the light structure of di!eologydoes not seem to be a weakness. The construction of the moment maps, associatedwith a closed 2-form on any di!eological space, shows the whole — and dealscorrectly with the — complexity of the various situations without exaggeratingtechnicalities. This is clearly shown in the few examples given in the previoussections of this chapter. So, even if it seems clear that di!eology is adequate forsuch a generalization of symplectic (or presymplectic) geometry, we still need a gooddefinition, or at least a serious discussion, about what is a symplectic di!eologicalspace? Or what does it mean to be symplectic in di!eology?

A relatively good answer has been given at least in the case of homogeneousspaces, that is, in the case of a closed 2-form # defined on a space X homogeneousfor its group of automorphisms Di!(X,#); see (art. 9.18) and (art. 9.19). In thiscase the situation is clear: a universal moment map distinguishes between being

12This di!erential has nothing to do with the derivative of the Dirac distributions in thesense of De Rham’s currents.

13which means, f(x) = f(x !) for all smooth real functions f if and only if x = x !.


presymplectic or symplectic. The homogeneous space (X,!) is symplectic if a uni-versal moment map µ! is a covering onto its image, or, which is equivalent, if thepreimages of its values are di!eologically discrete. Otherwise it is presymplectic,and the characteristics of µ! can be regarded as the characteristics of the 2-form!, by analogy with what happens in the special case of homogeneous manifolds(art. 9.26). The symplectic homogeneous case has been illustrated, in particular,by the Hilbert space H or the infinite dimensional projective space P (art. 9.33),even by the singular irrational tori (art. 9.30). The fact that these spaces are sym-plectic is now a well defined property, according to the definition given in (art. 9.19).

Why is the homogeneous case so important? Because for a closed 2-form! defined on a manifold M, being symplectic implies to be homogeneous underDi!(M,!), with moreover the universal moment maps µ! injective (art. 9.23).The homogeneity of M under the action of Di!(M,!) is a strong consequence oftwo things: firstly ! is invertible, has no kernel; and secondly, every symplecticmanifold is locally flat, that is, looks locally like some (R2n,!st). There existsonly one local model of symplectic manifold in each dimension, this is the famousDarboux theorem.

In di!eology, we have not a unique model to propose for all the covered cases:from singular tori to infinite projective space. But we can replace advantageouslythe local model of symplectic manifolds by the strong requirement of homogeneity.We remark that we could replace the global homogeneity by a local homogeneity, butthis would lead to unwanted subtleties, for now. Our question is then, Do we wantto preserve the fundamental property of homogeneity for symplectic di!eologicalspaces? The answer to this question is not that obvious.

If we want to preserve the property of homogeneity, the problem is just solvedby (art. 9.19). But the example of the cone orbifold Cm (art. 9.32) is a real culturalor sociological obstacle. Many mathematicians want to consider the cone orbifold,equipped with the 2-form described in (art. 9.32), as symplectic, because it is thequotient of the symplectic space (R2,!st) by a finite group preserving the standardsymplectic form!st. Unfortunately the cone orbifold is not homogeneous, the originis fixed by every di!eomorphism. However, even according to di!eology, regardingCm as a symplectic di!eological space makes sense: the cone orbifold is generated bysymplectic plots — in this case the only plot "m : R2 ! Cm. Let us remark that thecorner orbifold (art. 9.31) does not satisfy this property, and cannot be symplecticeven according to this definition. Yael Karshon suggested, in private discussions, todefine symplectic di!eological spaces as di!eological spaces generated by symplecticplots, that is, plots P such that !(P) is symplectic. Although this makes sense, itis a way we have not explored yet. In particular, the relationship between such aspace and the moment maps needs to be clarified, as well as the relationship withthe action of the group of automorphisms and the nature of its orbits. The ideabehind this is to consider symplectic reductions as symplectic spaces independentlyof the regularity of the distribution of the moment map strata.

Another approach should consist of only requiring the universal moment mapsto be injective or at least to have discrete preimages, since the preimages of a uni-versal moment map seem to realize the characteristics of !. Let us note that thecone orbifold satisfies this property, but not the corner orbifold, which is satisfac-tory. Unfortunately, the space R2 equipped with the 2-form ! = (x2 + y2)dx! dy

DISCUSSION ON SYMPLECTIC DIFFEOLOGY 353

Figure 9.3. Distribution of examples.

also satisfies this property, and it is clearly not symplectic. It seems that we haveto either abandon this approach, or to go deeper into it.

So, what remains? If we want to include the cone orbifold in the categoryof symplectic di!eological spaces we have to give up homogeneity by the group ofautomorphisms. Or, if we do not want to give up homogeneity, we abandon thecone orbifold and certainly many other examples of di!eological spaces, generatedby symplectic plots. For now, the set of examples and theorems in symplecticdi!eology is not large enough to make a reasonable choice. It is why there is nogeneral definition of symplectic di!eological spaces here. But, we can just dealwith these spaces, equipped with closed 2-forms, by trying to get the maximum ofinformation as we did in the examples above (see Figure 9.3): the nature of thecharacteristics of the universal moment map, the injectivity or not of the universalmoment map, the transitivity of the group of automorphisms or the nature of itsorbits, the computation of the universal holonomy, and the Souriau class, etc. Wehave all the tools needed to treat the examples given by the literature or by thephysicists, and maybe to define what the word symplectic means in di!eology isnot that important after all. . .

A few more words. Why do we need the 2-form to be nondegenerate? From aphysicist’s point of view, we do not need that, actually it is the opposite. A pair(X,!), with ! a closed 2-form, represents what we could call a dynamical struc-ture. The characteristics of !, that is, the connected components of the preimagesof the universal moment map, describe the dynamics of the system, together withthe partition into orbits by the group of automorphisms. This is what physicistsare interested in: they want equations describing the evolution of their systems.The fact that, in classical mechanics, the space of characteristics is symplectic isfortuitous, a consequence of the presymplectic nature of the dynamical structuresinvolved. Then, the group of automorphisms of the space of characteristics is tran-sitive and, by equivariance, the image of the moment map is one coadjoint orbit.In the general case of infinite dimension spaces, orbifolds, singular reductions etc.,


there is no reason for the automorphisms to be transitive, and the image of theuniversal moment map is just some union of coadjoint orbits, and then, maybe, notsymplectic. The picture is clear, and that is what we have to deal with. Anothercritical question is of the symplectic reduction of a subspace W ! X: the reducedspace may be simply defined as the space of characteristics of the dynamical 2-form ! restricted to W, or its regularization, that is, the image of the universalmoment map of the restriction. It is actually not di!erent from the general casewhere the subspace coincides with the whole space. The task is then to describethis image and what, from the 2-form, passes to the quotient or to the image? Forthe structure of the space of characteristics we get two di!eologies: the first one isthe natural quotient di!eology on the space of characteristics, the second one is thedi!eology induced on the image of the universal moment map by the ambient spaceof momenta. They may coincide or not, but they both have their role to play.

Symplectic Diﬀeology - The Hebrew...

Documents

Transcript of Symplectic Diﬀeology - The Hebrew...