INFINITE DIMENSIONAL HAMILTONIANSYSTEMS WITH SYMMETRIES

Rudolf Schmid ∗†

Abstract

We give a survey of infinite dimensional Hamiltonian systems withinfinite dimensional Lie groups as symmetry groups and discuss con-crete examples from soliton equations, plasma physics, fluid mechanicsand quantum field theory. We present some new results of applicationsto BRST symmetries and g-symplectic structures.

1 Introduction

Infinite dimensional Hamiltonian systems arise in many areas in pure and ap-plied mathematics and mathematical physics. This paper is a survey of somebasic properties of infinite dimensional manifolds and infinite dimensional Liegroups with a class of examples of infinite dimensional Hamiltonian systems.We give no proofs here but plenty of references for more details and proofs.In the 2nd section we define the notion of Hamilton’s equations on Poissonmanifolds and give examples in section 3. In section 4 we point out somedifferences between infinite dimensional Lie groups and finite dimensionalones and in section 5 we give examples of infinite dimensional Lie groups

∗Department of Mathematics, Emory University, Atlanta, Georgia 30322,rudolf@mathcs.emory.edu Supported in part by the Emory University Research Grant#2-50027 and NSF grant # DMS-9303215.†Lecture given at the International Conference ”Differential Geometry, Hamiltonian

Systems and Operator Theory”, Feb. 7-11, 1994, University of the West Indies, Jamaica.

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together with some of their applications. Section 6 and 7 discuss diffeomor-phism groups and some of their subgroups and some applications of themare mentioned. Section 8 deals with the group of Fourier integral operatorsand it application to the KdV equation. In section 9 to 11 we present somenew results about the BRST symmetries in quantum field theory, namelya solution to the so called Wess - Zumino consistency condition, using thenotion of g- symplectic structures.

2 Hamilton’s equations on Poisson manifolds

A short introduction and ”crash course” to geometric mechanics can be foundin Marsden [41]. For the general theory of infinite dimensional manifolds andglobal analysis see e.g. Bourbaki [13], Lang [35], Palais [49].

A Poisson manifold is a manifold P (in general infinite dimensional)equipped with a bilinear operation ., . ,called Poisson bracket, on the func-tion space C∞(P ) such that :(i) (C∞(P ), ., .) is a Lie algebra; i.e. ., . : C∞(P ) × C∞(P ) → C∞(P )is bilinear, skew symmetric and satisfies the Jacobi identity F,G, H +H,F, G+ G,H, F = 0 for all F,G,H ∈ C∞(P );(ii)., . satisfies the Leibniz rule; i.e. ., . is a derivation in each factor:F ·G,H = F · G,H+G · F,H.The notion of Poisson manifolds was rediscovered many times under differ-ent names, starting with Lie, Dirac, Pauli and others. The name Poissonmanifold was coined by Lichnerowicz [37].

For any H ∈ C∞(P ) we define the Hamiltonian vector field XH by

XH(F ) = F,H, F ∈ C∞(P ) .

It follows from (ii) that indeed XH defines a derivation on C∞(P ), hencea vector field on P . Hamilton’s equations of motion for the Hamiltonianfunction H (energy function) are then defined by the flow (integral curves)of the vector field XH i.e.

F = XH(F ) = F,H , ˙ =d

dt. (2.1)

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3 Examples of Poisson manifolds and Hamil-

ton’s equations

3.1 Finite dimensional classical mechanics

There are many well known texts on the geometric treatment of classicalmechanics, e.g. Abraham and Marsden [1], Arnold [8], Choquet-Bruhat,DeWitt-Morette and Dillard-Bleick [17], Goldstein [25], Guillemin and Stern-berg [26], Marle and Libermann [38], Marmo, Saletan, Simoni and Vitali [39],Marsden and Ratiu [43], Marsden [41], Souriau [61].

For finite dimensional classical mechanics we take P = IR2n with coordi-nates (q1, . . . , qn, p1, . . . , pn) and with the standard Poisson bracket for anytwo functions F (qi, pi) , H(qi, pi)

F,H =n∑i=1

∂F

∂pi

∂H

∂qi− ∂H

∂pi

∂F

∂qi. (3.1)

Then Hamiltons equations are

qi =∂H

∂pi, pi = − ∂H

∂qi, i = 1 . . . n . (3.2)

3.1.1 Harmonic oscillator

As a concrete example we consider the harmonic oscillator: Here P = IR2

and the Hamiltonian ( energy ) is H(q, p) = 12(q2 + p2). Then Hamiltons

equations areq = p , p = −q . (3.3)

3.2 Infinite dimensional classical field theory

For details on infinite dimensional classical systems see eg. Abraham , Mars-den and Ratiu [2], Binz and Sniatycki [10], Bleecker [11], Chernoff and Mars-den [15], Choquet-Bruhat and DeWitt-Morette II [17], Eguchi, Gilkey andHanson [21], Marsden [40], Schmid [53], Temam [64].

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Let V be a Banach space and V ∗ its dual space with respect to a pairing〈., .〉 : V × V ∗ → IR. On P = V × V ∗ we have the Poisson bracket

F,H = 〈δFδπ,δH

δϕ〉 − 〈δH

δπ,δF

δϕ〉 , ϕ ∈ V, π ∈ V ∗ , (3.4)

where δFδπ∈ V , δF

δϕ∈ V ∗ are the ”duals” under the pairing 〈., .〉 of the

partial gradients D1F (π) ∈ V ∗ , D2F (ϕ) ∈ V ∗∗ ' V . The correspondingHamilton’s equations are

ϕ =δH

δπ, π = − δH

δϕ. (3.5)

In finite dimensions, if V ' IRn so V ∗ ' IRn and P = V ×V ∗ ' IR2n , andthe pairing is the standard inner product in IRn, then the Poisson bracket(3.4) and Hamilton’s equations (3.5) are identical with (3.1) and (3.2).

3.2.1 Wave equations

As a concrete example we consider the wave equations. Let V = C∞(IR3) andV ∗ = Den(IR3) (densities) defined via the L2 pairing 〈ϕ, π〉 =

∫ϕ(x)π(x)dx.

Take the Hamiltonian to be H(ϕ, π) =∫

(12π2 + 1

2|∇ϕ|2 +F (ϕ)) dx, where F

is some function on V . Then Hamiltons equations 3.5 become

ϕ = π, π = ∇2ϕ− F ′(ϕ) (3.6)

which imply the wave equation

∂2ϕ

∂t2= ∇2ϕ− F ′(ϕ). (3.7)

Different choices of F give different wave equations, e.g. for F = 0 we getthe linear wave equation ∂2ϕ

∂t2= ∇2ϕ; for F = 1

2mϕ we get the Klein Gordon

equation ∇2ϕ− ∂2ϕ∂t2

= mϕ .

3.3 Cotangent bundles

The finite dimensional examples of Poissson brackets 3.1 and Hamilton’sequations 3.2 and the infinite dimensional examples 3.4 and 3.5 are the local

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versions of the general case where P = T ∗Q is the cotangent bundle (phasespace) of a manifold Q (configuration space), i.e. if Q in an n-dimensionalmanifold then T ∗Q is a 2n-Poisson manifold locally isomorphic to IR2n whosePoisson bracket is locally given by 3.1 and Hamilton’s equations are locallygiven by 3.2. If Q is an infinite dimensional Banach manifold then T ∗Q isa Poisson manifold locally isomorphic to V × V ∗ whose Poisson bracket isgiven by 3.4 and Hamilton’s equations are locally given by 3.5.

3.4 Symplectic manifolds

All the examples above are special cases of symplectic manifolds (P, ω).That means P is equipped with a symplectic structure ω which is aclosed, (weakely) nondegenerate 2-form on the manifold P . Then for anyH ∈ C∞(P ) the corresponding Hamiltonian vector field XH is defined bydH = ω(XH , . ) and the Poisson bracket is given by

F,H = ω(XF , XH) , F,H ∈ C∞(P ). (3.8)

For example, on IR2n the canonical symplectic structure ω is given by ω =∑ni=1 dpi ∧ dqi = dθ , where θ =

∑ni=1 pi ∧ dqi . The same formula holds

locally in T ∗Q for any finite dimensional Q (Darboux’s Lemma). For theinfinite dimensional example P = V × V ∗ the symplectic form ω is givenby ω((ϕ1, π1), (ϕ2, π2)) = 〈ϕ1, π2〉 − 〈ϕ2, π1〉. Again these two formulas areidentical if V = IRn.

Remarks:A) If P is a finite dimensional symplectic manifold then P is even dimen-sional.B) If the Poisson bracket ., . is nondegenerate then ., . comes form asymplectic form ω, i.e. ., . is given by 3.8.

3.5 The Lie-Poisson bracket

Not all Poisson brackets are of the from given in the above examples 3.1, 3.4and 3.8. An important class of Poisson bracket is the so called Lie Poissonbracket. It is defined on the dual of any Lie algebra. Let G be a Lie groupwith Lie algebra g = TeG ' left invariant vector fields on G and let

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[., .] denote Lie bracket (commutator) on g. Let g∗ be the dual of a g withrespect to a pairing 〈., .〉 : g∗×g→ IR. Then for any F,H ∈ C∞(g∗), µ ∈ g∗

the Lie Poisson bracket is defined by

F,H(µ) = ±〈µ, [δFδµ,δH

δµ]〉 , (3.9)

where δFδµ, δHδµ∈ g are the ”duals” of the gradients DF (µ), DH(µ) ∈ g∗∗ ' g

under the pairing 〈., .〉 . Note that the Lie-Poisson bracket is degenerate ingeneral, e.g. for G = SO(3) the vector space g∗ is 3 dimensional, so thePoisson bracket 3.9 cannot come from a symplectic structure. This Lie Pois-son bracket can also be obtained in a different way by taking the cannonicalPoisson bracket on T ∗G (locally given by 3.1 and 3.4) and then restrict it tothe fiber at the identity T ∗eG = g∗. In this sense the Lie Poisson bracket 3.9is induced from the canonical Poisson bracket on T ∗G discussed in section3.3. It is induced by the symmetry of left multiplication as we will discussin the next section.

3.5.1 Rigid body

A concrete example of the Lie Poisson bracket is given by the rigid body:Here G = SO(3) is the configuration space of a free rigid body . Identifyingthe Lie algebra (so(3), [., .]) with (IR3,×), where × is the vector product onIR3 and g∗ = so(3)∗ ' IR3 the Lie Poisson bracket translates into

F,H(m) = −m · (∇F ×∇H). (3.10)

We have for any F ∈ C∞(so(3)∗), dFdt

(m) = ∇F · m = F,H(m) = −m ·(∇F ×∇H) = ∇F · (m×∇H) hence m = m×∇H. With the Hamiltonian

H = 12(m2

1

I21

+m2

2

I22

+m2

3

I23

) we get Hamiltons equation as

m1 =I2 − I3

I2I3

m2m3, m2 =I3 − I1

I3I1

m3m1 , m3 =I1 − I2

I1I2

m1m2 . (3.11)

These are Euler’s equations for the free rigid body.

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3.6 Reduction by symmetries

The examples we have discussed so far are all canonical examples of Poissonbrackets, defined either on a symplectic manifold or on the dual of a Liealgebra. Different Poisson brackets can arise from symmetries. Assume aLie group G is acting in a Hamiltonian way on P . That means that themaps ϕg : P → P, g ∈ G are canonical transformations, hence generate aHamiltonian vector field ξF for any ξ ∈ g and a momentum map J : P → g∗,J(x)(ξ) = F (x), which is Ad∗ equivariant. If a Hamiltonian system XH

is invariant under a this Lie group action, i.e. H(ϕg(x)) = H(x), then weobtain a reduced Hamiltonian system on a reduced phace space. We recallthe Marsden-Weinstein reduction theorem [44]:

Theorem 1 (Reduction Theorem) For a Hamiltonian action of a Liegroup G on a Poisson manifold (P, ., .), there is an equivariant momen-tum map J : P → g∗ and for every regular µ ∈ g∗ the reduced pase spacePµ ≡ J−1(µ)/Gµ carries an induced Poisson structure ., .µ. For any G-invariant Hamiltonian H on P the integral curves of the vector field XH

project onto integral curves of the induced vector field XHµ on the reducedspace Pµ.

3.6.1 Rigid body

The rigid body discussed above can be viewd as an example of this reductiontheorem. If P = T ∗G and G is acting on T ∗G by the cotangent lift ofthe left translation lg : G → G , lg(h) = gh , then the momentum mapJ : T ∗G → g∗ is given by J(αg) = T ∗eRg(αg) and the reduced phase space(T ∗G)µ = J−1(µ)/Gµ is isomorphic to the coadjoint orbit Oµ through µ ∈ g∗.In particular T ∗G/G ' g∗ and the induced Poisson bracket ., .µ on Oµis identical with the Lie Poisson bracket restricted to the coadjoint orbitOµ ⊂ g∗.

We discuss some infinite dimensional examples of reduced Hamiltoniansystems in sections 7 and 8.

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4 Infinite dimensional Lie groups

A general theory of infinite dimensional Lie groups is hardly developed. EvenBourbaki [14] only developes a theory of infinite dimensional manifolds, butall of the important theorems about Lie groups are stated for finite dimen-sional ones.

An infinite dimensional Lie group G is a group and an infinite dimensionalmanifold with smooth group operations

m : G×G → G ; m(g, h) = g ·h , C∞ ; i : G → G ; i(g) = g−1 , C∞ . (4.1)

Such a Lie group G is locally diffeomorphic to an infinite dimensional vectorspace. This can either be a Banach space whose topology is given by anorm ‖.‖, a Hilbert space whose topology is given by an inner product 〈., .〉or a Frechet space whose topology is given by a metric but not by a norm.Depending on the choice of the topology on G we talk about Banach-, Hilbert-or Frechet Lie groups respectively.

The Lie aglebra g of G is defined as g= left invariant vector fields on G ' TeG, where the isomorphisme is given (as in finite dimensions) by

ξ ∈ TeG 7→ Xξ(g) = TeLg(ξ), (4.2)

and the Lie bracket on g is induced by the Lie bracket of left invariant vectorfields [ξ, η] = [Xξ, Xη](e) , ξ, η ∈ g .

These definitions in infinite dimensions are identical with the definitionsin finite dimensions. The big difference although is that infinite dimensionalmanifolds , hence Lie groups , are not locally compact. For Frechet Liegroups we have the additional nontrivial difficulty of the question how todefine differentiability of functions defined on a Frechet space, hence thevery definition of a Frechet manifold is not canonical. These problems arediscussed eg. in Adams, Ratiu and Schmid [3], Hamilton [27], Keller [32],Michor [46], Omori [48]. This problem does not arise for Banach and HilbertLie groups; the differential calculus extends in a straightforward manner fromIRn to Banach and Hilbert spaces, but not to Frechet spaces.

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4.1 Finite versus infinite dimensional Lie groups

The non local compactness of infinite dimensional Lie groups causes somedeficiencies of the Lie theory in infinite dimensions. We summarize someclassical results in finite dimensions which are NOT true in general in infi-nite dimensions:

1) The exponential map exp : g→ G is defined as follows: To each ξ ∈ gwe assign the corresponding left invariant vector field Xξ defined by 4.2. Wetake the flow ϕξ(t) of Xξ and define exp(ξ) = ϕξ(1) . The exponential map isa local diffeomorphism from a neighberhood of zero in g onto a neighberhoodof the identity in G, hence exp defines canonical coordinates on the Lie groupG.This is not true in infinite dimensions.

2) If f1, f2 : G1 → G2 are smooth Lie group homomorphisms (i.e. fi(g ·h) = fi(g) · fi(h), i = 1, 2 ) with Tef1 = Tef2 . Then locally f1 = f2 .This is not true in infinite dimensions.

3) If H is a closed subgroup of G then H is a Lie subgroup of G.This is not true in infinite dimensions.

4) For any finite dimensional Lie algebra g there exists a connected Liegroup G whose Lie albegra is g; i.e. such that g ' TeG.This is not true in infinite dimensions.

The classical finite dimensional examples of Lie groups are the matrixgroups GL(N), SL(n), O(n), SO(n), U(n), SU(n), Sp(n).

5 Examples of infinite dimensional Lie groups

5.1 Abelian gauge group G = (C∞(M),+)

Let M be a finite dimensional manifold and let G = C∞(M). With groupoperations m(f, g) = f + g, i(f) = −f, e = 0 , G is an abelian C∞ FrechetLie group with Lie algebra g = TeC

∞(M) ' C∞(M), with trivial bracket[ξ, η] = 0, and exp = id. If we complete these spaces in the Ck(M) -norm,k <∞ then G is a Banach Lie group, and if we complete in the Hs-Sobolev

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norm with s > 12dimM then G is a Hilbert Lie group.

5.1.1 Application of G = (C∞(M),+) to Maxwell’s equations

Let E,B be the electric and magnetic fields on IR3, then Maxwell’s equationsfor a charge density ρ are:

E = curl B, B = −curl E, div B = 0, div E = ρ . (5.3)

Let A be the magnetic potential such that B = − curl A. As configurationspace we take V = V ec(IR3), vector fields (potentials) on IR3 , so A ∈ V ,and as phase space we have P = T ∗V ' V ×V ∗ 3 (A,E), with the standardL2 pairing < A,E >=

∫A(x)E(x)dx, and canonical Poisson bracket given

by 3.4 , which becomes

F,H(A,E) =∫

(δF

δA

δH

δE− δH

δA

δF

δE)dx . (5.4)

As Hamiltonian we take the total electro-magnetic energy H(A,E) =12

∫(|curl A|2 + |E|2)dx. Then Hamiltons equations in the canonical vari-

ables A and E are A = δHδE

= E ⇒ B = −curl E and E = − δHδA

=−curl curl A = curl B. So the first two equations of Maxwell’s equations5.3 are Hamilton’s equations, the third one we get automatically from thepotential div B = −div curl A = 0 and the 4th equation divE = ρ weobtain through the following symmetry (gauge invariance): The Lie groupG = (C∞(IR3),+) acts on V by ϕ · A = A+∇ϕ, ϕ ∈ G , A ∈ V . The liftedaction to V ×V ∗ becomes ϕ · (A,E) = (A+∇ϕ,E), and has the momentummap J : V × V ∗ → g∗ 'charge densities

J(A,E) = div E . (5.5)

With g = C∞(IR3) and g∗ = Den(IR3) we identify elements of g∗ withcharge densities. The Hamiltonian H is G invariant, i.e. H(ϕ · (A,E)) =H(A + ∇ϕ,E) = H(A,E). Then the reduced pase space for ρ ∈ g∗ is(V × V ∗)ρ = J−1(ρ)/G = (E,B)|div E = ρ, div B = 0 and the reducedHamiltonian is

Hρ(E,B) =1

2

∫(|E|2 + |B|2)dx . (5.6)

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The reduced Poisson bracket becomes for any function F,H on (V × V ∗)ρ

F,Hρ(E,B) =∫

(δF

δE· curl δH

δB− δH

δE· curl δF

δB)dx , (5.7)

and a straightforward computation shows that

F = F,Hρρ ⇔E = curl B , B = −curl Ediv B = 0 , div E = ρ

(5.8)

So Maxwell’s equations 5.3 are Hamilton’s equations on this reduced phasespace with respect to the reduced Poisson bracket.

Details can be found in Marsden, Weinstein, Ratiu, Schmid and Spencer[45], or Schmid [53].

5.2 Abelian gauge group G = (C∞(M, IR− 0), ·)

Let M be a finite dimensional manifold and let G = C∞(M, IR− 0), withgroup operations m(f, g) = f ·g, i(f) = f−1, e = 1. For k <∞, Ck(M, IR−0) is open in C∞(M, IR) and if M is compact then Ck(M, IR − 0) is aBanach Lie group. If s > 1

2dim M then Hs(M, IR − 0) is closed under

multiplication and if M is compact then Hs(M, IR − 0) is a Hilbert Liegroup. See e.g. Palais [49] for details.

5.3 Gauge groups G = (Ck(M,G), ·)

We generaliz Example 5.2 by replacing IR − 0 with any finite dimen-sional Lie group G. Let G = Ck(M,G) with pointwise group opera-tions m(f, g)(x) = f(x) · g(x) , x ∈ M and i(f)(x) = (f(x))−1 where· and −1 are the operations in G. If k < ∞ then Ck(M,G) is a Ba-nach Lie group. Let g denote the Lie algebra of G, then the Lie al-gebra of G = Ck(M,G) is g = Ck(M, g), with pointwise Lie bracket[ξ, η](x) = [ξ(x), η(x)] , x ∈ M , the latter bracket being the Lie bracketin g. The exponential map exp : g → G defines the exponential mapEXP : g = Ck(M, g) → G = Ck(M,G), EXP (ξ) = exp ξ, which is alocal diffeomorphism. The same holds for Hs(M,G if s > 1

2dim M . See e.g.

Schmid [54] for details and further references.

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Applications of these infinite dimensional Lie groups are in gauge theoriesand quantum field theory (see e.g Cotta-Ramusion [18] or Lawson [36]), wherethey appear as groups of gauge transformations. We refere to L. Sibner [59]and R. Sibner [60] in these proceedings for applications to Yang-Mills theory.In section 9 and 10 we will discuss applications to BRST symmetries.

5.4 Loop groups G = Ck(S1, G)

As a special case of example 5.3 we take M = S1, the circle. Then G =Ck(S1, G) = Lk(G) is called a loop group and g = Ck(S1, g) = lk(g) itsloop algebra. They find applications in the theory of affine Lie algebras,Kac-Moody Lie algebras (central extensions), completely integrable systems,soliton equations (Toda, KdV, KP), quantum field theory. Central extensionsof Loop algebras are examples of infinite dimensional Lie algebras which neednot have a corresponding Lie group. See e.g. Kac [30], Segal and Pressley[50], Schmid [53] for details, applications and further references.

6 Diffeomorphism groups

Among the most important ”classical” infinite dimensional Lie groups are thediffeomorphism groups of manifolds. Their differential structure is not theone of a Banch Lie group as defined above. Nevertheless they have importantapplications.

Let M be a compact manifold (the noncompact case is technicallymuch more complicated and is in progress with J. Eichhorn) and let G =Diff∞(M) be the group of all smooth diffeomorphism on M , with groupoperations m(f, g) = f g, i(f) = f−1, e = idM . For C∞ diffeomorphismsDiff∞(M) is a Frechet manifold and there are nontrivial problems with thenotion of smooth maps between Frechet spaces. There is no canonical ex-tension of the differential calculus from Banach spaces (same as for IRn) toFrechet spaces. One possibility is to generalize the notion of differentability,see e.g. Keller [32], Hamilton [27], Michor [46]. For example, if we use the socalled C∞Γ differentiability , then G = Diff∞(M) becomes a C∞Γ Lie groupwith C∞Γ differentiable group operations. These notion of differentiability are

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difficult to apply to concrete examples. Another possibility is to completeDiff∞(M) in the Banach Ck - norm, 0 ≤ k < ∞, or in the Sobolev Hs

-norm, s > 12dim M . Then Diffk(M) and Diff s(M) become Banach and

Hilbert manifolds respectively, (details can be found in Palais [49], Ebin andMarsden [20] or Elliasson [22]). Then we consider the inverse limits of theseBanach - and Hilbert Lie groups respectively:

Diff∞(M) = lim←Diffk(M), (6.1)

becomes an ILB Lie group, or with the Sobolev topologies

Diff∞(M) = lim←Diff s(M) (6.2)

becomes an ILH Lie group , Omori [48]. Nevertheless, the group operationsare not smooth, but have the following differentiability properties. If we equipthe diffeomorphism group with the Sobolev Hs -topology, then Diff s(M),becomes a C∞ Hilbert manifold if s > 1

2dim M and the group multiplication

m : Diff s+k(M)×Diff s(M)→ Diff s(M) (6.3)

is Ck differentiable , hence for k = 0, m is only continuous on Diff s(M).The inversion

i : Diff s+k(M)→ Diff s(M) (6.4)

is Ck differentiable, hence for k = 0, i is only continuous on Diff s(M). Thesame differentiability properties of m and i hold in the Ck topology. Thissituation leads to the notion of nested Lie groups, Adams, Ratiu and Schmid[3], Omori [48].

The Lie algebra of Diff∞(M) is given by g = TeDiff∞(M) ' V ec∞(M)

the space of smooth vector fields on M . Note that the space V ec(M) of allvector fields is a Lie algebra only for C∞ vector fields, but not for Ck orHs vector fields if k < ∞, s < ∞, becoause one looses derivatives by takingbrackets.

The exponential map on the diffeomorphism group is given as follows: Forany vector field X ∈ V ec∞(M) take its flow ϕt ∈ Diff∞(M), then defineEXP : V ec∞(M) → Diff∞(M) : X 7→ ϕ1 , the flow at time t = 1. Theexponential map EXP is NOT a local diffeomorphism, it is not even locallysurjective.

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6.1 Applications of Diff∞(M)

6.1.1 General relativity

In general relativity the diffeomorphism group plays the role of a symmetrygroup of coordinate transformations. Let (M, g) be a Lorentz 4 -manifold.Then the vacuum Einstein’s field equations are Ric(g) = 0. These are invari-ant under coordinate transformations i.e. under the action of Diff∞(M) .Moreover , Einstein’s field equations are a Hamiltonian system on the spaceP = metrics on M/Diff∞(M) , (Fischer and Marsden [24], Marsden ,Ebin and Fischer [42], Binz [9]).

7 Subgroups of Diff∞(M)

Several subgroups of Diff∞(M) have important applications:

7.1 Group of volume preserving diffeomorphisms

Let µ be a volume on M and G = Diff∞µ (M) = f ∈ Diff∞(M) | f ∗µ =µ the group of volume preserving diffeomorphisms. Diff∞µ (M) is aclosed subgroup of Diff∞(M) with Lie algebra g = V ec∞µ (M) = X ∈V ec∞(M) | divµX = 0 the space of divergence free vector fields on M .V ec∞µ (M) is a Lie subalgebra of V ec∞(M). Details and proofs of these factscan be found in Ebin and Marsden [20].

Remark: We cannot apply the finite dimensional theorem that ifV ec∞µ (M) is Lie algebra then there exists a Lie group whose Lie albegrait is; nor that if Diff∞µ (M) ⊂ Diff(M) is a closed subgroup then it is a Liesubgroup.

7.1.1 Applications: Fluid dynamics

Euler’s equations for an incompressible fluid

∂u

∂t+ u · ∇u = −∇p , div u = 0 (7.5)

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are equivalent to the equations of geodesics on Diff∞µ (M), (Arnold [7], Ebinand Marsden [20], Marsden, Ebin and Fischer [42]).

7.2 Symplectomorphism group

Let ω be a symplectic 2-from on M and G = Diff∞ω (M) = f ∈Diff∞(M) | f ∗ω = ω the group of canonical transformations (or sym-plectomorphisms). Diff∞ω (M) is a closed subgroup of Diff∞(M) with Liealgebra g = V ec∞ω (M) = X ∈ V ec∞(M) | LXω = 0 the space of locallyHamiltonian vector fields on M . V ec∞ω (M) is a Lie subalgebra of V ec∞(M).

Remark: same as under 7.1).

7.2.1 Applications: Plasma physics

Maxwell-Vlasov’s equations for a plasma density f(x, v, t) generating theelectric and magnetic fields E and B are

∂f∂t

+ v · ∂f∂x

+ (E + v ×B)∂f∂v

= 0

∂B∂t

= −curl E∂E∂t

= curl B − Jf , Jf = current density

div E = ρf , ρf = charge density

div B = 0 .

(7.6)

This coupled non-linear system of evolution equations is a Hamiltonian sys-tem F = F,Hρf on the reduced phace space

MV = (T ∗Diff∞ω (IR6)× T ∗V )/C∞(IR6) (7.7)

(V the same space as in the example of Maxwell’s equations) with respect tothe following reduced Poisson bracket, which is induced via gauge symmetryfrom the canonical Poisson bracket on T ∗Diff∞ω (IR6)× T ∗V

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F,Gρf (f, E,B) =∫f δF

δf, δGδfdxdv

+∫

( δFδE· curl δG

δB− δG

δE· curl δF

δB)dxdv

+∫

( δFδE· ∂f∂v

δGδf− δG

δE· ∂f∂v

δFδf

)dxdv

+∫fB · ( ∂

∂vδFδf× ∂

∂vδGδf

)dxdv ,

(7.8)

and with Hamiltonian

H(f, E,B) =1

2

∫v2f(x, v, t)dv +

1

2

∫(|E|2 + |B|2)dx . (7.9)

Details can be found in Marsden, Weinstein, Ratiu, Schmid and Spencer [45],or Schmid [53]

More complicated plasma models are formulated as Hamiltonian systems.For example, for the two fluid model the phase space is a coadjoint orbitsof the semidirect product () of the group G = Diff∞(IR6) (C∞(IR6) ×C∞(IR6)). For the MHD model: G = Diff∞(IR6) (C∞(IR6)× Ω2(IR3)).

8 Fourier integral operators and the KdV

equation

The Lie group of invertible Fourier integral operators is a symmetry groupfor the Korteweg - deVries (KdV) equation

ut + 6uux + uxxx = 0 . (8.1)

Gardner found that with the bracket

F,G =∫ 2π

0

δF

δu

∂

∂x

δG

δudx (8.2)

and Hamiltonian

H(u) =∫ 2π

0(u3 +

1

2u3x)dx (8.3)

u satisfies the KdV equation 8.1 if and only if u = u,H. The questionis, where does this bracket and Hamiltonian come from ? In Adams, Ratiu

16

and Schmid [3] we showed that this bracket is the Lie-Poisson bracket on acoadjoint orbit of Lie group G = FIO of invertible Fourier integral operatorson the circle S1. We briefly summarize:

A Fourier integral operators on a compact manifold M is a operator

A : C∞(M)→ C∞(M) (8.4)

locally given by

A(u)(x) = (2π)−n∫ ∫

eiϕ(x,y,ξ)a(x, ξ)u(y)dydξ (8.5)

where ϕ(x, y, ξ) is a phase function with certain properties and the symbola(x, ξ) belongs to a certain symbol class. A pseudodifferential operator is aspecial kind of Fourier integral operators, locally of the form

P (u)(x) = (2π)−n∫ ∫

ei(x−y)·ξp(x, ξ)u(y)dydξ . (8.6)

Denote by FIO and ΨDO the groups under composition (operator prod-uct) of invertible Fourier integral operators and invertible pseudodifferentialoperators on M respectively. For an introduction to the theory of Fourierintegral operators and pseudodifferential operators see e.g. Treves [65]. Wehave the following theorems, Adams, Ratiu and Schmid [4] and [5]:

Theorem 2 a) ΨDO is a smooth infinite dimensional ILH - Lie group.b) FIO is a smooth infinite dimensional ILH - Lie group.c) FIO is a smooth infinite dimensional principal fiber bundle overDiff∞ω (T ∗M − 0) with structure group ΨDO.

Theorem 3 For M = S1 the Gardner bracket 8.2 is the Lie Poisson bracketon the coadjoint orbit of FIO through the Schroedinger operator P ∈ ΨDO.Complete integrability of the KdV eqaution follows from the infinite systemof conserved integral in involution given by Hk = Trace P k; in particular theHamiltonian 8.3 equals H = H2.

17

9 Wess-Zumino Consistency Condition for

BRST Symmetries

This is a quantum field theory problem in local Lie algebra cohomology.References for a general geometric treatment of quantum field theory are e.g.Eguchi, Gilkey and Hanson [21], Faddeev and Slavnov [23], Itzykson andZuber [29], Nash [47], Ramond [51].

The Lie group G involved is the group of gauge transformations on aprincipal G-bundle (P,M,G) i.e. G = C∞(P,G) as discussed in section 5.3.The Lie algebra g is the Lie algebra of infinitesimal gauge transformations,i.e. g = C∞(P, g) (g the Lie algebra of G). We have the following bicomplexC∗loc = Cq,p,∆q,p∈N, where Cq,p = Cq(g,Ωp

loc(P, g)) is the space of q-cochainson g with values in Ωp

loc(P, g), local g-valued p-forms on P (Anderson andThompson [6], Schmid [55], [56] and [57]) . The total differential ∆ = δloc +(−1)pd is the so called BRST operator and is defined as follows (Schmid[55], [56] and [57])

δloc : Cq,p(g,Ωloc) −→ Cq+1,p(g,Ωloc) (9.1)

is the Chevalley-Eilenberg coboundary operator (Chevalley and Eilenberg[16])

(δΦ)(ξ0, · · · , ξq) =∑qi=0(−1)iρ

′(ξi)Φ(ξ0, · · · , ξi, · · · , ξq)

+∑i<j(−1)i+jΦ([ξi, ξj], · · · , ξi, · · · , ξj, · · · , ξq) .

(9.2)

andd : Cq,p(g,Ωloc) −→ Cq,p+1(g,Ωloc) (9.3)

is the induced exterior derivative from d : Ωploc(P, g)→ Ωp+1

loc (P, g). We have

∆2 = δlocd+ dδloc = δ2loc = d2 = 0 . (9.4)

The induced cohomology H∗BRST (g,Ωloc) is called the BRST cohomology.Gauge anomalies (chiral) are elements in H1

BRST (g,Ωloc).

Details about BRST symmetries can be found in various interpretationsin Bonora and Cotta-Ramusino [12], Dubois-Violette [19], Henneaux [28],

18

Kastler and Stora [31], Kugo and Ojima [34], Schmid [54], [55], [56], [57],Stasheff and Teitelboim [63], Zumino [67].

The so called Wess-Zumino consistency condition for an element ω ∈ C∗locmeans that there exists an α ∈ C∗loc such that

δlocω + dα = 0 . (9.5)

Any solution of 9.5 of the form

ω = δlocβ + dγ, β, γ ∈ C∗loc (9.6)

is considered to be trivial, since then δlocω = δ2locβ+δlocdγ so δlocω−d(δlocγ) =

0. We restrict ourselves to the subalgebra Ωpinv(P,g) of G -invariant g-

forms. The consistency condition 9.5 produces the so called descent equa-tions, Stasheff [62], Zumino [67] . If δlocω + dα = 0 then taking δloc of (*)we get δloc

2ω + δlocdα = 0 hence δlocdα = 0 = −dδlocα. So by the Poincarelemma there exists a local form β such that δlocα = dβ, or δlocα+dβ = 0. Bydefinition δloc[ω] = [α]. If ω is trivial, i.e. ω = δlocβ + dγ then δlocdγ = −dα,and dα = −dδlocγ, hence α is of the form α = δlocγ + dλ, that is [α] = 0. Weget the descent equations

δlocω + dω1 = 0

δlocω1 + dω2 = 0

.

.

.

δlocωk−1 + dωk = 0

(9.7)

where k is the smallest integer such that [ω] ∈ Hkloc(LieG) with δlocω = 0.

In order to solve this consistency condition we introduce the notion ofLie algebra valued symplectic structures, called g-symplectic structures forshort.

10 The g - symplectic structure on orbits

In order to compute a solution to the consistency condition for the anomalywe combine the previous construction with symplectic geometry and Hamil-

19

tonian systems. We first formulate the ideas and results for the finite dimen-sional case and apply them in the next section to the infinite dimensionalsituation of the previous sections.

In Schmid [58] we introduced the notion of Lie algebra valued symplecticstructures, called g structures for short. For a principal G - bundle (P, π,M)we denote by Ωk(P,g) the space of Lie aglebra g - valued k-forms on P , calledg - forms for short. For g - forms the usual Cartan calculus holds like forclassical real valued forms. For instance, if f is a g - function on P ( 0 -form),f : P → g, then df is a g - one form on P . If ϕ : P → P is a smooth mapand α ∈ Ωk(P,g) , then the pull back ϕ∗α ∈ Ωk(P,g). The Lie derivative LXwith respect to a vector field X, the inner product operator iX and exteriorderivative d are defined analogous to the classical real valued case and wehave the Cartan formula LXα = diXα + iXdα for any α ∈ Ωk(P,g) and thePoincare lemma.

We define g - valued symplectic forms as follows:

Definition 1 A g - symplectic structures on P is a g - form ω ∈ Ω2(P,g)which is closed and nondegenerate.

A g - symplectic form ω on P induces a linear injective map

ω(p)[ : TpP → L(TpP,g) , ω(p)[(v) · w = ω(p)(v, w). (10.8)

A vector field X on P is called g - Hamiltonian if there exists a g - functionf : P → g such that df = iXω, or equivalent ω(p)[X(p) = df(p). A g - vectorfield X is locally g - Hamiltonian if and only if its flow ϕt is g - symplectic,i.e ϕ∗tω = ω.

Denote by Op be the G - orbit of the right action Rg : Op → Op , g ∈ Gthrough p, i.e. Op = R(p,G) with the diffeomorphism Rp : G→ Op; Rp(g) =R(p, g) . Let θ ∈ Ω1(G, g) be the right invariant Maurer Cartan form on G.

Theorem 4 If G is semi simple, then every G - orbit Op is a g - symplecticmanifold.

Proof: Let p ∈ P and define the g - 1 form Θp on the orbit Op by Θp := Rp∗θand the g - 2 form ωp on Op by ωp := dΘp. 2

20

It follows from the definition that the g - 1 form Θp and the g - symplecticform ωp are G - invariant. The g - Poisson bracket for any two g - functionsf, g : Op → g such that df(q), dg(q) ∈ ω[(p)(TqP ) (i.e. Xf , Xg exist) isdefined by

f, g = ω(p)(Xf (q), Xg(q)) ∈ g . (10.9)

This makes C∞(Op,g) into a Lie algebra.

10.1 The canonical momentum map on Op

For every ξ ∈ g the fundamental vector field ξP on Op defined by

ξP (q) =d

dt |t=0

Rexp tξ(q) (10.10)

is locally g -Hamiltonian. The corresponding Hamiltonian is locally given by

H(x) = −1

2[x, x · ξ] . (10.11)

The right action of G on Op has a natural g -momentum map J : Op →L(g,g) given by

J(q) = adη TRq (10.12)

where η = R∗pXt(g), q = p · g .

11 A solution to the consistency condition

We now combine the ideas of sections 9 and 10. The costruction of g-symplectic orbits and the momentum map is generalized from the finitedimensional case to the infinite dimensional situation in gauge theory asdescribed in section 9, as follows (we use the same notation with script sym-bols): We consider the principal G bundle (P , π,M), where P = Ω∗(P,g)and G acts on P by gauge transformations. M = P/G is the orbit space(under the usual assumptions on the action and topologies , see e.g. Law-son [36]). Then for A ∈ Ω∗(P,g) the canonical 1-form ΘA on the orbit OA

21

induced from the Maurer - Cartan form on G (as described in Theorem 4)becomes a map

ΘA : OA → Ω1(P,g) ' C0,1loc . (11.13)

The momentum map becomes a map

J : OA → L(LieG, LieG) = C1,0loc . (11.14)

Theorem 5 The momentum map J satisfies the consistency condition forthe canonical 1-form (Maurer-Cartan) Θ

δlocΘA + dJ = 0. (11.15)

Proof: We have δlocΘA ∈ C1,1loc and dJ(A) ∈ C1,1

loc and we get for any ξ ∈ LieG

δlocΘA(ξ) = dJ(A)(ξ) + LZJ(ξ)ΘA,

where ZJ(ξ) is the induced fundamental vector field. We conclude thatLZJ(ξ)

ΘA = 0. 2

Details will appear in a forthcoming paper.

References

[1] Abraham, R., Marsden, J.E. Foundations of Mechanics, 2nd ed.,Addison-Wesley, Reading Mass. 1978.

[2] Abraham, R., Marsden, J.E., Ratiu, T.S. Manifolds, Tensor Analysis,and Applications. Addison-Wesley, Reading, Mass. 1983.

[3] Adams, M., Ratiu, T.S., Schmid, R. The Lie Group Structure of Diffeo-morphism Groups and Invertible Fourier Integral Operators, with Appli-cations, MSRI Publications 4, V. Kac ed., Springer - Verlag, New York,1985.

[4] Adams, M., Ratiu, T., Schmid, R. A Lie group Structure for Pseudodif-ferential Operators, Math. Ann. 273 (1986), 529-551.

22

[5] Adams, M., Ratiu, T., Schmid, R. A Lie Group Structure for FourierIntegral Operators, Math. Ann. 275 (1986), 19-41.

[6] Anderson, I.M., Thompson, G. The Inverse Problem of the Calculuisof Variations for Ordinary Differential Equations, Memoirs AMS 473,AMS, Providence, R.I., 1992.

[7] Arnold, V.I. Sur la geometrie differentielle des groups de Lie de dimen-sion infinie et ses applications a l’hydrodynamique des fluids parfaits,Ann. Inst. Fourier, Grenoble 16 1966, 319-361.

[8] Arnold, V.I. Mathematical Methods of Classical Mechanics. 2nd ed,Grad. Text in Math. 60, Springer - Verlag, 1989.

[9] Binz, E. Einstein’s Evolution Equations for the Vacuum , Formulated onthe Space of Differentials of Immersions, Lecture Notes in Math. 1037,Springer - Verlag, Berlin, 1983, 2-37.

[10] Binz, E., Sniatycki, J. Geometry of Classical Fields, North-Holland, Am-sterdam, 1988.

[11] Bleecker, D. Gauge Theory and Variational Principles, Addison-Wesley,Reading, Mass., 1981.

[12] Bonora, L., Cotta-Ramusino, P. Some Remarks on BRS Transforma-tions, Anomalies and the Cohomology of the Lie Group of Gauge Trans-formations , Comm. Math. Phys. 87 (1983), 589-603.

[13] Bourbaki, N. Varietes Differentielles et Analytiques, Fascicule des Re-sultats, Herman ,Paris, 1971.

[14] Bourbaki, N. Lie Groups and Lie Algebras, Herman, Paris, 1975.

[15] Chernoff, P., Marsden, J.E. Properties of Infinite Dimensional Hamil-tonian Systems, Lecture Notes in Math. 425, Springer - Verlag, NewYork, 1974.

[16] Chevalley, C., Eilenberg, S. Cohomology Theory of Lie Groups and LieAlgebras , Trans. Amer. Math. Soc. 63 (1948), 85-124.

23

[17] Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard-Bleick, M. Analysis,Manifolds and Physics, 2nd ed., North Holland, Amsterdam 1982; PartII Applications, North Holland,Amsterdam, 1989.

[18] Cotta-Ramusino, P. Geometry of Gauge Orbits and Ghost Fields , Proc.Geometry and Physics, Florence, 1982, Pitagora, Bologna, 1983, 229-238.

[19] Dubois-Violette, M. The Weil-B.R.S. Algebra of a Lie Algebra and theAnomalous Term in Gauge Theories . Geom. Phys. 3 (1986), 525-565.

[20] Ebin, D., Marsden, J.E. Groups of Diffeomorphisms and the Motion ofan Incompressible Fluid, Ann. Math. 92, (1970), 102-163.

[21] Eguchi, T., Gilkey, P.B., Hanson, A.J. Gravitation, Gauge Theories andDifferential Geometry, Phys. Reports 66 (1980) , 213-393.

[22] Elliasson, H. Geometry of Manifolds of Maps, J. Diff. Geom. 1 (1967) ,169-194.

[23] Faddeev, L.D., Slavnov, A.A. Gauge Fields, Introduction to QuantumTheory, Benjamin/Cummings, Reading. Mass. 1980.

[24] Fischer, A., Marsden, J.E. The Einstein Equation of Evolution - a Ge-ometric Approach, J. Math. Phys. 13 (1972) , 546-568 .

[25] Goldstein, H. Classical Mechanics, 2nd ed., Addison- Wesley, Reading,Mass., 1980.

[26] Guillemin, V., Sternberg, S. Symplectic Techniques in Physics, Cam-bridge University Press 1984.

[27] Hamilton, R. The Inverse Function Theorem of Nash and Moser, Bull.AMS 7 (1982), 65-222.

[28] Henneaux, M. Classical Foundations of BRST Symmetry, Biblionopolis,Naples, 1988.

[29] Itzykson, C., Zuber, J.B. Quantum Field Theory, McGraw-Hill, NewYork, 1980.

24

[30] Kac, V. Infinite Dimensional Lie Algebras, 2nd. ed. Cambridge Univ.Press, 1985.

[31] Kastler, D., Stora, R. A Differential Geometric Setting for BRS Trans-formations and Anomalies I & II , J. Geom. Phys. 3 (1986), 437-482,483-505.

[32] Keller, H.H. Differential Calculus in Locally Convex Spaces, LectureNotes in Math. 417, Springer - Verlag, Berlin, 1974.

[33] Kostant, B., Sternberg, S. Symplectic Reduction, BRS Cohomology, andInfinite Dimensional Clifford Algebras , Ann. Phys. 176 (1987), 49-113.

[34] Kugo, T., Ojima, I. Local Covariant Operator Formalism of Non-AbelianGauge Theories and Quark Confinement Problem , Suppl. Progr. Theor.Phys. 66 (1979), 1-130.

[35] Lang, S. Differential Manifolds, Addison- Wesley, Reading, Mass., 1972.

[36] Lawson, B. The Theory of Gauge Fields in Four Dimensions, AMS,Providence, R.I., 1985.

[37] Lichnerowicz, A. Les Varietes de Poisson et Leurs Algebres de Lie As-sociees, J. Diff. Geom. 12 (1977) , 253-300.

[38] Marle, C., Libermann P. Symplectic Geometry and Analytical Mechan-ics, Reidel, Boston, 1987.

[39] Marmo, G., Saletan, E.J., Simoni, A., Vitale, B. Dynamical Systems,John Wiley, Chichester, 1985.

[40] Marsden, J.E. Applications of Global in Mathematical Physics, Publishor Perish, Waltman Mass., 1974.

[41] Marsden, J.E. Lectures on Mechanics. Cambridge Univ. Press, Cam-bridge, 1992.

[42] Marsden, J.E., Ebin, G.D., Fischer, A. Diffeomorphism Groups, Hy-drodynamics and Relativity, Proc. 13th Biennial Sem. Canadian Math.Congress. Vanstone, J.R. ed. Montreal , 1972 , 135-279 .

25

[43] Marsden, J.E., Ratiu, T. Mechanics and Symmetry, Springer - Verlag,1994.

[44] Marsden, J.E., Weinstein, A. Reduction of Symplectic Manifolds withSymmetry, Reports on Math. Phys. 5 (1974) , 121-130 .

[45] Marsden, J.E., Weinstein, A., Ratiu, T., Schmid, R., Spencer,R.G.Hamiltonian Systems with Symmetry, Coadjoint Orbits and PlasmaPhysics, Proc. IUTAM-ISIMM Symposium on modern developments inanalytical mechanics, Atti Accad. Sci. Torino, Suppl. 117 (1983) , 289-340 .

[46] Michor, P. Manifolds of Differential Mappings, Shiva Publ., Orpington,Kent, 1980.

[47] Nash, C. Relativistic Quantum Fields, Academic Press, London, NewYork, 1978.

[48] Omori, H. Infinite Dimensional Lie Transformation Groups, LectureNotes in Math. 427, Springer - Verlag, Berlin, 1974.

[49] Palais, R. Foundations of Global Nonlinear Analysis, Addison-Wesley,Reading, Mass., 1968.

[50] Pressley, A., Segal, G. Loop Groups and their Representations, OxfordUniversity Press, 1987.

[51] Ramond, P. Field Theory , a Modern Primer, Benjamin/Cummings,Reading, Mass., 1981.

[52] Ratiu, T. Schmid R. The differentiable structure of three remarkablediffeomorphism groups, Math. Zeitschr. 177 (1981) , 81-100 .

[53] Schmid, R. Infinite Dimensional Hamiltonian Systems, Lecture Notes3, Biblionopolis, Naples, 1987.

[54] Schmid, R. The Geometry of BRS Transformations , Illinois J. Math.34 (1990), 87-97.

[55] Schmid, R. Local Cohomology in Gauge Theories BRST Transforma-tions and Anomalies, Diff. Geom. and Appl. (1994), to appear.

26

[56] Schmid, R. BRST Cohomology and Anomalies, Geometrical and Alge-braic Aspects of Nonlinear Field Theory, S. De Filippo et al (eds.), North- Holland, 1989, 159-172.

[57] Schmid, R. A Few BRST Bicomplexes, Int. J. Mod. Phys. A (Proc.Suppl.) 3A (1993), 375-378.

[58] Schmid, R. A Solution to the BRST Consistency Condition and g -Symplectic Orbits, Erwin Schrodinger International Institute for Math-ematical Physics, preprint 41, 1993.

[59] Sibner, L. Non-Minimal Solutions of the Higgs Equations, in these pro-ceedings.

[60] Sibner, R. Constructing Monopoles, in these proceedings.

[61] Souriau, J.M. Structure des Systemes Dynamiques, Dunod, Paris, 1970.

[62] Stasheff, J. The De Rham Bar Construction as a Setting for the Zumino,Faddee’v, etc. Descent Equations, Proc. Symp. Anomalies, Geometry,Topology, Argonne Natl. Lab. 1984. W.E. Bardeen, A. R. White (eds.),World Scientific, Singapore (1985), 220-233.

[63] Stasheff, J., Teitelboim, C. Existence, Uniqueness and Cohomology ofthe Classical BRST Charge with Ghosts of Ghosts , Comm. Math. Phys.120 (1989), 379-.

[64] Temam R. Infinite Dimensional Dynamical Systems in Mechanics andPhysics, Springer - Verlag, New York, 1988.

[65] Treves, F. Introduction to Pseudodifferential and Fourier Integral Oper-ators I & II, Plemun Press, New York, 1980.

[66] Weinstein, A.The local structure of Poisson manifolds, J. Diff. Geom.18 (1983) , 523-557 .

[67] Zumino, B. Anomalies, Cocycles and Schwinger Terms, Proc. Symp.Anomalies, Geometry, Topology, Argonne Natl. Lab. 1984, W.E.Bardeen, A.R. White (eds.), World Scientific 1985.

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