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Equivariant characteristic forms

on the bundle of connections

R. Ferreiro Perez

Insituto de Fisica Aplicada, CSICC/ Serrano 144, 28006-MadridE-mail: roberto@iec.csic.es

Abstract

The characteristic forms on the bundle of connections of a principalbundle P → M of degree equal to or less than dimM , determine thecharacteristic classes of P , and those of degree k+dimM determine certaindifferential k-forms on the space of connections A on P .

The equivariant characteristic forms provide canonical equivariant ex-tensions of these forms, and therefore canonical cohomology classes inA/Gau0P .

More generally, for any closed β ∈ Ωr(M) and f ∈ IG

k , with 2k + r ≥

dimM a cohomology class in A/Gau0P is obtained. We show that theseclasses coincide with some classes previously defined by Atiyah and Singer.

1 Introduction

Let π : P → M , be a principal G-bundle and let p : C(P ) → M be its bundle ofconnections. For anyWeil polynomial f ∈ IG

k , the characteristic form associatedto f and to the canonical connection A on the bundle C(P )×M P → C(P ) (seebelow for the details) is a closed and AutP -invariant 2k-form on C(P ), cf (F)(e.g., see [13]), where F is the curvature of A. As C(P ) is an affine bundle,the map p∗ : H∗(M) → H∗(C(P )) is an isomorphism. The cohomology class inM corresponding to cf (F) under this isomorphism is the characteristic class ofP associated to f (e.g., see [14]). So, the characteristic forms on C(P ) deter-mine the characteristic classes in M , but the characteristic forms contain moreinformation than the characteristic classes; for example, if 2k > n the charac-teristic class vanishes, although the corresponding form does not necessarily, asdimC(P ) > dimM .

The principal aim of this paper is to provide a geometrical interpretation ofthe higher-degree characteristic forms, which is based on the following construc-tion. Let E → N be any bundle over a compact, oriented n-manifold withoutboundary. Given a form α ∈ Ωn+k(E), we define a k-form on the space of

1

sections of E, Γ(E), by setting,

([α])s(X1, . . . , Xk) =

∫

M

s∗(iXk. . . iX1

α)

for any section s ∈ Γ(E) and vertical vector fields along s, X1, . . . , Xk ∈TsΓ(E) ≃ Γ(N, s∗V (E)). The map : Ωn+k(E) → Ωk(Γ(E)) commutes withthe exterior differential and with the action of automorphisms (see Proposi-tion 11 below) and hence, if α is closed, exact, or invariant under a subgroupG ⊂ AutvE, then the form [α] enjoys the same property.

Applying this construction to the bundle C(P ) → M , for any characteristicform cf (F) with 2k > n, we obtain a closed and GauP -invariant (2k − n)-formin the space A = Γ(M,C(P )) of connections on P . More generally, as provedin [14], the space of GauP -invariant forms on C(P ) is generated by forms of thetype cf (F) ∧ p∗β, with β ∈ Ω∗(M). So, given f ∈ IG

k and a closed β ∈ Ωr(M),such that 2k + r ≥ n, we have a closed and GauP -invariant (2k + r − n)-formon A given by,

ωf,β = [cf (F) ∧ p∗β] ∈ Ω2k+r−n(A). (1)

AsA is an affine space, these forms are exact, and the cohomology classes definedby them on A, vanish; but in gauge theories—because of gauge symmetry—it isusually more interesting to consider the quotient space A/GauP instead of thespace A itself. As the forms (1) are GauP -invariant, we are led to think thatthey could be related to differential forms on this quotient space. The problemis that these forms are not generally projectable with respect to the naturalquotient map A → A/GauP .

Another way to relate the cohomology of a manifold and the cohomology ofits quotient by the action of a Lie group is by means of equivariant cohomology,e.g. see [19]. Below, we show that by applying the usual construction of equiv-ariant characteristic classes (e.g. see [9, 10, 12]) to the canonical connection, itprovides with canonical AutP -equivariant extensions of the characteristic forms.With this result and extending the map to equivariant differential forms in anobvious way, we obtain GauP -equivariant extensions of the forms (1); see The-orem 16 below. These extensions determine cohomology classes in the quotientspace A/Gau0P . We also prove that such classes coincide with those defined in[5] by Atiyah and Singer.

As is well known (e.g. see [4]), an equivariant extension of an invariantsymplectic two-form is equivalent to a moment map for it. Hence if the form(1) is of degree two on A, then the GauP -equivariant extension that we obtaindefines a canonical moment map for the symplectic action of the gauge groupon A, and we show that this symplectic forms and moment maps coincide withthose defined in [3, 16, 21].

2

2 The bundle of connections and the canonical

connection

If π : P → M is a principal G-bundle, its bundle of connections is an affinebundle p : C(P ) → M modelled over the vector bundle T ∗M ⊗ adP , such thatthere is a bijection between connections on P and the sections of C(P ) (e.g.see [13, 17, 22]). The natural projection p : P = C(P ) ×M P → C(P ) onto thefirst factor induces a principal G-bundle structure over C(P ), and we have thecommutative diagram

Pp

−→ Pπ ↓ ↓ π

C(P )p

−→ M

The bundle P has a canonical connection A ∈ Ω1(P, g) characterized by,

A(σA(x),u)(X) = Au(p∗X), (2)

for every connection A on P , x ∈ M , u ∈ π−1(x), X ∈ T(σA(x),u)P, and whereσA : M → C(P ) is the section corresponding to A. The canonical connectionenjoys the following properties (e.g. see [13]):

(1) A is invariant under the natural action of the group AutP of automor-phisms of P .

(2) For every connection A on P , we have σ∗A(A) = A, where σA : P → P is

defined by σA(u) = (σA(x), u), with x ∈ M , u ∈ π−1(x).

Remark 1 It can be shown (see [13]) that the bundle p : P → C(P ) is isomor-phic to J1P → (J1P )/G and, under this identification, the canonical connectionA corresponds to the structure form of J1P .

Let F be the curvature of A. If f ∈ IGk is a Weil polynomial of degree k

for G, we define the characteristic form associated to f as the 2k-form on C(P )defined by cf (F) = f(F, . . . ,F). This form has the following properties:

(3) cf (F) is closed.

(4) cf (F) is invariant under the action of the group AutP on C(P ).

(5) For every connection A on P we have σ∗A(cf (F)) = f(FA, . . . , FA).

As a consequence of (3) and (5) and the fact that the space of connections isan affine space, we obtain the well-known result of Chern-Weil theory that thecohomology class [f(FA, . . . , FA)] ∈ H2k(M) is independent of the connection A,and is it called the characteristic class associated to f . In other words, the mapσ∗A is an inverse of p∗ : H∗(M) → H∗(C(P )), and under this isomorphism the

cohomology class of cf (F) corresponds to the characteristic class of P associatedto f (e.g. see [14]).

3

3 Equivariant Characteristic forms

First, we recall the definition of equivariant cohomology in the Cartan model(e.g. see [8, 19]). Suppose that we have a left action of a connected Lie group Gon a manifold N , i.e. a homomorphism ρ : G → Diff(N). We have an inducedLie algebra homomorphism

LieG → X(N)

X 7→ XN =d

dt

∣∣∣∣t=0

ρ(exp(−tX))

LetΩG(N) = (S•(LieG∗)⊗ Ω•(N))

G= P•(LieG,Ω•(N))G

be the space of G-invariant polynomials on LieG with values in Ω•(N). Wedefine the following graduation: deg(α) = 2k + r if α ∈ Pk(LieG,Ωr(N)).Hence the space of G-equivariant differential q-forms is

ΩqG(N) =

⊕

2k+r=q

(Pk(LieG,Ωr(N)))G .

Let dc : ΩqG(N) → Ωq+1

G (N) be the Cartan differential,

(dcα)(X) = d(α(X))− iXNα(X), ∀X ∈ LieG.

As is well known, on Ω•G(N) we have (dc)

2= 0. Moreover, the equivariant

cohomology (in the Cartan model) of N with respect of the action of G isdefined as the cohomology of this complex, i.e.,

HqG(N) =

ker(dc : Ω

qG(N) → Ωq+1

G(N)

)

Im(dc : Ω

q−1G (N) → Ωq

G(N)) .

Definition 2 Given a closed and G-invariant form ω ∈ Ωq(M), an equivariantdifferential form ω# ∈ Ωq

G(M) is said to be a G-equivariant extension of ω if

dcω# = 0 and ω#(0) = ω.

In general, there could be obstructions to the existence of equivariant exten-sions (e.g., see [25]) but, as we will see, the classical construction of equivariantcharacteristic classes really provides canonical equivariant extensions for theforms we are dealing with.

Next, let us recall the relationship between equivariant cohomology and thecohomology of the quotient space. If the action of G on N is free and N/Gis a manifold, then N → N/G is a principal G-bundle. Let A be a connec-tion on this bundle. The following map is a generalization of the Chern-Weilhomomorphism:

ΦA : Ω∗G(N) → (Ω∗(N))basic ≃ Ω∗(N/G)

α 7→ (α(FA))hor

4

where βhor denotes the horizontal component of β with respect to the connectionA. We have

Proposition 3 If α ∈ Ω∗G(N), then ΦA(dcα) = d(ΦA(α)).

Proof. We refer the reader to [8, Theorem 7.34].

Theorem 4 The induced map in cohomology

ΦA : H∗G(N) → H∗(N/G)

is independent of the connection A chosen, and is denoted by

ΦN : H∗G(N) → H∗(N/G).

Proof. The result quickly follows by working on the bundle of connections.We use the notations introduced in Section 2 by setting P = N , M = N/G anddenoting by p : P → M the quotient map. Let α ∈ Ωq

G(P ) be an equivariant

q-form such that dcα = 0. Recall that p∗(α) belongs to ΩqG(P) as p is a G-

equivariant map. By Proposition 3, the form ΦA(p∗α) ∈ Ω∗(C(P )) is closed,

and from the formula (2) we obtain

σ∗A(ΦA(p

∗α)) = ΦA(α).

Again the result follows as the space of connections is contractible.

Remark 5 If G is compact and connected ΦN is an isomorphism (e.g. see [19]).

The definition of equivariant characteristic classes of Berline and Vergne(see [9, 10, 12]) can be introduced as follows. Let π : P → M a principal G-bundle and let us further assume that a Lie group G acts (on the left) on P byautomorphisms of this bundle. Let A be a connection on P , which is invariantunder the action of G.

For every f ∈ IGk the G-equivariant characteristic form associated to f and

A, cf (FGA) ∈ Ω2k

G (M), is defined by

cf (FGA)(X) = f

(FA −A(XP ),

(k. . ., FA −A(XP ))

=

k∑

i=1

(−1)k−i(ki

)f(FA,

(i. . ., FA, A(XP ),(k−i. . . , A(XP ))

for every X ∈ LieG.

Proposition 6 We have

(1) cf (FGA ) is a G-equivariant extension of cf (FA).

(2) The equivariant cohomology class cGf (P ) = [cf (FGA)] ∈ H2k

G (M) is inde-

pendent of the G-invariant connection A, and is called the G-equivariantcohomology class of P associated to f .

5

Proof. See [12, 10]).Applying the construction of equivariant characteristic forms to the bun-

dle P → C(P ) with the AutP -invariant connection A, we obtain the AutP -equivariant characteristic form cf (F

AutP ) ∈ Ω2kAutP (C(P )), with is an AutP -

equivariant extension of cf (F). If G ⊂ AutP is any subgroup of the automor-phism group, we have the corresponding G-equivariant characteristic form

cf (FG) = cf (F

AutP )|LieG .

Using the Formula (2) and the Property (5) of Section 2 it is easy to see that

Proposition 7 If G acts on π : P → M by automorphisms of π, and A is a

G-invariant connection on P , then we have

σ∗A(cf (F

G)) = cf (FGA ).

Remark 8 So, we obtain the analog situation to that of the ordinary charac-teristic classes; see the last paragraph in Section 2. Moreover, Proposition 7provides a very simple proof of Proposition 6–(2), as the space of G-invariantconnections is an affine subspace; more precisely, if A is a G-invariant connec-tion, σ∗

A is an inverse of p∗ : H∗G(M) → H∗

G(C(P )) (hence this map is an isomor-phism). Under this isomorphism the G-equivariant cohomology class of cf (F

G)corresponds to the G-equivariant characteristic class associated to f . Moreover,as in the case of ordinary characteristic classes, the equivariant characteristicforms contain more information than their corresponding characteristic classes.For example, in Section 5 we will use this forms in the case G = GauP to findequivariant extensions of the forms (1).

The analog of Proposition 4 for the equivariant characteristic classes, is

Proposition 9 Assume that G acts freely on P and M , and that the quotient

bundle πG : P/G → M/G exists, then

ΦM (cGf (P )) = cf (P/G).

Proof. We sketch the proof only, as it is basically the standard construc-tion of a connection in a quotient bundle (e.g. see [16, 5.2.3]). We have acommutative diagram

PqP−→ P/G

π ↓ ↓ πG

MqM−→ M/G

(3)

where qP and qM denote the quotient maps. Let A1 be a connection onM → M/G and A2 a G-invariant connection on P → M . Then it is defined aconnection A3 on P/G → M/G in the following way: If X ∈ X(M/G), using A1

we can lift X to a horizontal vector field X1 ∈ X(M). Using A2 we can lift X1

to X2 ∈ X(P ). As A2 is G-invariant X2 is also G-invariant and projects to avector field X ∈ X(P/G). By the commutativity of (3) X projects onto X , and

6

is the horizontal lift of X . It can be seen that the curvature of this connectionis given by

q∗P (FA3) = FA2

−A2((π∗FA1

)P ),

and hence,ΦA1

(cf (FGA2

)) = cf (FA3).

4 Forms in Γ(E) induced by forms in E

Let E → M be a locally trivial bundle over an oriented, connected, and com-pact n-manifold without boundary M . We denote by Γ(E) the space of globalsections of E, and we assume that it is not empty. We consider Γ(E) as a dif-ferential manifold; for the details of its infinite-dimensional structure see [20].For any s ∈ Γ(E) there is an identification TsΓ(E) ≃ Γ(M, s∗V (E)). We definea map

: Ωn+k(E) → Ωk(Γ(E))

by the formula

([α])s(X1, . . . , Xk) =

∫

M

s∗(iXk. . . iX1

α), (4)

for any α ∈ Ωn+k(E), s ∈ Γ(E), X1, . . . , Xk ∈ TsΓE ≃ Γ(M, s∗V (E)). Ifα ∈ Ωq(E), with q < n we define [α] = 0. There is another equivalentdefinition of , which is as follows: Consider the evaluation map

ev : M × Γ(E) → E

(x, s) 7→ s(x)

Pulling α ∈ Ωn+k(E) back via the evaluation map, we obtain a form ev∗α ∈Ωn+k(M × Γ(E)). Integrating over the fiber of M × Γ(E) → Γ(E) we obtain aform ∫

M

ev∗α ∈ Ωk(Γ(E)), (5)

which is seen to coincide with [α].If φ ∈ AutvE is a vertical automorphism, then we denote by φ ∈ Diff(Γ(E))

the diffeomorphism induced on Γ(E), i.e. φ(s) = φ s for every s ∈ Γ(E). Atthe Lie algebra level, every vertical vector field X ∈ Xv(E) determines a vector

field on Γ(E), X ∈ X(Γ(E)) by X(s) = X s, s ∈ Γ(E), but not all the vectorfields on Γ(E) are of this type, although those of this type suffice to determinethe differential forms on Γ(E). Precisely, we have

Proposition 10 Let β ∈ Ωp(Γ(E)). We have β = 0 if and only if iXβ = 0 for

every X ∈ Xv(E).

7

Proof. It follows from the fact that for every Xs ∈ TsΓ(E), there exists an

extension X ∈ Xv(E) of Xs, and we have X(s) = Xs.

Proposition 11 For every α ∈ Ωn+k(E), φ ∈ AutvE, and X ∈ Xv(E) we have

a) [dα] = d[α],

b) [φ∗α] = φ∗[α],

c) [iXα] = iX[α],

d) [LXα] = LX[α].

Proof. b) For every Y ∈∧k TsΓ(E), we have

(φ∗[α]

)

s(Y ) = [α]φs(φ∗Y ) =

∫

M

(φ s)∗(iφ∗Y α)

=

∫

M

s∗φ∗(iφ∗Y α) =

∫

M

s∗(iY φ∗α) = [φ∗α]s(Y ).

c) It follows from the very definition of [α].d) If φt is the flow of X , then φt is the flow of X and using b) we have:

[LXα] =

[limt→0

φ∗tα− α

t

]= lim

t→0

[φ∗tα]− [α]

t

= limt→0

φ∗

t ([α])−[α]

t= L

X[α].

a) By induction over k. If k < −1, by definition we have [α] = [dα] = 0.For k = −1, i.e., α ∈ Ωn−1(E), by definition we have [α] = 0, and

([dα])s =

∫

M

s∗dα =

∫

M

d(s∗α) = 0.

Now, assume k ≥ 0. Using c) and d), for every X ∈ Xv(E) we have

iX[dα] = [iXdα] = [LXα]− [diXα]

= LX[α]− di

X [α] = i

Xd[α],

where we have used that [diXα] = d [iXα] by virtue of the induction hy-pothesis, as iXα ∈ Ωn+k−1(E). Since X is arbitrary, from the Proposition 10we obtain [dα] = d[α].

Remark 12 The results of Proposition 11 suffice for our purposes althoughthey also hold for non-vertical automorphisms and vector fields (see [1]).

8

Now, assume that G is a subgroup of AutvE. Given α ∈ Pq(LieG,Ωn+k(E)),the composition

LieGα

−→ Ωn+k(E)−→ Ωk(Γ(E))

defines an element [α] of Pq(LieG,Ωk(Γ(E))); that is, for X ∈ LieG we have,

([α]) (X) = [α(X)].

By Proposition 11-b) if α is G-invariant, [α] is also G-invariant, and so themap extend to a map between G-equivariant differential forms,

: Ωn+kG

(E) → ΩkG(Γ(E)).

Proposition 13 For every α ∈ Ωn+kG (E) we have

[dcα] = dc[α].

Hence, we have an induced map in equivariant cohomology : Hn+kG

(E) →

HkG(Γ(E)).

Proof. If α ∈ Ωn+kG (C(P )) and X ∈ LieG, then from Proposition 11 we

have

([dcα]) (X) = [dcα(X)] = [d(α(X))]−[iXα(X)]

= d[α(X)]− iX[α(X)] = (dc[α]) (X).

5 Applications

In this section we combine the results of Sections 3 and 4. As remarked in theIntroduction, in Gauge theories GauP -invariant forms are specially interesting,so we focus on these forms. In [14] it is proved that the space of GauP -invariantforms is generated by the forms of type cf (F)∧p

∗β, with f ∈ IGk and β ∈ Ωr(M).

We assume that β is closed and 2k + r ≥ n.By applying the map to cf (F) ∧ p∗β we obtain

ωf,β = [cf (F) ∧ p∗β] ∈ Ω2k+r−n(A).

By Proposition 11 this form is closed and GauP -invariant.The space of connections A is an affine space modelled over Ω1(M, adP ).

Hence, we have the identification TAA ≃ Ω1(M, adP ) for every A ∈ A. Also,C(P ) is an affine bundle modelled over the vector bundle T ∗M ⊗ adP . So, forevery a ∈ Ω1(M, adP ) ⊂ Γ(C(P ), T ∗M ⊗ adP ) we have a vertical vector fieldXa ∈ Xv(C(P )).

Lemma 14 For every a, b ∈ Ω1(M, adP ), we have

iXaF = p∗a,

iXbiXa

F = 0.

9

Proof. It follows from the formula (5.8) in [13].Taking this lemma into account, it si easy to obtain the expression of ωf,β .

We have

Proposition 15 Let q = 2k + r − n. For a1, . . . , aq ∈ Ω1(M, adP ) we have:

(ωf,β)A(a1, . . . , aq) =(kq

) ∫

M

f(a1, . . . , aq, FA,(n−k−r. . . . . . , FA) ∧ β

By virtue of Proposition 6, the GauP -equivariant characteristic form cf (FGauP )

is an equivariant extension of cf (F). Also p∗β is a closed GauP -equivariantdifferential form, because it is closed and basic. So cf (F

GauP )∧ p∗β is a GauP -equivariant extension of cf (F) ∧ p∗β. We thus obtain the following

Theorem 16 The GauP -equivariant form

ω#f,β = [cf(F

GauP ) ∧ p∗β] ∈ Ω2k+r−nGauP (A) (6)

is a GauP -equivariant extension of ωf,β.

So, we have found a canonical GauP -equivariant extension of ωf,β, as statedin the Introduction.

Let X ∈ gauP = Ω0(M, adP ) ≃ Ω0Ad(P, g), and XP ∈ X(P) the vector field

corresponding to the action of GauP in P. We have

A(σA(x),u)(XP) = Au(p∗XP) = Au(XP ) = X(u).

So, A(XP) = p∗X , and hence

cf (FGauP )(X) = f

(F− p∗X, (k. . .,F− p∗X

)(7)

=k∑

i=1

(−1)i(ki

)f(F, (k−i. . . ,F, p∗X, (i. . ., p∗X).

Using (7) and Proposition 15 it is easy to obtain the expression for ω#f,β. In

the Example 19 of subsection 5.2, we detail such expression in a simple case.Also, for f(X) = Tr(expX) we obtain the equivariant forms defined in [21, sec.6] as a particular case.

5.1 Forms in A/Gau0P

Let Gau0P ⊂ GauP be the subgroup of gauge transformations acting as theidentity on the fiber over a fixed point x0 ∈ M . Then Gau0P acts freely on Aand the quotient A/Gau0P is well defined (e.g. see [23, 15]).

By virtue of Theorem 4, the Gau0P -equivariant differential form ω#f,β deter-

mines a cohomology class

ΦA(ω#f,β) ∈ H2k+r−n(A/Gau0P ).

10

To obtain a representative of this class it is necessary a connection on the bundleA → A/Gau0P . The construction of this connection is a standard fact in gaugetheories (e.g. see [15]): Given a Riemannian metric g in M there is a connectionin A → A/Gau0P given by the decomposition

TAA ≃ Ω1(M, adP ) ≃ Im(dA)⊕ ker(d∗A).

The space Im(dA) is the tangent space to the orbits, and ker(d∗A) is the horizontalcomplement. The expression of its corresponding connection form is

A = GAd∗A, (8)

where GA = (d∗A dA)−1 is the Green function of the Laplacian,

∆0A = d∗A dA : Ω0(M, adP ) → Ω0(M, adP ).

We denote by F the curvature of A.Next, we relate our classes ΦA(ω

#f,β) to the constructions in [5]. Consider

the principal G-bundle P ×A → M ×A. The group Gau0P acts on P and onA. Taking the quotient, we obtain a principal G-bundle

Q = (P ×A) /Gau0P → M × (A/Gau0P ).

If f ∈ IGk and [β] ∈ Hr(M), we have a class

cf (Q) ∧ [β] ∈ H2k+r(M × (A/Gau0P )).

Integrating over M we obtain the class

µf ([β]) =

∫

M

cf (Q) ∧ [β] ∈ H2k+r−n(A/Gau0P ).

Theorem 17 For every f ∈ IGk and every closed β ∈ Ωr(M) with 2k + r ≥ n,

we have

ΦA(ω#f,β) = µf ([β]).

Proof. The evaluation map extend to morphism of principal G-bundles

P ×Aev−→ P

↓ ↓ π

M ×Aev−→ C(P )

where ev(u,A) = (σA(x), u) for u ∈ π−1(x). Hence, ev∗(A) is a connection on

P×A → M×A, ev∗(cf (F)) gives its characteristic classes, and ev∗(cf (FGau0P )),

f ∈ IGk , are its Gau0P -equivariant characteristic forms. By Proposition 9 these

equivariant characteristic forms determine the characteristic classes of the quo-tient bundle Q → M × (A/Gau0P ), i.e., we have ΦM×A(ev

∗(cf (FGau0P ))) =

11

cf (Q). As the connection on M × A → M × (A/Gau0P ) is given by the con-nection (8), the cohomology class µf ([β]), is represented by the form

(∫

M

cf (ev∗(F)− F) ∧ β

)

hor

. (9)

From (5) and (7) we have

ω#f,β(X) =

∫

M

ev∗(cf (FGau0P (X)) ∧ p∗β) =

∫

M

cf (ev∗(F)−X) ∧ β.

So ΦA(ω#f,β) is also represented by the form (9).

Remark 18 The construction of the classes µf ([β]) appears in [5] in order tocompute the Chern character of the index of families of Dirac operators andto apply them to the study of anomalies in gauge theories. These classes alsoappear in other constructions in gauge theories, like the definition of Donaldsoninvariants (see [16]), Topological Quantum Field Theory ([11, 6]), etc.

It is remarkable that we obtain these classes only by studying GauP -invariantforms on C(P ) and its equivariant extensions.

5.2 Moment maps

Example 19 Let M be a surface, and G = U(k). If f(X) = 18π2 tr(X

2),X ∈ g, then the corresponding characteristic class onM vanishes by dimensionalreasons, but the characteristic form cf(F) ∈ Ω4(C(P )) does not. From ourconstructions, this form defines a closed and GauP -invariant 2-form on A, ωf =[cf(F)]. By Proposition 15 and formula (7) for a, b ∈ Ω1(M, adP ), and X ∈gauP = Ω0(M, adP ), we have

cf (F) = 18π2 tr(F ∧ F),

cf(FGauP )(X) = 1

8π2 tr(F ∧ F)− 14π2 tr(p

∗X · F) + 18π2 tr((p

∗X)2),

(ωf )A(a, b) = 14π2

∫

M

tr(a ∧ b),

(ω#f (X))A(a, b) = 1

4π2

∫

M

tr(a ∧ b)− 14π2

∫

M

tr(X · FA).

Hence, [cf (F)] coincides with the natural symplectic structure on A definedin [3]. Moreover, in the case of a 2-form, it is equivalent to give an equivariant

extension of the form and a moment map (e.g. see [4]). Hence ω#f defines a

canonical moment map m for this symplectic structure, given by,

m : A → (gauP )∗,

mA(X) = − 14π2

∫

M

tr(X · FA).

12

Under the pairing

Ω2(M, adP )× Ω0(M, adP ) → R (10)

(η,X) 7→ 〈η,X〉 = − 14π2

∫

M

tr(X · η).

this moment map corresponds to the curvature FA, and it thus coincides withthat defined in [3].

Also, we have m−1(0) = A ∈ A : FA = 0, and by symplectic reductionwe obtain the moduli space of flat connections, and our form gives rise to thesymplectic structure on this space.

More generally, let (M,σ) be a symplectic 2n-manifold. Then the form

1(n−1)!cf (F) ∧ σn−1 ∈ Ω2n+2(C(P ))

defines a symplectic structure on A, and the equivariant extension provides amoment map for it, which, in particular, coincides with that obtained in [16,Prop 6.5.8] and [21, sec. 3].

6 Additional remarks

1. For the Berline-Vergne definition of equivariant characteristic classes itis necessary to have a G-invariant connection. Note however that, inde-pendently of the existence of G-invariant connections, the G-equivariantcharacteristic forms exist always on C(P ) (since the canonical connectionis G-invariant), and the existence of G-invariant connections is needed onlyin order to obtain G-equivariant classes on M . We hope that our construc-tion could be useful in the study of equivariant characteristic classes fornon-compact Lie groups, where the existence of invariant connections isnot guaranteed in general, and the analysis is much more involved; e.g.,see [18].

Moreover, in Section 5 we have used GauP -equivariant characteristic forms.From the classical point of view of equivariant characteristic classes thisprocedure is meaningless as this group acts trivially on M and also thereare no GauP -invariant connections.

2. The map can be extended to forms in the jet bundle, in the followingway: If α ∈ Ωn+k(JrE), we have the form [α] ∈ Ωk (Γ(JrE)). Pullingthis form back via the r-jet prolongation map

jr : Γ(E) → Γ(Jr(E))

s 7→ jrs

we obtain a form (jr)∗([α]) ∈ Ωk(Γ(E)). For a more detailed study ofthis map, its extension to equivariant cohomology, and its relation withthe variational bicomplex see [1].

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The usefulness of this map is that it provides a general procedure to obtainresults about (equivariant) differential forms and cohomology classes onthe infinite dimensional manifold Γ(E) by working on a finite dimensionaljet bundle.

3. In this paper we have obtained results about connections on principalbundles, but they can be applied to other geometrical problems; for ex-ample, to the space of Riemannian metrics (see [2]). Precisely, let M bean n-manifold MM its bundle of Riemannian metrics and FM its framebundle. The map that assigns to any metric its Levi-Civita connection,defines a DiffM -invariant map

LC: J1MM → C(FM).

Pulling the characteristic forms back via LC we obtain differential formson J1MM (the Pontryagin forms), and also DiffM -equivariant extensionsof these forms. By applying the map , we obtain forms and Diff+M -equivariant forms on the space MetM of Riemannian metris on M , and socohomology classes on the quotient that can be proved to coincide withthose defined in [5] and in [7] for n = 2.

Finally, in the case of a surface (n = 2) we can follow the lines of Example19: The first Pontryagin form defines a canonical symplectic form onMetM , and its equivariant extension a canonical moment map m. It canbe proved that m−1(0) = Metcons is the space of metrics of constantcurvature, and hence, by symplectic reduction, we obtain the moduli spaceof metrics of constant curvature. This space is basically the Teichmullerspace of the surface, and we obtain a symplectic form on it, in a canonicalway. But the Teichmuller space is well known to be a Kahler (hencesymplectic) manifold with the Weil-Petersson metric (e.g. see [24]). Itcan be proved that both structures are the same (up to an scalar factor)(see [2]).

References

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[2] Equivariant characteristic forms in the bundle of metrics, in preparation.

[3] M. F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces,Phil. Trans. R. Soc. Lond. A 308 (1982) 523–615.

[4] —, The moment map and equivariant cohomology, Topology 23 (1984),1–28.

[5] M.F. Atiyah, I. Singer, Dirac operators coupled to vector potentials, Proc.Natl. Acad. Sci. USA 81 (1984), 2597–2600.

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[17] P.L. Garcia Perez, Gauge algebras, curvature and symplectic structure, J.Differential Geom. 12 (1977), 209–227.

[18] E. Getzler, The equivariant Chern Character for non-compact Lie groups,Adv. Math. 109 (1994), no.1, 88–107

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Orbit Manifold in Yang-Mills Theory, Commun. Math. Phys. 79 (1981)457–472

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