Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 429–437

LAGRANGIAN DENSITIES INVARIANT UNDER

THE FULL LIE ALGEBRA OF THE INFINITESIMAL

AUTOMORPHISMS OF A PRINCIPAL BUNDLE

P. L. García Pérez and J. Muñoz Masqué

Abstract. First order Lagrangian densities on the bundle of connections of a principal bundle invariant under the full Lie algebra of the infinitesimal automorphisms of the principal bundle are classified.

Keywords. Bundle of connections, infinitesimal automorphism of a principal bundle, invariant Lagrangian densities.

MS classification. 58E30, 49S05, 53C05, 57R20.

0. Introduction

Let π : P ⟶ M be a principal bundle and let K = K(P) ⟶ M be the bundle of connections of P. In gauge theories a fundamental result (now known as Utiyama’s theorem, see [U] and [G]) is the determination of the Lagrangian densities on J^{1}(K) which are invariant under the gauge algebra of P.

The aim of this work is to determine all the Lagrangian densities on J^{1}(K) not only gauge invariant but also invariant under the full Lie algebra of the infinitesimal automorphisms of P. Let us briefly describe our goal from a more technical point of view.

Let us denote by X ⟼ X_{K} the natural representation of G-invariant vector fields of P on K, and let Y ⟼ \overline{Y} stand for the 1-jet prolongation of vector fields (see below for more details concerning the notations). Let us fix a volume form v on the base manifold M. A Lagrangian density Lv, where L is a differentiable function on J^{1}(K), is said to be gauge invariant iff L_{\overline{X}_{K}} (Lv) = 0 for every G-invariant π-vertical vector field X on P. As X is vertical, L_{\overline{X}_{K}} (v) = 0 and the above condition thus means \overline{X}_{K}(L) = 0. We are presently interested in determining all the Lagrangian densities Lv such that L_{\overline{X}_{K}} (Lv) = 0 for every G-invariant vector field X on P, not necessarily π-vertical. As is well-known ([K], §2.9, Prop. 6), a G-invariant vector field X on P is projectable onto M, and denoting by X′ its projection, the above equation is equivalent to saying \overline{X}_{K}(L) + (div X′)L = 0, where div X′ stands for the divergence of X′ with respect to v.

As a consequence of our results, it can be proved that the condition “\overline{X}_{K}(L) = 0 for every G-invariant vector field X on P ” implies that L must be a constant. Nevertheless,

Typeset by AMS-TEX

Authors supported by D.G.I.C.Y.T (Spain) grants #P B88 − 0379 and #P B89 − 0004, respectively.

This paper is in final form and no version of it will be submitted for publication elsewhere.

we shall prove that non-trivial invariant Lagrangian densities do exist whenever n = dim M is even, thus showing the importance of considering densities in dealing with the invariance under the full Lie algebra of infinitesimal automorphisms. Actually, each of such invariant Lagrangian densities is associated with a Weil polynomial p of degree \frac{n}{2}; let us call it L_{p}v. The action functional defined by L_{p}v is not interesting from the variational point of view, in the sense that every section s_{Γ} is critical, s_{Γ} : M ⟶ K being the section induced by a connection Γ on P, but because of this one can obtain theorems of Gauss-Bonnet type ([MS], Appendix C, p.311) for such functionals; for instance, if M is compact, a constant c_{p} ∈ IR exists such that ∫_{M}(j^{1}s_{Γ})^{∗}(L_{p}v) = c_{p} for every connection Γ on P.

In the abelian case (i.e., for U(1)-bundles), up to scalar factors, a unique Weil polynomial of degree \frac{n}{2} exists, p(t) = t^{\frac{n}{2}} , and we have L_{p}v = P f(R_{hi}) dx_{1} ∧ ⋯ ∧ dx_{n}, where P f(R_{hi}) stands for the Pfaffian of the curvature mapping coordinates, R_{hi}. Hence, in the general case the functions L_{p} can be understood as “generalized Pfaffians in the non-abelian setting”.

Let us denote by aut_{v}(P) the Lie algebra of G-invariant vector fields on P having constant divergence with respect to v (i.e., div X′ ∈ IR) and let aut_{v}^{0}(P) be the ideal of divergence-free vector fields in aut_{v}(P). Also note that [aut_{v}(P) , aut_{v}(P)] ⊂ aut_{v}^{0}(P). Of course, the gauge algebra of P is contained in each aut_{v}^{0}(P), and from our previous discussion it follows that if a Lagrangian density Lv is gauge invariant, then it is also invariant under the larger algebra aut_{v}^{0}(P). Moreover, as a consequence of our results one can conclude that a Lagrangian density Lv is invariant under the full Lie algebra of infinitesimal automorphisms if and only if it is invariant under aut_{v}(P). (We should more properly speak of invariance under the sheaf of germs S(U) = aut_{v}(P_{U}), because unlike the gauge algebra and the full algebra of infinitesimal automorphisms —which are the global sections of suitable vector bundles— the sheaf S may have no global section with non-vanishing divergence.) This basic fact measures the difference between gauge invariance and invariance under the full Lie algebra of infinitesimal automorphisms.

1. Preliminaries and notations

Most of notations have been taken from [M], where we send the reader for further information.

1.1. ad P and aut P. Let G be the structure group of π : P ⟶ M and let be 𝔤 the Lie algebra of G. Let us denote by π_{G} : Ad P ⟶ M the bundle associated to P by the conjugation action of G on itself, and by π_{𝔤} : ad P ⟶ M the bundle associated to P by the adjoint representation of G on 𝔤. We set |u, σ| = orbit of (u, σ) ∈ P × G in Ad P = (P × G)/G, and similarly |u, A| = orbit of (u, A) ∈ P × 𝔤 in ad P = (P × 𝔤)/G. The rule |u, σ| · |u, τ| = |u, σ · τ| induces a Lie group structure on the fibres of Ad P, and similarly the rule [|u, A|, |u, B|] = |u, [A, B]| induces a Lie algebra structure on the fibres of ad P. Note that for every x ∈ M, the fibre π_{𝔤}^{−1}(x) = (ad P)_{x} is the Lie algebra of the Lie group π_{G}^{−1}(x) = (Ad P)_{x}.

Moreover, Γ(M, ad P) endowed with the corresponding Lie algebra structure, can be identified with the gauge algebra of P. Actually, we can associate to each section ξ : M ⟶ ad P a G-invariant vertical vector field X_{ξ} on P by setting X_{ξ}(u) = A_{u}^{∗} , where A^{∗} stands for the fundamental vector field associated to A ∈ 𝔤 and ξ(π(u)) = |u, A|.

An automorphism of P is a diffeomorphism ϕ : P ⟶ P such that ϕ(uσ) = ϕ(u)σ, for every u ∈ P, σ ∈ G. Each ϕ induces a diffeomorphism ϕ′ : M ⟶ M such that π ∘ ϕ = ϕ′ ∘ π. Assume a vector field X on P generates the 1-parameter group ϕ_{t}. Then, X is G-invariant if and only if every ϕ_{t} is an automorphism of P. Accordingly, we denote by aut P the Lie algebra of G-invariant vector fields of P, and we think of aut P as being the Lie algebra of the infinite Lie group Aut P (cf. [GS]). There is a unique vector bundle T_{G}(P) ⟶ M such that aut P_{U} = Γ(U, T_{G}(P)) for every open subset U ⊂ M, and the following exact sequence of vector bundles on M holds ([A]): 0 ⟶ ad P ⟶ T_{G}(P) ⟶ T (M) ⟶ 0 (the so-called, Atiyah sequence).

1.2. Local expression of G-invariant vector fields. Assume n = dim M, m = dim G and let (U; x_{1}, ..., x_{n}), (A_{1}, ..., A_{m}) be an open coordinate domain of M trivializing P (i.e., P_{U} = π^{−1}(U) ≈ U × G), and a basis of 𝔤, respectively. We define a 1-parameter group of gauge transformations by ϕ_{t}^{j}(x, σ) = (x, exp(tA_{j})σ), x ∈ U, σ ∈ G, 1 ≤ j ≤ m. The corresponding vector field will be denoted by Ã_{j}. We have, [Ã_{j}, Ã_{k}] = − ∑_{i=1}^{m} c_{jk}^{i}Ã_{i}, where c_{jk}^{i} are the structure constants of 𝔤 in the basis A_{1}, ..., A_{m}. More generally, for every differentiable mapping g : U ⟶ 𝔤 and every 1parameter group ϕ′_{t} of U we can define a 1-parameter family of automorphisms of P_{U} by setting ϕ_{t}(x, σ) = (ϕ′_{t}(x), exp(t g(x))σ), and the vector field X that ϕ_{t} induces is

X = ∑_{i=1}^{n} f_{i}(∂/∂x_{i}) + ∑_{j=1}^{m} g_{j}Ã_{j} (f_{i}(x) = (∂ϕ′_{i}/∂t)(0, x) , g(x) = ∑_{j=1}^{m} g_{j}(x)A_{j}, x ∈ U) .

This is the local expression for an arbitrary G-invariant vector field. Also note that if ξ : U ⟶ ad P is the section defined by g : U ⟶ 𝔤 (i.e., ξ(x) = |(x, 1), g(x)|), the corresponding gauge field X_{ξ} is given by X_{ξ} = ∑_{j=1}^{m} g_{j}Ã_{j}. In particular, if ξ_{j} : U ⟶ ad P is the section defined by the constant map A_{j} : U ⟶ 𝔤, then X_{ξ_{j}} = Ã_{j}.

2. The representation of aut P on J^{1}(K)

2.1. Representing aut P in 𝔛(K). The group Aut P acts on the connections of P by pulling back connection forms: ϕ · ω = (ϕ^{−1})^{∗}ω ([KN], Prop.6.2.(b)). It is checked that (ϕ^{−1})^{∗}ω_{|π^{−1}(ϕ′(x))} only depends on ω_{|π^{−1}(x)}. Hence, Aut P also acts on K = K(P). Assume ϕ_{t} is the 1-parameter group of a G-invariant vector field X. We denote by ϕ_{t}^{K} the 1-parameter group induced by ϕ_{t} acting on K, and by X_{K} the corresponding vector field. In this way, we obtain a representation of aut P on K, X ⟼ X_{K}.

2.2. Local expression of X ⟼ X_{K}. We recall that the sections of K can be identified with the splittings of the Atiyah sequence as follows: s_{Γ} : T (M) ⟶ T_{G}(P) , s_{Γ}(X) = X^{∗}, the horizontal lift of X with respect to Γ. With the same notations as above, we can introduce coordinates on K, (K_{U} ; x_{i}, y_{ij}), 1 ≤ i ≤ n, 1 ≤ j ≤ m , by setting s_{Γ}(∂/∂x_{i}) = ∂/∂x_{i} + ∑_{j=1}^{m}(y_{ij} ∘ s_{Γ})Ã_{j}. Locally, ϕ_{t} = (ϕ′_{t}, L_{exp(tg)}) (here, L stands for left translation), and according to our definitions,

ϕ_{t}^{K}(s_{Γ}(x))(\frac{∂}{∂x_{i}}) = \frac{∂}{∂x_{i}} + ∑_{j=1}^{m} y_{ij} (ϕ_{t}^{K}(s_{Γ}(x))) Ã_{j},

0 = (ϕ_{−t}^{∗}ω_{Γ}) (ϕ_{t}^{K}(s_{Γ}(x)(∂/∂x_{i})_{ϕ′_{t}(x)})

= ω_{Γ} ((ϕ_{−t})_{∗}(∂/∂x_{i})_{(ϕ′_{t}(x),σ)} + ∑_{j=1}^{m} y_{ij} (ϕ_{t}^{K}(s_{Γ}(x))) (ϕ_{−t})_{∗}(Ã_{j})_{(ϕ′_{t}(x),σ)}) .

Taking derivatives with respect to t and letting t = 0, the above equation yields:

0 = ([X, ∂/∂x_{i}]_{(x,σ)} + (X_{K}y_{ij})(s_{Γ}(x))(Ã_{j})_{(x,σ)} + y_{ij}(s_{Γ}(x))[X, Ã_{j}]_{(x,σ)}) .

Taking into account that ω_{Γ} (∂/∂x_{i} + ∑_{j=1}^{m}(y_{ij} ∘ s_{Γ})Ã_{j}) = 0, and developing the Lie brackets in the previous equation we obtain,

0 = ω_{Γ} (∑_{j=1}^{m} (∑_{h=1}^{n} \frac{∂f_{h}}{∂x_{i}} y_{hj} − \frac{∂g_{j}}{∂x_{i}} + X_{K}(y_{ij}) − ∑_{k,l=1}^{m} c_{kl}^{j}g_{k}y_{il}) Ã_{j}) ,

thus proving the vector field in brackets is horizontal. Since it is also vertical, it must vanish. Hence,

X_{K} = ∑_{i=1}^{n} f_{i} \frac{∂}{∂x_{i}} + ∑_{i=1}^{n} ∑_{j=1}^{m} (\frac{∂g_{j}}{∂x_{i}} + ∑_{k,l=1}^{m} c_{kl}^{j}g_{k}y_{il} − ∑_{h=1}^{n} \frac{∂f_{h}}{∂x_{i}} y_{hj}) \frac{∂}{∂y_{ij}} .

2.3. Jet prolongation. Let us denote by (x_{i}, y_{ij}, y_{ij,h}), 1 ≤ h, i ≤ n, 1 ≤ j ≤ m, the natural coordinate system induced by (x_{i}, y_{ij}) on J^{1}(K); i.e., y_{ij,h}(j_{x}^{1}s_{Γ}) = (∂(y_{ij} ∘ s_{Γ})/∂x_{h})(x). The general formulas for vector field jet prolongation lead us to the following expression for the prolongation of X_{K} to J^{1}(K):

\overline{X}_{K} = ∑_{i=1}^{n} f_{i} \frac{∂}{∂x_{i}} + ∑_{i=1}^{n} ∑_{j=1}^{m} (\frac{∂g_{j}}{∂x_{i}} + ∑_{k,l=1}^{m} c_{kl}^{j}g_{k}y_{il} − ∑_{h=1}^{n} \frac{∂f_{h}}{∂x_{i}} y_{hj}) \frac{∂}{∂y_{ij}}

+ ∑_{i,u=1}^{n} ∑_{j=1}^{m} (\frac{∂^{2}g_{j}}{∂x_{i}∂x_{u}} − ∑_{h=1}^{n} (\frac{∂f_{h}}{∂x_{i}} y_{hj,u} + \frac{∂f_{h}}{∂x_{u}} y_{ij,h} + \frac{∂^{2}f_{h}}{∂x_{i}∂x_{u}} y_{hj})

+ ∑_{k,l=1}^{m} c_{kl}^{j}(g_{k}y_{il,u} + \frac{∂g_{k}}{∂x_{u}} y_{il})) \frac{∂}{∂y_{ij,u}} .

Remark. If in the above formulas we consider the particular case f_{i} = 0, we obtain the usual representation of the gauge algebra on the bundle of connections and its corresponding jet prolongation (cf. [G]).

2.4. The curvature mapping. The curvature form Ω_{Γ} of a connection Γ on P is a tensorial 2-form of type ad G and hence it can be viewed as a 2-form on M with values in ad P ([KN], II, Example 5.2.); i.e., Ω_{Γ} ∈ Γ(M, ⋀^{2} T^{∗}(M) ⊗ ad P). Accordingly, we can define a map of fibred manifolds over M, called the curvature mapping,

Ω : J^{1}(K) ⟶ ⋀^{2} T^{∗}(M) ⊗ ad P, Ω (j_{x}^{1}(s_{Γ})) = Ω_{Γ}(x).

Let (U; x_{1}, ..., x_{n}), (A_{1}, ..., A_{m}), ξ_{j} : U ⟶ ad P be as in §1.2. We can define a coordinate system (x_{i}, R_{hij}), 1 ≤ h < i ≤ n, 1 ≤ j ≤ m, on ⋀^{2} T^{∗}(U) ⊗ (ad P)_{U} as follows: η_{2} = ∑_{1≤h<i≤n} ∑_{j=1}^{m} R_{hij}(η_{2})dx_{h} ∧ dx_{i} ⊗ ξ_{j}, η_{2} ∈ ⋀^{2} T^{∗}(U) ⊗ (ad P)_{U}. For h ≥ i we shall also set: R_{hij} = −R_{ihj}. By considering in J^{1}(K) the coordinate system introduced in §2.3, it is checked that the equations for Ω are:

R_{ilj} ∘ Ω = y_{lj,i} − y_{ij,l} + ∑_{1≤h<k≤m} c_{hk}^{j}(y_{lh}y_{ik} − y_{ih}y_{lk}).

Utiyama’s theorem can be stated geometrically by saying (cf. [G]) that: a Lagrangian function L : J^{1}(K) ⟶ IR is gauge invariant if and only if L factors through Ω by means of a function \overline{L} : ⋀^{2} T^{∗}(M)⊗ad P ⟶ IR which is invariant under the natural representation of the gauge algebra on ⋀^{2} T^{∗}(M) ⊗ ad P. Let us recall how this representation is defined ([M], §5). For every σ ∈ G, u ∈ P, A ∈ 𝔤, ω_{2} ∈ ⋀^{2} T_{x}^{∗}(M), x = π(u) we have: |u, σ| · (ω_{2} ⊗ |u, A|) = ω_{2} ⊗ |u, Ad_{σ}(A)|. Let us denote by V (N) the vertical bundle of a fibred manifold N ⟶ M. As (ad P)_{x} is the Lie algebra of (Ad P)_{x} (cf. §1.1), we can define a homomorphism of vector bundles over ⋀^{2} T^{∗}(M) ⊗ ad P,

ρ : (⋀^{2} T^{∗}(M) ⊗ ad P ) ×_{M} ad P ⟶ V (⋀^{2} T^{∗}(M) ⊗ ad P )

by setting: ρ(η_{2}, A_{x}) = (^{∗}A_{x})_{η_{2}}, where η_{2} ∈ ⋀^{2} T_{x}^{∗}(M) ⊗ (ad P)_{x}, A_{x} ∈ (ad P)_{x} and ∗A_{x} is the vector field tangent to the fibre of ⋀^{2} T^{∗}(M) ⊗ ad P over x generating the 1-parameter group L_{exp(t A_{x})}.

The image of ρ is an involutive differential system M with singularities; actually, ρ(ξ_{k}) = ∑ c_{kl}^{j}R_{hil}(∂/∂R_{hij}) (cf.[G], formula (5.1)), and the determination of all gauge invariant Lagrangians is reduced to calculating the ring R of first integrals of M. In spite of the singularities, it is proved in [M] that R is “differentiably finitely-generated”.

3. aut P -invariant Lagrangian density associated to a Weil polynomial

Definition. Assume n = dim M is even, and let p be an invariant polynomial of degree \frac{n}{2} with respect to the coadjoint representation of G on 𝔤^{∗}; i.e., p ∈ S^{n/2}(𝔤)^{G}. For each k ∈ IN, the polynomial p gives rise to a function \overline{p}_{k} : ⨁^{k} ad P ⟶ IR by setting \overline{p}_{k}(|u, A_{1}|, ..., |u, A_{k}|) = p(A_{1}, ..., A_{k}). Let \overline{λ}_{p} : ⋀^{2} T^{∗}(M) ⊗ ad P ⟶ ⋀^{n} T^{∗}(M) be the mapping defined by:

\overline{λ}_{p}(η_{2})(X_{1}, ..., X_{n}) = ∑_{σ∈S_{n}} ϵ_{σ}\overline{p}_{n/2} (η_{2}(X_{σ(1)}, X_{σ(2)}), ..., η_{2}(X_{σ(n−1)}, X_{σ(n)})) ,

where η_{2} ∈ ⋀^{2} T_{x}^{∗}(M) ⊗ (ad P)_{x}, X_{i} ∈ T_{x}(M), 1 ≤ i ≤ n. Once a volume form v on M has been fixed, \overline{λ}_{p} = \overline{L}_{p}v for a certain differentiable function \overline{L}_{p} : ⋀^{2} T^{∗}(M)⊗ad P ⟶ IR. Let Ω : J^{1}(K) ⟶ ⋀^{2} T^{∗}(M) ⊗ ad P be the curvature mapping (cf. §2.4). The composition L_{p}v = (\overline{L}_{p} ∘ Ω)v is said to be the Lagrangian density associated with the Weil polynomial p.

Theorem 3.1. The Lagrangian density associated with a Weil polynomial of degree \frac{n}{2} is aut P -invariant.

Proof. As L_{p} factors through Ω by its very definition, from Utiyama’s theorem it follows that L_{p}v is gauge invariant. Let C_{U} be the set of G-invariant vector fields X whose projection X′ onto M has a compact support contained in the open subset U ⊂ M. Taking into account that aut P = Γ(M, T_{G}(P)) (see §1.1), the following properties can be easily verified:

(1) C_{M} is dense in aut P.

(2) Every X ∈ C_{M} is a finite sum of vector fields X_{i} ∈ C_{U_{i}} so that U_{i} is an open coordinate domain trivializing P.

(3) If P is trivial on U and X^{h} stands for the horizontal lift of a vector field X ∈ 𝔛(U) with respect to the flat connection on P_{U}, then for every X ∈ aut P the difference X − X′^{h} belongs to the gauge algebra; i.e., (X − X′^{h}) ∈ Γ(U, ad P).

Since L_{p}v is gauge invariant, by applying (1), (2) and (3) successively it is clear that it will be sufficient to prove that

(A) For every X ∈ 𝔛(U) with compact support, L\overline{X}{h \atop K} (L_{p}v) = 0.

Note that the 1-parameter group generated by the vector field X^{h} is Φ_{t}(x, σ) = (ϕ_{t}(x), σ), x ∈ U, σ ∈ G, where ϕ_{t} is the 1-parameter group generated by X.

As ad P is a bundle associated to P, ⋀^{2} T^{∗}(U) ⊗ ad P_{U} = ⋀^{2} T^{∗}(U) ⊗ 𝔤. Hence, the sections of ⋀^{2} T^{∗}(U) ⊗ ad P_{U} can be viewed as 𝔤-valued 2-forms on U. For a differentiable mapping ϕ : M′ ⟶ U and η_{2} ∈ ⋀^{2} T^{∗}(U) ⊗ 𝔤, x = ϕ(x′), we can thus define ϕ^{∗}η_{2} ∈ ⋀^{2} T_{x′}^{∗}(M′) ⊗ 𝔤 by: (ϕ^{∗}η_{2})(X′_{1}, X′_{2}) = η_{2}(ϕ_{∗}X′_{1}, ϕ_{∗}X′_{2}). Moreover, we have:

(B) The group of diffeomorphisms of U acts on the set of fibred mappings over U, \overline{λ} : ⋀^{2} T^{∗}(U) ⊗ ad P_{U} ⟶ ⋀^{n} T^{∗}(U), \overline{λ} = \overline{L}v, as follows: (ϕ · \overline{λ})(η_{2}) = \overline{L}(ϕ^{∗}η_{2})(ϕ^{−1})^{∗} (v_{ϕ^{−1}(x)}). Then, \overline{λ}_{p} is invariant under this action.

Proof of (B). Assume p = ∑ c_{j_{1},...,j_{n/2}}A_{j_{1}}^{∗} ⋯ A_{j_{n/2}}^{∗}, where A_{1}^{∗}, ..., A_{m}^{∗} is the dual basis of

A_{1}, ..., A_{m} and the dots stand for the symmetric product. For every η_{2} = ∑_{j=1}^{m} ω_{2}^{j}⊗ξ_{j}(x) (recall: ξ_{j}(x) = |(x, 1), A_{j}|, §1.2), ω_{2}^{j} ∈ ⋀^{2} T_{x}^{∗}(M), we have, \overline{λ}_{p}(η_{2}) = ∑ c_{j_{1},...,j_{n/2}}ω_{2}^{j_{1}} ∧

... ∧ ω_{2}^{j_{n/2}} = \overline{L}_{p}(η_{2})v. Hence, \overline{λ}_{p}(ϕ^{∗} η_{2}) = ∑ c_{j_{1},...,j_{n/2}} (ϕ^{∗} ω_{2}^{j_{1}}) ∧ ... ∧ (ϕ^{∗} ω_{2}^{j_{n/2}}) =

\overline{L}_{p}(ϕ^{∗} η_{2})v_{ϕ^{−1}(x)}, and thus, (ϕ^{−1})^{∗} (\overline{λ}_{p}(ϕ^{∗}η_{2})) = \overline{λ}_{p}(η_{2}) = \overline{L}_{p}(ϕ^{∗}η_{2}) (ϕ^{−1})^{∗}v_{ϕ^{−1}(x)} = (ϕ · \overline{λ}_{p})(η_{2}).

According to (B), in order to prove (A) we only need to state the following

Lemma 3.2. Assume U is an open coordinate domain of M on which P is trivial. A mapping of fibred manifolds over U, \overline{λ} : ⋀^{2} T^{∗}(U) ⊗ ad P_{U} ⟶ ⋀^{n} T^{∗}(U), \overline{λ} = \overline{L}v, is invariant under every 1-parameter group of diffeomorphisms of U (i.e., ϕ_{t} · \overline{λ} = \overline{λ}) if and only if for every X ∈ 𝔛(U) we have, L\overline{X}{h \atop K} ((\overline{L} ∘ Ω)v) = 0.

Proof of Lemma 3.2. Let us consider a coordinate system (x_{1}, ..., x_{n}) on U such that v = dx_{1} ∧ .... ∧ dx_{n}.We have X^{h} = ∑_{i=1}^{n} f_{i}(∂/∂x_{i}), and in this case the formula for \overline{X}_{K} in §2.3 reads as:

\overline{X}_{K}^{h} = ∑_{i=1}^{n} f_{i} \frac{∂}{∂x_{i}} − ∑_{h,i=1}^{n} ∑_{j=1}^{m} (\frac{∂f_{h}}{∂x_{i}} y_{hj}) \frac{∂}{∂y_{ij}}

− ∑_{h,i,u=1}^{n} ∑_{j=1}^{m} (\frac{∂f_{h}}{∂x_{i}} y_{hj,u} + \frac{∂f_{h}}{∂x_{u}} y_{ij,h} + \frac{∂^{2}f_{h}}{∂x_{i}∂x_{u}} y_{hj}) \frac{∂}{∂y_{ij,u}} .

Hence, L\overline{X}{h \atop K} ((\overline{L} ∘ Ω)v) = 0 for every X ∈ 𝔛(U) if and only if,

\frac{∂L}{∂x_{i}} = 0 ; ∑_{j=1}^{m} y_{hj} (\frac{∂L}{∂y_{ij,u}} + \frac{∂L}{∂y_{uj,i}} ) = 0, 1 ≤ h, i, u ≤ n,

∑_{u=1}^{n} ∑_{j=1}^{m} (y_{hj} \frac{∂L}{∂y_{ij}} + y_{hj,u} \frac{∂L}{∂y_{ij,u}} + y_{uj,h} \frac{∂L}{∂y_{uj,i}} ) = 0, 1 ≤ h ≠ i ≤ n,

L = ∑_{j=1}^{m} (y_{ij} \frac{∂L}{∂y_{ij}} + ∑_{u=1}^{n} (y_{ij,u} \frac{∂L}{∂y_{ij,u}} + y_{uj,i} \frac{∂L}{∂y_{uj,i}} )) , 1 ≤ i ≤ n,

where we have set: L = \overline{L} ∘ Ω.

Moreover, as a computation shows, from the equations of the curvature mapping (cf. §2.4) we can conclude that the above four equations for L are equivalent to the following three equations for \overline{L},

(I)  \frac{∂\overline{L}}{∂x_{i}} = 0

(II)  ∑_{a<i} ∑_{b=1}^{m} R_{ahb} \frac{∂\overline{L}}{∂R_{aib}} = ∑_{i<a} ∑_{b=1}^{m} R_{ahb} \frac{∂\overline{L}}{∂R_{iab}} , 1 ≤ h ≠ i ≤ n

(III)  \overline{L} = ∑_{a<i} ∑_{b=1}^{m} R_{aib} \frac{∂\overline{L}}{∂R_{aib}} + ∑_{i<a} ∑_{b=1}^{m} R_{iab} \frac{∂\overline{L}}{∂R_{iab}} , 1 ≤ i ≤ n

Furthermore, by taking derivatives in the invariance condition ϕ_{t} ·(\overline{L}v) = \overline{L}v and letting t = 0, we obtain: −X̃_{2}(\overline{L})v − \overline{L} (∑_{i=1}^{n}(∂f_{i}/∂x_{i})) v = 0, where X̃_{2} stands for the vector field induced by ϕ_{t} on ⋀^{2} T^{∗}(M) ⊗ 𝔤, whose local expression is

X̃_{2} = ∑_{i=1}^{n} f_{i} \frac{∂}{∂x_{i}} − ∑_{h,i=1}^{n} ∑_{j=1}^{m} (∑_{a=1}^{n} R_{aij} \frac{∂f_{a}}{∂x_{h}} + ∑_{b=1}^{n} R_{hbj} \frac{∂f_{b}}{∂x_{i}} ) \frac{∂}{∂R_{hij}} ,

thus finishing the proof of the lemma and completing that of the theorem.

4. Determining the aut P -invariant Lagrangian densities

Theorem 4.1. Let π : P ⟶ M be a principal fibre bundle with structure group G. Assume both G and M are connected, and M is oriented by a volume form v. We have:

(i) If n = dim M is odd, the the unique aut P -invariant Lagrangian density is the zero density. (ii) If n = dim M is even, then the only aut P -invariant Lagrangian densities are those associated with Weil polynomials of degree \frac{n}{2}. (iii) If M is compact, a constant c_{p} ∈ IR exists only depending on p ∈ S^{n/2}(𝔤^{∗})^{G} such that for every connection Γ on P,

∫_{M} (j^{1}s_{Γ})^{∗} (\overline{L}_{p} ∘ Ω) v = c_{p}.

Proof. Let Lv be an aut P -invariant Lagrangian density. Since Lv is gauge invariant, from Utiyama’s theorem we obtain L = \overline{L} ∘ Ω, where \overline{L} : ⋀^{2} T^{∗}(M) ⊗ ad P ⟶ IR is a differentiable function invariant under the natural representation of the gauge algebra. As G is connected, \overline{L} is also Ad P -invariant. Moreover, as Lv is aut P -invariant, \overline{L} satisfies equations (I), (II), (III) in the proof of Lemma 3.2. Equation (I) thus shows that \overline{L} only depends on the variables (R_{hij}), 1 ≤ h < i ≤ n, 1 ≤ j ≤ m. First, let us consider the following elementary result:

Lemma 4.2. Let N be a differentiable manifold. If f ∈ C^{∞}(IR^{n} × N) verifies f = ∑_{i=1}^{n} x_{i}(∂f/∂x_{i}), then ∂f/∂x_{i} ∈ C^{∞}(N), 1 ≤ i ≤ n.

Proof of Lemma 4.2. As in the usual case we have: f(tx, y) = t f(x, y) for every t ∈ IR, x ∈ IR^{n}, y ∈ N. Hence, f(tx, y) = t f(x, y) = ∑_{i=1}^{n} tx_{i}(∂f/∂x_{i})(tx, y). Thus, f(x, y) = ∑_{i=1}^{n} x_{i}(∂f/∂x_{i})(tx, y), and letting t = 0, we obtain f = ∑_{i=1}^{n} x_{i}f_{i} for some f_{i} ∈ C^{∞}(N). Since ∂f_{h}/∂x_{i} = 0, we have ∂f/∂x_{i} = f_{i}.

Taking this lemma into account, equation (III) proves that \overline{L} depends linearly on each of the n systems of variables (R_{aib}, R_{ihb}), 1 ≤ a ≤ i − 1, i + 1 ≤ h ≤ n, 1 ≤ b ≤ m, the index i being fixed. Consequently, \overline{L} is a finite sum of monomials

λ_{hij} R_{h_{1}i_{1}j_{1}} ⋯ R_{h_{k}i_{k}j_{k}}, h_{a} < i_{a}, 1 ≤ a ≤ k,

such that,

1) λ_{hij} ≠ 0; 2) the indices h_{1}, i_{1}, ..., h_{k}, i_{k} are pairwise different; 3) {1, ..., n} = {h_{1}, i_{1}, ..., h_{k}, i_{k}}. Hence, n = 2k, thus forcing n to be even.

Moreover, equation (II) implies that the coefficients λ_{hij} should be skew-symmetric in the indices h, i; i.e., λ_{hij} = ϵ_{h,i}c_{j}, where ϵ_{h,i} stands for the sign of the permutation (1, 2, ..., n − 1, n) ⟼ (h_{1}, i_{1}, ..., h_{k}, i_{k}). With the coefficients c_{j} we can construct a polynomial p as in the proof of Theorem 3.1. Since \overline{L} is Ad P -invariant, it is clear that p is ad G-invariant and by virtue of the above construction we have \overline{L} = \overline{L}_{p}.

This proves the first two parts of the statement. As for the third, by using the same notations as in [KN], XII, §1, it is easy to see that for every connection Γ on P we have: (j^{1}s_{Γ})^{∗} (\overline{L}_{p} ∘ Ω) v = \overline{p}(Ω_{Γ}), where \overline{p}(Ω_{Γ}) is the unique n-form on M such that, π^{∗} (\overline{p}(Ω_{Γ})) = p(Ω_{Γ}) (cf. [KN], XII, Th.1.1). Hence, ∫_{M}(j^{1}s_{Γ})^{∗} (\overline{L}_{p} ∘ Ω) v = ∫_{M} \overline{p}(Ω_{Γ}), and it is known ([KN], XII, Th.1.1.(2)) that \overline{p}(Ω_{Γ}) is a closed n-form whose cohomology class in H^{n}(M; IR) does not depend on the connection chosen.

References

[A] Atiyah, M.F., Complex analytic connections in fibre bundles, Trans. Amer.Math. Soc. 85 (1957), 181–207.

[G] García, Pedro L., Gauge algebras, curvature and symplectic geometry, J. Differential Geometry 12 (1977), 209–227.

[GS] Guillemin, V. and Sternberg, S., Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984.

[K] Koszul, J. L., Lectures on fibre bundles and differential geometry, Tata Insititute of Fundamental Research, Bombay, 1960.

[KN] Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry I, II, Wiley, New York, 1963, 1969.

[M] Muñoz Masqué, J., Explicit determination of Utiyama Lagrangians (Catto, S. and Rocha, A., eds.), Proc. XX Int. Conf. on Diff. Geom. Methods in Theoretical Phys., Volume 1, New York 1991., World Scientific, Singapore.

[MS] Milnor, J. W. and Stasheff, J. D., Characteristic Classes (Annals of Mathematics Studies 76), Princeton University Press and University of Tokyo Press, Princeton, N. J., 1974.

[U] Utiyama, R., Invariant Theoretical Interpretation of Interaction, Physical Review 101 (1956), 1597–1607.

P. L. García Pérez. Departamento de Matemáticas, Universidad de Salamanca, Plza. de la Merced 1-4, 37008–Salamanca. Spain

J. Muñoz Masqué. C.S.I.C, Instituto de Electrónica de Comunicaciones, C/Serrano 144, 28006–Madrid. Spain E-mail: VCTQU01@CC.CSIC.ES

Received 22 October 1992