DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 461–468

GAUGE INVARIANT VARIATIONALLY TRIVIAL U(1)-PROBLEMS

M. CASTRILLÓN-LÓPEZ

Abstract. A geometrical characterization of gauge invariant variationally trivial lagrangians on the bundle of connections of U(1)-principal bundles is given.

1. Introduction

The mathematical notion of connection can be seen from a physical point of view as the potential of a given field theory. Many of those theories come from variational principles over a principal fiber bundle π : P → M, and therefore the bundle of connections CP arises as the natural place in which the action functional is defined. More precisely, if one is dealing with first order variational problems over a ground manifold M with fixed volume form v, a Lagrangian is a mapping from the 1-jet prolongation of the bundle CP to the real numbers.

On the other hand, the notion of gauge invariance plays an important role in field theories (in particular the theory of electromagnetism, with the structure group U(1)). The invariance of a variational functional means that for any gauge transformation Φ : P → P, the Euler-Lagrange form E(L) verifies (cf. [6])

(Φ_{C}^{(2)})^{∗} E(L) = E((Φ_{C}^{(1)})^{∗}L) = E(L) ,

where Φ_{C}^{(2)} stands for the 2-jet prolongation of the induced mapping Φ_{C} : CP → CP (see §2.2 below). The most usual way for obtaining this invariance is taking a Lagrangian such that (Φ_{C})^{∗}L = L, or equivalently

L_{X_{C}^{(1)}}L = X_{C}^{(1)}(L) = 0 ,

for any gauge vector field X. An infinitesimal gauge transformation is a G-invariant vector field (that is, an infinitesimal automorphism on P) which is π-vertical. The set of all gauge vector fields is denoted by gauP. In fact, the flow of a vector field in gauP is an one-parameter family of automorphisms of P inducing the identity over M. Thus, a Lagrangian density L is said to be gau (resp. aut) invariant if the Lie derivative of Lv by the fundamental representation (see §2.2 below) of all gau (resp. aut) vector fields vanishes.

1991 Mathematics Subject Classification. Primary 58E30; Secondary 17B66, 22E65, 53C07.

Key words and phrases. Gauge invariance, multivector field, variational triviality.

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

One of the classical problems in the core of the variational calculus is that determining whether two Lagrangians L, L′ are variationally equivalent; that is, L, and L′ produce the same variational problem. This is equivalent to study the variational triviality of the difference L − L′. Although general conditions for this problem are known (see, for example [7]), the goal of this paper deals with the important subset of gauge invariant lagrangians over the bundle of connections of any U(1)-principal bundle. In this case we have a nice geometrical characterization, namely, one can establish an one-to-one correspondence between the set of gauge invariant varationally trivial Lagrangians and the set of ’divergence free’ multivector fields, that is, sections of the bundle ⋀^{•} T M such that the Lie derivative of the volume form vanishes.

Notice that, when the bundle P is trivial (i.e., P = M × U(1)), the bundle of connections is canonically isomorphic to the cotangent bundle, thus characterizing all gauge invariant Lagrangians on T^{∗}M which are variationally trivial.

2. Preliminaries

2.1. The bundle of connections. Let π : P → M a principal bundle with structure group G. The action of G on P induces a natural action on T P such that the quotient T P/G has the structure of vector bundle over M. The Atiyah sequence (see for example [1]) is the exact sequence of vector bundles over M

0 → adP → T P/G → T M → 0 ,

where adP stands for the adjoint bundle adP; that is, the bundle associated to P by the adjoint representation of G on its Lie algebra 𝔤. It is well known that there exists a one-to-one correspondence between connections on P and splittings of the Atiyah sequence. Accordingly, the bundle of connections p : CP → M is defined as the subbundle of Hom(T M, T P/G) determined by all ℝ-linear mappings λ : T_{x}M → (T P/G)_{x} such that π_{x} ∘ λ = Id_{T_{x}M} (c.f. [4]). Connections on P can thus be identified to global sections of p : CP → M. Is not difficult to prove that CP is an affine bundle modelled over the vector bundle Hom(T M, adP) ≅ T^{∗}M ⊗ adP. A different approach for CP is the following (see for example [2]). Let π_{1} : J^{1}P → M be the 1-jet bundle of local sections of π. The group G acts on J^{1}P by j_{x}^{1}s · g = j_{x}^{1}(R_{g} ∘ s), where s is a section of π defined on a neighbourhood of x ∈ M, g ∈ G and stands for the right translation. the quotient (J^{1}P)/G exists as a fibered manifold over M and can be identified to the bundle of connections CP.

2.2. The fundamental representation. An automorphism of a principal Gbundle is a diffeomorphism Φ : P → P such that Φ∘R_{g} = R_{g} ∘Φ ∀g ∈ G, where R_{g} stands for the right translation by g on P. Every automorphism induces a unique diffeomorphism ϕ : M → M such that π ∘ Φ = ϕ ∘ π. If ϕ is the identity map, then Φ is said to be a gauge transformation.

It is straightforward to check that a vector field on P is G-invariant if and only if its flow is a uniparametric group of aumorphisms (that is, it is an infinitesimal automorphism). The algebra of all G-invariant vector fields is denoted by autP. Infinitesimal gauge transformations are those G-invariant vector fields which are π_{∗}-vertical, and accordingly, they are denoted by gauP.

GAUGE INVARIANT VARIATIONALLY TRIVIAL U(1)-PROBLEMS 463

Each automorphism Φ acts on connections as follows. Let Γ be a connection on P and let ω_{Γ} be its connection form. Then, Γ′ = Φ(Γ) is the connection corresponding to the connection form ω_{Γ′} = (Φ^{−1})^{∗}ω_{Γ}. For each Φ, this action induces a unique diffeomorphism

Φ_{C} : CP → CP

such that Φ_{C} ∘ σ_{Γ} = σ_{Φ(Γ)}, where σ_{Γ} stands for the global section of CP induced by any connection Γ. In other words, we have the natural representation

AutP → Diff(CP)
Φ ↦ Φ_{C} .

The derived representation of the algebra autP of infinitesimal automorphisms is denoted by

autP → 𝔛(CP)
X ↦ X_{C} ,

and is called the ’fundamental representation’.

2.3. Coordinates. From now, consider that the structure group G is the abelian group U(1).

Let (V; q^{1}, ..., q^{n}) a coordinate open domain in M such that π is trivializable over it, that is, π^{−1}(V) ≅ V × U(1). A vector field X on π^{−1}(V) is U(1)-invariant if and only if it can be written as

X = f^{i} (q^{1}, …, q^{n}) \frac{∂}{∂q^{i}} + g (q^{1}, …, q^{n}) · ξ, f^{i}, g ∈ C^{∞}(V) , (1)

where ξ is the fundamental vector field on P defined by i ∈ 𝔲(1) ≅ iℝ. Note that each U(1)-invariant vector field X on P is π-projectable and its projection is given by

π_{∗}X = f^{i} (q^{1}, …, q^{n}) \frac{∂}{∂q^{i}} .

In particular, X is π-vertical (and hence it is a gauge vector field) if and only if f^{i} = 0.

If p : CP → M is the bundle of connections of P, a trivialization π^{−1}(V) ≅ V × U(1) induces a coordinate system in p^{−1}(V) ≅ C(V × U(1)) as follows. Every connection form on V ×U(1) can be uniquely written as ω_{Γ} = (dθ +π^{∗}ω)⊗i, where ω is an arbitrary 1-form on V and θ is the angle coordinate on U(1). Therefore the bundle of connections C(V ×U(1)) can be identified to T^{∗}V and a natural coordinate system (p^{−1}(V); q^{1}, …, q^{n}, p_{1}, …, p_{n}) is induced by taking ω = p_{i}(ω) (dq^{i})_{x}, for every covector ω ∈ T_{x}^{∗}V.

The fundamental representation is then given by ([5, III.B]):

X_{C} = f^{i} \frac{∂}{∂q^{i}} − (\frac{∂g}{∂q^{i}} + \frac{∂f^{h}}{∂q^{i}} p_{h}) \frac{∂}{∂p_{i}} , (2)

where the local expression of X ∈ autP is given by the formula (1). In particular, for gauge vector fields we have

X_{C} = −\frac{∂g}{∂q^{i}} \frac{∂}{∂p_{i}} , X ∈ gauP . (3)

2.4. Jet prolongation. Let p_{1} : J^{1} (CP) → M be the first jet prolongation of the bundle of connections. Each coordinate open domain (V; q^{1}, …, q^{n}) in M induces a system of coordinates (q^{i}, p_{i}; p_{i,j}), i, j = 1, …, n, in p_{1}^{−1}(V) ⊆ J^{1} (CP), which are defined by

p_{i,j} (j_{x}^{1}ω) = \frac{∂ (p_{i} ∘ ω)}{∂q^{j}}(x), i, j = 1, …, n, (4)

where (p^{−1}(V); q^{i}, p_{i}) is the coordinate system on the bundle of connections introduced in §2.3. Given the coordinate expression (formula (2)) of the fundamental representation, from general formulas for jet prolongation we obtain

(5)  X_{C}^{(1)} = f^{i} \frac{∂}{∂q^{i}} − (\frac{∂g}{∂q^{i}} + \frac{∂f^{h}}{∂q^{i}} p_{h}) \frac{∂}{∂p_{i}}

− (\frac{∂^{2}g}{∂q^{i}∂q^{j}} + \frac{∂f^{h}}{∂q^{i}} p_{h,j} + \frac{∂f^{h}}{∂q^{j}} p_{i,h} + \frac{∂^{2}f^{h}}{∂q^{i}∂q^{j}} p_{h}) \frac{∂}{∂p_{i,j}} ,

which is the jet prolongation of the algebra of infinitesimal automorphisms.

2.5. Lie derivative with respect to a multi-vector field. We recall (cf. [3, p.79]) that a decomposable multi-vector field χ = X_{1} ∧ … ∧ X_{k} ∈ ⋀^{k} 𝔛(M), induces a C^{∞}(M)-linear graded endomorphism i_{χ}: Ω^{•}(M) → Ω^{•}(M) of degree −k by setting

i_{χ}ω_{r} = i_{X_{1}} ∘ … ∘ i_{X_{k}}ω_{r} ∈ Ω^{r−k}(M), ω_{r} ∈ Ω^{r}(M) . (6)

For an arbitrary multi-vector field χ we can define i_{χ} by extending the above formula by linearity. Then the Lie derivative L_{χ}: Ω^{•}(M) → Ω^{•}(M) with respect to χ ∈ ⋀^{k} 𝔛(M) is given by the formula

L_{χ} = i_{χ} ∘ d − (−1)^{k}d ∘ i_{χ} = [i_{χ}, d] . (7)

We remark that L_{χ} is a graded operator of degree −k + 1.

3. Invariant Lagrangians

3.1. Gauge invariance. A Lagrangian function L: J^{1}(CP) → ℝ is said to be gauge invariant if

X̃^{(1)} (L) = 0, ∀X ∈ gauP.

As we are dealing with the abelian structure group U(1), the Utiyama theorem states that a Lagrangian is gauge invariant if and only if it can be written as

L = \overline{L} ∘ Ω ,

where Ω: J^{1} (CP) → ⋀^{2} T^{∗}M ⊗ adP is the mapping which assigns to each 1-jet of connections its curvature as a two form on M; i.e., Ω (j_{x}^{1}ω_{Γ}) = (Ω_{Γ})_{x} and \overline{L}: ⋀^{2} T^{∗}M ⊗ adP → ℝ is an arbitrary differentiable function (cf. [4]). Remark that, as U(1) is abelian, the bundle adP is canonicaly isomorphic to M × 𝔲(1); if we identify 𝔲(1) ≅ iℝ, then ⋀^{2} T^{∗}M ⊗ adP ≅ ⋀^{2} T^{∗}M ⊗ ℝ = ⋀^{2} T^{∗}M and we can simplify the statement of the theorem. Let us introduce coordinates on the bundle p_{2} : ⋀^{2} T^{∗}M → M. For every 2-covector w_{2} ∈ ⋀^{2} T_{x}^{∗}M we set

w_{2} = p_{ij} (w_{2}) (dq^{i} ∧ dq^{j})_{x} , i < j .

Then, (q^{i}, p_{ij}) is a coordinate system on p_{2}^{−1}(V). We set p_{ij} = −p_{ji} for i ≥ j. With respect to these coordinates and those defined in (4), the equations of Ω are

p_{ij} ∘ Ω = p_{i,j} − p_{j,i} . (8)

Hence gauge invariant Lagrangians can be locally written as

L = \overline{L} (q^{h}, p_{i,j} − p_{j,i}) .

3.2. autP -invariance. Let us asume that M is oriented by a volume form v_{n}. A Lagrangian density Lv_{n}, L ∈ C^{∞}(J^{1} (CP)), is said to be invariant under the Lie algebra of all infinitesimal automorphisms of P (in short, autP -invariant) if

L_{X_{C}^{(1)}} (Lv_{n}) = 0 , ∀X ∈ autP .

As L_{X_{C}^{(1)}} (Lv_{n}) = (X_{C}^{(1)} (L)) v_{n}+L div (π_{∗}X) v_{n} = 0, where the divergence is taken with respect to v_{n}, the invariance condition is equivalent to the following:

X_{C}^{(1)} (L) + L div (π_{∗}X) = 0, ∀X ∈ autP . (9)

4. Variationally trivial Lagrangians

A Lagrangian density Lv_{n}, L ∈ C^{∞}(J^{1}(CP)), is said to be variationally trivial if every diferential is a solution to the Euler-Lagrange equations of the given Lagrangian; that is, for every connection ω defined on a neighbourhood of π^{−1}(x), x ∈ M we have

(10) \frac{∂L}{∂p_{i}} (j_{x}^{1}ω) − \frac{∂}{∂q^{k}} (\frac{∂L}{∂p_{i,k}} ∘ j^{1}ω) (x) = 0 , identically.

Lemma 1. A Lagrangian density Lv_{n} on CP is variationally trivial if and only if locally the following partial differential equations are satisfied:

\frac{∂^{2}L}{∂p_{i,h}∂p_{j,k}} + \frac{∂^{2}L}{∂p_{i,k}∂p_{j,h}} = 0, ∀h, i, j, k = 1, …, n. (11)

\frac{∂L}{∂p_{i}} = \frac{∂^{2}L}{∂q^{k}∂p_{i,k}} + \frac{∂^{2}L}{∂p_{j}∂p_{i,k}} p_{j,k}, ∀i = 1, …, n. (12)

Proof. Expanding the equation (10) we obtain

\frac{∂L}{∂p_{i}} (j_{x}^{1}ω) = \frac{∂^{2}L}{∂q^{k}∂p_{i,k}} (j_{x}^{1}ω) + \frac{∂^{2}L}{∂p_{j}∂p_{i,k}} (j_{x}^{1}ω) \frac{∂f_{j}}{∂q^{k}} (x) +

\frac{∂^{2}L}{∂p_{j,h}∂p_{i,k}} (j_{x}^{1}ω) \frac{∂^{2}f_{j}}{∂q^{h}∂q^{k}} (x) ,

where ω = f_{j}dq^{j}. Once the 1-jet of ω at x has been fixed, the second partial derivatives (∂^{2}f_{j}/∂q^{h}∂q^{k}) (x) can be arbitrarily taken. Hence equation (11) follows and also equation (12).

Notation: Given a multi-vector field χ_{k} ∈ Γ(M, ⋀^{2k} T M) of even degree 2k, we denote by χ̄_{k}: ⋀^{2} T^{∗}M → ℝ the mapping given by

χ̄_{k}(w_{2}) = i_{χ_{k}}(w_{2} ∧ …^{(k} ∧ w_{2}), w_{2} ∈ ⋀^{2} T^{∗}M ,

where i_{χ} is the total insertion operator (cf. §2.5).

The geometrical characterization of gauge invariant, variationally trivial Lagrangians is stated in the following Theorem 1. Let π : P → M be a U(1)-principal fiber bundle over an orientable n-dimensional manifold M with fixed volume form v_{n}. A first order gauge invariant Lagrangian density Lv_{n} on CP is variationally trivial if and only L can be written as follows:

L = χ̄ ∘ Ω ,

where Ω: J^{1} (CP) → ⋀^{2} T^{∗}M ⊗ adP ≅ ⋀^{2} T^{∗}M is the curvature mapping (see §3.1) and

χ̄ = ∑_{k=0}^{[n/2]} χ̄_{k} ,

each χ_{k} being a multi-vector field of degree 2k, such that (cf. §2.5)

L_{χ_{k}}v_{n} = 0 .

Proof. As Lv_{n} is gauge invariant according to Utiyama’s theorem (§3.1) we have

L = \overline{L} ∘ Ω ,

where \overline{L} is an arbitrary differentiable function on ⋀^{2} T^{∗}M. The variational triviality of L tells us that the above composition should satisfy locally the equations (11)– (12). Taking into account the equations of Ω (see formula (8)) we have

\frac{∂^{2}\overline{L}}{∂p_{ih}∂p_{jk}} ∘ Ω + \frac{∂^{2}\overline{L}}{∂p_{ik}∂p_{jh}} ∘ Ω = 0, ∀h, i, j, k = 1, …, n,

and

\frac{∂^{2}\overline{L}}{∂q^{k}∂p_{ik}} ∘ Ω = 0, ∀i = 1, …, n,

as ∂L/∂p_{i}, ∂^{2}L/∂p_{j}∂p_{i,k} vanish for L does not depend on p_{i}. Since Ω is a surjective mapping we obtain the corresponding equations for \overline{L} on the bundle ⋀^{2} T^{∗}M; i.e.,

\frac{∂^{2}\overline{L}}{∂p_{ih}∂p_{jk}} + \frac{∂^{2}\overline{L}}{∂p_{ik}∂p_{jh}} = 0, ∀h, i, j, k = 1, …, n . (13)

\frac{∂^{2}\overline{L}}{∂q^{k}∂p_{ik}} = 0, ∀i = 1, …, n . (14)

From (13) we obtain ∂^{2}\overline{L}/∂p_{ih}^{2} = 0. Hence for each variable p_{ih} we have \overline{L} = a+bp_{ih}, where a, b are differentiable functions depending on q^{j} and the rest of the variables p_{kl}. By recurrence we conclude that \overline{L} is a polynomial in the variables p_{ih} with coefficients in C^{∞}(M); that is,

\overline{L} = ∑_{k} f^{i_{1}...i_{2k}}p_{i_{1}i_{2}} ⋯ p_{i_{2k−1}i_{2k}}, f^{i_{1}...i_{2k}} ∈ C^{∞}(M) . (15)

For k = h in (13) we conclude that the functions f^{i_{1}...i_{2k}} with a repeated index, must vanish. For arbitrary indices, again equation (13) tells us that f^{i_{1},... ,j,... ,l,... ,i_{2k}} + f^{i_{1},... ,l,... ,j,... ,i_{2k}} = 0. Since transpositions generate the group of permutations, we obtain

f^{i_{π(1)},... ,i_{π(2k)}} = (sgnπ) f^{i_{1},... ,i_{2k}} ,

for every permutation π ∈ S_{2k}. Therefore,

\overline{L} = ∑_{k=0}^{[n/2]} ∑_{i_{1}<...<i_{2k}} ∑_{π∈S_{2k}} (sgnπ) f^{i_{1},... ,i_{2k}}p_{i_{π(1)}i_{π(2)}} ⋯ p_{i_{π(2k−1)}i_{π(2k)}} .

The, it is straightforward to check that \overline{L} = χ̄ = ∑_{k=0}^{[n/2]} χ̄_{k}, where

χ_{k} = (−1)^{\binom{2k}{2}}2^{k} ∑_{i_{1}<...<i_{2k}} f^{i_{1},... ,i_{2k}} \frac{∂}{∂q^{i_{1}}} ∧ … ∧ \frac{∂}{∂q^{i_{2k}}} .

Let us check that the equation (14) implies that each multi-vector field χ_{k} is divergence free. Let (V; q^{1}, …, q^{n}) be a coordinate open domain such that v_{n} = dq^{1} ∧ … ∧ dq^{n}. The Lie derivative of the volume form with respect to the multivector field χ_{k} is

(L_{χ_{k}}v_{n}) = c_{k} · (−1)^{k+1} ∑_{i_{1}<...<i_{2k}} d (f^{i_{1},... ,i_{2k}}i_{\frac{∂}{∂q^{i_{1}}} ∧...∧ \frac{∂}{∂q^{i_{2k}}}} (dq^{1} ∧ … ∧ dq^{n}))
= −c_{k} ∑_{i_{1}<...<i_{2k}}(−1)^{i_{1}+...+i_{2k}}d (f^{i_{1},... ,i_{2k}}dq^{j_{1}} ∧ … ∧ dq^{j_{n−2k}}) ,

where c_{k} = (−1)^{\binom{2k}{2}}2^{k} and j_{1} < … < j_{n−2k} is the complementary subset of i_{1} < … < i_{2k} in {1, …, n}. Hence

c_{k}^{−1} (L_{χ_{k}}v_{n}) = − ∑_{i_{1}<...<i_{2k}}(−1)^{i_{1}+...+i_{2k}} ∑_{α} \frac{∂f^{i_{1},... ,i_{2k}}}{∂q^{α}} dq^{α} ∧ dq^{j_{1}} ∧ … ∧ dq^{j_{n−2k}} .

If j_{0} < … < j_{n−2k} is the complementary subset of i_{2} < … < i_{2k} in {1, …, n}, then the coefficient of dq^{j_{0}} ∧ dq^{j_{1}} ∧ … ∧ dq^{j_{n−2k}} in the above formula is

(−1)^{i_{2}+...+i_{2k}} ∑_{α} \frac{∂f^{α,i_{2},... ,i_{2k}}}{∂q^{α}} ,

where for (not necessarily increasing) pairwise different indices i_{1}, …, i_{2k} we have set f^{i_{1},... ,i_{2k}} = (sgnπ)f^{π(i_{1}),... ,π(i_{2k})}, π being the unique permutation of the set {i_{1}, …, i_{2k}} such that π (i_{1}) < … < π (i_{2k}). Accordingly, L_{χ_{k}}v_{n} = 0 if and only if

∑_{α} \frac{∂f^{α,i_{2}... ,i_{2k}}}{∂q^{α}} = 0, i_{2}, ..., i_{2k} fixed. (16)

In the other hand, from the formula (15) we have

\frac{∂^{2}\overline{L}}{∂q^{α}∂p_{jα}} = ∑_{k=0}^{[n/2]} \frac{∂f^{i_{1}...i_{2k}}}{∂q^{α}} \frac{∂p_{i_{1}i_{2}} ⋯ p_{i_{2k−1}i_{2k}}}{∂p_{jα}}

= ∑_{k=0}^{[n/2]} k \frac{∂f^{j,α,i_{3}...i_{2k}}}{∂q^{α}} p_{i_{3}i_{4}} ⋯ p_{i_{2k−1}i_{2k}} .

The above expression vanishes if and only if

∑_{α} \frac{∂f^{j,α,i_{3}...i_{2k}}}{∂q^{α}} = 0 ,

which is equivalent to (16), thus finishing the proof.

Remark. Given a variationally trivial Lagrangian L, the action

∫_{M} L(j_{x}^{1}ω)v_{n}

does not depend on the choice of the connection ω taken. Hence, variationally trivial Lagrangians produce Gauss-Bonnet type theorems.

References

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Dpto. Geometría y Topología, Facultad de CC. Matemáticas UCM Avda Complutense S/N. 28040 Madrid, SPAIN E-mail: mcastri@mat.ucm.es