International Conference on

Secondary calculus and cohomological Physics Moscow, August 1997

GAUGE INVARIANCE ON PRINCIPAL SU(2)-BUNDLES

M. Castrillón López

Departamento de Geometría y Topología, Universidad Complutense de Madrid Ciudad Universitaria, s/n, 28040–Madrid, Spain E-mail: mcastri@sungt1.mat.ucm.es

J. Muñoz Masqué

CSIC, Instituto de Física Aplicada C/ Serrano 144, 28006-Madrid, Spain E-mail: jaime@iec.csic.es

Abstract. Given a principal SU (2)-bundle π: P → M, the structure of differential forms on J^{1} (P) which are invariant under the natural representation of the gauge algebra of P, is analyzed.

1. Introduction

Given a principal SU (2)-bundle π: P → M, we study the structure of the algebra of differential forms on J^{1} (P) (the 1-jet bundle of local sections of π) which are invariant under the natural representation of the Lie algebra of all infinitesimal automorphisms of P and also under the subalgebra of π-vertical infinitesimal automorphisms, the so-called gauge algebra of P. In [HM1], [HM2] the structure of the invariant differential forms with respect to the above two algebras has been determined in the case of a principal U (1)-bundle. In the non-abelian case however this structure seems to be much more complex. Because of this we proceed to analyze the invariant differential forms on the 1-jet bundle of sections of P as a first step to obtain the invariant differential forms on the bundle of connections. This is a well-known technique in gauge theories (e.g., see [A2], [B], [G], [GS], [K]) which is based on the fact that the bundle of connections can be identified to the quotient bundle of J^{1} (P) modulo de action of the structure group, so that we recover the action of the gauge algebra and also the algebra of all infinitesimal automorphisms of P on connections as a quotient action of these algebras on J^{1} (P) by infinitesimal contact transformations.

Key words and phrases: Automorphism of a principal bundle, bundle of connections, contact forms, gauge algebra, gauge group, infinitesimal contact transformation, invariant differential forms,jet bundle.

2. Preliminaries and Notations 2.1. Automorphisms

An automorphism of a principal G-bundle π: P → M is an equivariant diffeomorphism Φ: P → P; i.e.,

Φ (u · g) = Φ (u) · g, ∀u ∈ P, ∀g ∈ G.

The set of all automorphisms of P is a group denoted by AutP. An automorphism Φ ∈ AutP induces a diffeomorphism on the ground manifold ϕ: M → M, such that π ∘ Φ = ϕ ∘ π. If ϕ is the identity map then Φ is said to be a gauge transformation. The set of all gauge transformations is a normal subgroup GauP ⊂ AutP, and we have an exact sequence of groups

1 → GauP → AutP → DiffM.

In the case of the trivial bundle pr_{1}: P = M × G → M, every Φ ∈ AutP can be written as

Φ (x, g) = (ϕ (x) , ψ (x) · g) ,

with x ∈ M, g ∈ G, ψ ∈ C^{∞} (M, G). Hence

Gau (M × G) ≃ C^{∞} (M, G) .

2.2. G-invariant vector fields

A vector field X ∈ 𝔛 (P) is G-invariant if and only if

R_{g} · X = X, ∀g ∈ G.

Let Φ_{t} be the flow of X. Then,

X is G-invariant ⇔ Φ_{t} ∈ AutP, ∀t ∈ ℝ.

The G-invariant vector fields are a Lie algebra denoted by autP. G-invariant vector fields are π-projectable. The gauge algebra of P is the ideal gauP of π-vertical vector fields in the Lie algebra autP; i.e.,

gauP = {X ∈ autP | π_{∗}X = 0} .

The group G acts naturally on T (P) by setting:

X · g = (R_{g})_{∗} (X) , ∀X ∈ T (P), ∀g ∈ G.

The quotient manifold T (P) /G exists and we have

autP ≃ Γ (M, T (P)/G) .

If adP = (P × 𝔤) /G is the bundle associated to P by the adjoint representation of G on 𝔤 (cf. [A1]), then

gauP ≃ Γ (M, adP) .

gauge invariance on principal su(2)-bundles 3

By passing to the quotient modulo G in the exact sequence of vector bundles over P,

0 → V (P) → T (P) ⟶^{π_{∗}} π^{∗}T M → 0,

where V (P) stands for the vertical subbundle, we obtain an exact sequence of vector bundles over M (the so-called Atiyah sequence, [A1], [Bl], [G], [GS], [K]),

0 → adP → T (P)/G ⟶^{π_{∗}} T M → 0.

2.3. SU (2) notations

We consider the standard basis of the Lie algebra 𝔰𝔲(2) normalized by the factor \frac{1}{2}; i.e.,

B_{1} = \frac{1}{2} ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) , B_{2} = \frac{1}{2} ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , B_{3} = \frac{1}{2} ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) ,

with i = \sqrt{−1}. Note that 2 i B_{a}, 1 ≤ a ≤ 3 are the Pauli matrices. We have

[B_{1}, B_{2}] = B_{3}, [B_{2}, B_{3}] = B_{1}, [B_{3}, B_{1}] = B_{2}.

As is well-known (e.g., see [NS]), SU (2) can be identified to the 3-sphere; i.e.,

SU (2) ≃ S^{3} ⊂ ℂ^{2}.

Let (y^{0} + iy^{1}, y^{2} + iy^{3}) be the standard coordinates in ℂ^{2}. Then, a matrix g ∈ SU (2) is uniquely written as

\{ \begin{matrix}g = ( \begin{matrix}y^{0}(g) + iy^{1}(g) & y^{2}(g) + iy^{3}(g) \\ −y^{2}(g) + iy^{3}(g) & y^{0}(g) − iy^{1}(g) \end{matrix} ) \\ y^{0}(g)^{2} + y^{1}(g)^{2} + y^{2}(g)^{2} + y^{3}(g)^{2} = 1 \end{matrix}

3. The bundle of connections 3.1. Connections and splittings

A connection Γ on a principal G-bundle π: P → M gives an splitting

σ_{Γ}: T M → T (P)/G, π_{∗} ∘ σ_{Γ} = 1_{T M}

of the Atiyah sequence (cf. [A1], [K], [MV]), by setting σ_{Γ}(X) = X^{∗}, where X^{∗} stands for the Γ-horizontal lift of X to P, as the horizontal lift of a vector field X ∈ 𝔛(M) is G-invariant (see [KN]). The bundle of connections,

p: C (P) → M

is the bundle of all ℝ-linear maps

λ: T_{x}M → (T (P)/G)_{x} such that π_{∗} ∘ λ = 1_{T_{x}M}

Connections on P are the global sections of p: C (P) → M. Moreover, C (P) is an affine bundle modelled over the vector bundle

Hom (T M, adP) ≃ T^{∗}M ⊗ adP.

3.2. Coordinates on C (P)

Let (U; x^{1}, …, x^{n}) be a coordinate open domain in M such that a given principal SU (2)- bundle π: P → M is trivial over U. For every B ∈ 𝔰𝔲 (2), we define

φ_{t}^{B}: π^{−1} (U) ≃ U × SU(2) → π^{−1} (U)

by the formula:

φ_{t}^{B} (x, g) = (x, exp (tB) · g)

and we denote by B̃ the infinitesimal generator of φ_{t}^{B}. It follows that B̃ is a π-vertical SU(2)- invariant vector field; i.e., B̃ ∈ gauP. Furthermore, (B̃_{1}, B̃_{2}, B̃_{3}) is a basis of Γ (U, adπ^{−1} (U)). Hence for each σ_{Γ} there exist unique functions A_{j}^{a} (Γ) ∈ C^{∞} (U) such that,

σ_{Γ} (\frac{∂}{∂x^{j}} ) = \frac{∂}{∂x^{j}} − A_{j}^{a} (Γ) B̃_{a}, 1 ≤ j ≤ n.

The functions

(x^{j}; A_{j}^{a}), 1 ≤ j ≤ n, 1 ≤ a ≤ 3

are a coordinate system on p^{−1} (U) = C (π^{−1}U), which will be called the natural coordinate system induced by the coordinate system (U; x^{1}, …, x^{n}) on the bundle of connections.

4. AutP acting on C (P)

Each Φ ∈ AutP acts on the connections of P as follows: given Γ, Γ′ = Φ (Γ) is the connection corresponding to the connection form

ω_{Γ′} = (Φ^{−1})^{∗}ω_{Γ}.

If Ψ ∈ AutP, then

(Ψ ∘ Φ) (Γ) = Ψ (Φ (Γ)) .

For each Φ ∈ AutP there exists a unique diffeomorphism

Φ_{C}: C (P) → C (P)

such that p∘Φ_{C} = φ∘p, where φ ∈ Diff M is the diffeomorphism induced by Φ. Furthermore,

Φ_{C} ∘ σ_{Γ} = σ_{Φ(Γ)},

for every connection Γ. We obtain a group homomorphism

AutP → Diff C (P) , Φ ↦ Φ_{C}.

If Φ_{t} is the flow of a G-invariant vector field X ∈ autP, then (Φ_{t})_{C} is a one-parameter group in C (P). We set

X_{C} = infinitesimal generator of (Φ_{t})_{C} .

In this way we obtain a Lie algebra representation of the Lie algebra of all infinitesimal automorphisms of P on the vector fields of its bundle of connections,

autP → 𝔛 (C (P)) , X ↦ X_{C}.

Note that X and X_{C} both are projectable onto the same vector field of M. By using a coordinate domain (U; x^{1}, …, x^{n}) in M and the basis (B̃_{1}, B̃_{2}, B̃_{3}), each X ∈ autπ^{−1}(U) can be written as

X = f_{j} \frac{∂}{∂x^{j}} + g^{a}B̃_{a}, f_{j}, g^{a} ∈ C^{∞}(U).

The local expression of the above representation is as follows:

X_{C} = f_{j} \frac{∂}{∂x^{j}} − 𝔖_{123} (\frac{∂g^{1}}{∂x^{j}} + \frac{∂f_{i}}{∂x^{j}} A_{i}^{1} + g^{3}A_{j}^{2} − g^{2}A_{j}^{3}) \frac{∂}{∂A_{j}^{1}} ,

where 𝔖_{123} stands for cyclic sum over the indices 1, 2, 3. In particular, if X is an infinitesimal gauge transformation then f_{j} = 0. Hence

X_{C} = −𝔖_{123} (\frac{∂g^{1}}{∂x^{j}} + g^{3}A_{j}^{2} − g^{2}A_{j}^{3}) \frac{∂}{∂A_{j}^{1}} .

5. The identification (J^{1}P) /G ≃ C (P)

Let π: P → M be an arbitrary principal G-bundle and let π_{1}: J^{1}P → M be the 1-jet bundle of local sections of π. The group G acts (on the right) on J^{1}P by

j_{x}^{1}s · g = j_{x}^{1} (R_{g} ∘ s) ,

where s is a local section of π, g ∈ G and R_{g} stands for the right translation. The quotient (J^{1}P) /G exists as a fibred differentiable manifold over M which can be identified to C (P). Let us describe this identification. Let

q: J^{1}P → C (P)

be the mapping defined as follows. Each local section s defines a retract

Γ_{s(x)}: T_{s(x)}P → V_{s(x)}P = ker (π_{∗})_{s(x)}

of the inclusion

V_{s(x)}P ⊂ T_{s(x)}P,

by setting

Γ_{s(x)} (X) = X − s_{∗}π_{∗} (X) .

For every u ∈ π^{−1}(x) there exists a unique g ∈ G such that u = s(x) · g and we define Γ_{u}: T_{u}P → V_{u}P as

Γ_{u} = (R_{g})_{∗} ∘ Γ_{s(x)} ∘ (R_{g^{−1}})_{∗}.

In this way we obtain a “connection Γ at x”; that is, an element of C (P) which only depends on j_{x}^{1}s. Hence we set

q(j_{x}^{1}s) = Γ.

5.1. AutP acting on J^{1}P

Let X be a π-projectable vector field on P, let X′ be its projection onto M and let Φ_{t}, ϕ_{t} be the flows of X, X′, respectively. A flow Φ_{t}^{(1)} can be defined on J^{1}P by the formula

Φ_{t}^{(1)} (j_{x}^{1}s) = j_{ϕ_{t}(x)}^{1} (Φ_{t} ∘ s ∘ ϕ_{−t}) .

If X is π-vertical (i.e., X′ = 0 or even ϕ_{t} = id_{M}) then

Φ_{t}^{(1)} = J^{1}(Φ_{t}).

We denote by X^{(1)} the infinitesimal generator of the flow Φ_{t}^{(1)} which is called the infinitesimal contact transformation associated to X (or also the natural lift of X to the 1-jet bundle). The mapping X ↦ X^{(1)} is a Lie algebra monomorphism and X^{(1)} is π_{10}-projectable onto X, where π_{10} is the canonical projection,

π_{10}: J^{1}P → P, π_{10} (j_{x}^{1}s) = s(x).

For every Φ ∈ AutP we have

q ∘ J^{1}(Φ) = Φ_{C} ∘ q.

Hence for every X ∈ autP the vector field X^{(1)} is q-projectable and its projection is X_{C}; i.e.,

q_{∗} ∘ X^{(1)} = X_{C} ∘ q.

Therefore Proposition. The representation

autP → 𝔛 (C (P)) , X ↦ X_{C}.

can be obtained “by projecting” the natural representation of the algebra autP on J^{1}P by infinitesimal contact transformations by means of the identification (J^{1}P) /G ≃ C (P).

6. Gauge invariance in J^{1}P 6.1. Contact forms on J^{1}P

Let π: P → M be a principal SU (2) -bundle. We define a 𝔰𝔲(2)-valued 1-form θ on J^{1}P as follows. For every Y ∈ T_{j_{x}^{1}s}(J^{1}P) we have

q(j_{x}^{1}s) ((π_{10})_{∗}Y) ∈ V_{s(x)}P.

If B^{∗} ∈ 𝔛(P) is the fundamental vector field (cf. [KN]) associated to B ∈ 𝔰𝔲(2) we have a vector bundle isomorphism

P × 𝔰𝔲(2) → V (P),

given by

(u, B) ↦ B_{u}^{∗}.

Hence there exists a unique B ∈ 𝔰𝔲(2) such that

q(j_{x}^{1}s) ((π_{10})_{∗}Y) = B_{s(x)}^{∗},

and we set

θ(Y) = B.

In the standard basis we have

θ = θ^{a} ⊗ B_{a},

where θ^{1}, θ^{2}, θ^{3} are global ordinary 1-forms on J^{1}P called the standard contact forms. The following properties of the form θ are easily checked:

1. For every Φ ∈ GauP, we have

J^{1}(Φ)^{∗}θ = θ.

2. Hence for every X ∈ gauP,

L_{X^{(1)}}θ^{a} = 0, 1 ≤ a ≤ 3,

3. For every B ∈ 𝔰𝔲(2), let B^{•} be the fundamental vector field associated to B under the action of SU (2) on J^{1}P. Then,

L_{B^{•}}θ = [θ, B] .

6.2. Invariant differential forms

A differential form ω_{r} on J^{1}P of degree r ∈ ℕ is said to be gauP -invariant if for every X ∈ gauP we have

L_{X^{(1)}}ω_{r} = 0.

We denote by I_{gauP} the set of gauP -invariant differential forms. (Usually, gauP -invariant differential forms are called gauge invariant forms.)

Example. The standard contact forms are gauge invariant.

Note that I_{gauP} is a ℤ-graded algebra over Ω^{•}(M).

A differential form ω_{r} on C (P) of degree r ∈ ℕ is said to be autP -invariant if for every X ∈ autP we have

L_{X_{C}}ω_{r} = 0.

We denote by I_{autP} the set of autP -invariant differential forms. Note that I_{gauP} is a ℤgraded algebra over Ω^{•}(M) and that I_{autP} ⊂ I_{gauP} is a subalgebra.

8 m. castrillón lópez & j. muñoz masqué

6.3. Statement of the main result

Theorem. The algebra of gauge invariant forms on J^{1}P is generated over π_{1}^{∗}Ω^{•}(M) by the forms

(θ^{a}, dθ^{a}), 1 ≤ a ≤ 3,

that is,

I_{gauP} = π_{1}^{∗}Ω^{•}(M) [θ^{a}, dθ^{a}]_{1≤a≤3} .

The only autP -invariant differential forms on J^{1}P are:

I_{autP} = ℝ [θ^{a}, dθ^{a}]_{1≤a≤3} .

6.4. Conclusions

If π: P → M is a principal SU (2)-bundle, we have seen that the algebra of gauge invariant (resp. autP -invariant) differential forms on J^{1} (P) is differentiably generated over the graded algebra of differential forms on M (resp. over the real numbers) by the standard structure forms. In a previous paper ([EM, Theorem 3]) it was proved that if π: P → M is an arbitrary principal G-bundle with G and M connected, then on J^{1} (P) the unique gauge invariant Lagrangian densities are the differential n-forms on M. This result was originally motivated by Utiyama’s theorem ([U]; also see [B], [Bl], [G] for the geometric interpretation of Utiyama’s theorem) which stated the structure of gauge invariant Lagrangian densities on J^{1} (C (P)). In the case G = SU (2) Theorem 3 in [EM] can be obtained as a consequence of the theorem above remarking that contact forms on the 1-jet bundle are not horizontal over the base manifold.

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