The Diffiety Institute Preprint Series

Preprint DIPS–10/98 June 19, 1995

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The “three-line” theorem for the Vinogradov C-spectral sequence of the Yang–Mills equations

by

Dmitri GESSLER

Preprint SISSA 71/95/FM

The “three-line” theorem

for the Vinogradov C-spectral sequence of the Yang–Mills equations

Dmitri GESSLER

Abstract. The Vinogradov C-spectral sequence for the Yang– Mills equations is considered and the “three-line” theorem for the term E_{1} of the C-spectral sequence is proved: E_{1}^{p,q} = 0 if p > 0 and q < n − 2, where n is the dimension of spacetime.

1. Introduction

Homological methods play an important role in the study of systems of differential equations. The C-spectral sequence (variational bicomplex) introduced by A. M. Vinogradov [11] contains important invariants of differential equations such as conservation laws, characteristic classes of families of solutions, etc. It provides a means for studying various aspects of Lagrangian formalism, the inverse problem of calculus of variations. The term E_{1} of the C-spectral sequence is an analog of the de Rham complex in the category of nonlinear partial differential equations (for a very enlightening discussion see [15]).

General methods for calculating this important spectral sequence are based on the Spencer type cohomology techniques. In [13], the “two-line” theorem estimating the term E_{1} for determined systems of differential equations is proved, a concrete method of calculating for the term E_{1}^{1,n−1}, which is related to the theory of conservation laws, is given, where n is the number of independent variables. Due to this method, a complete description of the set of conservation laws is possible for determined differential equations (see, for example, [14]). Further development of Vinogradov’s results is done in [10, 9], where the Janet sequence for involutive differential equations is used and a general approach to calculation of the horizontal cohomology is proposed. In [5], this approach is applied to the C-spectral sequence for overdetermined equations and a method of calculating the C-spectral sequence, generalizing Vinogradov’s one, is given. This method enables one to reduce the problem of computing the term E_{1}^{p,n−1}, p > 0, to the problem of finding the kernel of some differential operator. The terms \overline{H}^{q} = E_{1}^{0,q} also known as the horizontal de Rham cohomology, or the characteristic cohomology, are studied for exterior differential systems from a different point of view in [3].

1991 Mathematics Subject Classification. Primary 35A30; Secondary 58A20, 58G05.

Key words and phrases. nonlinear differential equations, spectral sequences, Spencer cohomology, Yang–Mills equations.

Now it has been realized that many fundamental questions of the local field theory can be reformulated as cohomological ones. In [2, 6], the analysis of the antifield-BRST method of quantization from the homological point of view is done, in particular, the characteristic (horizontal) cohomology for the Yang–Mills equations is considered and it is proved that \overline{H}^{q} = E_{1}^{0,q} = 0 if q < n−2, where n is the dimension of the spacetime. In this paper, we extend the results of [2] concerning the characteristic cohomology for the Yang–Mills equations and prove the “three-line” theorem, which is the first step in studying the C-spectral sequence. The method used in this paper is the same as in [5, 4].

This paper is organized as follows. Section 2 is a summary of the geometrical theory of nonlinear differential equations and the C-spectral sequence. In Section 3, the formally exact resolution of the universal linearization operator for the Yang–Mills equation is considered and the “three-line” theorem is proved.

2. Jet manifolds and infinitely prolonged differential equations

In this section, we define the basic concepts of the geometrical theory of differential equations and the theory of the C-spectral sequence ([1, 5, 8, 12, 13]).

2.1. Jets. Let M be a smooth manifold and π: E → M be a smooth fiber bundle over M, dim M = n, dim E = m + n. Denote by Γ(π) the set of all (local) sections of π.

Let π_{k}: J^{k} → M be the bundle of k-jets for π,

J^{k}(π) = { [f]_{x}^{k} | f ∈ Γ(π), x ∈ M },

where [f]_{x}^{k} denotes the k-jet of a local section f at x. Denote by J^{∞}(π) the manifold of infinite jets for π. This manifold is the inverse limit with respect to the following system of mappings

π_{k,l}: J^{k}(π) → J^{l}(π), π_{k,l} ([f]_{x}^{k}) = [f]_{x}^{l}, k ≥ l,
π_{k}: J^{k}(π) → M, π_{k} ([f]_{x}^{k}) = x.

By definition, one has the natural projections

π_{∞,k}: J^{∞}(π) → J^{k}(π), π_{∞}: J^{∞}(π) → M.

Let us choose a coordinate neighborhood U in M such that the restriction of the bundle E to U is trivial. Let x_{1}, …, x_{n} be local coordinates in the neighborhood U and u^{1}, …, u^{m} be coordinates along the fiber of π in π^{−1}(U). Each local section f ∈ Γ(π) is of the form f = (u^{1}(x_{1}, …, x_{n}), …, u^{m}(x_{1}, …, x_{n})). Define functions p_{σ}^{j} by

p_{σ}^{j} ([f]_{x_{0}}^{k}) = \frac{∂^{|σ|}u^{j}}{∂x_{1}^{σ_{1}} … ∂x_{n}^{σ_{n}}} |_{x=x_{0}},

where σ = (σ_{1}, …, σ_{n}), |σ| = σ_{1} + ⋯ + σ_{n}. Then the smooth functions (x^{i}, p_{σ}^{j}), 1 ≤ i ≤ n, 1 ≤ j ≤ m, 0 ≤ |σ| ≤ k, form local coordinates in J^{k}(π), 0 ≤ k ≤ ∞.

Let F_{k}(π) denote the algebra C^{∞} (J^{k}(π)). Then one has the system of embeddings

π_{k,l}^{∗}: F_{l}(π) → F_{k}(π), l ≤ k, π_{k}^{∗}: C^{∞}(M) → F_{k}(π).

The direct limit F(π) with respect to the system {π_{k,l}^{∗}} is called the algebra of smooth functions on J^{∞}(π). We identify F_{k}(π) and C^{∞}(M) with their images in F(π),

F(π) = ⋃_{k≥0} F_{k}(π).

In the same manner, one can define the module of i-forms Λ^{i}(π), i ≥ 0, on J^{∞}(π)

Λ^{i}(π) = ⋃_{k≥0} Λ^{i}(J^{k}(π))

and consider the graded algebra Λ^{∗}(π) = ∑_{i=0}^{∞} Λ^{i}(π) of differential forms on J^{∞}(π).

A vector field X on J^{∞}(π) is an ℝ-linear mapping X: F(π) → F(π) satisfying the Leibniz rule X(φψ) = φX(ψ) + ψX(φ), for any φ, ψ ∈ F(π), and preserving the filtration in the algebra F(π), i.e., fulfilling the condition that there exists r ≥ 0 such that

X (F_{k}(π)) ⊂ F_{k+r}(π)

for each k ≥ 0. Denote by D(π) the set of all vector fields on J^{∞}(π). Obviously, D(π) is an F(π)-module and a Lie algebra over ℝ. Locally, each X ∈ D(π) can be represented as

X = ∑_{i} a_{i} \frac{∂}{∂x_{i}} + ∑_{j,σ} b_{σ}^{j} \frac{∂}{∂p_{σ}^{j}} ,

where b_{σ}^{j} ∈ F_{|σ|+r}(π) for some r.

Let ξ_{i}: F_{i} → J^{∞}(π), i = 1, 2, be vector bundles over J^{∞}(π), P_{i} = Γ(ξ_{i}) be F(π)-modules of sections. An ℝ-linear differential operator Δ: P_{1} → P_{2} is called C-differential if it can be restricted to the manifolds of the form [f]^{∞}, f ∈ Γ(π). That is

Δ(φ)|_{[f]^{∞}} = 0 when φ|_{[f]^{∞}} = 0, φ ∈ P_{1}.

If vector bundles ξ_{1}, ξ_{2} are finite dimensional, m_{i} = dim ξ_{i}, then in local coordinates each C-differential operator Δ can be represented as an m_{2} × m_{1} matrix

Δ = (\begin{matrix}∑_{σ} a_{11}^{σ}D_{σ} & & & … & & ∑_{σ} a_{1m_{1}}^{σ}D_{σ} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ ∑_{σ} a_{m_{2}1}^{σ}D_{σ} & & & … & & ∑_{σ} a_{m_{2}m_{1}}^{σ}D_{σ} \end{matrix}) ,

where σ = (σ_{1}, …, σ_{n}), D_{σ} = (D_{1})^{σ_{1}} ∘ ⋯ ∘ (D_{n})^{σ_{n}}, and

D_{i} = \frac{∂}{∂x_{i}} + ∑_{j,σ} p_{σ+1_{i}}^{j} \frac{∂}{∂p_{σ}^{j}} , i = 1, …, n,

is the i-th total derivative. The set of C-differential operators from P_{1} to P_{2} is clearly an F(π)-module and is denoted by CDiff(P_{1}, P_{2}).

In any module CDiff(Q, R), there exists a filtration by the modules CDiff^{(k)}(Q, R) consisting of C-differential operators of order ≤ k. Consider the module of C-symbols

Csmbl(Q, R) = ∑_{k=0}^{∞} Csmbl^{(k)}(Q, R),
Csmbl^{(k)}(Q, R) = CDiff^{(k)}(Q, R)/ CDiff^{(k−1)}(Q, R)
= S^{k}(\overline{Λ}^{1})^{∗} ⊗ Q^{∗} ⊗ R.

By definition, one has the projections

Csmbl^{(k)} : CDiff^{(k)}(Q, R) → Csmbl^{(k)}(Q, R).

For each Δ ∈ CDiff(Q, R), Δ ≠ 0, define the order of Δ by

ord Δ = min{k | Δ ∈ CDiff^{(k)}(Q, R)},

and the C-symbol s_{0}(Δ) = Csmbl^{(ord Δ)}(Δ).

The representative object for the functor CDiff^{(k)}(P, ·), 0 ≤ k ≤ ∞, is called the module of horizontal k-jets and is denoted by \overline{J}^{k}(P). By definition, we have

CDiff^{(k)}(P, Q) = Hom_{F}(\overline{J}^{k}(P), Q).

For any C-differential operator Δ: P → Q denote by \overline{J}^{k}(Δ): \overline{J}^{k}(P) → Q the corresponding homomorphism.

The filtration in the module CDiff(P, Q) yields projections

\overline{π}_{k,k−1} : \overline{J}^{k}(P) → \overline{J}^{k−1}(P), k > 0,

and the following sequence is exact

0 → S^{k}\overline{Λ}^{1} ⊗ P → \overline{J}^{k}(P) → \overline{J}^{k−1}(P) → 0.

If Δ: P → Q is a C-differential operator of order k, then s_{0}(Δ) ∈ S^{k}(\overline{Λ}^{1})^{∗} ⊗ Q^{∗} ⊗ P is an F-homomorphism

s_{0}(Δ): S^{k}\overline{Λ}^{1} ⊗ P → Q.

The l-th prolongation s_{l}(Δ) of the C-symbol s_{0}(Δ) is, by definition, the composition

s_{l}(Δ): S^{k+l}\overline{Λ}^{1} ⊗ P ⟶^{σ⊗1} S^{l}\overline{Λ}^{1} ⊗ S^{k}\overline{Λ}^{1} ⊗ P ⟶^{1⊗s_{0}(Δ)} S^{l}\overline{Λ}^{1} ⊗ Q,

where σ: S^{k+l}\overline{Λ}^{1} → S^{l}\overline{Λ}^{1} ⊗ S^{k}\overline{Λ}^{1} is the natural inclusion.

Let ξ_{i}: F_{i} → M, i = 1, 2, be vector bundles, P_{i} = Γ(ξ_{i}), and Δ: P_{1} → P_{2} be an ℝ-linear differential operator, Δ ∈ Diff(P_{1}, P_{2}). By definition, put \overline{P_{i}} = F(π) ⊗_{C^{∞}(M)} P_{i} = Γ (π_{∞}^{∗}(ξ_{i})), where π_{∞}^{∗}(ξ_{i}) is the induced vector bundle over J^{∞}(π). A unique C-differential operator \overline{Δ}: \overline{P_{1}} → \overline{P_{2}} such that \overline{Δ}(1 ⊗ p) = 1 ⊗ Δ(p) is called the lifting of Δ.

Consider a vector field X ∈ D(M) on M. It is a first-order differential operator acting from C^{∞}(M) to C^{∞}(M). Then the lifting \overline{X} ∈ CDiff (F(π), F(π)) is a vector field on J^{∞}(π). Consider the submodule CD(π) ⊂ D(π) generated by vector fields of the form \overline{X}. Thus we have a distribution on J^{∞}(π) which is called the Cartan distribution. The Cartan distribution is completely integrable in the sense that it satisfies the Frobenius integrability condition

[CD(π), CD(π)] ⊂ CD(π).  (1)

In local coordinates, if X = ∑_{i} a_{i}∂/∂x_{i}, a_{i} ∈ C^{∞}(M), then \overline{X} = ∑_{i} a_{i}D_{i}, and

CD(π) = { ∑_{i} φ_{i}D_{i} | φ_{i} ∈ F(π)}.

A vector field X ∈ D(π) is called vertical if X(φ) = 0 for each φ ∈ C^{∞}(M) ⊂ F(π). Denote by D^{V} (π) ⊂ D(π) the F(π)-module of vertical vector fields. The module D^{V} is also a subalgebra of the Lie algebra D(π),

[D^{V} (π), D^{V} (π)] ⊂ D^{V} (π),  (2)

and D(π) splits into the direct sum

D(π) = CD(π) ⊕ D^{V} (π).

Dually, the module of 1-forms on J^{∞}(π) splits into the direct sum

Λ^{1}(π) = C^{1}Λ^{1}(π) ⊕ \overline{Λ}^{1}(π),  (3)

where

\overline{Λ}^{1}(π) = \{ω ∈ Λ(π) | ω(X) = 0 for any X ∈ D^{V} (π)\}

is the module of horizontal 1-forms, and

C^{1}Λ^{1}(π) = {ω ∈ Λ(π) | ω(X) = 0 for any X ∈ CD(π)}

is the module of Cartan forms.

Locally,

\overline{Λ}^{1}(π) = \{ ∑_{i} φ_{i}dx_{i} | φ_{i} ∈ F(π)\},

C^{1}Λ^{1}(π) = { ∑_{j,σ} φ_{j}^{σ}ω_{σ}^{j} | φ_{j}^{σ} ∈ F(pi)},

where ω_{σ}^{j} = dp_{σ}^{j} − ∑_{i} p_{σ+1_{i}}^{j}dx_{i}.

From (3) it follows that each Λ^{i}(π), i > 0, splits into

Λ^{i}(π) = ∑_{α+β=i} \overline{Λ}^{α}(π) ⊗ C^{β}Λ^{1}(π),

where

\overline{Λ}^{α}(π) = \underbrace{\overline{Λ}^{1}(π) ∧ ⋯ ∧ \overline{Λ}^{1}(π)}_{α times}, C^{β}Λ^{1}(π) = \underbrace{C^{1}Λ^{1}(π) ∧ ⋯ ∧ C^{1}Λ^{1}(π)}_{β times} .

The graded algebra \overline{Λ}^{∗}(π) is the lifting of the graded algebra Λ^{∗}(M) of differential forms on M. The lifting of the de Rham differential d: Λ^{i}(M) → Λ^{i+1}(M) is called the horizontal differential and is denoted by \overline{d}: \overline{Λ}^{i}(π) → \overline{Λ}^{i+1}(π). The complex

0 → F(π) ⟶^{\overline{d}} \overline{Λ}^{1}(π) ⟶^{\overline{d}} ⋯ ⟶^{\overline{d}} \overline{Λ}^{n}(π) → 0

is called the horizontal de Rham complex and its cohomology at the term \overline{Λ}^{i}(π) is denoted by \overline{H}^{i}(π). In local coordinates

\overline{d}(dx_{i}) = 0, \overline{d}(φ) = ∑_{i} D_{i}(φ) dx_{i}, φ ∈ F(π).

Let V (π): V (E) → E be the vector bundle of vertical vector fields on E, V (E) = \{v ∈ T (E) | π_{∗}v = 0\}. By definition, put κ = \overline{Γ(V (π))} = Γ (π_{∞,0}^{∗} (V (π))). Then there exists a map

� : κ → D^{V} (π), φ ↦ �_{φ},

where �_{φ} is called an evolutionary derivation and is defined by the formula

�_{φ}(ψ)| _{[f]^{∞}} = \frac{d}{dt}|_{t=0} (ψ|_{[f_{t}]^{∞}}) ,

where f ∈ Γ(π), φ ∈ κ, ψ ∈ F(π), and f_{t} is a 1-parameter family of sections of π such that d/dt|_{t=0}f_{t} = φ|_{[f]^{∞}}, f_{0} = f.

In local coordinates, if φ = (φ^{1}, …, φ^{m}), then

�_{φ} = ∑_{j,σ} D_{σ}(φ^{j})\frac{∂}{∂p_{σ}^{j}} .

2.2. Differential operators and equations. A system of nonlinear differential equations of order k imposed on sections of π: E → M is a submanifold Y^{k} ⊂ J^{k}(π). Denote by Y^{k+s} ⊂ J^{k+s}(π) the s-th prolongation of Y^{k}. We shall always assume that Y^{k} is formally integrable. Then Y^{s}, s ≥ k, is a smooth manifold, and π_{s,t}, s ≥ t ≥ k, maps Y^{s} onto Y^{t} surjectively. The inverse limit of the system of maps

π_{s,t}: Y^{s} → Y^{t}, s ≥ t ≥ k,

is called an infinitely prolonged differential equation, or simply a differential equation, and is denoted by Y = Y^{∞}. In a trivial way, the infinite jet manifold J^{∞}(π) is a differential equation of zero order with Y^{k} = J^{k}(π), k ≥ 0.

One can construct a calculus on a differential equation Y ⊂ J^{∞}(π) in the same way as for the jet manifold J^{∞}(π). Let F, Λ^{∗}, D denote the algebra of smooth functions, the graded algebra of differential forms and the module of vector fields on Y respectively.

As for the jet manifold J^{∞}(π), there exists the splitting of the modules of vector fields D and 1-form Λ^{1}

D = D^{V} ⊕ CD, Λ^{1} = \overline{Λ}^{1} ⊕ C^{1}Λ^{1}.  (4)

Let ξ: E′ → M be a vector bundle, F: Γ(π) → Γ(ξ) be a nonlinear differential operator. The operator F can be considered as a section of the induced vector bundle π_{∞}^{∗}(ξ) over J^{∞}(π). Define a smooth map

J(F): J^{∞}(π) → J^{∞}(ξ), [f]_{x}^{∞} ↦ [F (f)]_{x}^{∞}.

Let Y = Y(F) be the differential equation defined by F,

Y(F) = ker J(F) = {θ ∈ J^{∞}(π) | J(F)(θ) = 0} .

In local coordinates, if F = (F^{1}, …, F^{s}), F^{l} = F^{l}(x_{i}, p_{σ}^{j}), then Y(F) is the infinite prolongation of the system

\begin{cases}F^{1} = 0, \\ . & . & . & . & . & . & . \\ F^{s} = 0. \end{cases}

Denote by P = Γ (π_{∞}^{∗}(ξ)) the F(π)-module of sections of the induced vector bundle over J^{∞}(π). The universal linearization of F is a Cdifferential operator ℓ_{F} ∈ CDiff(κ, P) such that

ℓ_{F} (φ)|_{[f]^{∞}} = \frac{d}{dt}|_{t=0} F (f_{t}), φ ∈ κ, f ∈ Γ(π),

where f_{t} is a 1-parameter family of sections of π such that d/dt|_{t=0}f_{t} = φ|_{[f]^{∞}}, f_{0} = f. In local coordinates,

ℓ_{F} = (\begin{matrix}∑_{σ} \frac{∂F^{1}}{∂p_{σ}^{1}} D_{σ} & & & … & & & ∑_{σ} \frac{∂F^{1}}{∂p_{σ}^{m}} D_{σ} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ ∑_{σ} \frac{∂F^{s}}{∂p_{σ}^{1}} D_{σ} & & & … & & & ∑_{σ} \frac{∂F^{s}}{∂p_{σ}^{m}} D_{σ} \end{matrix}) .

Denote by I_{Y} ⊂ F(π) the ideal of functions vanishing on Y. For each F(π)-module R denote by [R] the restriction of R to Y, [R] = R/(I_{Y} · R). If R is a projective F(π)-module, then [R] is a projective F-module as well.

A differential equation Y = Y(F) is called regular with respect to the nonlinear differential operator F = (F^{1}, …, F^{s}) if

1. The ideal I_{Y} is generated by the functions D_{σ}(F^{l}), 1 ≤ l ≤ s, 0 ≤ |σ| < ∞;

2. The module of 1-forms Λ^{1} is projective. This definition does not depend on the choice of coordinates.

If the F-module Λ^{1} is projective, then from (4) it follows that modules \overline{Λ}^{1} and C^{1}Λ^{1} are also projective. From now on we assume that all modules under consideration are projective.

Let i: Y → J^{∞}(π) be the natural inclusion, i^{∗}: [C^{1}Λ^{1}(π)] → C^{1}Λ^{1} be the induced homomorphism, [ℓ_{F}]: [κ] → [P] be the restriction of the universal linearization to Y.

2.3. Adjoint operator. Let Y be a differential equation. For any F-module Q consider the following complex (S(Q), \overline{d}_{1}):

0 → CDiff(Q, F) ⟶^{\overline{d}_{1}} CDiff(Q, \overline{Λ}^{1}) ⟶^{\overline{d}_{1}} ⋯ ⟶^{\overline{d}_{1}} CDiff(Q, \overline{Λ}^{n}) → 0, (5)

where \overline{d}_{1}(Δ) = −\overline{d} ∘ Δ. The cohomology of complex (5) is described by Proposition 1 ([13]). 1. H^{i} (S(Q)) = 0 if i ≠ n;

2. H^{n} (S(Q)) = Hom_{F}(Q, \overline{Λ}^{n}).

By definition, put Q̂ = Hom_{F}(Q, \overline{Λ}^{n}) for any module Q.

Each C-differential operator ∇: Q_{1} → Q_{2} induces a homomorphism of complexes

S(∇): S(Q_{2}) → S(Q_{1}),

S(∇)(Δ) = Δ ∘ ∇, Δ ∈ CDiff(Q_{2}, \overline{Λ}^{i}), and an ℝ-linear map of the cohomology

∇^{∗}: Q̂_{2} → Q̂_{1}.

The operator ∇^{∗} is called the adjoint operator for ∇. Proposition 2 ([13]). 1. ∇^{∗} ∈ CDiff(Q̂_{2}, Q̂_{1}).

2. For all ∇_{1} ∈ CDiff(Q_{1}, Q_{2}), ∇_{2} ∈ CDiff(Q_{2}, Q_{3}),

(∇_{2} ∘ ∇_{1})^{∗} = ∇_{1}^{∗} ∘ ∇_{2}^{∗}.

3. If in local coordinates ∇ is an m_{1} × m_{2}-matrix

∇ = ∥∑_{σ} a_{ij}^{σ}D_{σ}∥,

then ∇^{∗} is the m_{2} × m_{1}-matrix

∇^{∗} = ∥∑_{σ}(−1)^{|σ|}D_{σ} ∘ a_{ji}^{σ}∥.

2.4. The C-spectral sequence. Consider the C-filtration in the de Rham complex on an equation Y

Λ = C^{0}Λ ⊃ C^{1}Λ ⊃ C^{2}Λ ⊃ ⋯,

where

C^{p}Λ = ∑_{\begin{matrix}α≥p \\ β≥0 \end{matrix}} C^{α}Λ^{1} ⊗ \overline{Λ}^{β}.

The spectral sequence {E_{r}^{p,q}, d_{r}^{p,q}} determined by this filtration is called the C-spectral sequence for the differential equation Y. As usual, p denotes the filtration and p + q denotes the degree.

Proposition 3 ([13]). 1. E_{0}^{p,q} = C^{p}Λ^{1} ⊗ \overline{Λ}^{q}; 2. E_{0}^{0,q} = \overline{Λ}^{q}, d_{0}^{0,q} = \overline{d};

3. The C-spectral sequence converges and the term E_{∞} = {E_{∞}^{p,q}} is attached to H_{de Rham}^{∗}(Y).

Keeping in mind Statement 2 of Proposition 3, denote

d_{0}^{p,q} = \overline{d}: C^{p}Λ^{1} ⊗ \overline{Λ}^{q} → C^{p}Λ^{1} ⊗ \overline{Λ}^{q+1}.

If Y = J^{∞}(π), then in local coordinates \overline{d} is uniquely defined by the following equalities

\overline{d}(f) = ∑_{i} D_{i}(f) dx_{i}, f ∈ F;

\overline{d}(dx_{i}) = 0, dx_{i} ∈ \overline{Λ}^{1};

\overline{d}(ω_{σ}^{j}) = dω_{σ}^{j} = ∑_{i} dx_{i} ∧ ω_{σ+1_{i}}^{j}, ω_{σ}^{j} ∈ C^{1}Λ^{1}.

2.5. The term E_{1} of the Vinogradov C-spectral sequence. Consider a sequence of C-differential operators

P_{1} ⟶^{Δ_{1}} P_{2} ⟶^{Δ_{2}} P_{3}, ord Δ_{1} = k_{1}, ord Δ_{2} = k_{2}.  (6)

Sequence (6) is called formally exact if for any s ≥ 0 the sequence of homomorphisms

\overline{J}^{k_{1}+k_{2}+s}(P_{1}) ⟶^{\overline{J} (Δ_{1})} \overline{J}^{k_{2}+s}(P_{2}) ⟶^{\overline{J} (Δ_{2})} \overline{J}^{s}(P_{3})

is exact.

The following lemma is a fact from the formal theory of differential equations.

Lemma 1. Consider a sequence of differential operators

P_{0} ⟶^{Δ_{0}} P_{1} ⟶^{Δ_{1}} ⋯ ⟶^{Δ_{l−1}} P_{l} → P_{l+1} = 0, ord Δ_{i} = k_{i}.  (7)

Suppose that for any m = 1, …, l and i ≥ 0 the sequence of the prolongations of the C-symbols

S^{k_{m−1}+i}\overline{Λ}^{1} ⊗ P_{m−1} ⟶^{s_{i}(Δ_{m−1})} S^{i}\overline{Λ}^{1} ⊗ P_{m} ⟶^{s_{i−km}(Δ_{m})} S^{i−k_{m}} \overline{Λ}^{1} ⊗ P_{m+1}

is exact, where s_{i}(Δ_{m}) = 0 and S^{i}\overline{Λ}^{1} = 0 if i < 0. Then sequence (7) is formally exact.

The following proposition estimates the term E_{1} of the C-spectral sequence.

Proposition 4 ([5]). Let Y = Y(F) be a regular differential equation. Assume that there exists a formally exact sequence of C-differential operators

P_{0} = [κ] ⟶^{[ℓ_{F}]} P_{1} ⟶^{Δ_{1}} ⋯ ⟶^{Δ_{k−1}} P_{k} → 0.

Then 1. E_{1}^{p,q} = 0 if p > 0 and q < n − k;

2. For any p > 0 there exists a complex of C-differential operators

0 → C^{p−1}\overline{Λ}^{1} ⊗ P̂_{k} ⟶^{(Δ_{k−1}^{∗})_{p−1}} C^{p−1}\overline{Λ}^{1} ⊗ P̂_{k−1} ⟶^{(Δ_{k−2}^{∗})_{p−1}} ⋯

⟶^{[ℓ_{F}^{∗}]_{p−1}} C^{p−1}\overline{Λ}^{1} ⊗ [κ̂] → 0 (8)

and the term E_{1}^{p,q} is a direct summand in the cohomology group of complex (8) at term C^{p−1}\overline{Λ}^{1} ⊗ P̂_{n−q}.

3. The “three-line” theorem for the C-spectral sequence of the Yang–Mills equations

3.1. Let V be an oriented ℝ-linear space, dim V = n, and η: V × V → ℝ be a nondegenerate metric. Denote by H: Λ^{k} → Λ^{n−k}, k = 0, …, n, the Hodge isomorphism. If e_{1}, …, e_{n} is a positively oriented orthonormal basis, η(e_{i}, e_{j}) = 0 if i ≠ j and η(e_{i}, e_{i}) = η_{ii} = ±1, then

H(e_{i_{1}} ∧ ⋯ ∧ e_{i_{k}}) = η_{i_{1}i_{1}} … η_{i_{k}i_{k}}ϵ_{i_{1}...i_{k}j_{1}...j_{n−k}}e_{j_{1}} ∧ ⋯ ∧ e_{j_{n−k}},

where {j_{1}, …, j_{n−k}} = {1, …, n} \ {i_{1}, …, i_{k}}.

Consider the following d-complex

0 → S^{l}V ⟶^{d} S^{l−1}V ⊗ V ⟶^{d} S^{l−2}V ⊗ Λ^{2}V ⟶^{d} ⋯ ⟶^{d} Λ^{l}V → 0,

d(f ⊗ θ) = ∑_{i=1}^{n} \frac{∂f}{∂e_{i}} ⊗ e_{i} ∧ θ, f ∈ S^{k}V, θ ∈ Λ^{l−k}V, 0 ≤ k ≤ n,

which is the de Rham complex with polynomial coefficients. By definition, put

δ = (1 ⊗ H) ∘ d ∘ (1 ⊗ H): S^{k}V ⊗ Λ^{i}V → S^{k−1}V ⊗ Λ^{i−1}V

and consider the δ-complex

0 ⟶^{S}^{l} V ⊗ Λ^{n}V ⟶^{δ} S^{l−1}V ⊗ Λ^{n−1}V ⟶^{δ} ⋯ ⟶^{δ} S^{l−n}V → 0,

which is clearly isomorphic to the d-complex. It is easy to see that

δ(f ⊗ θ) = η_{11} ⋯ η_{nn} ∑_{i=1}^{n} η_{ii} \frac{∂f}{∂e_{i}} ⊗ \frac{∂θ}{∂e_{i}} .

The following lemma is a fact from linear algebra.

Lemma 2. The d-complex (resp. δ-complex) is acyclic if l > 0 (resp. l > n).

Lemma 3. The linear map

d + δ: S^{l}V ⊗ Λ^{k−1}V ⊕ S^{l}V ⊗ Λ^{k+1}V → S^{l−1}V ⊗ Λ^{k}V

is an epimorphism.

Proof. Choose a positively oriented orthonormal basis e_{1}, …, e_{n} of V. Denote ϵ = η_{11} ⋯ η_{nn}. Consider an element e_{1}^{l} ⊗ θ ∈ S^{l}V ⊗ Λ^{k}V. Obviously, θ = e_{1} ∧ θ′ + θ′′, where ∂θ′/∂e_{1} = ∂θ′′/∂e_{1} = 0. Then we have

e_{1}^{l} ⊗ θ = e_{1}^{l} ⊗ e_{1} ∧ θ′ + e_{1}^{l} ⊗ θ′′

= d(\frac{e_{1}^{l+1}}{l + 1} ⊗ θ′) + δ(ϵη_{11} \frac{e_{1}^{l+1}}{l + 1} ⊗ e_{1} ∧ θ′′),

that is e_{1}^{l} ⊗ θ ∈ im(d + δ).

By induction, suppose that e_{1}^{i}f ⊗ θ ∈ im(d + δ), where i = m, m + 1, …, l and ∂f/∂e_{1} = 0. Consider e_{1}^{m−1}f ⊗ θ, ∂f/∂e_{1} = 0. As above

e_{1}^{m−1}f ⊗ θ = e_{1}^{m−1}f ⊗ e_{1} ∧ θ′ + e_{1}^{m−1}f ⊗ θ′′

= d(\frac{e_{1}^{m}}{m} f ⊗ θ′) − \frac{e_{1}^{m}}{m} (∑_{i=2}^{n} \frac{∂f}{∂e_{i}} ⊗ e_{i} ∧ θ′)

+ δ(ϵη_{11} \frac{e_{1}^{m}}{m} f ⊗ e_{1} ∧ θ′′) − \frac{e_{1}^{m}}{m} (ϵη_{11} ∑_{i=2}^{n} η_{ii} \frac{∂f}{∂e_{i}} ⊗ e_{1} ∧ \frac{∂θ′′}{∂e_{i}}).

Hence, by inductive hypothesis, e_{1}^{m−1}f⊗θ ∈ im(d+δ). Thus im(d+δ) = S^{l}V ⊗ Λ^{k}V and the lemma is proved.

Corollary 1. The following complex

S^{l}V ⊗ Λ^{k}V ⟶^{□} S^{l−2}V ⊗ Lambda^{n−k}V ⟶^{d} S^{l−3}V ⊗ Λ^{n−k+1}V ⟶^{d} ⋯

⟶^{d} S^{l−2−k}V ⊗ Λ^{n}V → 0,

where □ = d ∘ H ∘ d and 0 ≤ k ≤ n, is exact.

3.2. Yang–Mills equations. We choose spacetime to be the manifold M = ℝ^{n} with local coordinates x^{i}, i = 1, …, n, equipped with a flat metric η. By definition, put ∂_{i} = ∂/∂x^{i}. To define the Yang– Mills field consider a trivial fiber bundle over the spacetime with the structure group G, dim G = m, M × G → M. Let 𝔤 be the Lie algebra of the Lie group G and τ^{α}, α = 1, …, m, be a basis of 𝔤 with the structure constants c_{βγ}^{α}. Consider a trivial vector bundle ξ: 𝔤 × M → M. Sections of this bundle can be identified with infinitesimal gauge transformation. Denote the C^{∞}(M)-module of sections by Γ(ξ) = G. In local coordinates, ϵ ∈ G is written in the form ϵ = ϵ^{α}τ_{α}, ϵ^{α} ∈ C^{∞}(M).

We can represent a connection on the principal fiber bundle by a 1-form A taking values in 𝔤. This 1-form is called the Yang–Mills field and is a section of a vector bundle π: ξ ⊗ T^{∗}M → M over M with the fiber 𝔤 ⊗ T_{x_{0}}^{∗}M at point x_{0} ∈ M . In local coordinates one has

A = A_{i}^{α}dx^{i} ⊗ τ_{α}.

Consider the infinite jets bundle J^{∞}(π). The module κ can be identified with

κ = \overline{Λ}^{1}(π) ⊗ \overline{G},

where the F-module \overline{G} is the lifting of the C^{∞}(M)-module G up to J^{∞}(π). Then

κ̂ = ˆ \overline{Λ}^{1}(π) ⊗ \overline{G}^{∗} = \overline{Λ}^{n−1}(π) ⊗ \overline{G}^{∗}.

Using the Hodge isomorphism H: \overline{Λ}^{n−1}(π) → \overline{Λ}^{1} and the chosen basis τ_{α} of 𝔤, we identify

κ = κ̂ = \overline{Λ}^{1}(π) ⊗ \overline{G}.

The curvature of a connection is a 2-form F on M with values in 𝔤, which is called the Yang–Mills field strength. In local coordinates,

F = F_{ij}^{α}dx^{i} ∧ dx^{j} ⊗ τ_{α}, F_{ij}^{α} = ∂_{i}A_{j}^{α} − ∂_{j}A_{i}^{α} + c_{βγ}^{α}A_{i}^{β}A_{j}^{γ}.

The Yang–Mills field equations are given by a differential operator of the 2-nd order Φ: ξ → ξ given in local coordinates as follows

Φ(A)_{j}^{α} = ∂^{i}F_{ij}^{α} + c_{βγ}^{α}A^{βi}F_{ij}^{γ}, α = 1, …, m, j = 1, …, n. (9)

Equations (9) define an infinitely prolonged equation Y = Y(Φ) ⊂ J^{∞}(π). Let ℓ_{Φ} : κ → κ be the universal linearization of Φ. Since equations (9) are Euler–Lagrange equations, the C-differential operator ℓ_{Φ}^{∗}: κ̂ → κ̂ in our identification κ = κ̂ coincides with ℓ_{Φ}.

An infinitesimal gauge transformation is given by a differential operator R: G → κ. Let \overline{R}: \overline{G} → κ be the lifting of the operator R up to J^{∞}(π). In local coordinates the C-differential operator \overline{R} is written as follows

\overline{R}(ϵ)_{i}^{α} = D_{i}(ϵ^{α}) + c_{βγ}^{α}A_{i}^{β}ϵ^{γ},

where ϵ = (ϵ^{β}) ∈ \overline{G}, β = 1, …, m, i = 1, …, n, D_{i} = \overline{∂_{i}} is a total derivative. Since the Yang–Mills equations are invariant under infinitesimal gauge transformations, we have

[ℓ_{Φ}] ∘ [\overline{R}] = 0,  (10)

where [·] denotes the restriction of a C-differential operator to the equation Y. From (10) one immediately obtain

[\overline{R}]^{∗} ∘ [ℓ_{Φ}]^{∗} = [\overline{R}]^{∗} ∘ [ℓ_{Φ}] = 0.  (11)

Equality (11) is a corollary of the second Noether theorem. Thus one has the complex

\overline{G} ⊗ \overline{Λ}^{1} ⟶^{[ℓ_{Φ}]} \overline{G} ⊗ \overline{Λ}^{n−1} ⟶^{[\overline{R}]^{∗}} \overline{G} ⊗ \overline{Λ}^{n} → 0.  (12)

Theorem 1. Complex (12) is formally exact.

Proof. Put c_{βγ}^{α} = 0, that is consider the Abelian case. Clearly, the symbols of the C-differential operators ℓ_{Φ} and \overline{R}^{∗} do not change. Then one easily obtains

Φ(A)_{j}^{α} = ∂^{i}∂_{i}A_{j}^{α} − ∂^{i}∂_{j}A_{i}^{α}, \overline{R}(ϵ)_{i}^{α} = D_{i}(ϵ^{α}).

It is not difficult to see that the C-symbols of the C-differential operators ℓ_{Φ} and \overline{R}^{∗} are written as follows

s_{0}(ℓ_{Φ}) = 1 ⊗ (d ∘ H ∘ d): \overline{G} ⊗ S^{2}\overline{Λ}^{1} ⊗ \overline{Λ}^{1} → \overline{G} ⊗ \overline{Λ}^{n−1},
s_{0}(\overline{R}^{∗}) = 1 ⊗ d: \overline{G} ⊗ \overline{Λ}^{1} ⊗ \overline{Λ}^{n−1} → \overline{G} ⊗ \overline{Λ}^{n}.

The theorem follows now from Corollary 1 and Lemma 1.

Combining Theorem 1 and Proposition 4, one obtains the following theorem.

Theorem 2 (the “three-line” theorem). E_{1}^{p,q} = 0 if p > 0 and q < n − 2.

acknowledgements

The author is grateful to Professor A. M. Vinogradov for attention to this work and to A. M. Verbovetsky for useful discussions.

He also wishes to thank the Scuola Internatzionale Superiore di Studi Avanzati (SISSA) in Trieste, where this work was written, for warm hospitality and the Italian Ministero degli Affari Esteri for a fellowship.

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