The Diffiety Institute Preprint Series

Preprint DIPS–2/2008 September 24, 2008

Available via INTERNET: http://diffiety.ac.ru; http://diffiety.org

The Diffiety Institute

Secondary Calculus and the Covariant Phase Space

by

L. Vitagliano

Last revised DIPS-2/2008 20.01.2009 arXiv:0809.4164

Secondary Calculus and the Covariant Phase Space

L. VITAGLIANO

Abstract. The covariant phase space of a lagrangian field theory is the solution space of the associated Euler-Lagrange equations. It is, in principle, a nice environment for covariant quantization of a lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) “presymplectic structure” ω (as first noticed by Zuckerman in 1986) whose degeneracy (functional) distribution is naturally interpreted as the Lie algebra of gauge transformations. We propose a fully rigorous approach to the covariant phase space in the framework of jet spaces and (A. M. Vinogradov’s) secondary calculus. In particular, we describe the degeneracy distribution of ω. As a byproduct we rederive the existence of a Lie bracket among gauge invariant functions on the covariant phase space.

Introduction

Covariant phase space (CPS) is the solution space of a system of Euler–Lagrange partial differential equations^{1} (PDEs). It has been first noticed by Zuckerman in the 1986 [40] (see also [11, 12]) that there is a canonical, closed 2–form ω on such a functional space generalizes the symplectic form on the phase space of a regular Lagrangian system in mechanics. Moreover, the degeneracy distribution of ω is naturally interpreted as Lie algebra of gauge transformations [25]. Therefore, the CPS is, in principle, a nice environment to perform a covariant (canonical) quantization of a lagrangian theory. Namely, gauge invariant functions on the CPS possess a well defined Lie bracket induced by ω, which has been proved in [8] to coincide with the so–called Peierls bracket [29].

2000 Mathematics Subject Classification. 53C80, 57R99, 70S05. Key words and phrases. Covariant Phase Space, Secondary Calculus, Geometry of PDEs, Gauge

Theory.

1Notice that sometimes the name covariant phase space is referred to the quotient of the above mentioned solution space with respect to gauge transformations.

2 L. VITAGLIANO

List of Main Symbols

J^{∞}π  space of ∞–jets of local sections of the bundle π : E ⟶ M
diff(π, τ)  module of differential operators from π to τ
ℰ_{Φ}  PDE determined by the differential operator Φ
ℰ  ∞ prolongation of a PDE
𝒞  Cartan distribution
𝒞 D(ℰ)  module of horizontal vector fields on ℰ
𝒞 Λ(ℰ)  Cartan ideal of ℰ
𝒞^{p}Λ(ℰ)  pth exterior power of 𝒞 Λ(ℰ)
𝒞^{•}Λ(ℰ)  algebra generated by 𝒞 Λ^{1}(ℰ)
𝒞 E(ℰ)  𝒞 –spectral sequence of ℰ
\overline{Λ}(ℰ)  algebra of horizontal forms on ℰ
\overline{d}  horizontal de Rham differential
\overline{H}(ℰ)  horizontal de Rham cohomology of ℰ
d^{V}  vertical de Rham differential
V D(ℰ)  module of vertical vector fields on ℰ
D_{𝒞} (ℰ)  Lie algebra of symmetries of (ℰ , 𝒞)
Sym(ℰ)  Lie algebra of non–trivial symmetries of (ℰ , 𝒞)
V D_{𝒞} (ℰ)  Lie algebra of vertical symmetries of (ℰ , 𝒞)
κ  module of generating sections of higher symmetries of π
ℓ_{Φ}  universal linearization of the differential operator Φ
C^{∞}(M)^{•} space of secondary functions on the secondary manifold M
D(M)^{•}  space of secondary vector fields on M
Λ(M)^{•}  space of secondary differential forms on M
d  secondary de Rham differential
\overline{S}  horizontal Spencer differential
𝒞 Diff(P, Q) module of horizontal differential operators P ⟶ Q
\overline{J}^{∞}P  module of ∞ horizontal jets of elements of P
\overline{j}_{∞}  ∞ horizontal jet prolongation P ⟶ \overline{J}^{∞}P
h_{□}^{∞}  homomorphism \overline{J}^{∞}P ⟶ \overline{J}^{∞}Q associated to □ ∈ 𝒞 Diff(P, Q)
η_{Φ}  natural monomorphism V D(ℰ) ⟶ \overline{J}^{∞}κ|_{ℰ}
η_{Φ}^{∗}  natural epimorphism 𝒞 Diff(κ|_{ℰ} , \overline{Λ}(ℰ)) ⟶ 𝒞 Λ^{1}(ℰ) ⊗ \overline{Λ}(ℰ)
∫  natural projection \overline{Λ}^{n}(ℰ) ⟶ \overline{H}^{n}(ℰ)
E(ℒ)  left hand side of the Euler–Lagrange equations
P  covariant phase space
ω  canonical, closed, secondary 2–form on P
Δ_{1}  compatibility operator for ℓ_{E(ℒ)}
Ω  linear map D(M)^{•} ⟶ Λ^{1}(M)^{•} associated to ω

In turn, Peierls bracket is at the basis of the global approach to quantum field theory [13]. Despite its conceptual relevance, the CPS is, in general, a complicated functional space, which is difficult to handle with analytic methods. Indeed, most of the literature about it (see [30] and references therein) comes from the physicists community and it is rarely completely rigorous from a mathematical point of view. For instance, it seems to be very hard to rigorously perform, in full generality, a symplectic reduction of the CPS to get rid of gauge (non–physical) degrees of freedom. On the other hand, A. M. Vinogradov developed a whole theory, the so–called secondary calculus (see [37] and references therein, and [38] for a short introduction), which properly formalizes in cohomological terms the idea of a (local) functional differential calculus on the space of solutions of a generic system of PDEs (for this reason, roughly speaking, the word “secondary” in this paper could be considered as a synonym of “functional”). Thus, secondary calculus appears to be a suitable setting to rigorously investigate the CPS and its properties. The aim of the paper is to describe rigorously the CPS, its canonical 2–form and some their properties within secondary calculus. As a byproduct it will become transparent the analogy between the CPS and the phase space of constrained mechanical systems. The paper is divided into two parts. In order to make it as self–consistent as possible we review, in the first part, those aspects of secondary calculus that are needed for a suitable formalization of the CPS. In Sections 1.1, 1.2 and 1.3 we briefly describe the geometry and the main properties of jet spaces and differential equations, and relevant structures on them. In Section 1.4 we define secondary vector fields and differential forms, and summarize the main formulas of first order secondary calculus. In Sections 1.5 and 1.6 we review the main technical aspects of secondary calculus and how to handle the relevant cohomologies. The second part of the paper is devoted to the CPS and to original results on the subject. In Section 2.1 we introduce the CPS for a general lagrangian field theory (any number of variable and any order) and rederive the existence of a canonical 2– form ω on it completing the proof by Zuckerman [40]. In Section 2.2 we propose a “symplectic version” of the first Noether theorem, which makes it evident the analogy with hamiltonian mechanics. In Section 2.3 we describe the degeneracy distribution of ω and propose, and motivate, a new (and very natural) definition of (infinitesimal) gauge symmetries in field theory. In Section 2.4 we describe gauge invariant secondary functions on the CPS and show that they are endowed with a canonical Lie bracket (such bracket formalizes rigorously the Peierls bracket [29]). In Section 2.5 we outline a possible path through a “secondary symplectic reduction” of the CPS. Applications to concrete lagrangian theories will be presented somewhere else. Most of the (almost) trivial computations will be performed in some details to emphasize similarities between secondary calculus and standard calculus on manifolds.

Notations and Conventions. In this section we collect notations and conventions about some general constructions in differential geometry that will be used in the following. Let N be a smooth manifold. We denote by C^{∞}(N) the ℝ–algebra of smooth, ℝ– valued functions on N. We will always understand a vector field X on N as a derivation X : C^{∞}(N) ⟶ C^{∞}(N). The value of X at the point x ∈ M will be denoted by X_{x}. We denote by D(N) the C^{∞}(N)–module of vector fields over N, by Λ(M) = ⨁_{k} Λ^{k}(N) the graded ℝ–algebra of differential forms over N and by d : Λ(N) ⟶ Λ(N) the de Rham differential. If F : N_{1} ⟶ N is a smooth map of manifolds, we denote by F^{∗} : Λ(N) ⟶ Λ(N_{1}) its pull–back. Let α : W ⟶ N be a vector bundle and F : N_{1} ⟶ N a smooth map of manifolds. The C^{∞}(N)–module of smooth sections of α will be denoted by Γ(α). For s ∈ Γ(α) and x ∈ N we put, sometimes, s_{x} := s(x). The zero section of α will be denoted by o : N ∋ x ⟼ o_{x} := 0 ∈ α^{−1}(x) ⊂ W. The vector bundle on N_{1} induced by α via F will be denote by F^{∘}(α) : F^{∘}(W) ⟶ N:

\xymatrix{F^{∘}(W) \ar[r] \ar[d]_{F^{∘}(α)} & W \ar[d]^{α} \\ N_{1} \ar[r]^{F} & N} .

For any section s ∈ Γ(α) there exists a unique section, which we denote by F^{∘}(s) ∈ Γ(F^{∘}(α)), such that the diagram

\xymatrix{F^{∘}(W) \ar[r] & W \\ N_{1} \ar[r]^{F} \ar[u]^{F^{∘}(s)} & N \ar[u]_{s}}

commutes. If i_{L} : L ↪ N is the embedding of a submanifold then we put α|_{L} := i_{L}^{∘}(α), Γ(α)|_{L} := Γ(α|_{L}) and for s ∈ Γ(α), s|_{L} := i_{L}^{∘}(s). s|_{L} will be referred to as the restriction to L of s. Let F : N_{1} ⟶ N be as above. A vector field along F is an ℝ–linear map X : C^{∞}(N) ⟶ C^{∞}(N_{1}) such that the following Leibnitz rule holds: X(fg) = F^{∗}(f)X(g)+ F^{∗}(g)X(f), f, g ∈ C^{∞}(N). Vector fields along F identify with sections of the induced bundle F^{∘}(τ_{N}) : F^{∘}(T N) ⟶ N_{1}, τ_{N} : T N ⟶ N being the tangent bundle to N. Let ζ : A ⟶ N be a fiber bundle. We denote by νζ : V ζ ⟶ A the vertical (with respect to ζ) tangent bundle to A and by V_{a}ζ := (νζ)^{−1}(a) its fiber over a ∈ A. Notice that V ζ ⊂ T A, the tangent manifold to A. If ζ_{1} : A_{1} ⟶ N_{1} is another fiber bundle, F : A_{1} ⟶ A a morphism of fiber bundles and T F : T A_{1} ⟶ T A the associated tangent map, then (T F)(V ζ_{1}) ⊂ V ζ and, therefore, it is well defined the restriction V F : V ζ_{1} ⟶ V ζ of T F to V ζ_{1} and V ζ, and the diagram

\xymatrix{V ζ_{1} \ar[r]^{V F} \ar[d]_{νζ_{1}} & V ζ \ar[d]^{νζ} \\ A_{1} \ar[r]^{F} & A}

commutes. Let

\xymatrix{⋯ \ar[r] & K_{l−1} \ar[r]^{δ_{l−1}} & K_{l} \ar[r]^{δ_{l}} & K_{l+1} \ar[r]^{δ_{l+1}} & ⋯}

be a complex. Put K := ⨁_{l} K_{l} and δ := ⨁_{l} δ_{l}. We denote by H(K, δ) := ⨁_{l} H^{l}(K, δ), the cohomology space of (K, δ), H^{l}(K, δ) := ker δ_{l}/ im δ_{l−1}. If ω ∈ ker δ, then we denote by [ω] its cohomology class. Denote by ℕ the set of natural numbers and put ℕ_{0} := ℕ ∪ {0}. We will always understand the sum over repeated upper-lower (multi-)indexes. Our notations about multi-indexes are the following. Let n ∈ ℕ, 𝕀_{n} = {1, …, n} and 𝕄_{n} be the free abelian monoid generated by 𝕀_{n}. Even if 𝕄_{n} is abelian we keep for it the multiplicative notation. Thus if I = i_{1} ⋯ i_{l}, J = j_{1} ⋯ j_{m} ∈ 𝕄_{n} are (equivalence classes of) words, i_{1}, …, i_{l}, j_{1}, …, j_{m} ∈ 𝕀_{n}, we denote by IJ = i_{1} ⋯ i_{l}j_{1} ⋯ j_{m} their composition. If I = i_{1} ⋯ i_{l} ∈ 𝕄_{n} is a word, i_{1}, …, i_{l} ∈ 𝕀_{n}, denote by |I| := l its length. We denote by O the (equivalence class of the) empty word. An element I ∈ 𝕄_{n} is called an n-multi-index (or, simply, a multi–index) and |I| the length of the multi-index. For k ≤ ∞ let 𝕄_{n,k} ⊂ 𝕄_{n} be the subset made of multi-indexes of length ≤ k. If (x^{1}, …, x^{n}) are local coordinates on a manifold N, n = dim N, and I = i_{1} ⋯ i_{k} ∈ 𝕄_{n}, we put \frac{∂^{|I|}}{∂x^{I}} := \frac{∂^{k}}{∂x^{i_{1}}···∂x^{i_{k}}}. We stress that this notation is different from more popular ones (see, for instance, [3]).

1. Secondary Calculus

1.1. Jet Spaces and PDEs. Let π : E ⟶ M be a fiber bundle, dim M = n, dim E = m + n. For l ≤ k ≤ ∞, we denote by π_{k} : J^{k}π ⟶ M the bundle of kjets of local sections of π, and by π_{k,l} : J^{k}π ⟶ J^{l}π the canonical projection. For any local section p : U ⟶ E of π, U ⊂ M being an open subset, we denote by j_{k}p : U ⟶ J^{k}π its kth jet prolongation and by Γ_{p}^{k} := im j_{k}p its graph. For x ∈ U, put [p]_{x}^{k} := (j_{k}p)(x). Any system of adapted to π coordinates (…, x^{i}, …, u^{α}, …) on an open subset U of E gives rise to a system of jet coordinates (…, x^{i}, …, u_{I}^{α}, …) on π_{k,0}^{−1}(U) ⊂ J^{k}π, i = 1, …, n, α = 1, …, m, I ∈ 𝕄_{n,k}, where we put u_{O}^{α} := u^{α}, α = 1, …, m. If a local section p of π is locally given by

u^{α} = p^{α}(…, x^{i}, …), α = 1, …, m,  (1)

then j_{k}p is locally given by

u_{I}^{α} = (\frac{∂^{|I|}}{∂x^{I}} p^{α})(…, x^{i}, …), α = 1, …, m, I ∈ 𝕄_{n,k}.

Recall that J^{∞}π is, by definition, an inverse limit of the tower of projections

\xymatrix{M & E \ar[l]_{π} & ⋯ \ar[l] & J^{k}π \ar[l]_{π_{k,k−1}} & J^{k+1}π \ar[l]_{π_{k+1,k}} & ⋯ \ar[l]} .  (2)

Now, let k < ∞, τ_{0} : T_{0} ⟶ J^{k}π be a vector bundle, dim T_{0} = dim J^{k}π + p, and (…, x^{i}, …, u_{I}^{α}, …, v^{a}, …) adapted to τ_{0}, local coordinates on T_{0}. A (possibly nonlinear) differential operator of order ≤ k ‘acting on local sections of π, with values in τ_{0}’ (in short ‘from π to τ_{0}’) is a section Φ : J^{k}π ⟶ T_{0} of τ_{0}. For any local section p : U ⟶ E of π, Φ determines an ‘image’ section Δ_{Φ}p := Φ ∘ j_{k}p : U ⟶ T_{0} of the bundle \underline{τ}_{0} := π_{k} ∘ τ_{0} : T_{0} ⟶ M. If Φ is locally given by

v^{a} = Φ^{a}(…, x^{i}, …, u_{I}^{β}, …), a = 1, …, p,  (3)

and p is locally given by (1), then Δ_{Φ}p is locally given by

\{ \begin{matrix}u_{I}^{α} = (\frac{∂^{|I|}}{∂x^{I}} p^{α})(…, x^{i}, …) \\ v^{a} = Φ^{a}(…, x^{i}, …, (\frac{∂^{|J|}}{∂x^{J}} p^{β})(…, x^{j}, …), …) \end{matrix} ,

α = 1, …, m, I ∈ 𝕄_{n,k}, a = 1, …, p. This motivates the name ‘differential operator’ for Φ. Denote by diff_{k}(π, τ_{0}) the set of all differential operators of order ≤ k from π to τ_{0}. For Φ ∈ diff_{k}(π, τ_{0}) and l ≤ ∞ we define the lth prolongation of Φ as follows. Consider the space J^{l}\underline{τ}_{0} of l-jets of local sections of \underline{τ}_{0}, and local jet coordinates (…, x^{i}, …, u_{I,J}^{α}, …, v_{J}^{a}, …) on J^{l}\underline{τ}_{0}, J ∈ 𝕄_{n,l}. In J^{l}\underline{τ}_{0} consider the submanifold T_{0}^{(l)} made of jets of local sections of the form Δ_{Ψ}p, where Ψ ∈ diff_{k}(π, τ_{0}) and p is a local section of π. T_{0}^{(l)} is locally defined by

u_{I,J}^{α} = u_{IJ}^{α}, α = 1, …, m, I ∈ 𝕄_{n,k}, J ∈ 𝕄_{n,l}.

Thus (…, x^{i}, …, u_{I}^{α}, …, v_{J}^{a}, …), I ∈ 𝕄_{n,k+l}, J ∈ 𝕄_{n,l}, are local coordinates on T_{0}^{(l)}. T_{0}^{(l)} projects canonically onto J^{k+l}π and the projection τ_{0}^{(l)} : T_{0}^{(l)} ⟶ J^{k+l}π is a vector bundle. Moreover, coordinates (…, x^{i}, …, u_{I}^{α}, …, v_{J}^{a}, …) on T_{0}^{(l)} are adapted to τ_{0}^{(l)}. Finally, define the lth prolongation Φ^{(l)} : J^{k+l}π ⟶ T_{0}^{(l)} of Φ by putting Φ^{(l)}([p]_{x}^{k+l}) := [Δ_{Φ}p]_{x}^{l} ∈ T_{0}^{(l)}, for all local sections p of π and x ∈ M. Then Φ^{(l)} ∈ diff_{k+l}(π, τ_{0}^{(l)}). For Φ ∈ diff_{k}(π, τ_{0}) put ℰ_{Φ} := {θ ∈ J^{k}π | Φ(θ) = 0}. ℰ_{Φ} is called the (system of) PDE(s) determined by Φ. For l ≤ ∞ put also ℰ_{Φ}^{(l)} := ℰ_{Φ^{(l)}}. ℰ_{Φ}^{(l)} is locally determined by equations

(D_{J}Φ^{a})(…, x^{i}, …, u_{I}^{α}, …) = 0, a = 1, …, p, J ∈ 𝕄_{n,l},  (4)

where D_{j_{1}···j_{l}} := D_{j_{1}} ∘ ⋯ ∘ D_{j_{l}} and D_{j} := ∂/∂x^{j} + u_{Ij}^{α}∂/∂u_{I}^{α} is the jth total derivative, j, j_{1}, …, j_{l} = 1, …, m. In the following we put ∂_{α}^{I} := ∂/∂u_{I}^{α} and ∂_{α} := ∂/∂u^{α}, α = 1, …, m, I ∈ 𝕄_{n}. A local section p of π is a (local) solution of ℰ_{Φ} iff, by definition, Γ_{p}^{k} ⊂ ℰ_{Φ} or, which is the same, Γ_{p}^{k+l} ⊂ ℰ_{Φ}^{(l)} for some l ≤ ∞. Notice that ℰ_{Φ}^{(∞)} ⊂ J^{∞}π is an inverse limit of the tower of maps

\xymatrix{M & ℰ_{Φ} \ar[l]_{π_{k}} & ⋯ \ar[l] & ℰ_{Φ}^{(l)} \ar[l]_{π_{k+l,k+l−1}} & ℰ_{Φ}^{(l+1)} \ar[l]_{π_{k+l+1,k+l}} & ⋯ \ar[l]}  (5)

and consists of “formal solutions” of ℰ_{Φ}, i.e., possibly non-converging Taylor series fulfilling (4) for every l. The PDE ℰ_{Φ} is called formally integrable iff ℰ_{Φ}^{(l)} ⊂ J^{k+l}π is a (closed) submanifold for any l < ∞ and (5) is a sequence of fiber bundles. Let us stress that, basically, all relevant PDEs in Mathematical Physics are formally integrable and, therefore, in the following, we will only consider differential operators determining formally integrable PDEs. J^{∞}π and ℰ_{Φ}^{(∞)} are not finite dimensional smooth manifolds, in general. However, they are pro-finite dimensional smooth manifolds. We do not give here a complete definition of a pro-finite dimensional smooth manifold, which would take too much space. Rather, we will just outline it. Basically, a pro-finite dimensional smooth manifold is a(n equivalence class of) set(s) 𝒪 together with a sequence of smooth fiber bundles

\xymatrix{𝒪_{0} & 𝒪_{1} \ar[l]_{μ_{1,0}} & ⋯ \ar[l] & 𝒪_{k} \ar[l]_{μ_{k,k−1}} & 𝒪_{k+1} \ar[l]_{μ_{k+1,k}} & ⋯ \ar[l]}  (6)

and maps μ_{∞,k} : 𝒪 ⟶ 𝒪_{k}, 0 ≤ k < ∞, such that 𝒪 (together with the μ_{∞,k}’s) is an inverse limit of (6). It is associated to the sequence (6) a filtration of algebras

\xymatrix{C^{∞}(𝒪_{0}) \ar[r]^{μ_{1,0}^{∗}} & ⋯ \ar[r] & C^{∞}(𝒪_{k−1}) \ar[r]^{μ_{k,k−1}^{∗}} & C^{∞}(𝒪_{k}) \ar[r]^{μ_{k+1,k}^{∗}} & ⋯} .  (7)

We understand the monomorphisms μ_{l+1,l}^{∗}’s and interpret (7) as a sequence of subalgebras. Similarly, we understand the μ_{∞,l} ’s and interpret elements in C^{∞}(𝒪_{k}) as functions on 𝒪. Put C^{∞}(𝒪) := ⋃_{l∈ℕ_{0}} C^{∞}(𝒪_{k}). C^{∞}(𝒪) is interpreted as algebra of smooth functions on 𝒪. Differential calculus over 𝒪 may then be introduced as filtered differential calculus over C^{∞}(𝒪) [37]. Since the main constructions (smooth maps, vector fields, differential forms, linear jets and differential operators, etc.) of such calculus do not look very different from the analogous ones in finite-dimensional differential geometry we will not insist on this and refer to [37] for the rigorous definitions and the main results (see [31] and [33, 34] for a sketch of alternative approaches). Here we just recall the definition of finite dimensional vector bundle over 𝒪. This is, basically, a vector bundle over 𝒪_{k} for some k < ∞, pull-backed to 𝒪 via μ_{∞,k}. In more details, let τ_{0} : T_{0} ⟶ 𝒪_{k} be a (finite dimensional) vector bundle, k < ∞. For l ≥ 0 let τ_{l} := μ_{k+l,k}^{∘}(τ_{0}) : T_{l} := μ_{k+l,k}^{∘}(T_{0}) ⟶ 𝒪_{k+l} be the induced (by τ_{0} via μ_{k+l,k}) vector bundle and ν_{l+1,l} : T_{l+1} ⟶ T_{l} the canonical projection. Denote by T the pro-finite dimensional smooth manifold determined by the sequence of fiber bundles

\xymatrix{T_{0} & T_{1} \ar[l]_{ν_{1,0}} & ⋯ \ar[l] & T_{l} \ar[l]_{ν_{l,l−1}} & T_{l+1} \ar[l]_{ν_{l+1,l}} & ⋯ \ar[l]} .  (8)

The maps τ_{l} : T_{l} ⟶ 𝒪_{l+k}, l ≥ 0, determine a smooth map τ : T ⟶ 𝒪. Any such map is, by definition, a (finite-dimensional) vector bundle over 𝒪. Notice that it is associated to the sequence (8) of vector bundle morphisms a filtration of vector spaces

\xymatrix{Γ(τ_{0}) \ar[r]^{μ_{k+1,k}^{∘}} & ⋯ \ar[r] & Γ(τ_{l−1}) \ar[r]^{μ_{k+l,k+l−1}^{∘}} & Γ(τ_{l}) \ar[r]^{μ_{k+l+1,k+l}^{∘}} & ⋯} .

We understand the monomorphisms μ_{k+l+1,k+l}^{∘}’s and interpret (7) as a sequence of vector subspaces. Similarly, we understand the μ_{∞,k+l}’s and interpret elements in Γ(τ_{l}) as functions 𝒪 ⟶ T. Put Γ(τ) := ⋃_{l∈ℕ_{0}} Γ(τ_{l}). Γ(τ) is naturally a C^{∞}(𝒪)-module and it is interpreted as the module of smooth sections of τ. As an example, let 𝒪 = J^{∞}π, τ_{0} : T_{0} ⟶ J^{k}π be a vector bundle for some k < ∞ and τ := π_{∞,k}^{∘}(τ_{0}) : T := π_{∞,k}^{∘}(T_{0}) ⟶ J^{∞}π. Since Γ(τ_{l}) = diff_{k+l}(π, τ_{l}) for any l, we have the filtration diff_{k}(π, τ_{0}) ⊂ diff_{k+1}(π, τ_{1}) ⊂ ⋯ ⊂ diff_{k+l}(π, τ_{l}) ⊂ · · ·. Put diff(π, τ) := ⋃_{l∈ℕ_{0}} diff_{k+l}(π, τ_{l}) = Γ(τ). Elements in diff(π, τ) are called differential operators ‘acting on local sections of π, with values in τ_{0}’ (in short ‘from π to τ_{0}’). They are nothing but sections of the vector bundle τ : T ⟶ J^{∞}π. An important technical advantage of formally integrable PDEs is the following. Let ℰ ⊂ J^{∞}(π) be the ∞th prolongation of a formally integrable PDE, τ : T ⟶ J^{∞}(π) a vector bundle and τ|_{ℰ} : T |_{ℰ} ⟶ ℰ its restriction to ℰ. Then for any section s ∈ Γ(τ|_{ℰ}) there exists a section s̃ ∈ Γ(τ) such that s = s̃|_{ℰ}. In the following we will often use this property without further comments. Finally, let us mention here that a vector field on an pro-finite dimensional manifold does not generate a flow in general (see, for instance, [10]).

1.2. The Cartan Distribution and the 𝒞 -Spectral Sequence. Let π : E ⟶ M and τ : T ⟶ J^{∞}π be as in the previous section and Φ ∈ diff(π, τ). In the following we will simply write J^{∞} for J^{∞}π and ℰ for ℰ_{Φ}^{(∞)}. i_{ℰ} : ℰ ↪ J^{∞} will denote the inclusion. Notice that for Φ = 0, ℰ = ℰ_{Φ}^{(∞)} = J^{∞}. Recall that J^{∞} is endowed with the Cartan distribution 𝒞 which is defined as follows:

𝒞 : J^{∞} ∋ θ ⟼ 𝒞_{θ} ⊂ T_{θ}J^{∞},

where 𝒞_{θ} := T_{θ}Γ_{p}^{∞} for θ = [p]_{x}^{∞}, x ∈ M. Denote by 𝒞 D(J^{∞}) ⊂ D(J^{∞}) the C^{∞}(J^{∞})- submodule made of vector fields in the Cartan distribution, i.e., vector fields X ∈ D(J^{∞}) such that X_{θ} ∈ 𝒞_{θ} for all θ ∈ J^{∞}. The Cartan distribution is n-dimensional, it is locally spanned by total derivatives …, D_{i}, … and it is involutive, i.e., [X, Y] ∈ 𝒞 D(J^{∞}) for all X, Y ∈ 𝒞 D(J^{∞}). Moreover, n-dimensional integral submanifolds L ⊂ J^{∞} of 𝒞 are of the form L = Γ_{p}^{∞} for some local section p of π. Let ℰ ⊂ J^{∞} be as above. The Cartan distribution 𝒞 restricts to ℰ in the sense that 𝒞_{θ} ⊂ T_{θ}ℰ for any θ ∈ ℰ. Abusing the notation we still denote by 𝒞 the restricted to ℰ distribution and call it the Cartan distribution of ℰ. Also we denote by 𝒞 D(ℰ) ⊂ D(ℰ) the C^{∞}(ℰ)-submodule made of vector fields in 𝒞. Elements in 𝒞 D(ℰ) are called horizontal vector fields. In particular, total derivatives restrict to ℰ, i.e., there are unique local vector fields …, D_{i}^{ℰ} , … on ℰ such that i_{ℰ}^{∗} ∘ D_{i} = D_{i}^{ℰ} ∘ i_{ℰ}^{∗}, i = 1, …, n. Again 𝒞 is locally spanned by vector fields …, D_{i}^{ℰ} , …, it is involutive and n-dimensional integral submanifolds of it are graphs Γ_{p}^{∞} of infinite jet prolongations of local solutions p of ℰ_{Φ}. A spectral sequence is naturally associated to an involutive distribution and, in particular, to the Cartan distribution on (the infinite prolongation of) a PDE as follows. Denote by 𝒞 Λ(ℰ) ⊂ Λ(ℰ) the subset made of differential forms ω such that

ω(X_{1}, …, X_{k}) = 0 for all X_{1}, …, X_{k} ∈ 𝒞 D(ℰ),

where k is the degree of ω. 𝒞 Λ(ℰ) is a differential ideal in Λ(ℰ). Namely, it is an algebraic ideal and, moreover, it is differentially closed, i.e., dω ∈ 𝒞 Λ(ℰ) for any ω ∈ 𝒞 Λ(ℰ). 𝒞 Λ(ℰ) is called the Cartan ideal of ℰ. For any p ∈ ℕ, denote by 𝒞^{p}Λ(ℰ) the pth exterior power of 𝒞 Λ(ℰ). Thus, the sequence

Λ(ℰ) ⊃ 𝒞 Λ(ℰ) ⊃ 𝒞^{2}Λ(ℰ) ⊃ ⋯ ⊃ 𝒞^{p}Λ(ℰ) ⊃ ⋯

is a filtration of the de Rham complex (Λ(ℰ), d) of ℰ. The associated spectral sequence is denoted by 𝒞 E(ℰ) = {(𝒞 E_{r}^{p,q}(ℰ), d_{r}^{p,q})}_{r}^{p,q} and called the 𝒞 -spectral sequence of ℰ [39]. It is regular and converges to de Rham cohomologies of ℰ. The first column of the 0th term of 𝒞 E(ℰ),

\xymatrix{0 \ar[r] & 𝒞 E_{0}^{0,0}(ℰ) \ar[r]^{d_{0}^{0,0}} & 𝒞 E_{0}^{0,1}(ℰ) \ar[r]^{d_{0}^{0,1}} & ⋯ \ar[r] & 𝒞 E_{0}^{0,q}(ℰ) \ar[r]^{d_{0}^{0,q}} & ⋯} ,

is, by definition, the quotient complex Λ(ℰ)/𝒞 Λ(ℰ), which is also denoted by

\xymatrix{0 \ar[r] & C^{∞}(ℰ) \ar[r]^{\overline{d}} & \overline{Λ}^{1}(ℰ) \ar[r]^{\overline{d}} & ⋯ \ar[r] & \overline{Λ}^{q}(ℰ) \ar[r]^{\overline{d}} & ⋯} ,

and called the horizontal de Rham complex of ℰ. Its cohomology algebra 𝒞 E_{1}^{0,•}(ℰ) is denoted by \overline{H}(ℰ), and called horizontal de Rham cohomology algebra of ℰ. Recall, in particular, that \overline{d}-closed elements in \overline{Λ}^{n−1}(ℰ) are called conserved currents and cohomology classes in \overline{H}^{n−1}(ℰ) conservation laws of the PDE ℰ_{Φ}. In the following we will denote by 𝒞 Λ^{k}(ℰ) (resp. 𝒞^{p}Λ^{k}(ℰ), \overline{Λ}^{k}(ℰ), \overline{H}^{k}(ℰ)) the kth homogeneous component of 𝒞 Λ(ℰ) (resp. 𝒞^{p}Λ(ℰ), \overline{Λ}(ℰ), \overline{H}(ℰ)), k ≥ 0, and by 𝒞^{•}Λ(ℰ) := ⨁_{p} 𝒞^{p}Λ^{p}(ℰ) ⊂ Λ(ℰ) the C^{∞}(ℰ)-subalgebra generated by 𝒞 Λ^{1}(ℰ). Notice that 𝒞^{p}Λ(ℰ) is generated by 𝒞^{p}Λ^{p}(ℰ) as an ideal, p > 0. The 𝒞 -spectral sequence 𝒞 E(ℰ) contains very relevant “invariants” of the PDE ℰ_{Φ} (see, for instance, [9, 21]). Moreover, it formalizes in a coordinate-free manner variational calculus (on local sections of π) constrained by ℰ_{Φ} [39]. Therefore, it is a most fundamental construction in the geometric theory of differential equations. Finally, it is a very general construction. For instance, it may be defined exactly in the same way when ℰ is the infinite prolongation of a system of PDEs “imposed on general n-dimensional submanifolds of E”. However, in the present case, the fibered structure π_{∞}|_{ℰ} : ℰ ⟶ M of ℰ allows a more simple description (which is, in the general case, valid only locally), the variational bi-complex [39], which we briefly recall in the following.

The Cartan distribution and the fibered structure π_{∞}|_{ℰ} : ℰ ⟶ M of ℰ determine a splitting of the tangent bundle T ℰ ⟶ ℰ into the Cartan or horizontal part 𝒞 and the vertical (with respect to π_{∞}) part V π_{∞}|_{ℰ}. Accordingly, D(ℰ) splits into a direct sum: D(ℰ) = 𝒞 D(ℰ) ⊕ V D(ℰ), V D(ℰ) ⊂ D(ℰ) being the C^{∞}(ℰ)-submodule made of π_{∞}- vertical vector fields, i.e., vector fields Y ∈ D(ℰ) such that Y ∘ π_{∞}^{∗} = 0. In particular, V D(J^{∞}) is locally generated by vector fields …, ∂_{a}^{I}, . . .. Dually, Λ^{1}(ℰ) splits into the direct sum

Λ^{1}(ℰ) = 𝒞 Λ^{1}(ℰ) ⊕ \overline{Λ}^{1}(ℰ);  (9)

here and in what follows \overline{Λ}^{1}(ℰ) is identified with the C^{∞}(ℰ)-submodule in Λ^{1}(ℰ) generated by π_{∞}^{∗}(Λ^{1}(M)). In particular, 𝒞 Λ^{1}(J^{∞}) is locally generated by forms …, ω_{I}^{α} := du_{I}^{α} − u_{Ii}^{α}dx^{i}, … and \overline{Λ}^{1}(J^{∞}) is locally generated by forms …, dx^{i}, . . .. Similarly, 𝒞 Λ^{1}(ℰ) is locally generated by forms …, i_{ℰ}^{∗}(ω_{I}^{α}), … and \overline{Λ}^{1}(ℰ) is locally generated by forms …, i_{ℰ}^{∗}(dx^{i}), . . .. In view of splitting (9) Λ(ℰ) factorizes as Λ(ℰ) ≃ 𝒞^{•}Λ(ℰ) ⊗ \overline{Λ}(ℰ) (here and in what follows tensor products will be always over C^{∞}(ℰ), or C^{∞}(J^{∞}) for Φ = 0). In particular, there are projections 𝔭_{p,q} : Λ(ℰ) ⟶ 𝒞^{p}Λ^{p}(ℰ) ⊗ \overline{Λ}^{q}(ℰ) for any p, q ∈ ℕ_{0}. Correspondingly, the de Rham complex of ℰ, (Λ(ℰ), d), splits in a bi-complex (𝒞^{•}Λ(ℰ) ⊗ \overline{Λ}(ℰ), \overline{d}, d^{V}) (in the following diagram we drop for simplicity the postfix (ℰ)),

\xymatrix{ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ 0 \ar[r] & 𝒞^{p+1}Λ^{p+1} \ar[r]^{\overline{d}} \ar[u]^{d^{V}} & ⋯ \ar[r] & 𝒞^{p+1}Λ^{p+1} ⊗ \overline{Λ}^{q} \ar[u]^{d^{V}} \ar[r]^{\overline{d}} & 𝒞^{p+1}Λ^{p+1} ⊗ \overline{Λ}^{q+1} \ar[u]^{d^{V}} \ar[r]^{\overline{d}} & ⋯ \\ 0 \ar[r] & 𝒞^{p}Λ^{p} \ar[r]^{\overline{d}} \ar[u]^{d^{V}} & ⋯ \ar[r] & 𝒞^{p}Λ^{p} ⊗ \overline{Λ}^{q} \ar[u]^{d^{V}} \ar[r]^{\overline{d}} & 𝒞^{p}Λ^{p} ⊗ \overline{Λ}^{q+1} \ar[u]^{d^{V}} \ar[r]^{\overline{d}} & ⋯ \\ & ⋯ \ar[u] & ⋯ & ⋯ \ar[u] & ⋯ \ar[u] & ⋯ \\ 0 \ar[r] & C^{∞} \ar[r]^{\overline{d}} \ar[u]^{d^{V}} & ⋯ \ar[r] & _{\overline{Λ}_{q}} \ar[r]^{\overline{d}} \ar[u]^{d^{V}} & _{\overline{Λ}_{q+1}} \ar[r]^{\overline{d}} \ar[u]^{d^{V}} & ⋯ \\ & 0 \ar[u] & & 0 \ar[u] & 0 \ar[u]} ,

(10)

defined by

\overline{d}(ω ⊗ \overline{σ}) := (𝔭_{p,q+1} ∘ d)(ω ∧ \overline{σ}) and d^{V} (ω ⊗ \overline{σ}) := (𝔭_{p+1,q} ∘ d)(ω ∧ \overline{σ}),

where ω ∈ 𝒞^{p}Λ^{p}(ℰ) and \overline{σ} ∈ \overline{Λ}^{q}(ℰ), p, q ∈ ℕ_{0}. \overline{d} and d^{V} are called the horizontal and the vertical de Rham differential, respectively, and (10) is called the variational bicomplex. In the following we will often understand isomorphism Λ(ℰ) ≃ 𝒞^{•}Λ(ℰ)⊗\overline{Λ}(ℰ). As a bi-complex (10) determines two spectral sequences. One of them is the 𝒞 - spectral sequence while the other is the Leray-Serre spectral sequence of the fibration π_{∞}|_{ℰ} : ℰ ⟶ M [27]. In particular, for any p, there is a canonical isomorphisms of complexes

(𝒞 E_{0}^{p,•}(ℰ), d_{0}^{p,•}) ≃ (𝒞^{p}Λ^{p}(ℰ) ⊗ \overline{Λ}(ℰ), \overline{d}),  (11)

and the differential d_{1}^{p,•} : 𝒞 E_{1}^{p,•}(ℰ) ⟶ 𝒞 E_{1}^{p+1,•}(ℰ) is isomorphic to the map induced by d^{V} in the cohomology H(𝒞^{p}Λ^{p}(ℰ) ⊗ \overline{Λ}(ℰ), \overline{d}). Notice that the embedding i_{ℰ} : ℰ ↪ J^{∞} of the infinite prolongation ℰ of a PDE determines via pull-back both a morphism of spectral sequences and a morphism of bi-complexes that, abusing the notation, we denote by the same symbol

i_{ℰ}^{∗} : {(𝒞 E_{r}^{•,•}(J^{∞}), d_{r}^{•,•})} ⟶ {(𝒞 E_{r}^{•,•}(ℰ), d_{r}^{•,•})},
i_{ℰ}^{∗} : (𝒞^{•}Λ(J^{∞}) ⊗ \overline{Λ}(J^{∞}), \overline{d}, d^{V}) ⟶ (𝒞^{•}Λ(ℰ) ⊗ \overline{Λ}(ℰ), \overline{d}, d^{V}).

1.3. Higher Symmetries of PDEs. Denote by D_{𝒞} (ℰ) ⊂ D(ℰ) the subset made of vector fields preserving the Cartan distribution, i.e., vector fields X such that [X, Y] ∈ 𝒞 D(ℰ) for any Y ∈ 𝒞 D(ℰ). D_{𝒞} (ℰ) is clearly a Lie subalgebra in D(ℰ). Elements in D_{𝒞} (ℰ) are called (infinitesimal) symmetries of ℰ_{Φ}. The theory of infinitesimal symmetries of PDEs is fundamental in many respects [9]. Notice that, since the Cartan distribution is involutive, then 𝒞 D(ℰ) ⊂ D_{𝒞} (ℰ) and it is an ideal in D_{𝒞} (ℰ). Elements in 𝒞 D(ℰ) are called trivial symmetries of ℰ_{Φ}, in that horizontal vector fields “are symmetries of every PDE”. The quotient Lie algebra Sym(ℰ) := D_{𝒞} (ℰ)/𝒞 D(ℰ) is called the algebra of non-trivial higher symmetries of ℰ_{Φ}. Clearly, every equivalence class X = X + 𝒞 D(ℰ) ∈ Sym(ℰ), X ∈ D_{𝒞} (ℰ), has got one and only one vertical representative X^{V} ∈ V D(ℰ). Any vertical element in D_{𝒞} (ℰ) is called an evolutionary vector field. Thus Sym(ℰ) is isomorphic to the Lie algebra V D_{𝒞} (ℰ) of evolutionary vector fields. In order to effectively describe V D_{𝒞} (ℰ) and, therefore, Sym(ℰ) let us first consider the case ℰ = J^{∞}. It is easy to prove that any evolutionary vector field Y ∈ V D_{𝒞} (J^{∞}) is determined by its restriction to C^{∞}(E) ⊂ C^{∞}(J^{∞}). Moreover, every vertical vector field χ : C^{∞}(E) ⟶ C^{∞}(J^{∞}) along π_{∞,0} : J^{∞} ⟶ E (χ is vertical if χ∘π^{∗} = 0) extends to a unique evolutionary vector field �_{χ} ∈ V D_{𝒞} (J^{∞}). We conclude that V D_{𝒞} (J^{∞}) is in one to one correspondence with the C^{∞}(J^{∞})-module κ of vector fields along π_{∞,0} or, which is the same, the module of sections of the induced vector bundle π_{∞,0}^{∘}(νπ) : π_{∞,0}^{∘}(V π) ⟶ J^{∞}. Elements in κ are called generating sections of higher symmetries of π. Let us now come to the general case when ℰ is any. First of all consider the C^{∞}(ℰ)- module κ|_{ℰ} of vertical vector fields χ : C^{∞}(E) ⟶ C^{∞}(ℰ) along π_{∞,0}|_{ℰ} : ℰ ⟶ E or, which is the same, the module of sections of the induced vector bundle π_{∞,0}|_{ℰ}^{∘}(νπ) : π_{∞,0}|_{ℰ}^{∘}(V π) ⟶ ℰ. Elements in κ|_{ℰ} are called generating sections of higher symmetries of ℰ. Similarly as to above, a generating section χ ∈ κ|_{ℰ} extends to a unique vertical vector field �_{χ} : C^{∞}(J^{∞}) ⟶ C^{∞}(ℰ) along the inclusion i_{ℰ} : ℰ ↪ J^{∞}. If χ is locally given by χ = χ^{α}∂_{α}, where …, χ^{α}, … are local functions on ℰ, then �_{χ} is locally given by �_{χ} = D_{I}^{ℰ} χ^{α}∂_{α}^{I}|_{ℰ}. However, in general �_{χ} is not tangent to ℰ and, therefore, is not in V D_{𝒞} (ℰ). Generating sections χ such that �_{χ} ∈ V D_{𝒞} (ℰ) are the ones in the kernel of a suitable differential operator: the so-called universal linearization of ℰ, which we now define (notice that to the author’s knowledge the following definition never appeared in the literature before in the general form presented here - see also [34]). Let τ, Φ and ℰ be as in the previous section, and put \underline{τ} := π_{∞} ∘ τ : T ⟶ M. Since τ is a vector bundle, V τ ⟶ T is naturally isomorphic to the induced bundle τ^{∘}(τ) : τ^{∘}(T) ⟶ T, τ^{∘}(τ) being the (restriction to τ^{∘}(T) ⊂ T × T of the) projection (T × T ⟶ T) onto the first factor. Denote by ρ_{2} : τ^{∘}(T) ⟶ T the projection onto the second factor and by ρ′_{2} : V τ ⟶ T the map induced by ρ_{2} via the isomorphism V τ ≃ τ^{∘}(T). Consider the vertical tangent map V Φ : V π_{∞} ⟶ V \underline{τ}. Put o_{ℰ} := o ∘ i_{ℰ} : ℰ ⟶ T and notice, preliminarily, that o_{ℰ} = Φ ∘ i_{ℰ}. The short exact sequence of induced bundles 0 ⟶ o_{ℰ}^{∘}(V τ) ⟶ o_{ℰ}^{∘}(V \underline{τ}) ⟶ V π_{∞}|_{ℰ} ⟶ 0 splits naturally via the map V o|_{ℰ} : V π_{∞}|_{ℰ} ⟶ o_{ℰ}^{∘}(V \underline{τ}) well defined by putting V o|_{ℰ} (θ, ξ) := (θ, V o(ξ)), (θ, ξ) ∈ V π_{∞}|_{ℰ}. In particular, there is a canonical projection V_{Φ} : o_{ℰ}^{∘}(V \underline{τ}) ⟶ o_{ℰ}^{∘}(V τ). Define a map

L_{Φ} : V π_{∞}|_{ℰ} ⟶ T |_{ℰ}

by putting L_{Φ}(ξ) := (θ, ρ′_{2}(V)), where (θ, V) := V_{Φ}(θ, V Φ(ξ)) ∈ o_{ℰ}^{∘}(V τ), for all ξ ∈ V_{θ}π_{∞}, θ ∈ ℰ. L_{Φ} is a morphism of vector bundles. For any χ ∈ κ|_{ℰ} let ℓ_{Φ}χ ∈ Γ(τ|_{ℰ}) be defined by putting (ℓ_{Φ}χ)_{θ} := L_{Φ}((�_{χ})_{θ}), θ ∈ ℰ. ℓ_{Φ} : κ|_{ℰ} ⟶ Γ(τ|_{ℰ}) is a well defined linear differential operator called the universal linearization of Φ. Let us describe ℓ_{Φ} locally. Let (…, x^{i}, …, u_{I}^{α}, …) be local jet coordinates on J^{∞}, (…, x^{i}, …, u_{I}^{α}, …, v^{a}, …) adapted to τ local coordinates on T, and (…, e_{a}, …) the local basis of Γ(τ|_{ℰ}) associated to them. If Φ has local representation (3), …, Φ^{a}, … being local functions on J^{∞}, and χ = χ^{α}∂_{α} locally, then

ℓ_{Φ}χ = e_{a}(∂_{α}^{I}Φ^{a})|_{ℰ} D_{I}^{ℰ} χ^{α}

locally. Now let χ ∈ κ|_{ℰ}. It is easy to see that if ℓ_{Φ}χ = 0 then �_{χ} is tangent to ℰ and, therefore, it is in V D_{𝒞} (ℰ). Vice versa, any symmetry Y ∈ V D_{𝒞} (ℰ) is of the form �_{χ} for a unique χ ∈ κ|_{ℰ} such that ℓ_{Φ}χ = 0. We conclude that Sym(ℰ) is in one to one correspondence with ker ℓ_{Φ}. In particular, ker ℓ_{Φ} inherits from Sym(ℰ) the Lie algebra structure. The corresponding bracket is denoted by {·, ·} and called the higher Jacobi bracket of the equation ℰ_{Φ}. Finally, notice that, for any χ ∈ ker ℓ_{Φ}, the ‘insertion of’ and the ‘Lie derivative along’ �_{χ} ∈ V D(ℰ) commute with the horizontal de Rham differential \overline{d} : Λ(ℰ) ⟶ Λ(ℰ), i.e.,

i_{�_{χ}} ∘ \overline{d} + \overline{d} ∘ i_{�_{χ}} = L_{�_{χ}} ∘ \overline{d} − \overline{d} ∘ L_{�_{χ}} = 0.  (12)

In their turn Identities (12) imply

i_{�_{χ}} ∘ d^{V} + d^{V} ∘ i_{�_{χ}} = L_{�_{χ}} , L_{�_{χ}} ∘ d^{V} − d^{V} ∘ L_{�_{χ}} = 0,

d^{V} : Λ(ℰ) ⟶ Λ(ℰ) being the vertical de Rham differential.

1.4. Secondary Differential Forms and Vector Fields. Let ℰ be as in the previous section. As noticed above, n-dimensional integral submanifolds of the Cartan distribution 𝒞 over ℰ are in one-to-one correspondence with local solutions of ℰ_{Φ}. Thus, informally speaking, the pair (ℰ , 𝒞) encodes all the information about the “functional space M of solutions” of ℰ_{Φ} (in the following we will in fact identify (ℰ , 𝒞) with M). For instance, “local functional calculus” over such functional space may be formalized geometrically (and homologically) by using (ℰ , 𝒞) as a starting point and the associated 𝒞 -spectral sequence as the main structure. Such formalization has been named secondary calculus [37] by its discoverer, A. M. Vinogradov, and its simplest constructions will be briefly reviewed in this section. Suppose temporarily that M is a compact, orientable and oriented manifold without boundary. Then an element S = [ℒ] ∈ \overline{H}^{n}(ℰ) = 𝒞 E_{1}^{0,n}(ℰ), ℒ ∈ \overline{Λ}^{n}(ℰ), identifies with the (local) action functional

M ∋ p ⟼ S(p) := ∫_{M}(j^{∞}p)^{∗}(ℒ) ∈ ℝ,

and in the following we will denote by

∫ : \overline{Λ}^{n}(ℰ) ∋ ℒ ⟼ ∫ ℒ := [ℒ] ∈ \overline{H}^{n}(ℰ)

the projection. Thus ℒ may be interpreted as the lagrangian density of a lagrangian theory constrained by the PDE ℰ_{Φ}. As a natural generalization, we interpret \overline{H}(ℰ), not only its n-degree component, as space of local function(al)s on M. By considering all less-dimensional cohomologies rather than just top ones we have in mind the possibility of defining functionals by integration on less-dimensional submanifolds of M. Such possibility is crucial in variational calculus with boundary conditions (see [26]). Similarly, for p > 0, 𝒞 E_{1}^{p,•}(ℰ) is naturally interpreted as space of local differential p-forms on M. This informal arguments motivate the

Definition 1. Elements in \overline{H}(ℰ) = 𝒞 E_{1}^{0,•}(ℰ) =: C^{∞}(M)^{•} are called secondary functions on M. For p > 0, elements in H(𝒞^{p}Λ^{p}(ℰ) ⊗ \overline{Λ}(ℰ), \overline{d}) ≃ 𝒞 E_{1}^{p,•}(ℰ) =: Λ^{p}(M)^{•} are called secondary differential p-forms on M. We put also Λ(M)^{•} := ⨁_{p} Λ^{p}(M)^{•}.

Notice that elements in Λ(M)^{n} are sometimes referred to in the literature as variational forms [28]. We apply similar arguments to motivate the definition of secondary vector fields. First of all, notice that there exists a complex

\xymatrix{0 \ar[r] & V D(ℰ) \ar[r]^{\overline{S}} & ⋯ \ar[r] & V D(ℰ) ⊗ \overline{Λ}^{q}(ℰ) \ar[r]^{\overline{S}} & V D(ℰ) ⊗ \overline{Λ}^{q+1}(ℰ) \ar[r]^{\overline{S}} & ⋯} , (13)

somehow “dual” to complex (𝒞 Λ^{1}(ℰ) ⊗ \overline{Λ}(ℰ), \overline{d}) ≃ (𝒞 E_{0}^{1,•}(ℰ), d_{0}^{1,•}), well defined by putting

\overline{S}(X ⊗ \overline{ω}) := \overline{S}(X) ∧ \overline{ω} + X ⊗ \overline{d}\overline{ω},

X ∈ V D(ℰ), \overline{ω} ∈ \overline{Λ}(ℰ), where \overline{S}(X) ∈ V D(ℰ) ⊗ \overline{Λ}^{1}(ℰ) is the V D(ℰ)-valued horizontal 1-form defined by putting \overline{S}(X)(Y) := [Y, X]^{V}, and [Y, X]^{V} is the vertical component of [Y, X]. Complex (13) is called the (horizontal) Spencer complex of ℰ. As we will see later on in more details, 0-cohomology H^{0}(V D(ℰ) ⊗ \overline{Λ}(ℰ), \overline{S}) of the Spencer complex is given by V D_{𝒞} (ℰ). Now, let X ∈ Sym(ℰ) and �_{χ} ∈ V D_{𝒞} (ℰ) be the associated evolutionary vector field, χ ∈ κ|_{ℰ} being a generating section such that ℓ_{Φ}χ = 0. Suppose temporarily that �_{χ} generates a flow {A_{t}}_{t} of local diffeomorphisms of ℰ. Then for any t, A_{t} preserves the Cartan distribution and therefore the image A_{t}(L) of an n-dimensional integral submanifold L is an n-dimensional integral submanifold. We conclude that X generates a flow of solutions of ℰ_{Φ} and, therefore, may be interpreted as a (local) vector field on M. This makes it rigorous the assertion that tangent vectors to the solution space of a PDE are solutions of the associated linearized PDE. As a natural generalization, we interpret the whole H(V D(ℰ) ⊗ \overline{Λ}(ℰ), \overline{S}), not only its 0-degree component, as space of vector fields on M. This motivates the

Definition 2. Elements in H(V D(ℰ)⊗\overline{Λ}(ℰ), \overline{S}) =: D(M)^{•} are called secondary vector fields on M.

All standard operations with vector fields and differential forms have their secondary analogue. Namely, let ω ∈ Λ^{p}(M)^{q}, ω_{1} ∈ Λ^{p_{1}}(M)^{q_{1}}, ω_{2} ∈ Λ^{p_{2}}(M)^{q_{2}}, X ∈ D(M)^{r}, X_{1} ∈ D(M)^{r_{1}}, X_{2} ∈ D(M)^{r_{2}}. Then ω = [ω], ω_{1} = [ω_{1}] and ω_{2} = [ω_{2}] for some ω ∈ 𝒞^{p}Λ^{p}(ℰ) ⊗ \overline{Λ}^{q}(ℰ), ω_{1} ∈ 𝒞^{p_{1}}Λ^{p_{1}}(ℰ) ⊗ \overline{Λ}^{q_{1}}(ℰ) and ω_{2} ∈ 𝒞^{p_{2}}Λ^{p_{2}}(ℰ) ⊗ \overline{Λ}^{q_{2}}(ℰ) such that \overline{d}ω = \overline{d}ω_{1} = \overline{d}ω_{2} = 0. Similarly, X = [X], X_{1} = [X_{1}] and X_{2} = [X_{2}] for some X ∈ V D(ℰ) ⊗ \overline{Λ}^{r}(ℰ), X_{1} ∈ V D(ℰ) ⊗ \overline{Λ}^{r_{1}}(ℰ) and X_{2} ∈ V D(ℰ) ⊗ \overline{Λ}^{r_{2}}(ℰ) such that \overline{S}(X) = \overline{S}(X_{1}) = \overline{S}(X_{2}) = 0. The following operations are well defined: exterior product of differential forms:

ω_{1} ∧ ω_{2} := [(−1)^{q_{1}p_{2}}ω_{1} ∧ ω_{2}] ∈ Λ^{p_{1}+p_{2}}(M)^{q_{1}+q_{2}};

exterior differential of a differential form:

dω := [d^{V} ω] ∈ Λ^{p+1}(M)^{q};

commutator of vector fields:

[X_{1}, X_{2}] := [[[X_{1}, X_{2}]]] ∈ D(M)^{r_{1}+r_{2}};

insertion of a vector field into a differential form:

i_{X}ω := [(−1)^{r(p−1)}i_{X}ω] ∈ Λ^{p−1}(M)^{q+r};

Lie derivative of a differential form along a vector field:

L_{X}ω := (i_{X} ∘ d + d ∘ i_{X})ω ∈ Λ^{p}(M)^{q+r};

[[ · , ·]] being the Frölicher-Nijenhuis bracket of form-valued vector fields. Secondary analogue of the standard relations among the above operations hold. Indeed, let ω_{1}, ω_{2}, X, X_{1}, X_{2} be as above. The exterior product endows Λ(M)^{•} = ⨁_{p,q} Λ^{p}(M)^{q} with the structure of a bi-graded algebra. Namely, ω_{1} ∧ ω_{2} = (−1)^{p_{1}p_{2}+q_{1}q_{2}}ω_{2} ∧ ω_{1}. The exterior differential is a bi-graded derivation of bi-degree (1, 0). Namely, d(ω_{1} ∧ ω_{2}) = dω_{1} ∧ ω_{2} + (−1)^{p_{1}}ω_{1} ∧ dω_{2}. The commutator endows D(M)^{•} = ⨁_{r} D(M)^{r} with the structure of a graded Lie algebra, in particular, [X, [X_{1}, X_{2}]] = [[X, X_{1}], X_{2}] + (−1)^{rr_{1}}[X_{1}, [X, X_{2}]]. The ‘insertion of’ and the ‘Lie derivative along’ X are bi-graded derivations of bi-degree (−1, r) and (0, r) respectively. Namely, i_{X}(ω_{1} ∧ ω_{2}) = i_{X}ω_{1} ∧ ω_{2} + (−1)^{p_{1}+rq_{1}}ω_{1} ∧ i_{X}ω_{2} and L_{X}(ω_{1} ∧ ω_{2}) = L_{X}ω_{1} ∧ ω_{2} + (−1)^{rq_{1}}ω_{1} ∧ L_{X}ω_{2}. Moreover, [d, d] = [d, L_{X}] = [i_{X_{1}}, i_{X_{2}}] = 0, [d, i_{X}] = L_{X}, [i_{X_{1}}, L_{X_{2}}] = i_{[X_{1},X_{2}]}, [L_{X_{1}}, L_{X_{2}}] = L_{[X_{1},X_{2}]}, where [ · , · ] denotes the bi-graded commutator. Despite some time has passed since they were introduced [20, 39], to the author knowledge, no general techniques have been developed so far in order to effectively compute secondary differential form and vector field spaces, i.e., cohomologies of complexes (11) and (13), in full generality, other than the one based on the so-called compatibility complexes [34, 35] (and, possibly, the Koszul-Tate resolution [36]), which is reviewed in the next two sections.

1.5. Horizontal Calculus on PDEs. The Cartan distribution determines a “horizontal differential calculus” on ℰ. Informally speaking, the horizontal differential calculus is obtained replacing standard partial derivatives with total derivatives. For instance, a horizontal linear differential operator is one which is a linear combination of compositions of total derivatives. More rigorously, let τ : T ⟶ ℰ (resp. ρ : R ⟶ ℰ) be a finite dimensional vector bundle and P := Γ(τ) (resp. Q := Γ(ρ)) the C^{∞}(ℰ)-module of sections of τ (resp. ρ). In the following any such module will be called a smooth module. A linear differential operator □ : P ⟶ Q is called a horizontal (linear) differential operator iff, by definition, for any θ ∈ ℰ and any submanifold L ⊂ ℰ such that θ ∈ L and T_{θ}L ⊂ 𝒞_{θ} there exists a differential operator □_{θ}^{L} : P |_{L} ⟶ Q|_{L} such that (□p)(θ) = □_{θ}^{L}(p|_{L})(θ) for all p ∈ P. As examples, notice that horizontal vector fields, the horizontal de Rham differential \overline{d}, the Spencer differential \overline{S} and universal linearizations are horizontal differential operators. Indeed, Let …, e_{a}, … (resp. …, ε_{A}, …) be a local basis of P (resp. Q). Then a horizontal differential operator □ : P ⟶ Q is characterized as being one locally given by

□p = ε_{A}□_{a}^{A}^{I}D_{I}^{ℰ} p^{a}, …, □_{a}^{A}^{I}, … being local functions on ℰ,  (14)

for all p = p^{a}e_{a} local sections of τ, …, p^{a}, … local functions on ℰ. In particular, if ℰ = J^{∞} and ℱ ⊂ J^{∞} is the infinite prolongation of a PDE, then any horizontal differential operator □ : P ⟶ Q restricts to ℱ, i.e., there exists a unique (horizontal) differential operator □^{ℱ} : P |_{ℱ} ⟶ Q|_{ℱ} such that □^{ℱ} (p|_{ℱ} ) = (□p)|_{ℱ} for all p ∈ P.

Denote by 𝒞 Diff(P, Q) the set of all horizontal differential operators □ : P ⟶ Q. Clearly, 𝒞 Diff(P, Q) is a C^{∞}(ℰ)-module naturally isomorphic to 𝒞 Diff(P, C^{∞}(ℰ)) ⊗ Q and in what follows we will understand such isomorphism. Similarly, one may define horizontal jets of sections of vector bundles over ℰ just replacing partial derivatives with total derivatives in the standard definition. We refer to [36] for the details of the construction. Analogously to the standard case, one may also define (systems of horizontal) PDEs determined by linear horizontal differential operators and, in particular, formally integrable PDEs. Denote by \overline{τ}_{∞} : \overline{J}^{∞}τ ⟶ ℰ the bundle of horizontal infinite jets of sections of τ and put \overline{J}^{∞}P := Γ(\overline{τ}_{∞}). For any p ∈ P denote by \overline{j}_{∞}p ∈ \overline{J}^{∞}P its infinite horizontal jet prolongation. There is a canonical monomorphism of C^{∞}(ℰ)-modules h : 𝒞 Diff(P, Q) ∋ □ ⟼ h_{□} ∈ Hom(\overline{J}^{∞}P, Q), where h_{□} is the unique C^{∞}(ℰ)-linear map such that h_{□}(\overline{j}_{∞}p) = □p for all p ∈ P. Moreover h_{□} can be uniquely prolonged to a C^{∞}(ℰ)- linear map h_{□}^{∞} : \overline{J}^{∞}P ⟶ \overline{J}^{∞}Q such that h_{□}^{∞}(\overline{j}_{∞}p) = \overline{j}_{∞}(□p) for all p ∈ P. The following remarkable correspondence,

\overline{J}^{∞}κ ∋ \overline{j}_{∞}χ ⟼ �_{χ} ∈ V D(J^{∞}),  (15)

determines a well defined isomorphism of C^{∞}(J^{∞})-modules. The dual isomorphism is given by

𝒞 Λ^{1}(J^{∞}) ∋ ω ⟼ □_{ω} ∈ 𝒞 Diff(κ, C^{∞}(J^{∞})),  (16)

where □_{ω} : κ ⟶ C^{∞}(J^{∞}) is defined by putting □_{ω}χ := ω(�_{χ}), χ ∈ κ. Accordingly, there is a natural embedding η_{Φ} : V D(ℰ) ↪ \overline{J}^{∞}κ|_{ℰ} given by the composition

\xymatrix{V D(ℰ) \ar@{^{(}->}[r] \ar[rr]_{η_{Φ}} & V D(J^{∞})|_{ℰ} \ar[r]|{} & \overline{J}^{∞}κ|_{ℰ}} ,

and, dually, a natural projection η_{Φ}^{∗} : 𝒞 Diff(κ|_{ℰ} , C^{∞}(ℰ)) ↠ 𝒞 Λ^{1}(ℰ) given by the composition

\xymatrix{𝒞 Diff(κ|_{ℰ} , C^{∞}(ℰ)) \ar[r] \ar[r]|{} \ar[rr]_{η_{Φ}^{∗}} & 𝒞 Λ^{1}(J^{∞})|_{ℰ} \ar@{->>}[r] & 𝒞 Λ^{1}(ℰ)} ,

where the arrows “˜⟶” are the inverses of restrictions to ℰ of isomorphisms (15) and (16), respectively. Finally, notice that the sequence

\xymatrix{0 \ar[r] & V D(ℰ) \ar[r]^{η_{Φ}} & \overline{J}^{∞}κ|_{ℰ} \ar[r]^{h_{Φ}^{∞}} & \overline{J}^{∞}P} ,  (17)

where h_{Φ} := h_{ℓ_{Φ}}, and its dual

\xymatrix{𝒞 Diff(P, C^{∞}(ℰ)) \ar[r]^{h_{Φ}^{∞}^{∗}} & 𝒞 Diff(κ|_{ℰ} , C^{∞}(ℰ)) \ar[r]^{η_{Φ}^{∗}} & 𝒞 Λ^{1}(ℰ) \ar[r] & 0},  (18)

where h_{Φ}^{∞}^{∗}(Δ) := Δ ∘ ℓ_{Φ}, Δ ∈ 𝒞 Diff(P, C^{∞}(ℰ)), are exact. There exists a horizontal analogue of the concept of adjoint operator to a linear differential operator. Let R be a smooth module (see above). Put R^{†} := Hom(R, \overline{Λ}^{n}(ℰ)). R^{†} is a smooth module as well and it is called the adjoint module to R. Obviously, R^{††} identifies canonically with R. Denote by R^{†} × R ∋ (r^{†}, r) ⟼ ⟨r^{†}, r⟩ := r^{†}(r) ∈ \overline{Λ}^{n}(ℰ) the natural bi-linear pairing. For any local basis …, κ_{a}, … of R we denote by …, κ^{†}^{a}, … the local basis of R^{†} such that κ^{†}^{a} is the local homomorphism R ⟶ \overline{Λ}^{n}(ℰ) defined by putting ⟨κ^{†}^{a}, κ_{b}⟩ := δ_{b}^{a}\overline{d}^{n}x and \overline{d}^{n}x := \overline{d}x^{1} ∧ ⋯ ∧ \overline{d}x^{n}, a, b = 1, 2, . . ..

Proposition 3. Let r ∈ R (resp. r^{†} ∈ R^{†}), then r = 0 (resp. r^{†} = 0) iff ∫ ⟨r^{†}, r⟩ = 0 for all r^{†} ∈ R^{†} (resp. r ∈ R).

Proposition 3 may be referred to as the cohomological DuBois-Reymond theorem and will be used later on without further comments. Now let P, Q be as above and □ : P ⟶ Q a horizontal differential operator. It can be proved that there exists a unique differential operator (of the same order as □) □^{†} : Q^{†} ⟶ P^{†} such that

∫ ⟨q^{†}, □p⟩ = ∫ ⟨□^{†}q^{†}, p⟩  (19)

for all p ∈ P, q^{†} ∈ Q^{†}. □^{†} is called the adjoint operator to □ and (19) is called the (horizontal) Green formula [9, 20, 37]. Adjoint operators have the following properties. First, □^{††} = □. Second, let Δ : Q ⟶ R be another horizontal differential operator, then (Δ ∘ □)^{†} = □^{†} ∘ Δ^{†}. If □ is locally given by (14) then □^{†} is locally given by

□^{†}q^{†} = (−1)^{|I|}e^{†}^{a}D_{I}(□_{a}^{A}^{I}q_{A}^{†}),

for all q^{†} = q_{A}^{†}ε^{†}^{A} local elements of Q^{†}, …, q_{A}^{†}, … local functions on ℰ. As an example, notice that the adjoint module of \overline{Λ}^{q}(ℰ) is canonically isomorphic to \overline{Λ}^{n−q}(ℰ), and that the adjoint operator of the horizontal de Rham differential \overline{d} : \overline{Λ}^{q}(ℰ) ⟶ \overline{Λ}^{q+1}(ℰ) is the operator (−1)^{n−q−1}\overline{d} : \overline{Λ}^{n−q−1}(ℰ) ⟶ \overline{Λ}^{n−q}(ℰ), q = 0, …, n. Notice that the Green formula amounts to say that for any p ∈ P, q^{†} ∈ Q^{†} there exists �_{p,q^{†}} ∈ \overline{Λ}^{n−1}(ℰ) such that ⟨q^{†}, □p⟩−⟨□^{†}q^{†}, p⟩ = \overline{d}�_{p,q^{†}}. It can be proved [1] that �_{p,q^{†}} can be chosen of the form � (p, q^{†}), � : P × Q^{†} ⟶ \overline{Λ}^{n−1}(ℰ) being a (possibly non unique) horizontal bi-differential operator independent of p and q^{†}. Any such operator � is called a Legendre operator for □ [2]. The Green formula plays a central role in the theory of the 𝒞 -spectral sequence.

1.6. Formal Theory of Horizontal PDEs and Secondary Calculus. There exists a horizontal analogue of the Goldschmidt-Spencer formal theory of linear differential equations (see [15, 32] for a complete account of the classical theory - see also [16] - and [21, 35] for its horizontal analogue). Let Δ : P ⟶ P_{1} be a horizontal differential operator of order ≤ k between smooth modules.

Definition 4. A complex of horizontal differential operators between smooth modules

\xymatrix{0 \ar[r] & P \ar[r]^{Δ} & P_{1} \ar[r]^{Δ_{1}} & ⋯ \ar[r] & P_{q} \ar[r]^{Δ_{q}} & P_{q+1} \ar[r]^{Δ_{q+1}} & ⋯}  (20)

is called a compatibility complex for Δ iff the sequence of homomorphisms

\xymatrix{0 \ar[r] & \overline{J}^{∞}P \ar[r]^{h_{Δ}^{∞}} & \overline{J}^{∞}P_{1} \ar[r]^{h_{Δ_{1}}^{∞}} & ⋯ \ar[r] & \overline{J}^{∞}P_{q} \ar[r]^{h_{Δq}^{∞}} & \overline{J}^{∞}P_{q+1} \ar[r]^{h_{Δ_{q+1}}^{∞}} & ⋯}

is exact. Δ_{1} is called a compatibility operator for Δ.

The existence of a non trivial compatibility operator for Δ formalizes the fact that the equation Δp = 0 is overdetermined [32]. We stress that Definition 4 is slightly different from the one usually found in the literature (see, for instance, [15, 21]). However, it can be shown that, if Δ determines a formally integrable PDE, then the two coincide, and Definition 4 is the most suitable for our purposes.

Theorem 5 (Goldschmidt). Let Δ be a horizontal differential operator between smooth modules. If Δ determines a formally integrable horizontal PDE, then there exists a (non unique) compatibility complex (20) for Δ, such that Δ_{i} determines a formally integrable horizontal PDE for any i = 1, 2, . . ..

Any compatibility complex as in the above theorem will be said regular. Let Δ : P ⟶ P_{1} determine a formally integrable PDE and (20) be a regular compatibility complex for it. Then, the compatibility operator Δ_{1} has the following remarkable property.

Proposition 6. Let □ : P_{1} ⟶ Q be a horizontal differential operator such that □ ∘ Δ = 0. Then there exists a horizontal differential operator ∇ : P_{2} ⟶ Q such that □ = ∇ ∘ Δ_{1}. If Δ_{2} = 0 then ∇ is unique.

Thus, let Δ : P ⟶ P_{1} determine a formally integrable PDE and

\xymatrix{0 \ar[r] & P \ar[r]^{Δ} & P_{1} \ar[r]^{Δ_{1}} & ⋯ \ar[r] & P_{s−1} \ar[r]^{Δ_{s−1}} & P_{s} \ar[r] & 0}

be a finite length regular compatibility complex. In this situation we say that the compatibility length of Δ is ≤ s. Now let π : E ⟶ M be a fiber bundle, τ : T ⟶ J^{∞} a vector bundle, Φ ∈ diff(π, τ) and ℰ := ℰ_{Φ}^{(∞)}. Put P_{1} := Γ(τ)|_{ℰ}. Notice that if ℰ_{Φ} is a formally integrable PDE, then ℓ_{Φ} : κ|_{ℰ} ⟶ P_{1} determines a formally integrable, linear, horizontal PDE [34].

Theorem 7 (Spencer). Cohomology D(M)^{•} of complex (V D(ℰ) ⊗ \overline{Λ}(ℰ), \overline{S}) is canonically isomorphic to cohomology of any regular compatibility complex

\xymatrix{0 \ar[r] & κ|_{ℰ} \ar[r]^{ℓ_{Φ}} & P_{1} \ar[r]^{Δ_{1}} & ⋯ \ar[r] & P_{q} \ar[r]^{Δ_{q}} & P_{q+1} \ar[r]^{Δ_{q+1}} & ⋯}

for ℓ_{Φ}.

In the following we will only consider regular compatibility complexes. Isomorphism ker ℓ_{Φ} ≃ D(M)^{0} is given by

ker ℓ_{Φ} ∋ χ ⟼ �_{χ} ∈ V D_{𝒞} (ℰ) = D(M)^{0}.

We now describe isomorphism ker Δ_{1}/ im ℓ_{Φ} ≃ D(M)^{1} referring to [21] for the remaining homogeneous components. Let p ∈ P_{1} be such that Δ_{1}p = 0. Consider \overline{j}^{∞}p ∈ \overline{J}^{∞}P_{1}. Then h_{Δ_{1}}(\overline{j}^{∞}p) = Δ_{1}p = 0 and, therefore, Δ_{1} being a compatibility operator for ℓ_{Φ}, there exists j ∈ \overline{J}^{∞}κ such that \overline{j}^{∞}p = h_{Φ}(j|_{ℰ}). Let X := η_{0}^{−1}(j) ∈ V D(J^{∞}) (here 0 is the trivial differential operator) and Ω̃ := \overline{S}(X) ∈ V D(J^{∞}) ⊗ \overline{Λ}^{1}(J^{∞}). It is easy to prove, suitably using exactness of sequence (17), that Ω̃ restricts to ℰ, i.e., Ω := Ω̃|_{ℰ} ∈ V D(ℰ) ⊗ \overline{Λ}^{1}(ℰ) ⊂ V D(J^{∞})|_{ℰ} ⊗ \overline{Λ}^{1}(ℰ). Moreover, \overline{S}(Ω) = 0. Finally, the isomorphism ker Δ_{1}/ im ℓ_{Φ} ≃ D(M)^{1} maps p + im ℓ_{Φ}^{ℰ} ∈ ker Δ_{1}/ im ℓ_{Φ} to [Ω] ∈ H^{1}(V D(ℰ) ⊗ \overline{Λ}(ℰ), \overline{S}) = D(M)^{1}.

Corollary 8. Cohomology Λ^{1}(M)^{•} of complex (𝒞 Λ^{1}(ℰ) ⊗ \overline{Λ}(ℰ), \overline{d}) is canonically isomorphic to homology of the adjoint complex

\xymatrix{0 & κ|_{ℰ}^{†} \ar[l] & P_{1}^{†} \ar[l]_{ℓ_{Φ}^{†}} & ⋯ \ar[l] & P_{q}^{†} \ar[l]_{Δ_{q−1}^{†}} & P_{q+1}^{†} \ar[l]_{Δ_{q}^{†}} & ⋯ \ar[l]}

of any regular compatibility complex for ℓ_{Φ}.

Isomorphism Λ^{1}(M)^{n} ≃ coker ℓ_{Φ}^{†} is described as follows. Projection η_{Φ}^{∗} : 𝒞 Diff(κ|_{ℰ} , C^{∞}(ℰ)) ↠ 𝒞 Λ^{1}(ℰ) gives rise to a projection

η_{Φ}^{∗} ⊗ id_{\overline{Λ}(ℰ)} : 𝒞 Diff(κ|_{ℰ} , \overline{Λ}(ℰ)) ↠ 𝒞 Λ^{1}(ℰ) ⊗ \overline{Λ}(ℰ)

which, abusing the notation, we denote again by η_{Φ}^{∗}. Thus, let ω ∈ 𝒞 Λ^{1}(ℰ) ⊗ \overline{Λ}^{n}(ℰ) and □ ∈ 𝒞 Diff(κ|_{ℰ} , \overline{Λ}^{n}(ℰ)) be such that η_{Φ}^{∗}(□) = ω. Consider □^{†} : C^{∞}(ℰ) ⟶ κ|_{ℰ}^{†}. Isomorphism Λ^{1}(M)^{n} ≃ coker ℓ_{Φ}^{†} maps [ω] ∈ H^{n}(𝒞 Λ^{1}(ℰ) ⊗ \overline{Λ}(ℰ), \overline{d}) = Λ^{1}(M)^{n} to □^{†}1 + im ℓ_{Φ}^{†} ∈ coker ℓ_{Φ}^{†}. We now describe isomorphism Λ^{1}(M)^{n−1} ≃ ker ℓ_{Φ}^{†}/ im Δ_{1}^{†} referring again to [21] for the remaining homogeneous components. Let ω ∈ 𝒞 Λ^{1}(ℰ) ⊗ \overline{Λ}^{n−1}(ℰ) and □ ∈ 𝒞 Diff(κ|_{ℰ} , \overline{Λ}^{n}(ℰ)) be such that \overline{d}ω = 0 and η_{Φ}^{∗}(□) = ω. Then, it follows from exactness of sequence (18) that \overline{d} ∘ □ = Δ ∘ ℓ_{Φ} for some Δ ∈ 𝒞 Diff(P_{1}, \overline{Λ}^{n}(ℰ)). Consider Δ^{†} : C^{∞}(ℰ) ⟶ P_{1}^{†} and put p^{†} := Δ^{†}1 ∈ P_{1}^{†}. We have ℓ_{Φ}^{†}(p^{†}) = (ℓ_{Φ}^{†} ∘ Δ^{†})(1) = (Δ ∘ ℓ_{Φ})^{†}(1) = (\overline{d} ∘ □)^{†}(1) = (□^{†} ∘ \overline{d}^{†})(1) = (□^{†} ∘ d)(1) = 0. Thus p^{†} ∈ ker ℓ_{Φ}^{†}. Isomorphism Λ^{1}(M)^{n−1} ≃ ker ℓ_{Φ}^{†}/ im Δ_{1}^{†} maps [ω] ∈ H^{n−1}(𝒞 Λ^{1}(ℰ) ⊗ \overline{Λ}(ℰ), \overline{d}) = Λ^{1}(M)^{n−1} to p^{†} + im Δ_{1}^{†} ∈ ker ℓ_{Φ}^{†}/ im Δ_{1}^{†}. Notice that the above corollary describes to some extent the 1-st column of the 1-st term of the 𝒞 -spectral sequence of ℰ. The following theorem due to Verbovetsky [35] (see also [39] for the case s = 2) extends it to the remaining columns.

Theorem 9 (s-lines). Let ℰ ⊂ J^{∞} be the infinite prolongation of a formally integrable PDE ℰ_{Φ} and let the compatibility length of ℓ_{Φ} be ≤ s. Then 𝒞 E_{1}^{p,q}(ℰ) = 0 if p > 0 and q < n − s.

Example 10 (empty equation). If Φ = 0 then ℰ = J^{∞}, ℓ_{Φ} = 0 and its compatibility length is 0. In this case D(M)^{r} = 0 for r ≠ 0 and D(M)^{•} = D(M)^{0} ≃ κ. The exact sequence

\xymatrix{0 \ar[r] & D(M)^{0} \ar[r] & V D(J^{∞}) \ar[l]^{ψ} \ar[r]^{\overline{S}} & V D(J^{∞}) ⊗ \overline{Λ}^{1}(J^{∞})}  (21)

splits via the composition

[figure: V D(J^{∞}); η_{0}; ψ; \overline{J}^{∞}(κ); κ; ; D(M)^{0}] .  (22)

Similarly, Λ^{1}(M)^{q} = 0 for q ≠ n and Λ^{1}(M)^{•} = Λ^{1}(M)^{n} ≃ κ^{†}. The exact sequence

\xymatrix{𝒞 Λ^{1}(J^{∞}) ⊗ \overline{Λ}^{n−1}(J^{∞}) \ar[r]^{\overline{d}} & 𝒞 Λ^{1}(J^{∞}) ⊗ \overline{Λ}^{n}(J^{∞}) \ar[r] & Λ^{1}(M)^{n} \ar[l]^{ψ^{†}} \ar[r] & 0}

splits via the composition

[figure: Λ^{1}(M)^{n}; ; ψ^{†}; κ^{†}; 𝒞 Diff(κ, \overline{Λ}^{n}); η_{0}^{∗}; 𝒞 Λ^{1}(J^{∞}) ⊗ \overline{Λ}^{n}] .

Let Y ∈ V D(J^{∞}) and φ ∈ κ^{†} be locally given by Y = Y_{I}^{α}∂_{α}^{I} and φ = φ_{α}∂^{†α}, …, Y_{I}^{α}, …, φ_{α}, … being local functions on J^{∞}. Then, locally,

ψ(Y) = Y_{O}^{α}∂_{α} and ψ^{†}(φ) = φ_{α}ω_{O}^{α} ⊗ \overline{d}^{n}x.

Finally, notice that both diagrams (21) and (22) restrict to the infinite prolongation of a PDE and such restrictions preserve the exactness.

In the following we will understand the above isomorphisms D(M)^{0} ≃ κ and Λ^{1}(M)^{n} ≃ κ^{†}. In order not to make the notation to heavy we will also understand the monomorphism ψ^{†}. According to this convention κ^{†} is understood as a subset in 𝒞 Λ^{1}(J^{∞})⊗\overline{Λ}^{n}(J^{∞}). Moreover, if φ ∈ κ^{†} and χ ∈ κ, then i_{�_{χ}} ψ^{†}(φ) ∈ \overline{Λ}^{n}(J^{∞}) identifies with ⟨φ, χ⟩.

Example 11 (irreducible equations). A non-empty PDE ℰ_{Φ} is called ℓ-normal (or, in physical terms, irreducible) iff the compatibility length of ℓ_{Φ} is ≤ 1. In this case Δ_{1} may be chosen equal to 0, D(M)^{r} = 0 for r ≠ 0, 1, D(M)^{0} ≃ ker ℓ_{Φ} as above and

D(M)^{1} ≃ coker ℓ_{Φ}.

Similarly, Λ^{1}(M)^{q} = 0 for q ≠ n, n − 1, Λ^{1}(M)^{n} ≃ coker ℓ_{Φ}^{†} as above and

Λ^{1}(M)^{n−1} ≃ ker ℓ_{Φ}^{†}.

2. The Covariant Phase Space

2.1. Lagrangian Field Theories and the CPS. The calculus of variations is formalized in a coordinate-free way via the 𝒞 -spectral sequence.

Definition 12. A lagrangian (field) theory is the datum (π, S) of a fiber bundle π : E ⟶ M and an action S ∈ \overline{H}^{n}(J^{∞}). Any representative ℒ ∈ \overline{Λ}^{n}(J^{∞}) of the cohomology class S = ∫ ℒ is called a lagrangian density of the theory (π, S).

Recall that the space M of n-dimensional integral submanifolds of the Cartan distribution 𝒞 on J^{∞} is in one-to-one correspondence (via infinite jet prolongation) with the space of local sections of π. As above we will often identify M with the “store” (J^{∞}, 𝒞) of its elements. M is known in the Physics literature as the space of histories and an action S ∈ \overline{H}^{n}(J^{∞}) ⊂ C^{∞}(M)^{•} is a secondary function on it. Within secondary calculus, the Euler-Lagrange equations (whose solutions make it stationary the action) associated to the lagrangian theory (π, S) are easily obtained by applying to S the secondary de Rham differential d : C^{∞}(M)^{•} ⟶ Λ^{1}(M)^{•}. Indeed, according to the previous section, Λ^{1}(M)^{•} = Λ^{1}(M)^{n} ≃ κ^{†} and dS identifies with the element E(ℒ) := ℓ̃_{ℒ}^{†} 1 ∈ κ^{†} ⊂ 𝒞 Λ^{1}(J^{∞}) ⊗ \overline{Λ}^{n}(J^{∞}), where we put ℓ̃_{ℒ} := (η_{0}^{∗})^{−1}(d^{V} ℒ) : κ ⟶ \overline{Λ}^{n}(J^{∞}), ℒ being any lagrangian density. Locally, ℒ = L \overline{d}^{n}x for some local function L = L(…, x^{i}, …, u_{I}^{α}, …) on J^{∞} and

E(ℒ) = \frac{δL}{δu^{α}} ∂^{†}^{α},

\frac{δL}{δu^{α}} := (−1)^{|I|}D_{I}(∂_{α}^{I}L) being the so-called Euler-Lagrange derivatives of L, α = 1, …, m. Thus dS is naturally interpreted as the left hand side of the Euler-Lagrange equations ℰ_{E(ℒ)} of the theory (π, S). In the following we will always assume ℰ_{E(ℒ)} to be a formally integrable PDE. Let ℰ := ℰ_{E(ℒ)}^{(∞)} . The space P of n-dimensional integral submanifolds of the Cartan distribution on ℰ is in one-to-one correspondence with the space of (local) solutions of ℰ_{E(ℒ)} and is called, according to Physics literature, the (non-reduced) CPS of the theory (π, S) [8, 4, 11, 12, 25]. By definition

d^{V} ℒ − E(ℒ) = \overline{d}θ  (23)

for some θ ∈ 𝒞 Λ^{1}(J^{∞}) ⊗ \overline{Λ}^{n−1}(J^{∞}). Any such θ will be called a Legendre form [2] (notice that ℒ − θ is a so-called lepagean equivalent [22, 23] of ℒ). Equation (23) may be interpreted as the first variation formula for the action S. In this respect, the existence of a global Legendre form was first discussed in [24]. Any two Legendre forms θ.θ′ differ by a closed, and therefore exact, form \overline{d}λ, λ ∈ 𝒞 Λ^{1}(J^{∞}) ⊗ \overline{Λ}^{n−2}(J^{∞}) (see, for instance, [2, 25] for a local description of Legendre forms). Notice that, in view of isomorphism η_{0}^{∗}, identity (23) may be understood as the Green formula

ℓ̃_{ℒ} − ℓ̃_{ℒ}^{†} 1 = (\overline{d} ∘ � )( · , 1)

for the horizontal operator ℓ̃_{ℒ} , � being a Legendre operator for it.

Theorem 13 (Zuckerman). There is a closed secondary 2-form ω on P canonically determined by the corresponding lagrangian theory (π, S).

Proof. Using the language introduced so far we reproduce here the proof in [40] by adding the only missing point, that is the independence of ω of the choice of a lagrangian density. Thus, let θ be a Legendre form. Put

ω := −i_{ℰ}^{∗}(d^{V} θ) ∈ 𝒞 Λ^{2}(ℰ) ⊗ \overline{Λ}^{n−1}(ℰ).

Then

\overline{d}ω = −\overline{d}i_{ℰ}^{∗}(d^{V} θ)
= i_{ℰ}^{∗}(d^{V} \overline{d}θ)
= i_{ℰ}^{∗}(d^{V} (d^{V} ℒ − E(ℒ)))
= −d^{V} i_{ℰ}^{∗}(E(ℒ))
= 0.

Since ω is \overline{d}-closed we may take its cohomology class ω := [ω] ∈ Λ^{2}(P)^{n−1}. Now, ω is canonical, as proved in what follows.

(1) ω does not depend on the choice of θ. Indeed, let θ′ := θ + \overline{d}λ be another Legendre form, λ ∈ 𝒞 Λ^{1}(J^{∞}) ⊗ \overline{Λ}^{n−2}(J^{∞}) and ω′ := −i_{ℰ}^{∗}(d^{V} θ′). Then ω′ = −i_{ℰ}^{∗}(d^{V} θ + d^{V} \overline{d}λ) = ω + \overline{d}i_{ℰ}^{∗}(d^{V} λ), so that [ω] = [ω′].

(2) ω does not depend on the choice of ℒ. Indeed, let ℒ be a trivial lagrangian density, i.e., ℒ = \overline{d}ν for some ν ∈ \overline{Λ}^{n−1}(J^{∞}). Then S = 0, E(ℒ) = 0 and d^{V} ℒ − E(ℒ) = −\overline{d}d^{V} ν. This proves that −d^{V} ν is a Legendre form, so that ω = [i_{ℰ}^{∗}(d^{V} d^{V} ν)] = 0.

Finally, dω = [d^{V} ω] = 0. □

Notice that the above theorem can be generalized to the case of a lagrangian field theory subject to constraints in the form of (the infinite prolongation of) a PDE ℱ ⊂ J^{∞}, under suitable cohomological conditions on ℱ. Constrained lagrangian theories will be considered somewhere else. A general coordinate formula for ω may be found, for instance, in [25]. The expression of ω for specific lagrangian theories may be found, for instance, in [11, 12, 18, 25, 30]. However, we stress that, in general, there is no distinguished representative ω in ω.

2.2. “Symplectic Version” of I Noether Theorem. Let (π, S) be a lagrangian field theory and χ ∈ κ the generating section of an higher symmetry of π. In view of isomorphism κ ≃ D(M)^{•}, χ may be understood as a secondary vector field on M. By definition, χ is a Noether symmetry of (π, S) iff L_{χ}S = 0, or, which is the same, i_{χ}dS = 0. In terms of a lagrangian density ℒ the last equality reads as i_{�_{χ}} d^{V} ℒ = \overline{d}σ for some σ ∈ \overline{Λ}^{n−1}(J^{∞}). Using (23) one gets i_{�_{χ}} (E(ℒ) + \overline{d}θ) = \overline{d}σ. In view of isomorphism η_{0}^{∗}, this implies \overline{d}(σ − i_{�_{χ}} θ) = ⟨E(ℒ), χ⟩ and, pulling-back to ℰ,

\overline{d}i_{ℰ}^{∗}(σ − i_{�_{χ}} θ) = 0.

We have thus shown that j := i_{ℰ}^{∗}(σ − i_{�_{χ}} θ) ∈ \overline{Λ}^{n−1}(ℰ) is a conserved current of ℰ and this is, basically, the content of the first Noether theorem. Any such conserved current is called a Noether current of (π, S). The associated conservation law f := [j] ∈ \overline{H}^{n−1}(ℰ) ⊂ C^{∞}(P)^{•} is called a Noether charge. Notice that nor j nor f are uniquely determined by χ in general. It is well known that if χ ∈ κ is a Noether symmetry of (π, S), then χ|_{ℰ} ∈ κ|_{ℰ} is the generating section of a symmetry of ℰ, i.e., ℓ_{E(ℒ)} χ|_{ℰ} = 0. This can be easily proved by means of the following useful

Lemma 14. Let φ ∈ κ^{†}, ℱ := ℰ_{φ}^{(∞)} ⊂ J^{∞}. For any χ ∈ κ,

(L_{�_{χ}} φ)|_{ℱ} = ℓ_{φ}\overline{χ}, \overline{χ} := χ|_{ℱ}

In particular, (L_{�_{χ}} φ)|_{ℱ} ∈ κ|_{ℰ}^{†} and it does only depend on the values of χ on ℱ.

Proof. For any χ_{1} ∈ κ, put \overline{χ}_{1} := χ_{1}|_{ℱ} . Similarly, for a (local) function f on J^{∞}, put \overline{f} := f|_{ℱ} . Compute

i_{�_{χ_{1}}} L_{�_{χ}} φ = i_{[�_{χ_{1}},�_{χ}]} φ + L_{�_{χ}} i_{�_{χ_{1}}} φ
= i_{�_{{χ,χ_{1}}}} φ + L_{�_{χ}} ⟨φ, χ_{1}⟩
= ⟨φ, {χ, χ_{1}}⟩ + L_{�_{χ}} ⟨φ, χ_{1}⟩.

Since φ|_{ℱ} = 0, we have

(i_{�_{χ_{1}}} L_{�_{χ}} φ)|_{ℱ} = (L_{�_{χ}} ⟨φ, χ_{1}⟩)|_{ℱ} .  (24)

Now, let φ, χ and χ_{1} be locally given by φ = φ_{α}∂^{†α}, χ = χ^{β}∂_{β}, χ_{1} = χ_{1}^{γ}∂_{γ}, …, φ_{α}, …, χ^{β}, …, χ_{1}^{γ}, … local functions on J^{∞}. Then locally,

L_{�_{χ}} ⟨φ, χ_{1}⟩ = D_{I}χ^{β}∂_{β}^{I}(φ_{α}χ_{1}^{α})\overline{d}^{n}x = [D_{I}χ^{β}(∂_{β}^{I}φ_{α})χ_{1}^{α} + D_{I}χ^{β}(∂_{β}^{I}χ_{1}^{α})φ_{α}]\overline{d}^{n}x.

Since φ_{α}|_{ℱ} = 0, α = 1, …, m, we have locally

(L_{�_{χ}} ⟨φ, χ_{1}⟩)|_{ℱ} = D_{I}^{ℱ} \overline{χ}^{β}\overline{(∂_{β}^{I}φ_{α})}\overline{χ}_{1}^{α}\overline{d}^{n}x = ⟨ℓ_{φ}\overline{χ}, \overline{χ}_{1}⟩.  (25)

Using (25) into (24) we get

i_{�_{χ_{1}}}^{ℱ} (L_{�_{χ}} φ)|_{ℱ} = (i_{�_{χ_{1}}} L_{�_{χ}} φ)|_{ℱ} = ⟨ℓ_{φ}\overline{χ}, \overline{χ}_{1}⟩ = i_{�_{χ_{1}}}^{ℱ} ℓ_{φ}\overline{χ},

where i_{�_{χ_{1}}}^{ℱ} is the restriction to ℱ of the operator i_{�_{χ_{1}}} (see Section 1.5). From the arbitrariness of χ_{1} the result follows. □

Now, let χ ∈ κ be a Noether symmetry of the lagrangian theory (π, S), and ℒ a lagrangian density. Then, in view of Lemma 14,

ℓ_{E(ℒ)} χ|_{ℰ} = (L_{�_{χ}} E(ℒ))|_{ℰ} ,  (26)

and

L_{�_{χ}} E(ℒ) = L_{�_{χ}} (d^{V} ℒ − \overline{d}θ) = d^{V} (i_{�_{χ}} d^{V} ℒ) + \overline{d}(L_{�_{χ}} θ) = \overline{d}(L_{�_{χ}} θ − d^{V} σ).

This shows that the horizontal cohomology class [L_{�_{χ}} E(ℒ)] ∈ Λ^{1}(M)^{n} ≃ κ^{†} is zero (and so is its “restriction” to ℰ) and, therefore, ℓ_{E(ℒ)} χ|_{ℰ} = 0 (see the final comment in Example 10). The above remark proves that if χ is a Noether symmetry, then X := χ|_{ℰ} ∈ ker ℓ_{E(ℒ)} ≃ D(P)^{0} is a secondary vector field on P. Let f ∈ C^{∞}(P)^{n−1} be, as above, a Noether charge associated to χ.

Proposition 15. df = −i_{X}ω (see Equation 22 in [25]).

Proof. Let j, σ, θ and ω be as above. Then

df = [d^{V} j]
= [d^{V} i_{ℰ}^{∗}(σ − i_{�_{χ}} θ)]
= [i_{ℰ}^{∗}(d^{V} σ − d^{V} i_{�_{χ}} θ)]
= [i_{ℰ}^{∗}(d^{V} σ − L_{�_{χ}} θ + i_{�_{χ}} d^{V} θ)]
= [i_{ℰ}^{∗}(d^{V} σ − L_{�_{χ}} θ) + i_{�_{χ}|_{ℰ}} i_{ℰ}^{∗}(d^{V} θ)]
= [i_{ℰ}^{∗}(d^{V} σ − L_{�_{χ}} θ) − i_{�_{χ}|_{ℰ}} ω]
= i_{ℰ}^{∗}[d^{V} σ − L_{�_{χ}} θ] − i_{X}ω,

where we used that \overline{d}(d^{V} σ − L_{�_{χ}} θ) = 0. Now, d^{V} σ − L_{�_{χ}} θ ∈ 𝒞 Λ^{1}(J^{∞}) ⊗ \overline{Λ}^{n−1}(J^{∞}) and [d^{V} σ − L_{�_{χ}} θ] ∈ Λ^{1}(M)^{n−1}. But, according to Example 10, Λ^{1}(M)^{n−1} = 0. We conclude that df = −i_{X}ω. □

Notice that Proposition 15 resembles very closely the analogous result in hamiltonian mechanics. Moreover, if ℰ_{E(ℒ)} is an irreducible equation then d : C^{∞}(P)^{n−1} ⟶ Λ^{1}(M)^{n−1} is injective [21, 39] modulo obstructions in H^{n−1}(E) ⊂ \overline{H}^{n−1}(ℰ). Thus, df determines the “non-trivial conservation law” (see [9]) f + H^{n−1}(E) ∈ \overline{H}^{n−1}(ℰ)/H^{n−1}(E) and is interpreted as the generating section of it. Proposition 15 can be then understood as a way to compute the generating section of the non-trivial conservation law associated to a Noether symmetry.

2.3. “Symplectic Version” of (Infinitesimal) II Noether Theorem. First of all, recall that the operator ℓ_{E(ℒ)} : κ|_{ℰ} ⟶ κ|_{ℰ}^{†} is self-adjoint, i.e., ℓ_{E(ℒ)} = ℓ_{E(ℒ)}^{†} : κ|_{ℰ} ⟶ κ|_{ℰ}^{†}. This fact is key in the calculus of variations [39] and will be crucial in what follows (for a proof see, for instance, [9, 39] - see also [3] for an alternative approach). The usual definition of (infinitesimal) gauge symmetries of a lagrangian field theory is the following (see [25]).

Definition 16. A Noether gauge (or local) symmetry of the lagrangian theory (π, S) is a horizontal linear differential operator G : Q ⟶ κ such that G(ε) is a Noether symmetry for any ε ∈ Q.

We added the prefix “Noether” in the above definition of a “gauge symmetry” to distinguish it from an alternative (and, generally, inequivalent) definition that will be proposed below. Physicists say sometimes that G is a Noether symmetry depending on the arbitrary parameters ε. The second Noether theorem states that, in presence of gauge symmetries, there are relations among the Euler-Lagrange equations. Namely, for all ε ∈ Q,

0 = L_{G(ε)}S
= i_{G(ε)}dS
= ∫ i_{�_{G(ε)}} d^{V} ℒ
= ∫ ℓ̃_{ℒ} (G(ε))
= ∫ ⟨1, (ℓ̃_{ℒ} ∘ G)(ε)⟩
= ∫ ⟨(ℓ̃_{ℒ} ∘ G)^{†}(1), ε⟩
= ∫ ⟨G^{†}(ℓ̃_{ℒ}^{†} 1), ε⟩
= ∫ ⟨G^{†}(E(ℒ)), ε⟩,

and it follows from the arbitrariness of ε that G^{†}(E(ℒ)) = 0. These relations are traditionally called Noether identities. An “infinitesimal version” of the second Noether theorem can be formulated. First of all, notice that, since G(ε) is a Noether symmetry (so that G(ε)|_{ℰ} is the generating section of a symmetry of ℰ) for all ε, one also has 0 = ℓ_{E(ℒ)} G(ε)|_{ℰ} = (ℓ_{E(ℒ)} ∘ G^{ℰ})(ε|_{ℰ}) and, from the arbitrariness of ε,

ℓ_{E(ℒ)} ∘ G^{ℰ} = 0.  (27)

Identity (27) may be interpreted by saying that the linearized Euler-Lagrange equations admit “gauge symmetries”. Indeed, if χ ∈ κ|_{ℰ} is in the kernel of ℓ_{E(ℒ)} so is the “gauge transformed” element χ+G^{ℰ} (ϵ), for any arbitrary ϵ ∈ Q|_{ℰ}. In particular, the linearized Euler-Lagrange equations are, in a sense, “underdetermined”. By passing to the adjoint operators in (27) and using the self-adjointness of ℓ_{E(ℒ)} we get

(G^{ℰ})^{†} ∘ ℓ_{E(ℒ)} = 0.  (28)

This shows that there are relations among the linearized Euler-Lagrange equations and that they are, in a sense, “constrained”. Thus, “infinitesimal gauge symmetries correspond to infinitesimal constraints” via adjunction [25]. Identities (28) (and sometimes the operator (G^{ℰ})^{†} itself) are called infinitesimal Noether identities. Now let Δ_{1} : κ|_{ℰ}^{†} ⟶ P_{2} be a compatibility operator for ℓ_{E(ℒ)} . Consider also the adjoint operator Δ_{1}^{†} : P_{2}^{†} ⟶ κ|_{ℰ}^{†}. In particular, Δ_{1} ∘ ℓ_{E(ℒ)} = 0 and (using again the self-adjointness of ℓ_{E(ℒ)} ) ℓ_{E(ℒ)} ∘Δ_{1}^{†} = 0. In view of the last identity, if χ ∈ κ|_{ℰ} is in the kernel of ℓ_{E(ℒ)} so is the element χ + Δ_{1}^{†}ϑ, for any arbitrary ϑ ∈ P_{2}^{†}. Notice also that, in view of Proposition 6, all infinitesimal Noether identities (G^{ℰ})^{†} “are generated by Δ_{1}” in the sense that (G^{ℰ})^{†} = ∇ ∘ Δ_{1} for some horizontal differential operator ∇ : P_{2} ⟶ Q|_{ℰ}. Similarly, by passing to the adjoint operators, we see that all infinitesimal gauge symmetries G^{ℰ} are generated by Δ_{1}^{†}, i.e., G^{ℰ} = Δ_{1}^{†}∘∇^{†} for some ∇^{†} : Q|_{ℰ} ⟶ P_{2}^{†}. These simple remarks suggest a more natural definition of infinitesimal gauge symmetries.

Definition 17. A gauge symmetry of the lagrangian theory (π, S) is an element in the image of the adjoint operator Δ_{1}^{†} of a compatibility operator Δ_{1} for ℓ_{E(ℒ)} .

We will sometimes denote by 𝔤 := im Δ_{1}^{†} the set of gauge symmetries. Notice that, in view of Theorem 7, the above definition is independent of the choice of Δ_{1}. Moreover, while it is clear that im G^{ℰ} ⊂ 𝔤 for any Noether gauge symmetry G, to the author knowledge it has not been determined yet in full rigour and generality if 𝔤 is generated by the images of Noether gauge symmetries or not. Therefore, we prefer to adopt definition 17. This choice is strengthened even more by the results presented in the remaining part of this section. Consider the natural ℝ-linear map

Ω : D(P)^{•} ∋ X ⟼ Ω(X) := i_{X}ω ∈ Λ^{1}(P)^{•}.

Definition 18. The kernel ker Ω ⊂ D(P)^{•} is called the degeneracy distribution of ω and will be also denoted by ker ω. The secondary 2-form ω is said to be 1) weakly symplectic (or non-degenerate) iff ker ω = 0, 2) strongly symplectic (or, simply, symplectic) iff Ω is an isomorphism.

In order to better characterize ω it is desirable to describe its degeneracy distribution. First of all, notice that, since ω is closed, ker ω is a secondary involutive distribution, i.e., it is a graded Lie subalgebra in D(P)^{•}. Indeed, let X, Y ∈ ker ω then

Ω([X, Y ]) = i_{[X,Y ]}ω = [i_{X}, L_{Y}]ω = [i_{X}, [i_{Y}, d]]ω = 0,

i.e., [X, Y ] ∈ ker ω. Denote by Ω^{r} : D(P)^{r} ⟶ Λ^{1}(P)^{r+n−1} the restriction of Ω to D(P)^{r}, r = 0, …, n. Obviously, Ω^{r} = 0 for r > 1, independently of the lagrangian theory. For this reason, every degree r > 1 secondary vector field over P is said to be a trivial element in ker ω. Thus, non-trivial elements in ker ω must be searched in D(P)^{0} and D(P)^{1}. In the following we will “describe” such elements. Put tker ω := ker ω ∩ ⨁_{r>1} D(P)^{r}.

Theorem 19. Diagrams

[figure: D(P)^{0}; Ω^{0}; Λ^{1}(P)^{n−1}; ; 0; im Δ_{1}^{†}; ker ℓ_{E(ℒ)}; ; ker ℓ_{E(ℒ)} / im Δ_{1}^{†}; 0]  (29)

and

[figure: D(P)^{1}; Ω^{1}; Λ^{1}(P)^{n}; ; 0; ker Δ_{1}/ im ℓ_{E(ℒ)}; ; coker ℓ_{E(ℒ)}; κ|_{ℰ}^{†} / ker Δ_{1}; 0]  (30)

commute.

Proof. The vertical arrows in Diagram (29) are described in Section 1.6. Thus, let X = �_{χ} ∈ D(P)^{0}, χ ∈ κ|_{ℰ}, ℓ_{E(ℒ)} χ = 0. Let χ̃ ∈ κ be such that χ̃|_{ℰ} = χ. Now, Ω^{0}(X) = i_{X}ω = [i_{ℰ}^{∗}(−i_{�χ̃} d^{V} θ)], θ being a Legendre form. Put ˜□ := (η_{0}^{∗})^{−1}(−i_{�χ̃} d^{V} θ) ∈ 𝒞 Diff(κ, \overline{Λ}^{n−1}(J^{∞})) and □ := ˜□^{ℰ} ∈ 𝒞 Diff(κ|_{ℰ} , \overline{Λ}^{n−1}(ℰ)). Then, obviously, η_{E(ℒ)}^{∗} (□) = i_{ℰ}^{∗}(−i_{�χ̃} d^{V} θ). Show that \overline{d} ∘ □ = Δ_{χ} ∘ ℓ_{E(ℒ)} where Δ_{χ} ∈ 𝒞 Diff(κ|_{ℰ}^{†} , \overline{Λ}^{n}(ℰ)) is defined by putting Δ_{χ}φ := ⟨φ, χ⟩, φ ∈ κ|_{ℰ}^{†} (thus, Δ_{χ} is actually a C^{∞}(ℰ)-linear map). Indeed, let χ_{1} ∈ κ and put \overline{χ}_{1} := χ_{1}|_{ℰ}. Compute

(\overline{d} ∘ □)(\overline{χ}_{1}) = \overline{d}( ˜□χ_{1})|_{ℰ}
= \overline{d}(−i_{�_{χ_{1}}} i_{�χ̃} d^{V} θ)|_{ℰ}
= (−i_{�χ̃} i_{�_{χ_{1}}} d^{V} \overline{d}θ)|_{ℰ}
= (i_{�χ̃} i_{�_{χ_{1}}} d^{V} E(ℒ))|_{ℰ}
= (i_{�χ̃} L_{�_{χ_{1}}} E(ℒ))|_{ℰ} − (i_{�χ̃} d^{V} ⟨E(ℒ), χ_{1}⟩)|_{ℰ}
= ⟨ℓ_{E(ℒ)} \overline{χ}_{1}, χ⟩ − (L_{�χ̃} ⟨E(ℒ), χ_{1}⟩)|_{ℰ}
= ⟨ℓ_{E(ℒ)} \overline{χ}_{1}, χ⟩ − ⟨ℓ_{E(ℒ)} χ, \overline{χ}_{1}⟩
= (Δ_{χ} ∘ ℓ_{E(ℒ)} )(\overline{χ}_{1}),

where we used Identities (25) and (26). It follows from the arbitrariness of χ_{1} that \overline{d} ∘ □ = Δ_{χ} ∘ ℓ_{E(ℒ)} . Therefore, i_{X}ω corresponds to Δ_{χ}^{†}1 + im Δ_{1}^{†} ∈ ker ℓ_{E(ℒ)} / im Δ_{1}^{†} via isomorphism Λ^{1}(P)^{n−1} ≃ ker ℓ_{E(ℒ)} / im Δ_{1}^{†}. Finally, it is easy to see that Δ_{χ}^{†}1 = χ. Now, consider diagram (30) whose vertical arrows are described in Section 1.6 as well. Let φ ∈ κ|_{ℰ}^{†} and j ∈ \overline{J}^{∞}κ be such that Δ_{1}φ = 0 and \overline{j}_{∞}φ = h_{E(ℒ)} (j|_{ℰ}) ∈ \overline{J}^{∞}κ|_{ℰ}^{†}. Since \overline{J}^{∞}κ is pro-finitely generated by elements of the form \overline{j}_{∞}χ,^{2} χ ∈ κ, then j = ∑ f\overline{j}_{∞}χ for some (generally, infinite in number) …, f, … ∈ C^{∞}(J^{∞}) and …, χ, … ∈ κ. Consequently, φ = ∑ f|_{ℰ} ℓ_{E(ℒ)} χ|_{ℰ}. Put Z := (\overline{S} ∘ η_{0}^{−1})(j) = ∑ �_{χ} ⊗ \overline{d}f ∈ V D(J^{∞}) ⊗ \overline{Λ}^{1}(J^{∞}) and recall that 1) Z restricts to ℰ and 2) φ + im ℓ_{E(ℒ)} ∈ ker Δ_{1}/ im ℓ_{E(ℒ)} corresponds to Z := [\overline{Z}] ∈ D(P)^{1}, \overline{Z} ∈ V D(ℰ) ⊗ \overline{Λ}^{1}(ℰ) being the restriction of Z to ℰ, via isomorphism ker Δ_{1}/ im ℓ_{E(ℒ)} ≃ D(P)^{1}. Now, Ω^{1}(Z) = i_{Z}ω = [i\overline{Z}i_{ℰ}^{∗}(d^{V} θ)] = [i_{ℰ}^{∗}(i_{Z}d^{V} θ)]. Compute

i_{Z}d^{V} θ = ∑ \overline{d}f ∧ i_{�_{χ}} d^{V} θ
= \overline{d}ρ − ∑ f\overline{d}i_{�_{χ}} d^{V} θ
= \overline{d}ρ − ∑ fi_{�_{χ}} d^{V} \overline{d}θ
= \overline{d}ρ + ∑(fL_{�_{χ}} E(ℒ) − fd^{V} ⟨E(ℒ), χ⟩),

where ρ = ∑ fi_{�_{χ}} d^{V} θ ∈ 𝒞 Λ^{1}(J^{∞}) ⊗ \overline{Λ}^{n}(J^{∞}). Therefore,

i\overline{Z}i_{ℰ}^{∗}(d^{V} θ) = \overline{d}i_{ℰ}^{∗}(ρ) + ∑ i_{ℰ}^{∗}(fL_{�_{χ}} E(ℒ) + fd^{V} ⟨E(ℒ), χ⟩)
= \overline{d}i_{ℰ}^{∗}(ρ) + ∑ η_{E(ℒ)}^{∗} (f|_{ℰ} ℓ_{E(ℒ)} χ|_{ℰ}) + ∑ fd^{V} i_{ℰ}^{∗}⟨E(ℒ), χ⟩
= \overline{d}i_{ℰ}^{∗}(ρ) + η_{E(ℒ)}^{∗} (φ).

Finally, Ω^{1}(Z) = [η_{E(ℒ)}^{∗} (φ)] corresponds to φ^{†}(1) + ker Δ_{1} ∈ κ|_{ℰ}^{†} ∈ coker ℓ_{E(ℒ)} via isomorphism Λ^{1}(P)^{n} ≃ coker ℓ_{E(ℒ)} . It is easy to prove that φ^{†}(1) = φ and this concludes the proof. □

Some corollaries are in order.

Corollary 20. There is a natural isomorphism ker ω ≃ 𝔤 ⊕ tker ω.

Corollary 21. 𝔤 ⊂ ker ℓ_{E(ℒ)} is a Lie subalgebra (see, for instance, [7]).

Corollary 22. Let G : Q ⟶ κ be a Noether gauge symmetry. Then im G^{ℰ} ⊂ ker ω (see also [25]).

Corollary 23. The secondary 2-form ω is weakly symplectic iff it is strongly symplectic iff the Euler-Lagrange equations ℰ_{E(ℒ)} are irreducible.

Proof. In view of Theorem 19, Ω^{0} and Ω^{1} are isomorphisms iff ℰ_{E(ℒ)} is an irreducible PDE (see Example 11). In view of the 2-lines Theorem 9 (s = 2), irreducibility of ℰ_{E(ℒ)} implies, in its turn, that tker ω = 0. □

2This means that an element in \overline{J}^{∞}κ may be understood as a(n equivalence class of) formal infinite linear combination(s) of elements of the form \overline{j}_{∞}χ, χ ∈ κ. Notice that, in any case, all the following computations remain still valid.

2.4. Gauge Invariant Secondary Functions. Let N be a smooth manifold and σ ∈ Λ^{2}(N) a presymplectic structure on it. There is no Poisson structure on N associated to σ . However, a Poisson bracket may be introduced among “gauge invariant” functions on N, i.e., functions which are constant along the leaves of the degeneracy distribution of σ. This is precisely the Poisson bracket on the symplectic reduction of (N, σ). In this section we describe “gauge invariant secondary functions” on the CPS P and show that, similarly to the standard situation, ω induces a Lie bracket among them. Thus, the results presented in this section are propaedeutic to a “secondary symplectic reduction” of (P , ω) (see next section).

Definition 24. A secondary function f ∈ C^{∞}(P)^{•} is called gauge invariant iff L_{Y}f = 0 for all Y ∈ ker ω.

Let us describe gauge invariant elements in C^{∞}(P)^{n−1} and C^{∞}(P)^{n}.

Proposition 25. Any element in C^{∞}(P)^{n−1} is gauge invariant.

Proof. Recall that the map Ω^{0} : D(P)^{0} ⟶ Λ^{1}(P)^{n−1} is surjective (see Theorem 19). For any f ∈ C^{∞}(P)^{n−1}, let X ∈ D(P)^{0} be such that Ω(X) = df ∈ Λ^{1}(P)^{n−1} and Y ∈ ker ω. Then L_{Y}f = i_{Y}df = i_{Y}i_{X}ω = −i_{X}i_{Y}ω = 0. □

Now, let f_{1}, f_{2} ∈ C^{∞}(P)^{n−1} and X_{1}, X_{2} ∈ D(P)^{0} be such that Ω(X_{1}) = df_{1} and Ω(X_{2}) = df_{2}. Put {f_{1}, f_{2}} := −i_{X_{1}}i_{X_{2}}ω ∈ C^{∞}(P)^{n−1}.

Corollary 26. (C^{∞}(P)^{n−1}, { · , · }) is a well defined Lie algebra.

Proof. In view of Proposition 25, {f_{1}, f_{2}} is well defined for all f_{1}, f_{2} ∈ C^{∞}(P)^{n−1}, i.e., it is independent of the choice of X_{1}, X_{2}. Skew-symmetry and the Leibnitz rule follow (as in standard presymplectic geometry) from dω = 0 and the fact that, if Ω(X_{1}) = df_{1} and Ω(X_{2}) = df_{2}, then Ω([X_{1}, X_{2}]) = d{f_{1}, f_{2}}. □

Notice that the existence of a natural Lie bracket among conservation laws of an Euler-Lagrange equation was already known and may be also proved by “off shell” methods such as BRST ones (see, for instance, [5, 6]).

Proposition 27. An element F ∈ C^{∞}(P)^{n} is gauge invariant iff dF ∈ im Ω.

Proof. If dF = Ω(Z) for some Z ∈ D(P)^{1}, then F is gauge invariant (see the proof of Proposition 25). Vice versa, suppose L_{Y}F = 0 for all Y ∈ ker ω. Let F = ∫ ρ, ρ ∈ \overline{Λ}^{n}(ℰ). Recall that dF = [d^{V} ρ] ∈ Λ^{1}(P)^{n−1} corresponds to □^{†}1 + im ℓ_{E(ℒ)} ∈ coker ℓ_{E(ℒ)} via the isomorphism Λ^{1}(P)^{n−1} ≃ coker ℓ_{E(ℒ)} , □ : κ|_{ℰ} ⟶ \overline{Λ}^{n}(ℰ) being any horizontal differential operator such that η_{ℰ}^{∗}(□) = d^{V} ρ. In view of Theorem 19, dF ∈ im Ω if Δ_{1}(□^{†}1) = 0. Let χ = Δ_{1}^{†}ϑ ∈ κ|_{ℰ}, ϑ ∈ P_{1}^{†}, and Y = �_{χ}. Then

0 = L_{Y}F
= ∫ L_{�_{χ}} ρ

= ∫ i_{�_{χ}} d^{V} ρ
= ∫ □χ
= ∫ (□ ∘ Δ_{1}^{†})(ϑ)
= ∫ ⟨(□ ∘ Δ_{1}^{†})(ϑ), 1⟩
= ∫ ⟨ϑ, (Δ_{1} ∘ □^{†})(1)⟩.

It follows from the arbitrariness of ϑ that Δ_{1}(□^{†}1) = 0, and this concludes the proof. □

The Lie algebra (C^{∞}(P)^{n−1}, { · , · }) acts naturally on gauge invariant elements in C^{∞}(P)^{n}. Indeed, let F ∈ C^{∞}(P)^{n} be a gauge invariant element and f ∈ C^{∞}(P)^{n−1}. Put {f, F } := L_{X}F ∈ C^{∞}(P)^{n}, X ∈ D(P)^{0} being any secondary vector field such that Ω^{0}(X) = df. Exactly as above, {f, F } is well defined. Moreover, it holds the

Proposition 28. {f, F } is gauge invariant.

Proof. Recall that, in view of Proposition 27, dF ∈ im Ω^{1}, i.e., dF = i_{Z}ω for some Z ∈ D(P)^{1}. Show that d{f, F } ∈ im Ω^{1} as well and then apply Proposition 27 again. Indeed,

d{f, F } = dL_{X}F
= L_{X}dF
= L_{X}i_{Z}ω
= [L_{X}, i_{Z}]ω + i_{Z}L_{X}ω
= −i_{[X,Z]}ω + i_{Z}di_{X}ω
= Ω^{1}(−[X, Z]) + i_{Z}ddf
= Ω^{1}([Z, X]).
□

It is easy to prove that the action of C^{∞}(P)^{n−1} on gauge invariant elements in C^{∞}(P)^{n} is indeed a Lie-algebra representation.

Remark 29. Notice that if the Euler-Lagrange equations are irreducible, then Ω is an isomorphism, ker Ω = 0 and every element in C^{∞}(P)^{•} is trivially gauge invariant. In this case (C^{∞}(P)^{n−1}, { · , · }) acts on the whole C^{∞}(P)^{n}.

In [8] (see also [14]) it has been shown that the bracket described in full rigour in this section coincides with the Peierls bracket [29]. In its turn the Peierls bracket is at the basis of a covariant approach to quantization of field theories [13]. It is likely that the mathematically rigorous picture presented here will help to better understand, deal with and, possibly, generalize this complicated “functional” structure.

2.5. Perspectives: Secondary Symplectic Reduction. Most of the remarks in this section will be informal. From the physical point of view, gauge invariant functions on P are the true observables of the lagrangian theory and, therefore, play a special role. We shew in the last section that, basically, a Lie bracket is defined on gauge invariant functions. We may go even further and ask:

(1) are gauge invariant functions secondary functions on some secondary manifold P̃?

(2) if yes, is P̃ a symplectic reduction of the secondary “presymplectic manifold” (P , ω)?

In some more details, asking the last question amounts to wonder if there is an embedding of algebras π^{∗} : Λ(P̃)^{•} ↪ Λ(P)^{•} and a secondary two form ω̃ on P̃ such that 1) ker ω̃ = 0 and 2) ω = π^{∗}(ω̃). Finding an answer to the above questions would definitely establish the parallelism between secondary calculus on the CPS and standard theory of constrained (finite-dimensional) hamiltonian systems. Moreover, it would fix the bases of a mathematically rigorous, covariant, symplectic formalism for classical lagrangian field theories. Finally, it would represent a well founded starting point for a covariant quantization of gauge systems [17]. A possible route through the answers to the above questions is described below. First of all, there is a geometric counterpart of the degeneracy distribution of ω. Let

\xymatrix{0 \ar[r] & κ|_{ℰ} \ar[r]^{ℓ_{E(ℒ)}} & κ|_{ℰ}^{†} \ar[r]^{Δ_{1}} & P_{2} \ar[r]^{Δ_{2}} & ⋯}

be a compatibility complex for ℓ_{E(ℒ)} and

\xymatrix{0 & κ|_{ℰ}^{†} \ar[l] & κ|_{ℰ} \ar[l]_{ℓ_{E(ℒ)}} & P_{2}^{†} \ar[l]_{Δ_{1}^{†}} & ⋯ \ar[l]}

its adjoint complex. There is an associated complex of C^{∞}(ℰ)-modules:

\xymatrix{0 & \overline{J}^{∞}κ|_{ℰ}^{†} \ar[l] & \overline{J}^{∞}κ|_{ℰ} \ar[l]_{h_{E(ℒ)}^{∞}} & \overline{J}^{∞}P_{2}^{†} \ar[l]_{h_{1}^{∞}} & ⋯ \ar[l]} ,

where we put h_{E(ℒ)}^{∞} := h_{ℓ_{E(ℒ)}}^{∞} , h_{1}^{∞} := h_{Δ_{1}^{†}}^{∞} and so on. As discussed above, ker h_{E(ℒ)}^{∞} ⊂

\overline{J}^{∞}κ|_{ℰ} identifies with V D(ℰ) ⊂ V D(J^{∞})|_{ℰ} via the isomorphism \overline{J}^{∞}κ|_{ℰ} ≃ V D(J^{∞})|_{ℰ} that sends \overline{j}_{∞}χ to �_{χ} (see Section 1.5). In particular, ker h_{E(ℒ)}^{∞} has got a natural Lie algebra structure. Similarly, im h_{1}^{∞} identifies with the module of sections of an involutive distribution 𝒢 on ℰ made of vertical vector fields.

Proposition 30. im h_{1}^{∞} ⊂ ker h_{E(ℒ)}^{∞} is a Lie-subalgebra.

Proof. Let j_{1}, j_{2} ∈ \overline{J}^{∞}P_{2}^{†}. Then j_{1} = ∑ f_{1}\overline{j}_{∞}ϑ_{1} and j_{2} = ∑ f_{2}\overline{j}_{∞}ϑ_{2} for some …, f_{1}, f_{2}, … ∈ C^{∞}(J^{∞}) and …, ϑ_{1}, ϑ_{2}, … ∈ P_{2}^{†} (see Footnote 2.3, Section 2.3, p. 27). Moreover, h_{1}^{∞}(j_{1}), h_{1}^{∞}(j_{2}) correspond to vector fields X_{1} := ∑ f_{1}�_{Δ_{1}^{†}ϑ_{1}}, X_{2} := ∑ f_{2}�_{Δ_{1}^{†}ϑ_{2}}, respectively, via the isomorphism ker h_{E(ℒ)}^{∞} ≃ V D(ℰ). Compute

[X_{1}, X_{2}] = ∑[f_{1}�_{Δ_{1}^{†}ϑ_{1}}, f_{2}�_{Δ_{1}^{†}ϑ_{2}}]
= ∑(f_{1}(�_{Δ_{1}^{†}ϑ_{1}}f_{2})�_{Δ_{1}^{†}ϑ_{2}} − f_{2}(�_{Δ_{1}^{†}ϑ_{2}}f_{1})�_{Δ_{1}^{†}ϑ_{1}} + f_{1}f_{2}�_{{Δ_{1}^{†}ϑ_{1},Δ_{1}^{†}ϑ_{2}}}).

Now, recall that 𝔤 = im Δ_{1}^{†} ⊂ ker ℓ_{E(ℒ)} is a Lie subalgebra (see Corollary 21) so that {Δ_{1}^{†}ϑ_{1}, Δ_{1}^{†}ϑ_{2}} = Δ_{1}^{†}ϑ for some ϑ ∈ P_{2}^{†}. Put

j := ∑ f_{1}(�_{Δ_{1}^{†}ϑ_{1}}f_{2})\overline{j}_{∞}ϑ_{2} − f_{2}(�_{Δ_{1}^{†}ϑ_{2}}f_{1})\overline{j}_{∞}ϑ_{1} + f_{1}f_{2}\overline{j}_{∞}ϑ ∈ \overline{J}^{∞}P_{2}^{†}.

Then h_{1}^{∞}(j) ∈ im h_{1}^{∞} ⊂ ker h_{E(ℒ)}^{∞} corresponds to [X_{1}, X_{2}] ∈ V D(ℰ) via the isomorphism ker h_{E(ℒ)}^{∞} ≃ V D(ℰ). □

In the following we will understand isomorphism ker h_{E(ℒ)}^{∞} ≃ V D(ℰ). In view of Proposition 30, 𝒢 is an involutive distribution on ℰ. Namely, 𝒢 is the (involutive) distribution generated by evolutionary derivatives with generating sections in 𝔤 (such kinds of distributions have been recently considered in [19]). Notice that the horizontal Spencer differential \overline{S} : V D(ℰ) ⊗ \overline{Λ}(ℰ) ⟶ V D(ℰ) ⊗ \overline{Λ}(ℰ) “restricts” to im h_{1}^{∞} ⊗ \overline{Λ}(ℰ). Denote by \overline{s} : im h_{1}^{∞} ⊗ \overline{Λ}(ℰ) ⟶ im h_{1}^{∞} ⊗ \overline{Λ}(ℰ) the restricted differential. Clearly, 𝔤 ⊂ 𝔤_{1} := H^{0}(im h_{1}^{∞} ⊗ \overline{Λ}(ℰ), \overline{s}) = D(P)^{0} ∩ im h_{1}^{∞}. We now describe the quotient 𝔤_{1}/𝔤. Let □ : κ|_{ℰ} ⟶ Q_{2} be a compatibility operator for Δ_{1}^{†} : P_{2}^{†} ⟶ κ|_{ℰ}, and put k := h_{□}^{∞} : \overline{J}^{∞}κ|_{ℰ} ⟶ \overline{J}^{∞}Q_{2}. Then im h_{1}^{∞} = ker k, so that

𝔤_{1} = H^{0}(im h_{1}^{∞} ⊗ \overline{Λ}(ℰ), \overline{s}) = H^{0}(ker k ⊗ \overline{Λ}(ℰ), \overline{s}) = ker □.

We conclude that 𝔤_{1}/𝔤 = ker □/ im Δ_{1}^{†} ≃ H^{1}(im h_{1}^{∞} ⊗ \overline{Λ}(ℰ), \overline{s}) (see Theorem 7) and there is an exact sequence (of vector spaces)

\xymatrix{0 \ar[r] & 𝔤 \ar@{^{(}->}[r] & 𝔤_{1} \ar@{->>}[r] & H^{1}(im h_{1}^{∞} ⊗ \overline{Λ}(ℰ), \overline{s}) \ar[r] & 0}.

Thus H^{1}(im h_{1}^{∞} ⊗ \overline{Λ}(ℰ), \overline{s}) is the obstruction to 𝔤 being isomorphic to 0-cohomology of the complex (im h_{1}^{∞} ⊗ \overline{Λ}(ℰ), \overline{s}), that is, in a sense, the obstruction to “the algebraic description and the geometric description of gauge symmetries coinciding”. Despite the possible existence of such an obstruction, define the new distribution on ℰ, 𝒞̃ := ℰ + 𝒢. 𝒞̃ is, generally, infinite-dimensional. Moreover, it is an involutive distribution. Roughly speaking, integral submanifolds of 𝒞̃identify with “gauge equivalence classes” of solutions of the Euler-Lagrange equations. Therefore, it is natural to put P̃ := {maximal integral submanifolds of 𝒞̃} and interpret P̃ as the space of “physical states” of fields of the lagrangian theory (π, S).

A secondary calculus may be introduced on P̃, basically via the 𝒞̃-spectral sequence 𝒞̃E(ℰ), so that elements in 𝒞̃E_{1}(ℰ) =: Λ(P̃)^{•} are interpreted as (secondary) differential forms on P̃. The inclusion 𝒞 ⊂ 𝒞̃ induces a morphism 𝒞̃E(ℰ) ⟶ 𝒞 E(ℰ) of spectral sequences whose 1-st term we denote by π^{∗} : Λ(P̃)^{•} ⟶ Λ(P)^{•}. Now, we’d like to interpret P̃ as a “(symplectically) reduced CPS”. In order to be able to do this in a consistent and physically meaningful way at least the following two conditions should be fulfilled:

(1) the image of C^{∞}(P̃)^{•} := 𝒞̃E_{1}^{0,•}(ℰ) under π^{∗} should be made of gauge invariant (secondary) functions on P,

(2) a secondary 2-form ω̃ on P̃ should exist so that ker ω̃ = 0 and π^{∗}(ω̃) = ω.

If this was the case then, in the author’s opinion, (P̃, ω̃) could be “safely” referred to as the “symplectic reduction of (P , ω)” from the mathematical point of view, and as the “reduced CPS” [17, 25, 30] from the physical point of view. As suggested by the example in this section and by preliminary work by the author, typical homological algebra (and, possibly, homological perturbation theory) techniques seem to be necessary to investigate further in this direction and complete the above sketched program.

Conclusions

We proposed a fully rigorous approach to the geometry of the covariant phase space P, and the canonical, closed 2-form ω on it, in the framework of secondary calculus. In particular, we described the kernel of ω in terms of the compatibility operator for the linearized Euler-Lagrange equations thus revealing the precise relation between gauge symmetries and constraints in field theory [25]. We also described gauge invariant (secondary) functions on P and their Lie algebra structure. It is likely that such a Lie algebra is at the basis of a covariant canonical quantization of the theory [13]. A step forward in this direction would be to rigorously perform a symplectic reduction of (P , ω). The preliminary analysis presented in Section 2.4 suggests that this is possible, and should be done, within secondary calculus (or a slight generalization of it) and, in any case, by means of cohomological techniques. We stress that, in this paper, we basically worked “on shell”. The relationship with “off shell” methods (Koszul-Tate resolution and BRST complex [5, 6, 17] - see also [36]) should be carefully analyzed.

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Luca Vitagliano, DMI, Università degli Studi di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA) and Istituto Nazionale di Fisica Nucleare, GC Salerno, Italy

  E-mail address: lvitagliano@unisa.it