The Diffiety Institute Preprint Series

Preprint DIPS–07 July 10, 2002

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

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

On symmetries and cohomological invariants of equations possessing flat

representations

by

S. Igonin, P. Kersten, I. Krasil′shchik

On symmetries and cohomological invariants of equations possessing flat representations

S. IGONIN, P. KERSTEN, I. KRASIL′SHCHIK

Abstract. We study the equation E_{fc} of flat connections in a given fiber bundle and discover a specific geometric structure on this equation, which we call a flat representation. We generalize this notion to arbitrary PDE and prove that flat representations of an equation E are in 1-1 correspondence with morphisms φ: E → E_{fc}, E and E_{fc} being treated as submanifolds of infinite jet spaces. We show that flat representations include several known types of zero-curvature formulations of PDEs. In particular, the Lax pairs of the self-dual Yang–Mills equations and their reductions are of this type. With each flat representation φ we associate a complex C_{φ} of vector-valued differential forms such that H^{1}(C_{φ}) describes infinitesimal deformations of the flat structure, which are responsible, in particular, for parameters in Bäcklund transformations. In addition, each higher infinitesimal symmetry S of E defines a 1-cocycle c_{S} of C_{φ}. Symmetries with exact c_{S} form a subalgebra reflecting some geometric properties of E and φ. We show that the complex corresponding to E_{fc} itself is 0-acyclic and 1acyclic (independently of the bundle topology), which means that higher symmetries of E_{fc} are exhausted by generalized gauge ones, and compute the bracket on 0-cochains induced by commutation of symmetries.

Contents

Introduction 2 1. Basic facts 4 1.1. Notation agreements 4 1.2. Jets and equations 4 1.3. Nijenhuis bracket 7 1.4. Coverings 8 1.5. Horizontal cohomologies 8 2. The equation of flat connections 9 2.1. More on connections 9 2.2. A coordinate-free formulation 11 3. Symmetries of the equation of flat connections 12 4. Flat representations 16 4.1. Definition and the main property 16 4.2. Deformations of a flat representation 17 4.3. Symmetries and cocycles 18 4.4. A bracket on the equation of flat connections 19 5. Examples of flat representations 20

2000 Mathematics Subject Classification. 37K35, 37K25, 37K10, 58J10, 53C05. Key words and phrases. Flat connections, flat representations, differential complexes, symmetries,

Nijenhuis bracket, zero-curvature representations, self-dual Yang–Mills equations.

2 S. Igonin, P. Kersten, I. Krasil′shchik

5.1. Coverings as flat representations and lifting of symmetries 20 5.2. Flat representations and symmetries of the KdV equation 20 5.3. Flat representations of the self-dual Yang–Mills equations 22 Acknowledgements 23 References 23

Introduction

Nonlinear PDE in more than two independent variables with nontrivial invariance properties is a sort of challenge for those who are interested in geometry of differential equations and related topics. Evidently, chosen “at random”, such an equation will be of no interest, but natural (whatever it means) equations arising in physics and geometry may provide the researcher with the desired results. Thinking of the second source (geometry), a first idea is to consider equations describing geometrical structures (e.g., integrable distributions, complex structures, etc.) and, by definition, possessing a rich symmetry group. One of the simplest equations of this type is the equation E_{fc} describing flat connections in a certain locally trivial fiber bundle.

Let π: E → M, dim M = n, dim E = n + m, be a smooth locally trivial fiber bundle over a smooth manifold M. A connection ∇ in the bundle π is a C^{∞}(M)-linear correspondence that takes any vector field X on M to a vector field ∇X on E in such a way that π_{∗}(∇X)_{θ} = X_{π(θ)} for any point θ ∈ E. A connection is called flat if

∇[X, Y] = [∇X, ∇Y]

for any two vector fields X and Y on M.

Let U ⊂ M be a local chart, such that π becomes trivial over U, with local coordinates x_{1}, …, x_{n}, v^{1}, …, v^{m} in π^{−1}(U), x_{1}, …, x_{n} being coordinates in U. Then ∇ is determined by n×m arbitrary functions v_{i}^{α} = v_{i}^{α}(x_{1}, …, x_{n}, v^{1}, …, v^{m}) by the relations

∇(∂x_{i}) = ∂x_{i} + ∑_{α=1}^{m} v_{i}^{α}∂v^{α}, i = 1, …, n,

where and below ∂x_{i} = ∂/∂x_{i}, ∂v^{α} = ∂/∂v^{α}, etc., while the condition of flatness is expressed in the form

[∇(∂x_{i}), ∇(∂x_{j})] ≡ ∑_{α=1}^{m} (\frac{∂v_{j}^{α}}{∂x_{i}} + ∑_{β=1}^{m} v_{i}^{β} \frac{∂v_{j}^{α}}{∂v^{β}} ) ∂v^{α}

− ∑_{α=1}^{m} (\frac{∂v_{i}^{α}}{∂x_{j}} + ∑_{β=1}^{m} v_{j}^{β} \frac{∂v_{i}^{α}}{∂v^{β}} ) ∂v^{α} = 0,

i, j = 1, …, n. Thus, flat connections are distinguished by the following system of nonlinear equations

\frac{∂v_{j}^{α}}{∂x_{i}} + ∑_{β=1}^{m} v_{i}^{β} \frac{∂v_{j}^{α}}{∂v^{β}} = \frac{∂v_{i}^{α}}{∂x_{j}} + ∑_{β=1}^{m} v_{j}^{β} \frac{∂v_{i}^{α}}{∂v^{β}} ,  (1)

1 ≤ i < j ≤ n, α = 1, …, m. This is a system of mn(n − 1)/2 equations imposed on mn unknown functions v_{i}^{α}.

A strong motivation to study this equation is the following. If one assumes that the coefficients v_{i}^{α} of a connection are expressed in terms of some functions f_{k} in a special way, then the flatness is equivalent to a system E of differential equations on f_{k} such that each solution to E gives a flat connection. It is well known that many integrable nonlinear PDEs arise exactly in this way. Treating the equations E and E_{fc} with their differential consequences as (infinite-dimensional) submanifolds of the corresponding infinite jet spaces, we obtain a smooth map

φ: E → E_{fc},  (2)

which preserves the Cartan distribution. Such a map generates on E an additional geometric structure which we call a flat representation. In particular, the identical map id: E_{fc} → E_{fc} determines a canonical flat representation for E_{fc} itself.

Thus E_{fc} is a sort of universal space where integrable equations live (or at least a broad class of them). In particular, we show that coverings (generalized Lax pairs) of equations in two independent variables as well as the Lax pairs of the self-dual Yang– Mills equations and their reductions lead to flat representations. Therefore, studying the geometry of E_{fc} may help to obtain new knowledge on integrable systems, and in this paper we begin the study. Despite the fact that locally any flat connection can be obtained from a trivial one by means of a local automorphism of the bundle, there is no natural way to parameterize all flat connections, and this makes the geometry of E_{fc} nontrivial.

We show that each flat representation (2) leads to several complexes of vector-valued differential forms on E constructed by means of the Nijenhuis bracket. One subcomplex denoted by C_{φ} is of particular interest, since its 1-cocycles are infinitesimal deformations of the flat representation, the exact cocycles being trivial deformations. As usual, higher cohomology contains obstructions to prolongation of infinitesimal deformations to formal deformations [3]. In particular, zero-curvature representations and Bäcklund transformations dependent on a parameter provide an example of a deformed flat representation and determine, therefore, some 1-coholomolgy classes of C_{φ}.

In addition, each higher infinitesimal symmetry S of E defines a formal deformation and, therefore, a 1-cocycle c_{S} in C_{φ}. Symmetries with exact cocycles c_{S} form a Lie subalgebra with respect to the commutator. If morphism (2) is an embedding then it is precisely the subalgebra of those symmetries which are restrictions of some symmetries of the equation of flat connections. If (2) is coming from a covering of E then this subalgebra consists of symmetries which can be lifted to the covering equation.

The complex C_{id} corresponding to the identical flat representation id: E_{fc} → E_{fc} of E_{fc} itself is a new canonical complex associated with the bundle π: E → M. Symmetries of E_{fc} are in one-to-one correspondence with 1-cocycles of this complex.

We prove that

H^{0}(C_{id}) = H^{1}(C_{id}) = 0  (3)

independently of the topology of M and E. This implies that the higher symmetries of E_{fc} are in one-to-one correspondence with 0-cochains, which are sections of some vector bundle over E_{fc} of rank m. That is, symmetries are determined locally by m-tuples of smooth functions on E_{fc}. This is similar to the case of an infinite jet space. Due to their coordinate expression, it is natural to call them generalized gauge symmetries. The commutator of symmetries induces a bracket on 0-cochains, and we write down this bracket in explicit terms.

Note that only for a few multidimensional equations the complete description of symmetries is known. We mention the paper [1], which proves that symmetries of the vacuum Einstein equations in four spacetime dimensions are also freely determined by 5-tuples of smooth functions on the equation.

The paper is organized as follows. In Section 1, we review some facts on the geometry of PDE and the Nijenhuis bracket. In Section 2, we introduce the equation of flat connections and complexes of vector-valued forms on it. We prove (3) and deduce the description of symmetries of E_{fc} in Section 3. In Section 4, we introduce flat representations in general and prove the above facts on their deformations and relations with symmetries. The bracket on 0-cochains of C_{id} is computed in Subsection 4.4. Finally, in Section 5, we show that coverings of PDE and the Lax pair of the self-dual Yang-Mills equations lead to flat representations φ and study the map from symmetries to 1-cocycles of C_{φ} for these examples.

1. Basic facts

This is a brief review of the geometry of PDE. We refer to [2, 8] for more details.

1.1. Notation agreements. Let M be a smooth manifold and π be a smooth fiber bundle over M. Throughout the paper we use the following notation:

• D(M) denotes the C^{∞}(M)-module (and Lie algebra over ℝ) of vector fields on M. A distribution on M is a C^{∞}(M)-submodule of D(M).

• More generally, if P is an arbitrary C^{∞}(M)-module, then D(P) denotes the module of P -valued derivations:

D(P) = { X ∈ hom_{C^{∞}(M)}(C^{∞}(M), P) | X(fg) = fX(g) + gX(f), f, g ∈ C^{∞}(M)) }.

• Λ^{i}(M) is the module of i-differential forms on M while

Λ^{∗}(M) = ⨁_{i=1}^{dim M} Λ^{i}(M)

is the corresponding algebra (with respect to the wedge product ∧).

• Γ(π) (resp., Γ_{loc}(π)) is the set of all (resp., all local) sections of π. In particular, when π is a vector bundle, Γ(π) is a C^{∞}(M)-module.

1.2. Jets and equations. Let π: E → M be a fiber bundle and θ ∈ E, π(θ) = x ∈ M. Consider a local section f ∈ Γ_{loc}(π) whose graph passes through the point θ and the class [f]_{x}^{k} of all local sections whose graphs are tangent to the graph of f at θ with order ≥ k. The set

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

carries a natural structure of a smooth manifold and is called the manifold of k-jets of the bundle π. Moreover, the natural projections

π_{k}: J^{k}(π) → M, [f]_{x}^{k} ↦ x, π_{k,k−1} : J^{k}(π) → J^{k−1}(π), [f]_{x}^{k} ↦ [f]_{x}^{k−1},

are locally trivial bundles. If π is a vector bundle, then π_{k} is a vector bundle as well while all π_{k,k−1} carry a natural affine structure. So, we have an infinite sequence of bundles

⋯ → J^{k}(π) ⟶^{π_{k,k−1}} J^{k−1}(π) → ⋯ → J^{1}(π) ⟶^{π_{1}} J^{0}(π) = E ⟶^{π} M

and its inverse limit is called the manifold of infinite jets and denoted by J^{∞}(π). In an obvious way, one can consider the fiber bundles π_{∞}: J^{∞}(π) → M and π_{∞,k}: J^{∞}(π) → J^{k}(π). Using the notation similar to the above one, we can state that J^{∞}(π) consists of classes of the form [f]_{x}^{∞}.

Let f ∈ Γ_{loc}(π); then one can consider the sections j_{k}(f) ∈ Γ_{loc}(π_{k}), k = 0, 1, …, ∞, defined by j_{k}(f): x ↦ [f]_{x}^{k} and called the k-jet of f. Consider a point θ = [f]_{x}^{∞} ∈ J^{∞}(π) and the tangent plane to the graph of j_{∞}(f) at θ. One can show that this plane does not depend on the choice of f and is determined by the point θ only. Denote this plane by C_{θ}.

Let X ∈ D(M). Then by the above construction there exists a unique vector field CX ∈ D(J^{∞}(π)) such that CX_{θ} ∈ C_{θ} and (π_{∞})_{∗}CX_{θ} = X_{x} for any θ ∈ J^{∞}(π) and x = π_{∞}(θ). Thus we obtain a connection C in the bundle π_{∞} called the Cartan connection. The distribution CD ⊂ D(J^{∞}(π)) generated by the fields of the form CX, X ∈ D(M), is called the Cartan distribution.

Proposition 1. The Cartan distribution is integrable in the formal Frobenius sense, i.e., [CD, CD] ⊂ CD. The Cartan connection is flat, i.e., [CX, CY] = C[X, Y] for any X, Y ∈ D(M).

Remark 1. Consider a point θ_{k} ∈ J^{k}(π), k > 0. Then it is completely determined by the point θ_{k−1} = π_{k,k−1}(θ_{k}) = [f]_{x}^{k−1}, f ∈ Γ_{loc}(π), and the tangent plane to the graph of j_{k−1}f at θ_{k−1}. Consequently, sections of the bundle π_{k,k−1} : J^{k}(π) → J^{k−1}(π) are identified with horizontal n-dimensional distributions on J^{k−1}(π). In particular, sections of π_{1,0} : J^{1}(π) → E are just connections in the bundle π.

A k-th order differential equation imposed on sections of the bundle π is a submanifold E^{0} ⊂ J^{k}(π) of the manifold J^{k}(π). Any equation may be represented in the form

E^{0} = { θ ∈ J^{k}(π) | (Δ_{E})_{θ} = 0, Δ_{E} ∈ Γ(π_{k}^{∗}(ξ)) },

where ξ: E′ → M is a bundle and π_{k}^{∗}(ξ) is its pullback. The section Δ_{E} = Δ can be understood as a nonlinear differential operator acting from Γ(π) to Γ(ξ): Δ(f) = j_{k}(f)^{∗}(Δ_{E}) ∈ Γ(ξ), f ∈ Γ(π). Of course, neither ξ, nor Δ is unique.

The l-th prolongation of E^{0} is the set

E^{l} = { [f]_{x}^{k+l} | the graph of j_{k}(f) is tangent to E^{0} with order ≥ l at [f]_{x}^{k} ∈ E },

l = 0, 1, …, ∞. Restricting the maps π_{k+l} and π_{k+l,k+l−1} to E^{l} and preserving the same notation for these restrictions, we obtain the sequence of maps

⋯ → E^{l} ⟶^{π_{k+l,k+l−1}} E^{l−1} → ⋯ → E^{0} ⟶^{π_{k,0}} E ⟶^{π} M.

Without loss of generality, one can always assume π_{k} to be surjective. Imposing natural conditions of regularity, we shall also assume that all E^{l} are smooth manifolds, while the maps π_{k+l,k+l−1} : E^{l} → E^{l−1} are smooth fiber bundles (such equations are called formally integrable). Everywhere below the infinite prolongation E^{∞} will be denoted by E.

It can be shown that any vector field of the form CX, X ∈ D(M), is tangent to E. This means that the Cartan connection restricts from the bundle J^{∞}(π) → M to E → M and determines on E a horizontal n-dimensional integrable distribution, the Cartan distribution on E. We denote by CD_{E} ⊂ D(E) the submodule generated by the vector fields lying in the Cartan distribution on E. Its maximal integral manifolds are the infinite jets of solutions of E. A (higher) symmetry of E is a π-vertical vector field on E preserving the Cartan distribution, i.e., S ∈ D(E) is a symmetry if and only if

(a) S(f) = 0 for any function f ∈ C^{∞}(M) ⊂ C^{∞}(E);

(b) [S, Y] lies in CD_{E} whenever Y ∈ D(E) lies in CD_{E} (in fact, from (a) it follows

that [S, CX] = 0 for any X ∈ D(M)). The set sym E of all symmetries forms a Lie ℝ-algebra with respect to the commutation of vector fields. In the trivial case E = J^{∞}(π) we denote this algebra by sym π. A complete description of symmetries is given by the following

Theorem 1. Let π: E → M be a locally trivial vector bundle. Then there exists a one-to-one correspondence between sym π and sections of the pullback π_{∞,0}^{∗}(π^{v}), where π^{v}: T^{v}E → E is the vertical tangent bundle to E. If E^{0} ⊂ J^{k}(π) is an equation given by a differential operator Δ: Γ(π) → Γ(ξ) and satisfying the above formulated conditions, then sym E is in one-to-one correspondence with solutions of the equation

ℓ_{E}(φ) = 0,  (4)

where φ is a section of the pullback π_{∞,0}^{∗}(π^{v}) restricted to E while ℓ_{E} is the linearization of Δ also restricted to E.

Remark 2. We call φ the generating section of a symmetry. It may be considered as an element of the module Γ(π^{v}) ⊗_{C^{∞}(E)} C^{∞}(J^{∞}(π)). When π is a vector bundle, this module is isomorphic to Γ(π_{∞}^{∗}(π)) = Γ(π) ⊗_{C^{∞}(M)} C^{∞}(J^{∞}(π)).

Coordinates. Let U ⊂ M be a local chart with coordinates x_{1}, …, x_{n} and U × ℝ^{m} be a trivialization of π over U with coordinates v^{1}, …, v^{m} in ℝ^{m}. Then in π_{∞}^{−1}(U) the so-called adapted coordinates arise defined by

v_{σ}^{α}([f]_{x}^{k}) = \frac{∂^{k}f^{α}}{∂x_{i_{1}} … ∂x_{i_{k}}} |_{x} , f ∈ Γ_{loc}(π), x ∈ U, α = 1, …, m,  (5)

where σ = i_{1} … i_{k} is a multi-index, 1 ≤ i_{α} ≤ n. If |σ| ≤ k, then v_{σ}^{α} together with x_{1}, …, x_{n} constitute a local coordinate system in π_{k}^{−1}(U) ⊂ J^{k}(π).

In these coordinates, the Cartan connection is completely determined by its values on partial derivatives and the corresponding vector fields are the so-called total derivatives

C(∂x_{i}) ≡ D_{x_{i}} = ∂x_{i} + ∑_{σ,α} v_{σi}^{α} ∂v_{σ}^{α},  (6)

i = 1, …, n. The Cartan distribution is given by the system of Cartan forms

ω_{σ}^{α} = dv_{σ}^{α} − ∑_{i=1}^{n} v_{σi}^{α} dx_{i},  (7)

while the correspondence between Γ(π_{∞}^{∗}(π)) and sym π (see Theorem 1 and Remark rem:gen-sect) is expressed by the formula

φ = (φ^{1}, …, φ^{m}) ↦ �_{φ} = ∑_{σ,α} D_{σ}(φ^{α}) ∂v_{σ}^{α},  (8)

where D_{σ} = D_{x_{i_{1}}} ∘ ⋯ ∘ D_{x_{i_{k}}} and �_{φ} is called the evolutionary vector field with the generating section φ. Finally, if Δ: Γ(π) → Γ(ξ) is locally given by the relations

u^{s} = F^{s}(x_{1}, …, x_{n}, …, v_{σ}^{α}, …), s = 1, …, m′,

where m′ = dim ξ and u^{s} are coordinates in the fibers of the bundle ξ over U, then

ℓ_{Δ} = ∥∑_{σ} \frac{∂F^{s}}{∂v_{σ}^{α}} D_{σ}∥ , α = 1, …, m, s = 1, …, m′,  (9)

is its linearization. To get the expression for ℓ_{E} it suffices to rewrite (9) in internal coordinates on E.

1.3. Nijenhuis bracket. Let N be a smooth manifold and consider the ℤ-graded module D(Λ^{∗}(N)) = ⊕_{i=0}^{dim N}D(Λ^{i}(N)) over C^{∞}(N) of Λ^{∗}(N)-valued derivations. First note that for any vector field X ∈ D(N) and a derivation Ω: C^{∞}(N) → Λ^{i}(N) one can define a derivation X Ω: C^{∞}(N) → Λ^{i−1}(N) by

(X Ω)(f) = X Ω(f), f ∈ C^{∞}(N),

(the inner product, or contraction), where X Ω(f) is the usual inner product of a vector field and a differential form. Further, if ω ∈ Λ^{j}(N), then the inner product Ω ω ∈ Λ^{i+j−1}(N) is defined by induction on i + j:

X (Ω ω) = (X Ω) ω − (−1)^{i}Ω (X ω)

with an obvious base. Note that Ω is an operation of grading i − 1.

Now, let us define the Lie derivative L_{Ω} : Λ^{j}(N) → Λ^{j+i}(N) by

L_{Ω}(ω) = d(Ω ω) + (−1)^{i}Ω (dω).

Proposition 2. For any elements Ω ∈ D(Λ^{i}(N)) and Ω′ ∈ D(Λ^{i′}(N)) there exists a uniquely defined element [[Ω, Ω′]] ∈ D(Λ^{i+i′}(N)) satisfying the identity

[L_{Ω}, L_{Ω′}] = L_{[[Ω,Ω′]]},

where [L_{Ω}, L_{Ω′}] denotes the graded commutator.

The element [[Ω, Ω′]] is called the Nijenhuis (or Frölicher–Nijenhuis) bracket of Ω and Ω′. The module D(Λ^{∗}(N)) is a graded Lie algebra with respect to this bracket, i.e.,

[[Ω, Ω′]] + (−1)^{ii′}[[Ω′, Ω]] = 0,  (10)
[[Ω, [[Ω′, Ω′′]]]] = [[[[Ω, Ω′]], Ω′′]] + (−1)^{ii′}[[Ω′, [[Ω, Ω′′]]]],  (11)

where i, i′ and i′′ are the degrees of Ω, Ω′ and Ω′′, respectively. Note that for “decomposable” elements of D(Λ^{∗}(N)), i.e., for elements of the form Ω = ω ⊗ X, ω ∈ Λ^{∗}(N), X ∈ D(N), acting on f ∈ C^{∞}(M) by (ω ⊗ X)(f) = X(f)ω, one has

[[ω ⊗ X, ω′ ⊗ X′]] = ω ∧ ω′ ⊗ [X, X′] + ω ∧ X(ω′) ⊗ X′ − X′(ω) ∧ ω′ ⊗ X

+ (−1)^{i} dω ∧ (X ω′) ⊗ X′ + (−1)^{i}(X′ ω) ∧ dω′ ⊗ X, (12)

where X, X′ ∈ D(N), ω ∈ Λ^{i}(N), ω′ ∈ Λ^{i′}(N).

Let U ∈ D(Λ^{1}(N)). Then the operator ∂_{U} = [[U, ·]]: D(Λ^{i}(N)) → D(Λ^{i+1}(N)) is defined. If [[U, U]] = 0, then, due to properties (10) and (11), the operator ∂_{U} satisfies the condition ∂_{U} ∘ ∂_{U} = 0 and thus we obtain the complex

0 → D(N) ⟶^{∂_{U}} D(Λ^{1}(N)) → ⋯ → D(Λ^{i}(N)) ⟶^{∂_{U}} D(Λ^{i+1}(N)) → ⋯  (13)

with the corresponding cohomology groups H_{U}^{i}(N).

1.4. Coverings. Let ξ: N → M and ξ′: N′ → M be two fiber bundles endowed with flat connections ∇ and ∇′ respectively. Assume that N′ is fibered over N by some smooth surjection τ: N′ → N and ξ′ = ξ ∘ τ. We say that the pair (N′, ∇′) covers the pair (N′, ∇′) (or τ is a covering) if for any x ∈ M, X ∈ D(M) and θ′ ∈ N′ one has τ_{∗}(∇′X)_{θ′} = (∇X)_{τ(θ′)}. A covering τ is called linear if

(1) τ: N′ → N is a vector bundle;

(2) vector fields of the form C′X, preserve the space Lin(τ) ⊂ C^{∞}(N′), where Lin(τ) consists of smooth functions on N′ linear along the fibers of τ.

If N = E, ξ = π_{∞}: E → M and ∇ = C is the Cartan connection on E, we speak of a covering over E. In local coordinates, any covering structure over E is determined by a set of τ-vertical vector fields X_{1}, …, X_{n} on N′ such that the conditions

[D_{x_{i}} + X_{i}, D_{x_{j}} + X_{j}] = 0

hold for all 1 ≤ i < j ≤ n, where D_{x_{α}} are the total derivatives given by (6). In this case, if τ is finite-dimensional then the manifold N′ is locally also isomorphic to an infinitely prolonged differential equation called the covering equation with the Cartan connection ∇′ and the total derivatives D_{x_{i}} + X_{i}.

A detailed discussion of the theory of coverings over nonlinear PDE can be found in [2, 10].

1.5. Horizontal cohomologies. The notion of the Cartan connection can be generalized. Namely, consider an infinitely prolonged equation π_{∞}: E → M and two vector bundles ξ, ξ′ over M. Let Δ: Γ(ξ) → Γ(ξ′) be a linear differential operator. Consider the pullbacks π_{∞}^{∗}(ξ), π_{∞}^{∗}(ξ′) and denote the corresponding modules of sections by Γ(π, ξ), Γ(π, ξ′), respectively. Then there exists a unique linear differential operator CΔ: Γ(π, ξ) → Γ(π, ξ′) satisfying the local condition

j_{∞}(f)^{∗}CΔ(φ) = Δ(j_{∞}(f)^{∗}(φ))

for any f ∈ Γ_{loc}(π) such that θ = [f]_{x}^{∞} ∈ E and any φ ∈ Γ_{loc}(π, ξ) in the neighborhood of θ.

In particular, if ξ = ξ′: ⋀^{i} T^{∗}M → M is the i-th external power of the cotangent bundle, elements of the module Γ(π, ξ) ⊂ Λ^{i}(E) are called horizontal forms while the module itself is denoted by Λ_{h}^{i}(E). Taking the de Rham differential d: Λ^{i}(M) → Λ^{i+1}(M) and introducing the notation Cd = d_{h}, we obtain the complex

0 → C^{∞}(E) ⟶^{d_{h}} Λ_{h}^{1}(E) → ⋯ → Λ_{h}^{i}(E) ⟶^{d_{h}} Λ_{h}^{i+1}(E) → ⋯ → Λ_{h}^{n}(E),

the horizontal de Rham complex of E. Its cohomology at the i-th term is denoted by H_{h}^{i}(E) and called the horizontal cohomology. In the adapted coordinates we have

d_{h}(fdx_{i_{1}} ∧ … ∧ dx_{i_{k}}) = ∑_{i=1}^{n} D_{x_{i}}(f)dx_{i} ∧ dx_{i_{1}} ∧ … ∧ dx_{i_{k}}, f ∈ C^{∞}(E).  (14)

Proposition 3. If E = J^{∞}(π), π: E → M, then H_{h}^{i}(E) for i < n is isomorphic to the ith de Rham cohomology of the manifold E.

Remark 3. Let τ be a linear covering over E and P = Γ(τ). Then one can extend the horizontal differential to d_{h}: Λ_{h}^{i}(E) ⊗ P → Λ_{h}^{i+1}(E) ⊗ P and thus obtain horizontal cohomologies with coefficients in P.

For more details on horizontal cohomology see [9].

2. The equation of flat connections

2.1. More on connections. Let π: E → M be a fiber bundle. Consider the C^{∞}(E)- module D(M, E) of C^{∞}(E)-valued derivations C^{∞}(M) → C^{∞}(E). Due to the embedding π^{∗}: C^{∞}(M) → C^{∞}(E), for any vector field X ∈ D(E) one can consider its restriction X_{M} ∈ D(M, E) to M. A connection in the bundle π is a C^{∞}(M)-linear map ∇: D(M) → D(E) such that ∇(X)_{M} = X for any X ∈ D(M).

Remark 4. If M is finite-dimensional, this definition becomes equivalent to the following one: a connection in a bundle π is a C^{∞}(E)-linear map ∇: D(M, E) → D(E) such that ∇(X)_{M} = X for any X ∈ D(M, E). In particular, we use this fact to define the connection form below.

Consider the element Ū_{∇} ∈ D(Λ^{1}(E)) defined by

X Ū_{∇}(f) = (∇X_{M})(f),

where X ∈ D(E), f ∈ C^{∞}(E). The element U_{∇} ∈ D(Λ^{1}(E)) defined by

X U_{∇}(f) = X(f) − (∇X_{M})(f)

is called the connection form of ∇. Obviously, Ū_{∇} +U_{∇} = d, where d: C^{∞}(E) → Λ^{1}(E) is the de Rham differential

Recall that a connection ∇ is called flat, if [∇X, ∇Y] = ∇[X, Y] for any X, Y ∈ C^{∞}(M).

Proposition 4. For any Ω ∈ D(Λ^{∗}(E)) one has [[U_{∇}, Ω]] = [[Ū_{∇}, Ω]], and ∇ is flat if and only if [[Ū_{∇}, Ū_{∇}]] = [[U_{∇}, U_{∇}]] = 0.

From Proposition 4 and Subsection 1.3 it follows that to any flat connection ∇ we can associate the complex

0 → D(E) ⟶^{∂_{∇}} D(Λ^{1}(E)) → ⋯ → D(Λ^{i}(E)) ⟶^{∂_{∇}} D(Λ^{i+1}(E)) → ⋯,  (15)

where ∂_{∇} = [[U_{∇}, ·]] = [[Ū_{∇}, ·]], cf. (13), and the corresponding cohomology groups H_{∇}^{i}(E).

Remark 5. In any fiber bundle π: E → M one can consider π-vertical vector fields defined by

D^{v}(E) = { X ∈ D(E) | X_{M} = 0 }.

Dually, horizontal q-forms can be introduced as

Λ_{h}^{q}(E) = { ω ∈ Λ^{q}(E) | X_{1} … X_{q} ω = 0, X_{1}, …, X_{q} ∈ D^{v}(E) }.

Given a flat connection ∇, one can split the module D(E) into the direct sum D(E) = D^{v}(E) ⊕ ∇D(E), where ∇D(E) ⊂ D(E) is the submodule generated by the vector fields ∇X, X ∈ D(M). This splitting induces the dual one: Λ^{1}(E) = Λ_{h}^{1}(E) ⊕ Λ_{∇}^{1}(E), where Λ_{∇}^{1}(E) is the annihilator of ∇D(E). Finally we obtain the splitting Λ^{i}(E) = ⊕_{i=p+q}Λ^{p,q}(E), where Λ^{p,q}(E) = Λ_{∇}^{p}(E) ⊗ Λ_{h}^{q}(E) and Λ_{∇}^{p}(E) is defined in an obvious way.

Proposition 5. Let ∂_{∇} be the differential from (15). Then

∂_{∇}(D(Λ^{p,q}(E)) ⊂ D(Λ^{p,q+1}(E))

and thus the complexes

0 → D(Λ^{p}(E)) ⟶^{∂_{∇}} D(Λ^{p,1}(E)) → ⋯ → D(Λ^{p,q}(E)) ⟶^{∂_{∇}} D(Λ^{p,q+1}(E)) → ⋯ (16)

are defined.

The corresponding cohomology is denoted by H_{∇}^{p,q}(E).

Remark 6. In the sequel, we shall also deal with subcomplexes of complexes (15) and (16). As an example, let us indicate the complex

0 → D^{v}(E) ⟶^{∂_{∇}} D^{v}(Λ^{1}(E)) → ⋯ → D^{v}(Λ^{i}(E)) ⟶^{∂_{∇}} D^{v}(Λ^{i+1}(E)) → ⋯  (17)

of vertical Λ^{∗}(E)-valued derivations.

Coordinates. Let U ⊂ M be a local chart with the coordinates x_{1}, …, x_{n}, v^{1}, …, v^{m} in π^{−1}(U) ⊂ E (the case m = ∞ is included). Then, as it was noted above, a connection ∇ in π is given by the correspondence

∂x_{i} ↦ ∇_{x_{i}} = ∇(∂x_{i}) = ∂x_{i} + ∑_{α=1}^{m} v_{i}^{α} ∂v^{α}, i = 1, …, n, v_{i}^{α} ∈ C^{∞}(E).

By definition, one has

Ū_{∇} = ∑_{i=1}^{n} dx_{i} ⊗ ∇_{x_{i}}, U_{∇} = ∑_{α=1}^{m} (dv^{α} − ∑_{i=1}^{n} v_{i}^{α} dx_{i}) ⊗ ∂v^{α}.  (18)

Restricting ourselves to vertical complex (17), from (12) and (18) we get

∂_{∇}( ∑_{α} θ^{α} ⊗ ∂v^{a}) = ∑_{i=1}^{n} dx_{i} ∧ ∑_{α=1}^{m} (∇_{x_{i}}(θ^{α}) − ∑_{β=1}^{m} \frac{∂v_{i}^{α}}{∂v^{β}} θ^{β}) ⊗ ∂v^{α},  (19)

where θ^{α} ∈ Λ^{∗}(E).

2.2. A coordinate-free formulation. Fix a finite-dimensional bundle π: E → M and recall that connections (see Remark 1) in this bundle are identified with the sections of the bundle π_{1,0} : J^{1}(π) → E. Consider a point θ_{1} ∈ J^{1}(π_{1,0}). Then it can be represented as a class [∇]_{θ}^{1}, θ = (π_{1,0})(θ_{1}), of local sections of π_{1,0}, or local connections in the the bundle π in a neighborhood of the point π_{1}(θ) ∈ M.

Clearly, the value of [[Ū_{∇}, Ū_{∇}]] at θ is independent of the choice of a representative ∇ in the class θ_{1} = [∇]_{θ}^{1} and is determined by the point θ_{1} ∈ J^{1}(π_{1,0}) only. The equation of flat connections is defined to be the submanifold

E = {θ_{1} ∈ J^{1}(π_{1,0}) | [[Ū_{∇}, Ū_{∇}]]_{π_{1,0}(θ_{1})} = 0, θ_{1} = [∇]_{θ}^{1}}.

Indeed, by Proposition 4, a section s of π_{1,0} determines a flat connection if and only if s(E) ⊂ E.

In adapted coordinates this equation can be rewritten as

[[∑_{i} (D_{x_{i}} + ∑_{α} v_{i}^{α}D_{v^{α}}) ⊗ dx_{i}, ∑_{i} (D_{x_{i}} + ∑_{α} v_{i}^{α}D_{v^{α}}) ⊗ dx_{i}]] = 0,  (20)

or, equivalently,

[D_{x_{i}} + ∑_{α} v_{i}^{α}D_{v^{α}}, D_{x_{i}} + ∑_{α} v_{i}^{α}D_{v^{α}}] = 0, 1 ≤ i < j ≤ n,

where D_{x_{i}}, i = 1, …, n, D_{v^{α}}, α = 1, …, m, are the total derivatives corresponding to ∂x_{i} and ∂v^{α}, respectively, by the Cartan connection C in the fiber bundle (π_{1,0})_{∞}: J^{∞}(π_{1,0}) → E.

We shall now describe the element ∑_{i} (D_{x_{i}} +∑_{α} v_{i}^{α}D_{v^{α}})⊗dx_{i} appearing in equation (20) in a new, invariant way. To this end, consider the Cartan connection C^{π} in the bundle π_{∞}: J^{∞}(π) → M. This connection takes any vector field X ∈ M to the vector field C^{π}X on J^{∞}(π). Due to the projection π_{∞,0} : J^{∞}(π) → E, we can restrict C^{π}X to E (more precisely, we restrict the map C^{π}X: C^{∞}(J^{∞}(π)) → C^{∞}(J^{∞}(π)) to the subalgebra C^{∞}(E) ⊂ C^{∞}(J^{∞}(π))). This restriction, (C^{π}X)_{0}, is an element of the module D(E, J^{1}(π)), and, by Remark 4, since dim E < ∞, we can apply the Cartan connection C thus obtaining a vector field on J^{∞}(π_{1,0}). The sequence of maps

D(M) ⟶^{C^{π}} D(J^{∞}(π)) ⟶^{restriction to E} D(E, J^{1}(π)) ⟶^{C} D(J^{∞}(π_{1,0}))

results in a new connection C_{fc} : D(M) → D(J^{∞}(π_{1,0})) in the bundle

π̃ = π ∘ (π_{1,0})_{∞}: J^{∞}(π_{1,0}) → M.

For this connection, we shall use the notation Ū_{fc} = Ū_{C_{fc}} and ∂_{fc} = ∂_{C_{fc}}.

It is easily checked that in local coordinates one has

Ū_{fc} = ∑_{i} (D_{x_{i}} + ∑_{α} v_{i}^{α}D_{v^{α}}) ⊗ dx_{i}

and we have the following

Proposition 6. The equation of flat connections E ⊂ J^{1}(π_{1,0}) is distinguished by the condition [[Ū_{fc}, Ū_{fc}]]_{θ_{1}} = 0, θ_{1} ∈ J^{1}(π_{1,0}). All vector fields of the form C_{fc}X, X ∈ D(M), are tangent to E ⊂ J^{∞}(π_{1,0}) and thus the restriction of C_{fc} to the subbundle E → M of the bundle π̃: J^{∞}(π_{1,0}) → M is flat.

3. Symmetries of the equation of flat connections

From the last proposition it follows that the connection C_{fc} determines on E the complexes similar to (16). We are especially interested in the case p = 0 and thus obtain

0 → D(E) ⟶^{∂_{fc}} D(Λ_{h}^{1}(E)) ⟶^{∂_{fc}} ⋯ → D(Λ_{h}^{q}(E)) ⟶^{∂_{fc}} D(Λ_{h}^{q+1}(E)) → ⋯,  (21)

where ∂_{fc} = ∂_{C_{fc}}.

Consider now in D(E) the submodule V = V^{0} spanned by the vector fields D_{v^{α}}, α = 1, …, m, and the modules V^{q} = Λ_{h}^{q}(E) ⊗ V ⊂ D(Λ_{h}^{q}(E)).

Lemma 1. The modules V^{q} are preserved by the differential ∂_{fc}, i.e., ∂_{fc}(V^{q}) ⊂ V^{q+1}.

Proof. Indeed, by (19) one has

∂_{fc}(f dx_{i_{1}} ∧ … ∧ dx_{i_{q}} ⊗ D_{v^{α}}) = ∑_{i=1}^{n} D_{i}(f) dx_{i} ∧ dx_{i_{1}} ∧ … ∧ dx_{i_{q}} ⊗ D_{v^{α}}

− (−1)^{q} ∑_{i=1}^{n} ∑_{β=1}^{m} D_{v^{α}}(v_{i}^{β})f dx_{i} ∧ dx_{i_{1}} ∧ … ∧ dx_{i_{q}} ⊗ D_{v^{β}}, (22)

where f ∈ C^{∞}(E) and, as everywhere below, D_{i} = D_{x_{i}} +∑_{β} v_{i}^{β}D_{v^{β}} = C_{fc}∂x_{i} is the total derivative with respect to the connection C_{fc}. □

Consequently, we obtain the complex

0 → V^{0} = V ⟶^{∂_{fc}} V^{1} ⟶^{∂_{fc}} ⋯ → V^{q} ⟶^{∂_{fc}} V^{q+1} → ⋯  (23)

Let us now return back to the equation of flat connections written in the form (20). Since this equation is bilinear, its linearization in local coordinates is of the form

[[Ū_{fc}, ∑_{i,α} φ_{i}^{α}D_{v^{α}} ⊗ dx_{i}]] + [[∑_{i,α} φ_{i}^{α}D_{v^{α}} ⊗ dx_{i}, Ū_{fc}]] = 2[[Ū_{fc}, ∑_{i,α} φ_{i}^{α}D_{v^{α}} ⊗ dx_{i}]] = 0,

where φ = ∑_{i,α} φ_{i}^{α}D_{v^{α}} ⊗ dx_{i} is the generating section of a symmetry, φ_{i}^{α} ∈ C^{∞}(E), or

ℓ_{E}φ ≡ 2∂_{fc}φ = 0, φ ∈ V^{1}.  (24)

Proposition 7. The map

sym E_{fc} → V^{1}, S ↦ [[Ū_{fc}, S]]  (25)

is an isomorphism of the space of symmetries onto the space of 1-cocycles of complex (23).

Proof. Let S ∈ sym E_{fc}. Since S commutes with all total derivatives, we have [[Ū_{fc}, S]] ∈ V^{1}. Moreover, [[S, Ū_{fc}]] is a cocycle, since [[Ū_{fc}, Ū_{fc}]] = 0. The fact that (25) is an isomorphism onto the kernel of ∂_{fc} follows from the above coordinate consideration, since −[[Ū_{fc}, S]] = ∑_{i,α} S(v_{i}^{α})D_{v^{α}} ⊗ dx_{i} is nothing but the generating section of the symmetry S. □

Remark 7. It is not surprising that in (24) the symmetry generating section is an element of the module V^{1}. Indeed, in our case the bundle ξ = π_{1,0} : J^{1}(π) → E is affine and π_{0,1}^{v} (see Remark 2) is isomorphic to the pullback π_{1}^{∗}(π ⊗ τ^{∗}), where τ^{∗}: T^{∗}M → M is the cotangent bundle. Consequently, the module of generating sections is isomorphic to V^{1}.

Theorem 2. Complex (23) is 0-acyclic, i.e.,

ker (∂_{fc} : V → V^{1}) = 0,

and 1-acyclic, i.e.,

∂_{fc}(V) = ker (∂_{fc} : V^{1} → V^{2}).

Corollary 1 (description of symmetries). Any symmetry φ = (φ_{i}^{α}), i = 1, …, n, α = 1, …, m, of equation (1) is of the form

φ_{i}^{α} = D_{i}(f^{α}) − ∑_{β=1}^{m} v_{i}^{α,β}f^{β}  (26)

for arbitrary functions f^{α} ∈ C^{∞}(E).

In Subsection 4.4, we shall describe a Lie algebra structure of symmetries in terms of functions f^{α}.

Proof of Corollary 1. The result immediately follows from Theorem 2 in combination with (22) and (24). □

To prove Theorem 2, we shall need some auxiliary constructions. Consider the following smooth functions on E

v_{I}^{α,A} = D_{v^{A}}D_{I}(v^{α}), |I| > 0,  (27)

where D_{v^{A}} = D_{v^{α_{1}}} ∘ ⋯ ∘ D_{v^{α_{l}}}, D_{I} = D_{i_{1}} ∘ ⋯ ∘ D_{i_{k}} for the multi-indices A = α_{1} … α_{l} and I = i_{1} … i_{k} respectively. In particular, v_{∅}^{α,∅} = v^{α} and v_{i}^{α,∅} = v_{i}^{α}. It is easily seen that the functions x_{i}, v_{I}^{α,A} form a system of coordinates on E. Let F_{r} ⊂ C^{∞}(E) be the subalgebra of functions dependent on the variables

x_{i}, v_{I}^{α,A}, |I| ≤ r,

only. Then we have

F_{r} ⊂ F_{r+1}, ⋃_{r≥0} F_{r} = C^{∞}(E).

Evidently, for f ∈ F_{r} one obtains

D_{i}(f) ≡ ∑_{α,A,I} v_{Ii}^{α,A} \frac{∂f}{∂v_{I}^{α,A}} mod F_{r}.  (28)

We have also the filtration V_{r}^{q} ⊂ V_{r+1}^{q} in the modules V^{q}, where V_{r}^{q} ⊂ V^{q} consists of forms with coefficients in F_{r}. From (22) and (28) we get

∂_{fc}(f dx_{l_{1}} ∧ … ∧ dx_{l_{q}} ⊗ D_{v^{α}})

≡ ∑_{i=1}^{n} ∑_{I,β,A} v_{Ii}^{β,A} \frac{∂f}{∂v_{I}^{β,A}} dx_{i} ∧ dx_{l_{1}} ∧ … ∧ dx_{l_{q}} ⊗ D_{v^{α}} mod V_{r}^{q+1}. (29)

On the other hand, consider the infinite-dimensional vector bundle

π′: ℝ^{∞} × ℝ^{n} → ℝ^{n}

with the coordinates u^{α,A}, α = 1, …, m, A = α_{1} … α_{l} with α_{j} ≥ 0, along fibers and coordinates y_{i}, i = 1, …, n, in the base. The infinite jet bundle J^{∞}(π′) → ℝ^{n} of π′ has the adapted coordinates

u_{I}^{α,A} = D_{y_{I}}(u^{α,A})

along the fibers, where, as above, D_{y_{I}} denotes the composition of total derivatives.

To any element

s = ∑_{i,α} f_{i}^{α} dx_{i} ⊗ D_{v^{α}} ∈ V^{1}

and an integer a ≥ 0 we associate an m-tuple ω(s, a)^{1}, …, ω(s, a)^{m} of parameter-de-pendent horizontal 1-forms on J^{∞}(π′) as follows.

Let r(s) be the minimal integer such that all the coefficients f_{i}^{α} of s belong to F_{r(s)}. Replace the variables v_{I}^{β,A} with |I| ≥ r(s) − a in f_{i}^{α} by u_{I}^{α,A} and denote thus obtained functions by f̃_{i}^{α}. We treat f̃_{i}^{α} as functions on J^{∞}(π′) dependent on the parameters x_{1}, …, x_{n} and v_{I}^{β,A} with |I| < r(s) − a. Set

ω(s, a)^{α} = ∑_{i} f̃_{i}^{α} dy_{i}.

We say that an element s ∈ V^{1} is semilinear if the coefficients of s are linear with respect to the highest order coordinates v_{I}^{α,A}, |I| = r(s).

Lemma 2. The following facts are valid:

(1) If s ∈ V^{1} and ∂_{fc}(s) = 0 then d_{h}(ω(s, 0)^{α}) = 0 for each α and for all values of the parameters, where d_{h} is the horizontal de Rham differential on J^{∞}(π′).

(2) If s ∈ V^{1} is semilinear, r(s) ≥ 2, and ∂_{fc}(s) = 0, then d_{h}(ω(s, 1)^{α}) = 0.

Proof. (1) By (29) and (14), the equations d_{h}(ω(s, 0)^{α}) = 0 for all α = 1, …, m are equivalent to

∂_{fc}(s) = 0 mod V_{r(s)}^{2}.

(2) Let F̃_{r} ⊂ F_{r} be the F_{r−1}-submodule generated by v_{I}^{α,A}, |I| = r, and, respectively, let Ṽ_{r}^{1} ⊂ V_{r}^{1} be the subset of forms with coefficients in F̃_{r}. Then from the identity

[D_{i}, D_{v^{α}}] = − ∑_{β} v_{i}^{β,α} D_{v^{β}}  (30)

we obtain

D_{i}(v_{I}^{α,A}) ≡ v_{Ii}^{α,A} mod F̃_{|I|}.  (31)

By (22) and (31), the equation

∂_{fc}(s) = 0 mod V_{r(s)−1}^{2} + Ṽ_{r(s)}^{2}

for semilinear s implies d_{h}(ω(s, 1)^{α}) = 0. □

Lemma 3. Any cocycle s ∈ V^{1} with r(s) ≥ 1 is semilinear.

Proof. If there is a nonlinear coefficient, we can fix the values of x_{i}, v_{I}^{α,A}, |I| < r(s), such that some form ω(s, 0)^{α} is not linear with respect to u_{I}^{α,A}, |I| = r(s), for these values of the parameters. Proposition 3 implies

H_{h}^{1}(J^{∞}(π′)) = 0.  (32)

By Lemma 2 (1) and (32), we have ω(s, 0)^{α} = d_{h}(w′) for some function dependent only on u_{I}^{α,A} with |I| = r(s) − 1. Taking into account (14) and (6), we see that ω(s, 0)^{α} is linear. □

Lemma 4. For each cocycle s ∈ V^{1} there exists a cohomological cocycle with coefficients in F_{1}.

Proof. Assume that r(s) ≥ 2. By Lemma 3, Lemma 2 (2), and (32), we have

ω(s, 1)^{α} = d_{h}(w_{α})  (33)

for some functions w_{α} ∈ C^{∞}(J^{∞}(π′)) dependent also on the parameters x_{i}, v_{I}^{α,A} with |I| ≤ r(s) − 2. Moreover, we can assume that w_{α} depend only on the coordinates u_{I}^{α,A} with |I| ≤ r(s) − 1. Replace the variables u_{I}^{α,A} in w_{α} by v_{I}^{α,A} and denote the obtained functions by w̃_{α}. Consider the vector field

s′ = ∑_{α} w̃_{α}D_{v^{α}} ∈ V_{r(s)−1}^{0}.

Then from (33) and (29) we have r(s − ∂_{fc}(s′)) ≤ r(s) − 1. Proceeding in this way, by induction on r(s) one completes the proof. □

Proof of Theorem 2. The first statement, i.e., 0-acyclicity, is proved by direct computation. This means that it is sufficient to prove the second statement of Theorem 2 locally.

Let s ∈ V^{1} be a cocycle. By Lemmas 3 and 4, there is a cohomological cocycle

s′ = ∑_{i,β} f_{i}^{β} dx_{i} ⊗ D_{v^{β}} ∈ V^{1}

such that the functions f_{i}^{β} belong to F_{1} and are linear with respect to the coordinates v_{I}^{α,A}, |I| = 1. From d_{h}ω(s′, 0) = 0 we obtain

f_{i}^{β} = f_{i,β}(x_{1}, …, x_{n}, v^{1}, …, v^{m}) + ∑_{γ,A} f_{β}^{γ,A}(x_{1}, …, x_{n}, v^{1}, …, v^{m})v_{i}^{γ,A}.

By (22), the equation ∂_{fc}(s′) = 0 is equivalent to

D_{i}(f_{j}^{β}) − ∑_{γ=1}^{m} v_{i}^{β,γ} f_{j}^{γ} = D_{j}(f_{i}^{β}) − ∑_{γ=1}^{m} v_{j}^{β,γ} f_{i}^{γ}  (34)

for all β, i and j. Comparing equations (1) and (34), one shows that the functions f_{β}^{γ,A}(x_{1}, …, x_{n}, v^{1}, …, v^{m}) vanish for |J| > 1, that is,

f_{i}^{β} = f_{i,β}(x_{1}, …, x_{n}, v^{1}, …, v^{m}) + ∑_{γ=1}^{m} ∑_{δ=0}^{m} f_{β}^{γ,δ}(x_{1}, …, x_{n}, v^{1}, …, v^{m}) v_{i}^{γ,δ}. (35)

Finally, it is easily seen that if s′ ∈ V_{1}^{1} with coefficients (35) satisfies (34) then s′ = ∂_{fc}(F) for some F ∈ V_{0}^{0}. □

4. Flat representations

4.1. Definition and the main property. Consider a formally integrable infinitely prolonged differential equation ξ: E → M with the Cartan distribution CD ⊂ D(E), where M is the space of independent variables. A flat representation of E is given by a fibering π: M → N and a flat connection F in the composition bundle η = π∘ξ: E → N such that for each X ∈ D(N)

F(X) ∈ CD.  (36)

Let x_{1}, …, x_{n′} be local coordinates in N and y^{1}, …, y^{m} be coordinates along the fibers of the bundle π. Then any flat representation is determined by the correspondence

∂x_{i} ↦ D_{x_{i}} + ∑_{α=1}^{m} a_{i}^{α}D_{y^{α}}, i = 1, …, n′,  (37)

where D_{x_{i}}, D_{y^{α}} are the total derivatives on E and the functions a_{i}^{j} ∈ C^{∞}(E) satisfy the condition

[D_{x_{i}} + ∑_{α=1}^{m} a_{i}^{α}D_{y^{α}}, D_{x_{j}} + ∑_{α=1}^{m} a_{j}^{α}D_{y^{α}}] = 0, 1 ≤ i < j ≤ n′.  (38)

Let E_{fc} be the equation of flat connections in the bundle π and consider a morphism φ: E → E_{fc} such that the diagram

\begin{aligned}E & & \overline{φ  →} & E_{fc} \\ & [figure: ξ] & [figure] \\ & & M \end{aligned}  (39)

is commutative. Take a point θ ∈ E. Then φ(θ) is the infinite jet of some flat connection ∇_{θ} in the bundle π. This means that any tangent vector to N at the point η(θ) can be lifted by ∇ to M and then by the Cartan connection in E to a tangent vector to E at the point θ. Evidently, this procedure is independent of choice of ∇ and we obtain a flat representation for E.

The converse construction is also valid, i.e., any flat representation determines a morphism φ satisfying the commutativity condition (39). To establish this fact let us consider a point θ ∈ E of the equation E and a tangent vector w to N at the point η(θ). Then we can construct a vector θ_{F}(w) ∈ T_{ξ(θ)}M by

θ_{F}(w) = ξ_{∗}|_{θ}(F(w)) ∈ T_{ξ(θ)}M,  (40)

and one has π_{∗}(θ_{F}(w)) = w. Assume now that E ⊂ J^{∞}(ζ) for some bundle ζ over M and the connection F can be extended to some connection F̃ in the bundle π ∘ ζ_{∞} in such a way that F̃ also possesses property (36) (F̃ may not be flat). This can be always done. For a point θ ∈ E, consider the points x = ξ(θ) ∈ M, x′ = π(x) ∈ N and a neighborhood U′ ⊂ N of x′ in N. Set U = π^{−1}(U′) ⊂ M. Let θ = [f]_{x}^{∞}, f ∈ Γ_{loc}(ζ) and U_{f} = j_{∞}(f)(U) ⊂ J^{∞}(ζ). Then, using (40), for any point x̃ ∈ U and a vector field X on U′ we set

(∇_{θ}X)x̃ = θ̃F̃(X_{}_{x̃′}),

where θ̃ = j_{∞}(f)(x̃) and x̃′ = π(x̃). It is straightforward to check that ∇_{θ} is a local connection in π, its infinite jet at x depends on θ only and lies in E_{fc}. The map φ: θ ↦ [∇_{θ}]_{x}^{∞} is a morphism of E to E_{fc} and the corresponding flat representation coincides with F. Thus we obtain

Theorem 3. Let ξ: E → M be an infinitely prolonged equation and π: M → N be a fiber bundle. Then flat representations of E in the bundle π are in one-to-one correspondence with the morphisms of E to the equation of flat connections for π satisfying commutativity condition (39).

Remark 8. In local coordinates the construction of the morphism φ looks quite simple. Namely, let a flat representation be given by the functions a_{i}^{α}, see (37). Then we set φ^{∗}(v_{i}^{α}) = a_{i}^{α} and take all differential consequences of these relations, i.e.,

φ^{∗}(v_{i,σ}^{α}) = D_{σ}(a_{i}^{α}).

Due to relations (38), these formulas are consistent. Here v_{i}^{α} are the coordinates introduced at the end of Subsection 2.1.

4.2. Deformations of a flat representation. Integrable equations possess parametric families of flat representations. Let us study this structure.

A smooth family of flat connections F(ε) in the bundle η is called a deformation of a flat representation F^{0} if F(ε) meets (36) and F(0) = F^{0}. Below we also consider formal deformations, which are not smooth families, but formal power series in ε.

Set U = Ū_{F^{0}} ∈ D(Λ^{1}(E)). Consider the submodule V_{E} ⊂ CD of η-vertical Cartan fields and set V_{E}^{q} = Λ_{h}^{q}(E) ⊗ V ⊂ D(Λ_{h}^{q}(E)).

Let F(ε) be a (formal) deformation of F^{0} = F(0). From (36) it follows that Ū_{F(ε)} = U + U_{1}ε + O(ε^{2}), where U_{1} ∈ V_{E}^{1}. By Proposition 4, we have

[[U + U_{1}ε + O(ε^{2}), U + U_{1}ε + O(ε^{2})]] = 2[[U, U_{1}]]ε + O(ε^{2}) = 0.

Therefore, the infinitesimal part U_{1} ∈ V_{E}^{1} of any deformation satisfies the identity

[[U, U_{1}]] = 0.  (41)

For any vector field V ∈ V_{E} we can consider the formal deformation

e^{ε ad V} (U) = U + [[V, U]]ε + \frac{1}{2}[[V, [[V, U]]]]ε^{2} + …  (42)

with the infinitesimal part

[[V, U]].  (43)

Since such deformations exist independently of the equation structure, it is natural to call them as well as their infinitesimal parts [[V, U]], V ∈ V_{E}, trivial.

Formulas (41) and (43) prompt the following construction. It is easily seen that ∂_{U}(V_{E}^{q}) ⊂ V_{E}^{q+1}, and, therefore, we obtain a complex

0 → V_{E}^{0} = V_{E} ⟶^{∂_{U}} V_{E}^{1} ⟶^{∂_{U}} ⋯ → V_{E}^{q} ⟶^{∂_{U}} V_{E}^{q+1} → ⋯  (44)

By the construction and formulas (41), (43), its 1-cocycles are infinitesimal deformations, and exact 1-cocycles are trivial infinitesimal deformations. From the general theory of deformations [3], we obtain that obstructions to prolongation of infinitesimal deformations to formal ones belong to higher cohomology groups of (44).

Remark 9. Deformations of zero-curvature representations (finite-dimensional linear coverings) of PDE were studied analogously in [12]. In fact, the gauge complex of a zero-curvature representation introduced in [11, 12] is a partial case of the above construction, since according to Subsection 5.1 any finite-dimensional covering determines a flat representations.

4.3. Symmetries and cocycles. Let S ∈ D(E) be a symmetry of E. Since S commutes with the fields of the form C(X), X ∈ D(M), the power series e^{ε ad S}(U) is a formal deformation of F, and its infinitesimal part [[S, U]] ∈ V_{E}^{1} is a 1-cocycle of complex (44). Let us set c_{S} = [[U, S]]. Note that this cocycle is not generally exact, since S ∉ V_{E}.

Theorem 4. Symmetries S with exact cocycles c_{S} form a Lie subalgebra in sym E.

Proof. Let S_{1}, S_{2} be two symmetries such that

[[U, S_{1}]] = [[U, V_{1}]], [[U, S_{2}]] = [[U, V_{2}]], V_{1}, V_{2} ∈ V_{E}.  (45)

Using this and properties (10), (11) of the Nijenhuis bracket, we easily obtain

c_{[S_{1},S_{2}]} = [[U, [S_{1}, S_{2}]]] = [[U, [V_{1}, V_{2}] + [V_{1}, S_{2}] + [S_{1}, V_{2}]]].  (46)

Note that [V_{E}, V_{E}] ⊂ V_{E} and for each symmetry S one has [S, V_{E}] ⊂ V_{E}. Therefore, [V_{1}, V_{2}] + [V_{1}, S_{2}] + [S_{1}, V_{2}] ∈ V_{E} and the cocycle c_{[S_{1},S_{2}]} is exact. □

Theorem 5. Suppose that the morphism φ: E → E_{fc} corresponding to the flat representation F is an embedding such that φ(E) is a submanifold of E_{fc}. Then the cocycle c_{S} is exact if and only if there is a symmetry S′ ∈ D(E_{fc}) of E_{fc} such that

φ_{∗}(S) = S′|_{φ(E)}.  (47)

Remark 10. Recall that E is a submanifold of some infinite jet space J^{∞}(ζ). It is well known [2] that under rather weak regularity assumptions each symmetry of E is the restriction of some symmetry of J^{∞}(ζ). The above theorem provides a criterion for the similar property in the situation when E_{fc} is considered instead of J^{∞}(ζ). In the next section we give examples of both types of symmetries: with exact and non-exact c_{S}. Therefore, generally not every symmetry is the restriction of some symmetry of E_{fc}.

Proof. First suppose that (47) holds. By Theorem 2, there is V ∈ V such that

[[U, S′]] = [[U, V]].  (48)

Since V belongs to the Cartan distribution and φ is a morphism of differential equations, the vector field V is tangent to the submanifold φ(E). Denote by V′ ∈ V_{E} the preimage φ_{∗}^{−1}(V). Then from (47) and (48) it follows that

c_{S} = [[U, V′]],  (49)

i.e., the cocycle c_{S} is exact.

Conversely, if c_{S} is exact then (49) holds for some V′ ∈ V_{E}. Extend φ_{∗}(V′) ∈ D(φ(E)) to a vector field V ∈ V. Then the symmetry S′ of E_{fc} with the generating section [[U, V]] satisfies (47). □

4.4. A bracket on the equation of flat connections. The identity mapping

id: E_{fc} → E_{fc}

is a flat representation, and complex (23) coincides with (44) for this flat representation. From the results of Section 3 it follows that sym E_{fc} is isomorphic to V. Therefore, the commutator of symmetries induces a bracket on V. By formulas (45) and (46), it is given by

{V_{1}, V_{2}} = [V_{1}, V_{2}] + [V_{1}, S_{2}] + [S_{1}, V_{2}], V_{1}, V_{2} ∈ V,  (50)

where S_{i} is the symmetry with the generating section [[Ū_{fc}, V_{i}]]. Let us compute this bracket in coordinates.

In adapted coordinates, any element of V is of the form V_{f} = ∑_{i=1}^{n} f^{i}D_{i}, f^{i} ∈ C^{∞}(E_{fc}). Hence, locally, the bracket {V_{f}, V_{g}} generates a bracket {f, g} on vectorfunctions defined by {V_{f}, V_{g}} = V_{{f,g}}. Using (50), we compute {f, g}.

Let

S_{f} = ∑_{|I|>0} S_{I}^{α,A}∂v_{I}^{α,A}

be the symmetry of E_{fc} with the generating section [[Ū_{fc}, V_{f}]]. Then by direct computations we obtain that

S_{I}^{α,A} = D_{v^{A}}(S_{I}^{α,∅}),  (51)

where D_{v^{A}} is the composition of the total derivatives D_{v^{β}} corresponding to the multiindex A, while

S_{Ii}^{α,∅} = D_{i}(S_{I}^{α,∅}) + ∑_{β=1}^{m} v_{I}^{α,β}φ_{i}^{β},  (52)

where φ_{i}^{β} = S_{i}^{β,∅}, and, as we already know (see (26)),

φ_{i}^{β} = D_{i}(f^{β}) − ∑_{γ=1}^{m} v_{i}^{β,γ}f^{γ}.

Substituting these expressions to (50) and using straightforward computations, we obtain the following

Proposition 8. The commutator of symmetries determines the following bracket on the module of smooth m-dimensional vector-functions on E

{f, g}^{α} = S_{f}g^{α} − S_{g}f^{α} + V_{f}g^{α} − V_{g}f^{α}.  (53)

For any vector-function h = (h^{1}, …, h^{m}) the coefficients of the vector field S_{h} = ∑ S_{I}^{α,A}∂v_{I}^{α,A} in special coordinates v_{I}^{α,A} can be computed by the formulas

S_{I}^{α,A} = D_{v^{A}}(S_{I}^{α,∅}), S_{Ii}^{α,∅} = D_{i}(S_{I}^{α,∅}) + ∑_{β=1}^{m} v_{I}^{α,β}ζ_{i}^{β},

where ζ_{i}^{β} = S_{i}^{β,∅} and ζ^{β} = D_{i}(h^{β}) − ∑_{γ=1}^{m} v_{i}^{β,γ}h^{γ}, while V_{h} = ∑_{α=1}^{m} h^{α}D_{v^{α}}.

5. Examples of flat representations

5.1. Coverings as flat representations and lifting of symmetries. Consider an infinitely prolonged differential equation ξ: E → M with the Cartan connection

C: D(M) → D(E).

Let π: E → M be a fiber bundle of finite rank and consider its pullback with respect to ξ

\begin{matrix}E′ & ⟶^{ξ′} & E \\ ^{ξ^{∗}(π)}↓ & & ↓^{π} \\ E & ⟶^{ξ} & M \end{matrix}  (54)

Suppose that the bundle τ = ξ^{∗}(π): E′ → E is endowed with a covering structure given by a flat connection F: D(M) → D(E′), τ_{∗} ∘ F = C. This structure determines a flat representation not of the equation E itself, but of some trivial extension of E.

Namely, consider the distribution CD_{1} = τ_{∗}^{−1}(CD_{E}) ⊂ D(E′). Clearly, the differential ξ′_{∗} projects CD_{1} isomorphically onto D(E). Denote by C_{1} : D(E) → D(E′) the arising connection in the bundle ξ′: E′ → E. It is easily seen that this connection is flat, and, moreover, the pair (E′, C_{1}) is isomorphic to an infinitely prolonged equation with E as the space of independent variables and C_{1} as the Cartan connection.

Remark 11. In coordinates this construction looks as follows. Locally the bundle τ is trivial τ: M × W → M. Let x_{i}, w_{k} be coordinates in M and W respectively. Then the equation (E′, C_{1}) is obtained from (E, C) as follows. We assume that the dependent variables u^{j} in E are functions of not only x_{i}, but also w_{k}, and add the equations ∂u^{j}/∂w_{k} = 0.

The bundle π: E → M and the connection F constitute a flat representation of the equation (E′, C_{1}).

Consider complex (44) corresponding to this flat representation. By construction, the module V_{E} consists of τ-vertical vector fields on E′ and (44) in this case is the horizontal de Rham complex of the covering equation (E′, F) with coefficients in V_{E} (see Section 3). For arbitrary coverings this complex was studied in [5] in relation with parametric families of Bäcklund transformations.

By construction (54), each ξ-vertical vector field Y ∈ D(E) is canonically lifted to E′ as a ξ′-vertical vector field denoted by Ŷ ∈ D(E′). Evidently, for each symmetry S of E the vector field Ŝ is a symmetry of the equation (E′, C_{1}).

Theorem 6. Let S be a symmetry of E and consider complex (44) corresponding to the flat representation F of (E′, C_{1}). The cocycle cŜ is exact if and only if there is a symmetry S′ ∈ D(E′) of the covering equation (E′, F) such that τ_{∗}(S′) = S.

Proof. If cŜ = [[Ū_{F}, Ŝ]] = [[Ū_{F}, V]] for some V ∈ V_{E} then [[Ū_{F}, Ŝ − V]] = 0, i.e., the vector field S′ = Ŝ − V is a symmetry of (E′, F). Clearly, τ_{∗}(S′) = S. □

5.2. Flat representations and symmetries of the KdV equation. Let us illustrate the above construction by the following classical example of covering.

Consider the well-known overdetermined system

v_{x} = λ + u + v^{2},  (55)
v_{t} = u_{xx} + 2u^{2} − 2λu − 4λ^{2} + 2u_{x}v + v^{2}(2u − 4λ), λ ∈ ℝ,  (56)

associated with the Miura transformation of the KdV equation. Its compatibility condition is equivalent to the KdV equation

u_{t} = u_{xxx} + 6uu_{x}.  (57)

The infinite prolongation E of (57) has the natural coordinates

x, t, u_{k} = \frac{∂^{k}u}{∂x^{k}}, k = 0, 1, 2, …  (58)

The total derivatives restricted to E are written in these coordinates as follows

D_{x} = ∂x + ∑_{k≥0} u_{k+1} ∂u_{k},  (59)
D_{t} = ∂t + ∑_{k≥0} D_{x}^{k}(u_{3} + 6uu_{1}) ∂u_{k}.  (60)

Evidently, functions (58) and v form a system of coordinates for the infinite prolongation E′ = E × ℝ of system (55), (56). In these coordinates the total derivatives restricted to E′ are D̃_{x} = D_{x} + A∂/∂v, D̃_{t} = D_{x} + B∂/∂v, where A, B ∈ C^{∞}(E′) are the right-hand sides of (55) and (56) respectively. We have the trivial bundle τ: E′ → E, (x, t, u_{k}, v) ↦ (x, t, u_{k}), equipped with the covering structure τ_{∗}(D̃_{x}) = D_{x}, τ_{∗}(D̃_{t}) = D_{t}.

By the construction of the previous subsection, this covering determines the following flat representation of the KdV equation.

We assume that u in (57) is a function of x, t, v and add the condition ∂u/∂v = 0. The infinite prolongation of this trivially extended KdV equation is the manifold E × ℝ with coordinates (58) and v. The Cartan distribution CD_{1} on E × ℝ is 3dimensional. The total derivatives with respect to x, t are given by (59) and (60), while the total derivative with respect to v is just ∂v. Then a flat representations dependent on parameter λ is given by the flat connection

∂x ↦ D_{x} + A∂v ∈ CD_{1}, ∂t ↦ D_{t} + B∂v ∈ CD

in the bundle E × ℝ → ℝ^{2}.

The infinitesimal part of this parametric family is

\frac{∂A}{∂λ} dx + \frac{∂B}{∂λ} dt = dx ⊗ ∂v − (2u + 8λ + 4v^{2}) dt ⊗ ∂v.

This 1-cocycle is not exact, which reflects the well-known fact that the parameter here is essential.

According to Theorem 6, the cocycle c_{S} corresponding to a symmetry S of (57) is exact if and only if the symmetry is lifted to a symmetry of system (55), (56). It is well known which symmetries of (57) can be lifted. Namely, all (x, t)-independent symmetries (including the symmetries corresponding to the higher KdV equations) are lifted, while the Galilean symmetry is not. The scaling symmetry can be lifted only in the case λ = 0.

Remark 12. For an equation E in two independent variables existence of a nontrivial covering is an important indication of integrability and leads to Bäcklund transformations and the inverse scattering method [10]. For non-overdetermined equations in more than two independent variables there are no nontrivial finite-dimensional coverings (this was proved in [11] for linear coverings and in [4] for arbitrary coverings). In this case the notion of flat representation may serve as a good substitute, see a typical example in the next subsection.

5.3. Flat representations of the self-dual Yang–Mills equations. Consider four smooth vector-functions A_{i}: ℝ^{4} → 𝔤, i = 1, 2, 3, 4, where 𝔤 is a finite-dimensional Lie algebra. The self-dual Yang–Mills equations are

[∂x_{1} + A_{1} + λ(∂x_{3} + A_{3}), ∂x_{2} + A_{2} + λ(∂x_{4} + A_{4})] = 0,  (61)

which must hold for all values of the parameter λ (see, for example, [13]). Here x_{i} are coordinates in ℝ^{4}. Denote by E the infinite prolongation of (61) and by C the Cartan connection in the bundle ξ: E → ℝ^{4}.

Fix an action σ: 𝔤 → D(W) of 𝔤 on a finite-dimensional manifold W. Similarly to Subsection 5.1, we construct a trivial extention E′ = E × W of E as follows. Consider the flat connection

C_{1} = C ⊕ id: D(ℝ^{4} × W) → D(E′)

in the bundle ξ_{1} = ξ × id: E′ → ℝ^{4} × W.

Evidently, the pair (E′, C_{1}) is isomorphic to an infinitely prolonged equation with ℝ^{4}×W as the space of independent variables and C_{1} as the Cartan connection. Consider the bundle

π: ℝ^{4} × W → ℝ^{2}, (a, w) ↦ (x_{1}, x_{2}) ∈ ℝ^{2}, a = (x_{1}, x_{2}, x_{3}, x_{4}) ∈ ℝ^{4}, w ∈ W.

The bundle π and the flat connection F in the bundle π ∘ ξ_{1} given by

F(∂x_{i}) = C_{1}(∂x_{i} + σ(A_{i}) + λ(∂x_{i+2} + σ(A_{i+2}))), i = 1, 2,  (62)

constitute a λ-dependent family of flat representations for the (trivially extended) selfdual Yang–Mills equations (61).

The 1-cocycle C_{1}(∂x_{3}+σ(A_{3})) dx^{1}+C_{1}(∂x_{4}+σ(A_{4}))dx^{2} corresponding to the family of flat representations (62) is not exact, which says that the parameter in (62) is essential.

Let H: E → 𝔤 be a smooth function, then the formula

G_{H}(A_{i}) = D_{x_{i}}(H) − [H, A_{i}]  (63)

defines a higher symmetry of E. Since in the case when H depends on x_{i} only (does not depend on A_{i} and their derivatives) this is a classical gauge symmetry, it is natural to call (63) the generalized gauge symmetry corresponding to H.

By the definition in Section 4.3, the 1-cocycle corresponding to the symmetry G_{H} is

[[Ū_{F}, G_{H}]] = ∑_{i=1, 2}[D_{x_{i}} + σ(A_{i}) + λ(D_{x_{i+2}} + σ(A_{i+2})), G_{H}] dx^{i} =

− σ(G_{H}(A_{1} + λA_{3})) dx^{1} − σ(G_{H}(A_{2} + λA_{4})) dx^{2} = −[[Ū_{F}, σ(H)]] (64)

The last equality in (64) says that this cocycle is exact.

On the other hand, it can be shown that cocycles corresponding to classical conformal symmetries of (61) (see, for example, [14]) are not exact.

Remark 13. Some other symmetries of (61) (see [14] and references therein) have been extensively studied, but they are nonlocal and, therefore, lie out of scope of the present article.

Remark 14. Various integrable reductions of the self-dual Yang–Mills equations possess similar to (61) Lax pairs [13] and, therefore, admit flat representations.

Acknowledgements

The third author is grateful for hospitality to the University of Twente, where this paper was started and finished.

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S. Igonin, Independent University of Moscow, University of Twente, Faculty of Mathematical Sciences, P.O. Box 217, 7500 AE Enschede, the Netherlands

E-mail address: igonin@mccme.ru

P. Kersten, University of Twente, Faculty of Mathematical Sciences, P.O. Box 217, 7500 AE Enschede, the Netherlands

E-mail address: kersten@math.utwente.nl

I. Krasil′shchik, The Diffiety Institute, Independent University of Moscow Correspondence to: 1st Tverskoy-Yamskoy per. 14, Apt. 45, 125047 Moscow, Russia

E-mail address: josephk@diffiety.ac.ru