Diffiety

Let ${\displaystyle E}$ be an ${\displaystyle (m+e)}$-dimensional smooth manifold. Two ${\displaystyle m}$-dimensional submanifolds ${\displaystyle M}$, ${\displaystyle M'}$ of ${\displaystyle E}$ are tangent up to order ${\displaystyle k}$ at the point ${\displaystyle p\in M\cap M'\subset E}$ if one can locally describe both submanifolds as zeroes of functions defined in a neighbourhood of ${\displaystyle p}$, whose derivatives agree up to order ${\displaystyle k}$. One can show that being tangent up to order ${\displaystyle k}$ is a coordinate-invariant notion and an equivalence relation.[2] One says also that ${\displaystyle M}$ and ${\displaystyle M'}$ have same ${\displaystyle k}$-th order jet at ${\displaystyle p}$, and denotes their equivalence class by ${\displaystyle [M]_{p}^{k}}$ or ${\displaystyle j_{p}^{k}M}$. The ${\displaystyle k}$-jet space of ${\displaystyle k}$-submanifolds of ${\displaystyle E}$, denoted by ${\displaystyle J^{k}(E,m)}$, is defined as the set of all ${\displaystyle k}$-jets of ${\displaystyle m}$-dimensional submanifolds of ${\displaystyle E}$ at all points of ${\displaystyle E}$: $${\displaystyle J^{k}(E,m):=\{[M]_{p}^{k}~|~p\in M,~{\text{dim}}(M)=m,M\subset E\ {\text{ submanifold}}\}}$$ As any given jet ${\displaystyle [M]_{p}^{k}}$ is locally determined by the derivatives up to order ${\displaystyle k}$ of the functions describing ${\displaystyle M}$ around ${\displaystyle p}$, one can use such functions to build local coordinates ${\displaystyle (x^{i},u_{\sigma }^{j})}$ and provide ${\displaystyle J^{k}(E,m)}$ with a natural structure of smooth manifold.[2]

${\displaystyle M}$ and ${\displaystyle M'}$ have the same 1-jet at ${\displaystyle p\in E}$ while ${\displaystyle M}$ and ${\displaystyle M''}$ have the same 3-jet.

As a particular case, when ${\displaystyle E}$ has a structure of fibred manifold over an ${\displaystyle n}$-dimensional manifold ${\displaystyle X}$, one can consider submanifolds of ${\displaystyle E}$ given by the graphs of local sections of ${\displaystyle \pi :E\to X}$. Then the notion of jet of submanifolds boils down to the standard notion of jet of sections, and the jet bundle ${\displaystyle J^{k}(\pi )}$ turns out to be an open and dense subset of ${\displaystyle J^{k}(E,m)}$.[3]

A differential equation of order ${\displaystyle k}$ on the manifold ${\displaystyle E}$ is a submanifold of a jet space, ${\displaystyle {\mathcal {E}}\subset J^{k}(E,m)}$.

If one defines solutions as below, then this geometric definition of PDEs in local coordinates gives rise to expressions that are usually used to define PDEs and their solutions in mathematical analysis.

The ${\displaystyle k}$-jet prolongation of a submanifold ${\displaystyle M\subset E}$, is the embedding ${\displaystyle j^{k}(M):M\rightarrow J^{k}(E,m)}$ given by ${\displaystyle j^{k}(M)(p):=[M]_{p}^{k}.}$ Furthermore, say that ${\displaystyle M^{k}:={\text{im}}(j^{k}(M))}$ is a prolongation of the submanifold ${\displaystyle M}$.

Furthermore, one can define prolongations of equations, i.e. of submanifolds of Jet spaces. Given a differential equation ${\displaystyle {\mathcal {E}}\subset J^{l}(E,m)}$ of order ${\displaystyle l}$, one would like its ${\displaystyle k}$-th prolongation to be an equation of order ${\displaystyle k+l}$, i.e. a submanifold of the jet space ${\displaystyle J^{k+l}(E,m)}$. To achieve this, one first constructs the ${\displaystyle k}$-jet space ${\displaystyle J^{k}({\mathcal {E}},m)}$ of ${\displaystyle m}$-dimensional submanifolds of ${\displaystyle {\mathcal {E}}}$. As ${\displaystyle {\mathcal {E}}}$ is embedded in ${\displaystyle J^{l}(E,m)}$, one can always naturally embed ${\displaystyle J^{k}({\mathcal {E}},m)}$ into ${\displaystyle J^{k}(J^{l}(E,m),m)}$. But since the latter is the space of repeated jets of submanifolds of ${\displaystyle E}$, one can also always embed ${\displaystyle J^{k+l}(E,m)}$ into ${\displaystyle J^{k}(J^{l}(E,m),m)}$. As a result, when considering both ${\displaystyle J^{k+l}(E,m)}$ and ${\displaystyle J^{k}({\mathcal {E}},m)}$ as subspaces of ${\displaystyle J^{k}(J^{l}(E,m),m)}$, their intersection is well-defined. This is used for the definition of the prolongation of ${\displaystyle {\mathcal {E}}}$.

The ${\displaystyle k}$-th prolongation of a differential equation ${\displaystyle {\mathcal {E}}\subset J^{l}(E,m)}$ is defined as ${\displaystyle {\mathcal {E}}^{k}:=J^{k}({\mathcal {E}},m)\cap J^{k+l}(E,m)}$

Note however that such an intersection is not necessarily a manifold again. One therefore usually requires ${\displaystyle {\mathcal {E}}}$ to be nice enough such that at least its first prolongation is indeed a submanifold of ${\displaystyle J^{k+1}(E,m)}$.

It can also be shown that this definition still makes sense, even when ${\displaystyle k}$ goes to infinity.

An ${\displaystyle R}$-plane at a point ${\displaystyle \theta \in J^{k}(E,m)}$ is a subspace of the tangent space ${\displaystyle T_{\theta }(J^{k}(E,m))}$ of the form ${\displaystyle T_{\theta }(M^{k})}$, where ${\displaystyle M}$ is any submanifold of ${\displaystyle E}$ whose prolongation contains the point ${\displaystyle \theta }$.

The Cartan distribution on ${\displaystyle J^{k}(E,m)}$ is the distribution ${\displaystyle {\mathcal {C}}\subseteq T(J^{k}(E,m))}$ defined by

$${\displaystyle {\mathcal {C}}:J^{k}(E,m)\rightarrow TJ^{k}(E,m),\qquad \theta \mapsto {\mathcal {C}}_{\theta }\subset T_{\theta }(J^{k}(E,m))}$$

where ${\displaystyle {\mathcal {C}}_{\theta }}$ is the span of all ${\displaystyle R}$-planes at a point ${\displaystyle \theta \in J^{k}(E,m)}$.

The Cartan distribution is important in the algebro-geometric approach to differential equations because it allows to define generalized solutions of differential equations in purely geometric terms.

A generalized solution of a differential equation ${\displaystyle {\mathcal {E}}\subset J^{k}(E,m)}$ is defined to be an ${\displaystyle m}$-dimensional submanifold ${\displaystyle S\subset {\mathcal {E}}}$ that fulfills ${\displaystyle T_{\theta }S\subset {\mathcal {C}}_{\theta }}$ for all ${\displaystyle \theta \in S}$.

One can also look at the Cartan distribution of a submanifold of ${\displaystyle J^{k}(E,m)}$ without the need to consider it inside ${\displaystyle J^{k}(E,m)}$. To do so, one defines the restriction of the distribution to a submanifold of ${\displaystyle J^{k}(E,m)}$ as follows.

If ${\displaystyle {\mathcal {E}}\subset J^{k}(E,m)}$, then its Cartan Distribution is defined by ${\displaystyle {\mathcal {C}}({\mathcal {E}}):=\{{\mathcal {C}}_{\theta }\cap T_{\theta }({\mathcal {E}})~|~\theta \in {\mathcal {E}}\}}$

In this sense, the pair ${\displaystyle ({\mathcal {E}},{\mathcal {C}}({\mathcal {E}}))}$ encodes the information about the (generalized) solutions of the differential equation ${\displaystyle {\mathcal {E}}}$.

An elementary diffiety is a pair ${\displaystyle ({\mathcal {E}}^{\infty },{\mathcal {C}}({\mathcal {E}}^{\infty }))}$ where ${\displaystyle {\mathcal {E}}}$ is a ${\displaystyle k}$-th order differential equation, ${\displaystyle {\mathcal {E}}^{\infty }}$ its infinite prolongation and ${\displaystyle {\mathcal {C}}({\mathcal {E}}^{\infty })}$ its Cartan distribution.

Note that when considering a differential equation ${\displaystyle {\mathcal {E}}\subset J^{k}(E,m)}$, then one can show that the Cartan distribution ${\displaystyle {\mathcal {C}}({\mathcal {E}}^{\infty })}$ is exactly ${\displaystyle m}$-dimensional unlike in the case of finitely many prolongations.

A diffiety is a triple ${\displaystyle ({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))}$, consisting of a (generally infinite-dimensional) manifold ${\displaystyle {\mathcal {O}}}$, the algebra of its smooth functions ${\displaystyle {\mathcal {F}}({\mathcal {O}})}$ and a finite-dimensional distribution ${\displaystyle {\mathcal {C}}({\mathcal {O}})}$, which is locally of the form ${\displaystyle ({\mathcal {E}}^{\infty },{\mathcal {F}}({\mathcal {E}}^{\infty }),{\mathcal {C}}({\mathcal {E}}^{\infty }))}$, where ${\displaystyle ({\mathcal {E}}^{\infty },{\mathcal {C}}({\mathcal {E}}^{\infty }))}$ is an elementary diffiety and ${\displaystyle {\mathcal {F}}({\mathcal {E}}^{\infty })}$ denotes the algebra of smooth functions on ${\displaystyle {\mathcal {E}}^{\infty }}$. Here locally means a suitable localization with respect to the Zariski topology corresponding to the algebra ${\displaystyle {\mathcal {F}}({\mathcal {O}})}$.

The dimension of ${\displaystyle {\mathcal {C}}({\mathcal {O}})}$ is called dimension of the diffiety and its denoted by ${\displaystyle \mathrm {Dim} ({\mathcal {O}})}$, with a capital D (to distinguish it from the dimension of ${\displaystyle {\mathcal {O}}}$ as a manifold).

A morphism between two diffieties ${\displaystyle ({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))}$ and ${\displaystyle ({\mathcal {O}}',{\mathcal {F}}({\mathcal {O}}'),{\mathcal {C}}({\mathcal {O'}}))}$ consists of a smooth map ${\displaystyle \Phi$ :{\mathcal {O}}\rightarrow {\mathcal {O}}'} whose pushforward preserves the Cartan distribution, i.e. such that, for every point ${\displaystyle \theta \in {\mathcal {O}}}$, one has ${\displaystyle d_{\theta }\Phi ({\mathcal {C}}_{\theta })\subseteq {\mathcal {C}}_{\Phi (\theta )}}$.

Diffieties together with their morphisms define the category of differential equations.[3]

The Vinogradov ${\displaystyle {\mathcal {C}}}$-spectral sequence (or, for short, Vinogradov sequence) is a spectral sequence associated to a diffiety, which can be used to investigate certain properties of the formal solution space of a differential equation by exploiting its Cartan distribution ${\displaystyle {\mathcal {C}}}$.[4]

Given a diffiety ${\displaystyle ({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))}$, consider the algebra of differential forms over ${\displaystyle {\mathcal {O}}}$

${\displaystyle \Omega ({\mathcal {O}}):=\sum _{i\geq 0}\Omega ^{i}({\mathcal {O}})}$

and the corresponding de Rham complex:

${\displaystyle C^{\infty }({\mathcal {O}})\longrightarrow \Omega ^{1}({\mathcal {O}})\longrightarrow \Omega ^{2}({\mathcal {O}})\longrightarrow \cdots }$

Its cohomology groups ${\displaystyle H^{i}({\mathcal {O}}):={\text{ker}}({\text{d}}_{i})/{\text{im}}({\text{d}}_{i-1})}$ contain some structural information about the PDE; however, due to the Poincaré Lemma, they all vanish locally. In order to extract much more and even local information, one thus needs to take the Cartan distribution into account and introduce a more sophisticated sequence. To this end, let

${\displaystyle {\mathcal {C}}\Omega ({\mathcal {O}})=\sum _{i\geq 0}{\mathcal {C}}\Omega ^{i}({\mathcal {O}})\subseteq \Omega ({\mathcal {O}})}$

be the submodule of differential forms over ${\displaystyle {\mathcal {O}}}$ whose restriction to the distribution ${\displaystyle {\mathcal {C}}}$ vanishes, i.e.

${\displaystyle {\mathcal {C}}\Omega ^{p}({\mathcal {O}}):=\{w\in \Omega ^{p}({\mathcal {O}})\mid w(X_{1},\cdots ,X_{p})=0\quad \forall ~X_{1},\ldots ,X_{p}\in {\mathcal {C}}({\mathcal {O}})\}.}$

Note that ${\displaystyle {\mathcal {C}}\Omega ^{i}({\mathcal {O}})\subseteq \Omega ^{i}({\mathcal {O}})}$ is actually a differential ideal since it is stable w.r.t. to the de Rham differential, i.e. ${\displaystyle {\text{d}}({\mathcal {C}}\Omega ^{i}({\mathcal {O}}))\subset {\mathcal {C}}\Omega ^{i+1}({\mathcal {O}})}$.

Now let ${\displaystyle {\mathcal {C}}^{k}\Omega ({\mathcal {O}})}$ be its ${\displaystyle k}$-th power, i.e. the linear subspace of ${\displaystyle {\mathcal {C}}\Omega }$ generated by ${\displaystyle w_{1}\wedge \cdots \wedge w_{k},~w_{i}\in {\mathcal {C}}\Omega }$. Then one obtains a filtration

${\displaystyle \Omega ({\mathcal {O}})\supset {\mathcal {C}}\Omega ({\mathcal {O}})\supset {\mathcal {C}}^{2}\Omega ({\mathcal {O}})\supset \cdots }$

and since all ideals ${\displaystyle {\mathcal {C}}^{k}\Omega }$ are stable, this filtration completely determines the following spectral sequence:

${\displaystyle {\mathcal {C}}E({\mathcal {O}})=\{E_{r}^{p,q},{\text{d}}_{r}^{p,q}\}\qquad {\text{where}}\qquad E_{0}^{p,q}:={\frac {{\mathcal {C}}^{p}\Omega ^{p+q}({\mathcal {O}})}{{\mathcal {C}}^{p+1}\Omega ^{p+q}({\mathcal {O}})}},\qquad {\text{and}}\qquad E_{r+1}^{p,q}:=H(E_{r}^{p,q},d_{r}^{p,q}).}$

The filtration above is finite in each degree, i.e. for every ${\displaystyle k\geq 0}$

${\displaystyle \Omega ^{k}({\mathcal {O}})\supset {\mathcal {C}}^{1}\Omega ^{k}({\mathcal {O}})\supset \cdots \supset {\mathcal {C}}^{k+1}\Omega ^{k}({\mathcal {O}})=0,}$

so that the spectral sequence converges to the de Rham cohomology ${\displaystyle H({\mathcal {O}})}$ of the diffiety. One can therefore analyse the terms of the spectral sequence order by order to recover information on the original PDE. For instance:[5]

• ${\displaystyle E_{1}^{0,n}}$ corresponds to action functionals constrained by the PDE ${\displaystyle {\mathcal {E}}}$. In particular, for ${\displaystyle {\mathcal {L}}\in E_{1}^{0,n}}$, the corresponding Euler-Lagrange equation is ${\displaystyle {\text{d}}_{1}^{0,n}{\mathcal {L}}=0}$.
• ${\displaystyle E_{1}^{0,n-1}}$ corresponds to conservation laws for solutions of ${\displaystyle {\mathcal {E}}}$.
• ${\displaystyle E_{2}}$ is interpreted as characteristic classes of bordisms of solutions of ${\displaystyle {\mathcal {E}}}$.

Many higher-order terms do not have an interpretation yet.

As a particular case, starting with a fibred manifold ${\displaystyle \pi :E\to X}$ and its jet bundle ${\displaystyle J^{k}(\pi )}$ instead of the jet space ${\displaystyle J^{k}(E,m)}$, instead of the ${\displaystyle {\mathcal {C}}}$-spectral sequence one obtains the slightly less general variational bicomplex. More precisely, any bicomplex determines two spectral sequences: one of the two spectral sequences determined by the variational bicomplex is exactly the Vinogradov ${\displaystyle {\mathcal {C}}}$-spectral sequence. However, the variational bicomplex was developed independently from the Vinogradov sequence.[6][7]

Similarly to the terms of the spectral sequence, many terms of the variational bicomplex can be given a physical interpretation in classical field theory: for example, one obtains cohomology classes corresponding to action functionals, conserved currents, gauge charges, etc.[8]

Vinogradov developed a theory, known as secondary calculus, to formalise in cohomological terms the idea of a differential calculus on the space of solutions of a given system of PDEs (i.e. the space of integral manifolds of a given diffiety).[9][10][11][3]

In other words, secondary calculus provides substitutes for vector fields, differential forms, differential operators, etc., on a (generically) very singular space where these objects cannot be defined in the usual (smooth) way.[12]

Secondary calculus can also be related to the covariant Phase Space, i.e. the solution space of the Euler-Lagrange equations associated to a Lagrangian field theory.[13]