Diffiety

Not to be confused withDiffeology.

Inmathematics,adiffiety(/dəˈfaɪəˌtiː/) is ageometricalobject which plays the same role in the modern theory ofpartial differential equationsthatalgebraic varietiesplay foralgebraic equations,that is, to encode the space of solutions in a more conceptual way. The term was coined in 1984 byAlexandre Mikhailovich Vinogradovasportmanteaufromdifferential variety.

Intuitive definition

Inalgebraic geometrythe main objects of study (varieties) model the space of solutions of a system of algebraic equations (i.e. the zero locus of a set ofpolynomials), together with all their "algebraic consequences". This means that, applying algebraic operations to this set (e.g. adding those polynomials to each other or multiplying them with any other polynomials) will give rise to the same zero locus. In other words, one can actually consider the zero locus of the algebraicidealgenerated by the initial set of polynomials.

When dealing with differential equations, apart from applying algebraic operations as above, one has also the option todifferentiatethe starting equations, obtaining new differential constraints. Therefore, the differential analogue of a variety should be the space of solutions of a system of differential equations, together with all their "differential consequences". Instead of considering the zero locus of an algebraic ideal, one needs therefore to work with adifferential ideal.

Anelementary diffietywill consist therefore of theinfinite prolongation${\mathcal {E}}^{\infty }$of a differential equation${\mathcal {E}}\subset J^{k}(E,m)$,together with an extra structure provided by a specialdistribution.Elementary diffieties play the same role in the theory of differential equations asaffine algebraic varietiesdo in the theory of algebraic equations. Accordingly, just likevarietiesorschemesare composed of irreducibleaffine varietiesoraffine schemes,one defines a (non-elementary)diffietyas an object thatlocally looks likean elementary diffiety.

Formal definition

The formal definition of a diffiety, which relies on the geometric approach to differential equations and their solutions, requires the notions of jets of submanifolds, prolongations, and Cartan distribution, which are recalled below.

Jet spaces of submanifolds

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

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

For instance, for$k=1$one recovers just points in$E$and for$k=1$one recovers theGrassmannianof$n$-dimensional subspaces of$TE$.More generally, all the projections$J^{k}(E)\to J^{k-1}E$arefibre bundles.

As a particular case, when$E$has a structure offibred manifoldover an$n$-dimensional manifold$X$,one can consider submanifolds of$E$given by thegraphsof localsectionsof$\pi:E\to X$.Then the notion of jet of submanifolds boils down to the standard notion ofjetof sections, and thejet bundle$J^{k}(\pi )$turns out to be anopenanddensesubset of$J^{k}(E,m)$.

Prolongations of submanifolds

The$k$-jet prolongation of a submanifold$M\subseteq E$is

$$j^{k}(M):M\rightarrow J^{k}(E,m),\quad p\mapsto [M]_{p}^{k}$$

The map$j^{k}(M)$is a smoothembeddingand its image$M^{k}:={\text{im}}(j^{k}(M))$,called theprolongationof the submanifold$M$,is a submanifold of$J^{k}(E,m)$diffeomorphic to$M$.

Cartan distribution on jet spaces

A space of the form$T_{\theta }(M^{k})$,where$M$is any submanifold of$E$whose prolongation contains the point$\theta \in J^{k}(E,m)$,is called an$R$-plane(or jet plane, or Cartan plane) at$\theta$.TheCartan distributionon the jet space$J^{k}(E,m)$is thedistribution${\mathcal {C}}\subseteq T(J^{k}(E,m))$defined by$${\mathcal {C}}:J^{k}(E,m)\rightarrow TJ^{k}(E,m),\qquad \theta \mapsto {\mathcal {C}}_{\theta }\subset T_{\theta }(J^{k}(E,m))$$where${\mathcal {C}}_{\theta }$is the span of all$R$-planes at$\theta \in J^{k}(E,m)$.

Differential equations

Adifferential equationof order$k$on the manifold$E$is a submanifold${\mathcal {E}}\subset J^{k}(E,m)$;asolutionis defined to be an$m$-dimensional submanifold$S\subset {\mathcal {E}}$such that$S^{k}\subseteq {\mathcal {E}}$.When$E$is a fibred manifold over$X$,one recovers the notion ofpartial differential equations on jet bundlesand their solutions, which provide a coordinate-free way to describe the analogous notions ofmathematical analysis.While jet bundles are enough to deal with many equations arising in geometry, jet spaces of submanifolds provide a greater generality, used to tackle several PDEs imposed on submanifolds of a given manifold, such asLagrangian submanifoldsandminimal surfaces.

As in the jet bundle case, the Cartan distribution is important in the algebro-geometric approach to differential equations because it allows to encode solutions in purely geometric terms. Indeed, a submanifold$S\subset {\mathcal {E}}$is a solution if and only if it is anintegral manifoldfor${\mathcal {C}}$,i.e.$T_{\theta }S\subset {\mathcal {C}}_{\theta }$for all$\theta \in S$.

One can also look at the Cartan distribution of a PDE${\mathcal {E}}\subset J^{k}(E,m)$more intrinsically, defining$${\mathcal {C}}({\mathcal {E}}):=\{{\mathcal {C}}_{\theta }\cap T_{\theta }({\mathcal {E}})~|~\theta \in {\mathcal {E}}\}$$In this sense, the pair$({\mathcal {E}},{\mathcal {C}}({\mathcal {E}}))$encodes the information about the solutions of the differential equation${\mathcal {E}}$.

Prolongations of PDEs

Given a differential equation${\mathcal {E}}\subset J^{l}(E,m)$of order$l$,its$k$-th prolongation is defined as$${\mathcal {E}}^{k}:=J^{k}({\mathcal {E}},m)\cap J^{k+l}(E,m)\subseteq J^{k+l}(E,m)$$where both$J^{k}({\mathcal {E}},m)$and$J^{k+l}(E,m)$are viewed as embedded submanifolds of$J^{k}(J^{l}(E,m),m)$,so that their intersection is well-defined. However, such an intersection is not necessarily a manifold again, hence${\mathcal {E}}^{k}$may not be an equation of order$k+l$.One therefore usually requires${\mathcal {E}}$to be "nice enough" such that at least its first prolongation is indeed a submanifold of$J^{k+1}(E,m)$.

Below we will assume that the PDE isformally integrable,i.e. all prolongations${\mathcal {E}}^{k}$are smooth manifolds and all projections${\mathcal {E}}^{k}\to {\mathcal {E}}^{k-1}$are smooth surjective submersions. Note that a suitable version ofCartan–Kuranishi prolongation theoremguarantees that, under minor regularity assumptions, checking the smoothness of a finite number of prolongations is enough. Then theinverse limitof the sequence$\{{\mathcal {E}}^{k}\}_{k\in \mathbb {N} }$extends the definition of prolongation to the case when$k$goes to infinity, and the space${\mathcal {E}}^{\infty }$has the structure of aprofinite-dimensionalmanifold.

Definition of a diffiety

Anelementary diffietyis a pair$({\mathcal {E}}^{\infty },{\mathcal {C}}({\mathcal {E}}^{\infty }))$where${\mathcal {E}}\subset J^{k}(E,m)$is a$k$-th orderdifferential equationon some manifold,${\mathcal {E}}^{\infty }$its infinite prolongation and${\mathcal {C}}({\mathcal {E}}^{\infty })$its Cartan distribution. Note that, unlike in the finite case, one can show that the Cartan distribution${\mathcal {C}}({\mathcal {E}}^{\infty })$is$m$-dimensional andinvolutive.However, due to the infinite-dimensionality of the ambient manifold, theFrobenius theoremdoes not hold, therefore${\mathcal {C}}({\mathcal {E}}^{\infty })$is not integrable

Adiffietyis a triple$({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))$,consisting of

• a (generally infinite-dimensional) manifold${\mathcal {O}}$
• the algebra of its smooth functions${\mathcal {F}}({\mathcal {O}})$
• a finite-dimensional distribution${\mathcal {C}}({\mathcal {O}})$,

such that$({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))$is locally of the form$({\mathcal {E}}^{\infty },{\mathcal {F}}({\mathcal {E}}^{\infty }),{\mathcal {C}}({\mathcal {E}}^{\infty }))$,where$({\mathcal {E}}^{\infty },{\mathcal {C}}({\mathcal {E}}^{\infty }))$is an elementary diffiety and${\mathcal {F}}({\mathcal {E}}^{\infty })$denotes the algebra of smooth functions on${\mathcal {E}}^{\infty }$.Herelocallymeans a suitable localisation with respect to theZariski topologycorresponding to the algebra${\mathcal {F}}({\mathcal {O}})$.

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

Morphisms of diffieties

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

Diffieties together with their morphisms define thecategoryof differential equations.

Applications

Vinogradov sequence

TheVinogradov${\mathcal {C}}$-spectral sequence(or, for short,Vinogradov sequence) is aspectral sequenceassociated 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${\mathcal {C}}$.

Given a diffiety$({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))$,consider the algebra ofdifferential formsover${\mathcal {O}}$

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

and the correspondingde Rham complex:

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

Itscohomology groups$H^{i}({\mathcal {O}}):={\text{ker}}({\text{d}}_{i})/{\text{im}}({\text{d}}_{i-1})$contain some structural information about the PDE; however, due to thePoincaré 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

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

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

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

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

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

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

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

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

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

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

so that the spectral sequence converges to the de Rham cohomology$H({\mathcal {O}})$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:

• $E_{1}^{0,n}$corresponds toaction functionalsconstrained by the PDE${\mathcal {E}}$.In particular, for${\mathcal {L}}\in E_{1}^{0,n}$,the correspondingEuler-Lagrange equationis${\text{d}}_{1}^{0,n}{\mathcal {L}}=0$.
• $E_{1}^{0,n-1}$corresponds toconservation lawsfor solutions of${\mathcal {E}}$.
• $E_{2}$is interpreted ascharacteristic classesofbordismsof solutions of${\mathcal {E}}$.

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

Variational bicomplex

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

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

Secondary calculus

Vinogradov developed a theory, known assecondary 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).

In other words, secondary calculus provides substitutes for functions, vector fields, differential forms, differential operators, etc., on a (generically) very singular space where these objects cannot be defined in the usual (smooth) way on the space of solution. Furthermore, the space of these new objects are naturally endowed with the same algebraic structures of the space of the original objects.

More precisely, consider thehorizontal De Rham complex${\overline {\Omega }}^{\bullet }({\mathcal {O}}):=\Gamma (\wedge ^{\bullet }{\mathcal {C(O)}}^{*})$of a diffiety, which can be seen as the leafwisede Rham complexof the involutive distribution${\mathcal {C(O)}}$or, equivalently, theLie algebroid complexof the Lie algebroid${\mathcal {C(O)}}$.Then the complex${\overline {\Omega }}^{\bullet }({\mathcal {O}})$becomes naturally a commutativeDG algebratogether with a suitable differential${\overline {d}}$.Then, possibly tensoring with the normal bundle${\mathcal {V}}:=T{\mathcal {O}}/{\mathcal {C(O)}}\to {\mathcal {O}}$,its cohomology is used to define the following "secondary objects":

• secondary functionsare elements of the cohomology${\overline {H}}^{\bullet }({\mathcal {O}})=H^{\bullet }({\overline {\Omega }}^{\bullet }({\mathcal {O}}),{\overline {d}})$,which is naturally a commutative DG algebra (it is actually the first page of the${\mathcal {C}}$-spectral sequence discussed above);
• secondary vector fieldsare elements of the cohomology${\overline {H}}^{\bullet }({\mathcal {O}},{\mathcal {V}})=H^{\bullet }({\overline {\Omega }}^{\bullet }({\mathcal {O}}\otimes {\mathcal {V}}),{\overline {d}})$,which is naturally a Lie algebra; moreover, it forms a graded Lie-Rinehart algebra together with${\overline {H}}^{\bullet }({\mathcal {O}})$;
• secondary differential$p$-formsare elements of the cohomology${\overline {H}}^{\bullet }({\mathcal {O}},\wedge ^{p}{\mathcal {V}}^{*})=H^{\bullet }({\overline {\Omega }}^{\bullet }({\mathcal {O}}\otimes \wedge ^{p}{\mathcal {V}}^{*}),{\overline {d}})$,which is naturally a commutative DG algebra.

Secondary calculus can also be related to the covariantPhase Space,i.e. the solution space of the Euler-Lagrange equations associated to aLagrangian field theory.

See also

• Secondary calculus and cohomological physics
• Partial differential equations on Jet bundles
• Differential ideal
• Differential calculus over commutative algebras

Another way of generalizing ideas from algebraic geometry isdifferential algebraic geometry.

External links

• The Diffiety Institute(frozen since 2010)
• The Levi-Civita Institute(successor of above site)
• Geometry of Differential Equations
• Differential Geometry and PDEs