Diffiety

In mathematics, a diffiety (/dəˈfaɪəˌtiː/) is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equations, that is, to encode the space of solutions in a more conceptual way.

The term was coined in 1984 by Alexandre Mikhailovich Vinogradov as portmanteau from differential variety.

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 algebraic ideal generated by the initial set of polynomials.

Elementary diffieties play the same role in the theory of differential equations as affine algebraic varieties do in the theory of algebraic equations.

Accordingly, just like varieties or schemes are composed of irreducible affine varieties or affine schemes, one defines a (non-elementary) diffiety as an object that locally looks like an elementary diffiety.

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.

if one can locally describe both submanifolds as zeroes of functions defined in a neighbourhood of

, one recovers the notion of partial differential equations on jet bundles and their solutions, which provide a coordinate-free way to describe the analogous notions of mathematical 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 as Lagrangian submanifolds and minimal 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.

encodes the information about the solutions of the differential equation

Note that a suitable version of Cartan–Kuranishi prolongation theorem guarantees that, under minor regularity assumptions, checking the smoothness of a finite number of prolongations is enough.

Note that, unlike in the finite case, one can show that the Cartan distribution

However, due to the infinite-dimensionality of the ambient manifold, the Frobenius theorem does not hold, therefore

Here locally means a suitable localisation with respect to the Zariski topology corresponding to the algebra

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

and the corresponding de Rham complex: Its cohomology groups

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.

so that the spectral sequence converges to the de Rham cohomology

-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

However, the variational bicomplex was developed independently from the Vinogradov sequence.

[8][9] 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.

[10] 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).

[11][12][13][3] 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.

of a diffiety, which can be seen as the leafwise de Rham complex of the involutive distribution

becomes naturally a commutative DG algebra together with a suitable differential

, its cohomology is used to define the following "secondary objects": 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.