nLab diffiety

Idea

The concept of diffiety (Vinogradov 81) reflects the concept of partial differential equation (generally non-linear) in analogy to how the concept of algebraic variety reflects that of polynomial equation:

A diffiety is the solution-locus $\mathcal{E}\hookrightarrow J^\infty_\Sigma(E)$ of a partial differential equation $D \Phi = 0$ regarded as an ordinary equation $\tilde D = 0$ on the jet bundle $J^\infty_\Sigma(E)$ of some bundle $E \overset{fb}{\to} \Sigma$.

Here $\Sigma$ is the space of free variables of the PDE, $E \overset{fb}{\to} \Sigma$ is the bundle of dependent variables, and a differential operator

$$D \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(F)$$

on the space of smooth sections of $fb$ is represented by a bundle morphism

$$\tilde D \colon J^\infty_\Sigma(E) \to F$$

out of the jet bundle via jet prolongation $j^\infty_\Sigma \colon \Gamma_\Sigma(E) \to \Gamma_\Sigma(J^\infty_\Sigma(E))$ as $D \Phi = \tilde D (j^\infty_\Sigma(\Phi))$:

$\,$

$$\array{ & \text{diffiety} && \text{jet bundle} & \array{\text{differential} \\ \text{operator}} & \\ & \mathcal{E} &\overset{\tilde D = 0}{\hookrightarrow}& J^\infty_\Sigma(E) &\overset{\tilde D}{\longrightarrow}& F }$$

For instance in Lagrangian field theory the bundle in question is a field bundle $E \overset{fb}{\to} \Sigma$, the partial differential equation is the Euler-Lagrange equation $\delta_{EL}\mathbf{L} = 0$, and its diffiety solution locus $\mathcal{E}$ inside the jet bundle $J^\infty_\Sigma(E)$ is called the shell of the field theory.

In Marvan 86 it was observed that Vinogradov’s formally integrable diffieties are equivalently the coalgebras over the jet comonad acting on locally pro-manifold-bundles (over a base space $\Sigma$ of free variables). This statement generalizes to the synthetic differential geometry of the Cahiers topos $\mathbf{H}$ (Khavkine-Schreiber 17), where the jet comonad is realized as the comonad corresponding to base change along the de Rham shape projection $\Sigma \overset{\eta_\Sigma}{\longrightarrow} \Im \Sigma$. By comonadic descent this implies that over formally smooth base spaces $\Sigma$ formally integrable diffieties are equivalently the bundles over the de Rham shape $\Im \Sigma$:

$$PDEs_\Sigma(\mathbf{H}) \;=\; Diffeties_\Sigma(\mathbf{H}) \;\simeq\; J^\infty_\Sigma CoAlg\left(\mathbf{H}_{/\Sigma} \right) \;\simeq\; \mathbf{H}_{/\Im(\Sigma)}$$

(Khavkine-Schreiber 17, thorem 3.52, theorem 3.60)

This makes manifest how diffieties are the analog in differential geometry of concepts in algebraic geometry: For $\Sigma$ a suitable scheme then a quasicoherent module over its de Rham shape $\Im \Sigma$ (“crystal”) is called a D-module and represents an algebraic linear partial differential equation, while a relative scheme over $\Im \Sigma$ is called a D-scheme and represents a general algebraic partial differential equation. See also at D-geometry for more on this.

Related concepts

• variational calculus

References

General

• Alexandre Vinogradov, Geometry of nonlinear differential equations, Journal of Soviet Mathematics 17 (1981) 1624–1649 (doi:10.1007/BF01084594)

• Alexandre Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., Vol. 2, 1984, p. 21 (MR85m:58192, doi)

• Alexandre Vinogradov, Symmetries and conservation laws of partial differential equations: basic notions and results, Acta Appl. Math., Vol. 15, 1989, p. 3. MR91b:58282, doi

• Alexandre Vinogradov, Scalar differential invariants, diffieties and characteristic classes, in: Mechanics, Analysis and Geometry: 200 Years after, 379–414, MR92e:58244

• G. Cicogna, G. Gaeta, Lie-point symmetries in bifurcation problems, Annales de l’institut Henri Poincaré (A) Physique théorique, 56 no. 4 (1992), p. 375-414 numdam

• L. Vitagliano, Hamilton-Jacobi diffieties, arxiv/1104.0162

• Alexandre M. Vinogradov‘s webpage

• Joseph Krasil'shchik‘s webpage (with links to some papers) and wiki list of publications

Review

• Joseph Krasil'shchik, Alexander Verbovetsky, Homological methods in equations of mathematical physics, arxiv:math.DG/9808130, 150 p.

• Joseph Krasil'shchik, Alexander Verbovetsky, Geometry of jet spaces and integrable systems, J. Geom. Phys. (2010), doi, arXiv:1002.0077

See also

• wikipedia: diffiety, secondary calculus and cohomological physics

As jet coalgebras

Diffieties as coalgebras over the jet comonad are discussed in

• Michal Marvan, A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno (pdf)

• Michal Marvan, thesis, 1989 (pdf, web)

• Michal Marvan, On the horizontal cohomology with general coefficients, 1989 (web announcement, web archive)

• Michal Marvan, section 1.1 of On Zero-Curvature Representations of Partial Differential Equations, (1993) (web)

• Igor Khavkine, Urs Schreiber, Synthetic geometry of differential equations: I. Jets and comonad structure (arXiv:1701.06238)

Conferences

• diffiety school, diffiety institute, gdeq.org wiki

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  XXI Summer Diffiety School School on Geometry of PDEs, July 19 - 31, 2018