nLab
diffiety

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Variational calculus

Contents

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 J Σ (E)\mathcal{E}\hookrightarrow J^\infty_\Sigma(E) of a partial differential equation DΦ=0D \Phi = 0 regarded as an ordinary equation D˜=0\tilde D = 0 on the jet bundle J Σ (E)J^\infty_\Sigma(E) of some bundle EfbΣE \overset{fb}{\to} \Sigma.

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

D:Γ Σ(E)Γ Σ(F) D \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(F)

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

D˜:J Σ (E)F \tilde D \colon J^\infty_\Sigma(E) \to F

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

\,

diffiety jet bundle differential operator D˜=0 J Σ (E) D˜ F \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 EfbΣE \overset{fb}{\to} \Sigma, the partial differential equation is the Euler-Lagrange equation δ ELL=0\delta_{EL}\mathbf{L} = 0, and its diffiety solution locus \mathcal{E} inside the jet bundle J Σ (E)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 Σ\Sigma of free variables). This statement generalizes to the synthetic differential geometry of the Cahiers topos H\mathbf{H} (Khavkine-Schreiber 17), where the jet comonad is realized as the comonad corresponding to base change along the de Rham shape projection Ση ΣΣ\Sigma \overset{\eta_\Sigma}{\longrightarrow} \Im \Sigma. By comonadic descent this implies that over formally smooth base spaces Σ\Sigma formally integrable diffieties are equivalently the bundles over the de Rham shape Σ\Im \Sigma:

PDEs Σ(H)=Diffeties Σ(H)J Σ CoAlg(H /Σ)H /(Σ) 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 Σ\Sigma a suitable scheme then a quasicoherent module over its de Rham shape Σ\Im \Sigma (“crystal”) is called a D-module and represents an algebraic linear partial differential equation, while a relative scheme over Σ\Im \Sigma is called a D-scheme and represents a general algebraic partial differential equation. See also at D-geometry for more on this.

References

General

Review

See also

As jet coalgebras

Diffieties as coalgebras over the jet comonad are discussed in

Conferences

Revised on December 11, 2017 06:26:30 by Urs Schreiber (178.6.238.60)