nLab variational bicomplex

Idea

For $X$ a (spacetime) manifold and $E \to X$ a bundle (in physics called the field bundle) with jet bundle $Jet(E) \to X$, the variational bicomplex is essentially the de Rham complex $(\Omega^\bullet(Jet( E)),\mathbf{d})$ of $Jet(E)$ with differential forms $\Omega^n(Jet(E)) = \bigoplus_{h+v=n} \Omega^{h,v}(E)$ bigraded by horizontal degree $h$ (with respect to $X$) and vertical degree $v$ (along the fibers of $j_\infty E$)). Accordingly the differential decomposes as

$$\mathbf{d} = d + \delta \,,$$

where $\mathbf{d}$ is the de Rham differential on $Jet(E)$, $d$ is called the horizontal differential and $\delta$ is called the vertical differential.

With $E \to X$ thought of as a field bundle over spacetime/worldvolume, then $d$ is a measure for how quantities change over spacetime, while $\delta$ is the variational differential that measures how quantities change as the field configurations are varied.

Accordingly, much of classical mechanics and classical field theory on $X$ is formalized in terms of the variational bicomplex. For instance

• a field configuration is a section of $E$;

• a Lagrangian is an element $L \in \Omega^{n,0}(E)$;

• a local action functional is a map

  $$S : \Gamma(E) \to \mathbb{R}$$

  of the form

  $$S(\phi) = \int_X L(j^\infty \phi) \,,$$

• the Euler-Lagrange equation is

  $$E(L) := \delta L \mod im d = 0$$

• the covariant phase space is the locus

  $$\{ \phi \in \Gamma(E) | E(L)(j^\infty \phi) = 0 \}$$

• a conserved current is an element $\eta\in \Omega^{n-1,0}(E)$ that is horizontally closed on the covariant phase space

  $$d \eta = 0 \mod E(L)$$

• a symmetry is an evolutionary vector field $v$ such that

  $$v(L) = 0 \mod im d$$

• Noether's theorem asserts that every symmetry induces a conserved current.

Definition

Let $X$ be a smooth manifold and $p : E \to X$ some smooth bundle over $X$. Write $Jet(E) \to X$ for the corresponding jet bundle.

The bicomplex

The spaces of sections $\Gamma(E)$ and $\Gamma(Jet(E))$ canonically inherit a generalized smooth structure that makes them diffeological spaces: we have a pullback diagram of diffeological spaces

$$\array{ \Gamma(E) & \longrightarrow & * \\ \downarrow && \downarrow^{id} \\ [X,E] &\stackrel{p_*}{\longrightarrow}& [X,X] } \,.$$

This induces the evaluation map

$$X \times \Gamma(E) \to E \,.$$

and composed with the jet prolongation

$$j^\infty : \Gamma(E) \to \Gamma(Jet(E))$$

it yields a smooth map (homomorphism of diffeological spaces)

(1)$$e_\infty : X \times \Gamma(E) \stackrel{(id,j^\infty)}{\to} X \times \Gamma(Jet(E)) \stackrel{ev}{\to} Jet(E) \,.$$

Write

$$\Omega^{\bullet, \bullet}(X \times \Gamma(E))$$

for the cochain complex of smooth differential forms on the product $X \times \Gamma(E)$, bigraded with respect to the differentials on the two factors

$$\mathbf{d} \coloneqq d + \delta \,,$$

where the $\mathbf{d}$, $d$ and $\delta$, are the de Rham differentials of $X\times\Gamma(E)$, $X$ and $\Gamma(E)$, respectively.

Definition

The variational bicomplex of $E \to X$ is the sub–bi-complex of $\Omega^{\bullet, \bullet}(X \times \Gamma(E))$ that is the image of the pullback of forms along the map $e_\infty$ (1):

$$e_\infty^* : \Omega^{\bullet}(Jet(E)) \to \Omega^\bullet(X \times \Gamma(E)) \,.$$

We write

$$\Omega^{\bullet, \bullet}_{loc} \coloneqq im (e_\infty^*)$$

and speak of the bicomplex of local forms on sections on $E$.

The bicomplex structure on $\Omega^{\bullet, \bullet}_{loc}$ is attributed in (Olver 86) to (Takens 79). The above formulation as a sub-bicomplex of the evident bicomplex of forms on $X \times \Gamma(E)$ is due to [Zuckerman 87, p. 5].

More on the horizontal differential complex

Remark

In terms of a coordinate chart $(x^i, u^\alpha,u^\alpha_i,u^\alpha_{i j},\cdots)$ of $E$ covering a coordinate chart $(X^i)$ of $X$, the action of the horizontal differential on functions $f \in C^\infty(Jet(E))$ is given by the formula for the total derivative operation, but with concrete differentials substituted by the respective jet coordinates:

$$d_h f \;\coloneqq\; \sum_i \left( \frac{\partial f}{\partial x^i} + \frac{\partial f}{\partial u^\alpha}u^\alpha_i + \sum_j \frac{\partial f}{\partial u^\alpha_j} u^\alpha_{j i} + \cdots \right) d x^i \,.$$

(Anderson 89, p. 10).

More abstractly, the horizontal differential is characterized as follows:

Proposition

The horizontal differential takes horizontal forms to horizontal forms, and for all sections $\phi \in \Gamma(E)$ it respects pullback of differential forms along the jet prolongation $j_\infty \phi \in \Gamma(Jet(E))$

$$(j^\infty \phi)^\ast \circ d_h = d \circ (j^\infty \phi)^\ast$$

(where on the right we have the ordinary de Rham differential on the base space).

Yet more abstractly, the horizontal complex may be understood in terms of differential operators and the jet comonad as follows.

Remark

A horizontal differential $n$-form $\alpha$ on $Jet(E) \to X$ is equivalently a homomorphism of bundles over $X$

$$\alpha \colon Jet(E) \longrightarrow \wedge^n T^\ast X$$

from the jet bundle $Jet(E)$ to the exterior bundle $\wedge^n T^\ast X$. This in turn is, by the discussion there, equivalently a differential operator $\alpha \colon E \to \wedge^n T^\ast X$.

Now of course also the de Rham differential $d_X$ on $X$ is a differential operator $\wedge^n T^\ast X \to \wedge^n T^\ast X$. In view of this, the horizontal differential of the variational bicomplex is just the composition operation of differential operators, with horizontal forms regarded as differential operators as above.

By the fact that differential operators are the co-Kleisli morphisms of the Jet comonad, this means that the horizontal differential is

$$d_H \alpha \colon Jet(F) \longrightarrow Jet(Jet(F)) \stackrel{Jet(\alpha)}{\longrightarrow} Jet(\wedge^n T^\ast X) \stackrel{\tilde d_X}{\longrightarrow} \wedge^n T^\ast X \,.$$

(e.g. Krasil’shchik-Verbovertsky 98, around def. 3.27, Krasil’shchik-Vinogradov 99, ch 4, around def. 1.8)

Evolutionary vector fields

Vector fields on $J^\infty E$ also split into a direct sum of vertical and horizontal ones, respectively being annihilated by contraction with any horizontal $1$-forms or with any vertical $1$-forms, $\mathfrak{X}(J^\infty E) = \mathfrak{X}_H(J^\infty E) \oplus \mathfrak{X}_V(J^\infty E)$. A special kind of vertical vector field $v \in \mathfrak{X}_V(J^\infty E)$ is called an evolutionary vector field provided it satisfies $\mathcal{L}_v d = d \mathcal{L}_v$ and $\mathcal{L}_v = \iota_v \delta + \delta \iota_v$, we denote the subspace of evolutionary vector fields as $\mathfrak{X}_{ev}(J^\infty E) \subset \mathfrak{X}_V(J^\infty E)$.

Properties

Horizontal, vertical, and total cohomology

Let $E \to X$ be a smooth fiber bundle over a base smooth manifold $X$ of dimension $n.$ Write $J^\infty E \to X$ for the jet bundle of $E\to X$.

Write

$$\mathcal{F}^s(J^\infty E) \coloneqq I (\Omega^{n,s}(J^\infty E))$$

for the projection of $(n,s)$-forms to the image of the “interior Euler operator” (Anderson 89, p. 21 (50/318)).

Proposition

(Takens acyclicity theorem)

The cochain cohomology of the Euler-Lagrange complex

$$0 \to \mathbb{R} \to \Omega^{0,0}(J^\infty E) \stackrel{d_H}{\to} \Omega^{1,0}(J^\infty E) \stackrel{d_H}{\to} \cdots \stackrel{d_H}{\to} \Omega^{n,0}(J^\infty E) \stackrel{E}{\to} \mathcal{F}^1(J^\infty E) \stackrel{\delta_V}{\to} \mathcal{F}^2(J^\infty E) \stackrel{\delta_V}{\to} \cdots$$

is isomorphic to the de Rham cohomology of the total space $E$ of the given fiber bundle.

For smooth functions of locally bounded jet order this is due to (Takens 79). A proof is also in (Anderson 89, theorem 5.9).

For smooth functions of globally bounded order and going up to the Euler-Lagrange operator $E$, this is also shown in (Deligne 99, vol 1, p.188).

The fundamental variational formula

Definition

A source form is an element $\alpha$ in $\Omega^{n,1}_{loc}$ such that

$$\alpha_\phi(\delta \phi)$$

depends only on the 0-jet of $\delta \phi$.

Proposition

Let $L \in \Omega^{n,0}_{loc}$.

Then there is a unique source form $E(L)$ such that

$$\delta L = E(L) - d \Theta \,.$$

Moreover

• $E(L)$ is independent of changes of $L$ by $d$-exact terms:

  $$E(L) = E(L + d K) \,.$$

• $\Theta$ is unique up to $d$-exact terms.

This is [Zuckerman 87, theorem 3].

Here $E$ is the Euler-Lagrange operator .

Definition

Write

$$\Omega = \delta \Theta \,.$$

Remark

By prop. have

$$d \Omega = -\delta E(L) \,.$$

Proposition

$\delta E$ vanishes when restricted to vertical tangent vectors based in covariant phase space (but not necessarily tangential to it).

$$\delta E(L) |_{T_{E(L) = 0} \Gamma(E)} = 0 \,.$$

This is [Zuckerman 87, lemma 8].

Presymplectic covariant phase space

Corollary

The form $\Omega$ is a conserved current.

Proof

By remark and prop. .

Definition

For $\Sigma \subset X$ a compact closed submanifold of dimension $n-1$, one says that

$$\omega := \int_\Sigma \Omega \in \Omega^{0,2}_{loc}$$

is the presymplectic structure on covariant phase space relative to $\Sigma$.

Proposition

The 2-form $\omega$ is indeed closed

$$\delta \omega = 0$$

and in fact exact:

$$\theta := \int_\Sigma \Theta$$

is its presymplectic potential .

$$\delta \theta = \omega \,.$$

Symmetries

Let $L \in \Omega^{n,0}_{loc}$.

Definition

An evolutionary vertical vector field $v \in \mathfrak{X}_{ev}(J^\infty E)$ is a symmetry if

$$v(L) = 0 \mod im d \,.$$

Proposition

The presymplectic form $\omega$ from def. is annihilated by the Lie derivative of the vector field on $\Gamma(E)$ induced by a symmetry.

This appears as [Zuckerman 87, theorem 13].

Elementary formalization in differential cohesion

We discuss aspects of an elementary formalization in differential cohesion of the concept of the variational bicomplex .

under construction

Let $\mathbf{H}$ be a context of cohesion and differential cohesion, with

• flat modality denoted $\flat$;

• infinitesimal shape modality denoted $\Im$.

Choose

1. an object $\Sigma \in \mathbf{H}$, the base space (or spacetime or worldvolume);

2. an object $E \in \mathbf{H}_{/\Sigma}$, the field bundle,

3. an object $\mathbf{A} \in Stab(\mathbf{H}_{/\Sigma}) \stackrel{\Omega}{\to} \mathbf{H}_{/\Sigma}$, the differential coefficients.

Write

• $\mathbf{H}_{/\Sigma} \stackrel{\overset{\sum_\Sigma}{\longrightarrow}}{\stackrel{\overset{\Sigma^\ast}{\longleftarrow}}{\underset{\prod_\Sigma}{\longrightarrow}}} \mathbf{H}$ for the base change adjoint triple over $\Sigma$, the étale geometric morphism of the slice (infinity,1)-topos $\mathbf{H}_{/\Sigma}$;

• $\Gamma_X \coloneqq \flat \circ \prod_\Sigma \colon \mathbf{H}_{/\Sigma} \to \mathbf{H}$ for the external space of sections functor;

• $i \colon \Sigma \longrightarrow \Im(\Sigma)$ for the $\Sigma$-component of the unit of $\Im$;

• $Jet_\Sigma \coloneqq i^\ast i_\ast$ for the induced jet comonad;

• $\mathbf{H}_{/\Sigma} \stackrel{\overset{}{\longleftarrow}}{\underset{\iota}{\longrightarrow}} PDE(\mathbf{H})_{\Sigma}$ for the Eilenberg-Moore category of $Jet_\Sigma$-coalgebras (the objects are differential equations with variables in $\Sigma$, the morphisms are differential operators between these, preserving solution spaces), manifested as a topos of coalgebras over $\mathbf{H}$;

  the (non-full) direct image of this geometric morphism is the co-Kleisli category of the jet comonad and so for $\phi \colon free(E) \to free(F)$ a morphism in $PDE(\mathbf{H})_\Sigma$, we write $\tilde f \colon Jet(E) \to F$ for the corresponding co-Kleisli morphism in $\mathbf{H}_{/\Sigma}$;

We record the following simple fact, which holds generally since the jet comonad $Jet_\Sigma$ is a right adjoint (to the infinitesimal disk bundle functor), hence preserves terminal objects, and $\Sigma \in \mathbf{H}_{/\Sigma}$ is the terminal object:

Proposition

The essentially unique morphism

$$Jet(\Sigma) \stackrel{\simeq}{\longrightarrow} \Sigma$$

in $\mathbf{H}_{/\Sigma}$ in an equivalence.

Definition

The jet prolongation map

$$j \colon \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(Jet(E))$$

is the the Jet functor itself, regarded, in view of prop. , as taking sections to sections via

$$(\Sigma \stackrel{\sigma}{\to} E) \;\;\mapsto \;\; \left( \Sigma \stackrel{\simeq}{\to} Jet(\Sigma) \stackrel{Jet(\sigma)}{\longrightarrow} Jet\left(E\right) \right) \,.$$

Definition

For $E \in \mathbf{H}_{\Sigma}$ a bundle over $\Sigma$, then a horizontal $\mathbf{A}$-form on the jet bundle $Jet(E)$ is a morphism in $PDE(\mathbf{H})_{\Sigma}$ of the form

$$\alpha \colon \iota E \to \iota \mathbf{A} \,.$$

For $d \colon \iota Et_\Sigma\Sigma^\ast \mathbf{A}\to \iota Et_\Sigma\Sigma^\ast \mathbf{A}'$ a morphism in $\mathbf{PDE}(\mathbf{H})_{\Sigma}$, then the induced horizontal differential is the operation of horizontal forms sending $\alpha$ to the composite

$$d \alpha \colon \iota E \stackrel{\alpha}{\longrightarrow} \iota \mathbf{A} \stackrel{d}{\longrightarrow} \iota \mathbf{A}' \,.$$

Remark

Since all objects in def. are in the co-Kleisli category of the jet comonad, the morphism $\alpha$ there is equivalently a morphism in $\mathbf{H}_{/\Sigma}$ of the form

$$\tilde \alpha \colon Jet(E) \longrightarrow \mathbf{A} \,.$$

For the special case that $E = \Sigma$ in def. , then $Jet_{\Sigma}(\Sigma)\simeq\Sigma$ and so a horizontal $\mathbf{A}$-form on $\Sigma$ we call just a an $\mathbf{A}$-form.

Proposition

The horizontal differential of def. commutes with pullback of horizontal differential forms $\alpha$ along the jet prolongation, def. , of any field section $\sigma \in \Gamma_X(E)$.

In detail: for

• $d \colon \iota Et_\Sigma\Sigma^\ast \mathbf{A} \longrightarrow \iota Et_\Sigma\Sigma^\ast \mathbf{A}'$ a morphism,

• $\alpha \colon \iota E \to \iota Et_\Sigma\Sigma^\ast \mathbf{A}$ a horizontal $\mathbf{A}$-form on $Jet(E)$, def. ;

• $\sigma \in \Gamma_\Sigma(E)$ a field section,

then there is a natural equivalence

$$j(\sigma)^\ast (d \alpha) \simeq d (j(\sigma)^\ast \alpha) \,.$$

Proof

Since all objects are in the direct image $free\colon \mathbf{H} \to PDE(\mathbf{H})_\Sigma$, this is an equivalence of morphisms in the co-Kleisli category of the jet comonad, hence is equivalently an equivalence of co-Kleisli composites of morphisms in $\mathbf{H}$.

As such, the left hand side of the equality is given in $\mathbf{H}$ by the composite morphism

$$\Sigma \stackrel{\simeq}{\to} Jet(\Sigma) \stackrel{Jet(\sigma)}{\longrightarrow} Jet(E) \stackrel{}{\to} Jet(Jet(E)) \stackrel{Jet(\tilde \alpha)}{\longrightarrow} Jet(\mathbf{A}) \stackrel{\tilde d}{\longrightarrow} \mathbf{A}' \,,$$

thought of as bracketed to the right. By naturality of the Jet-counit this is equivalently

$$Jet(\Sigma) \stackrel{\simeq}{\to} Jet(Jet(\Sigma)) \stackrel{Jet(Jet(\sigma))}{\longrightarrow} Jet(Jet(E)) \stackrel{Jet(\tilde \alpha)}{\longrightarrow} Jet(\mathbf{A}) \stackrel{\tilde d}{\longrightarrow} \mathbf{A}' \,,$$

By functorality of $Jet(-)$ this is equivalent to

$$Jet(\Sigma) \stackrel{\simeq}{\to} Jet(Jet(\Sigma)) \stackrel{Jet ( \tilde \alpha \circ Jet(\sigma) )}{\longrightarrow} Jet(\mathbf{A}) \stackrel{\tilde d}{\to} \mathbf{A}'$$

which is the right hand side of the equivalence to be proven.

Related concepts

• local Lagrangian, Euler-Lagrange form, source form, Lepage form

• variational calculus

• secondary differential calculus

• variational sequence

• BV-BRST complex

• BV-BRST variational bicomplex

• presymplectic current, covariant phase space

References

The variational bicomplex was introduced independently in

• Alexandre Vinogradov, A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints , Sov. Math. Dokl. 19 (1978) 144–148.

• W. M. Tulczyjew, The Euler-Lagrange resolution , in Lecture Notes in Mathematics 836 22–48 (Springer-Verlag, New York 1980).

• T. Tsujishita, On variation bicomplexes associated to differential equations, Osaka J. Math. 19 (1982), 311–363.

See also

• Floris Takens, A global version of the inverse problem of the calculus of variations J. Diff. Geom. 14 (1979) 543-562

• Pierre Deligne et al. Quantum Fields and Strings

An introduction is in

• Ian Anderson, Introduction to the variational bicomplex, in Mathematical aspects of classical field theory, Contemp. Math. 132 (1992) 51–73, gBooks

A careful discussion that compares the two versions (one over smooth functions globally of finite jet order, one over smooth functions locally of finite jet order) is in

• G. Giachetta, L. Mangiarotti, Gennadi Sardanashvily, Cohomology of the variational bicomplex on the infinite order jet space, Journal of Mathematical Physics 42, 4272-4282 (2001) (arXiv:math/0006074)

Textbook accounts include

• Peter Olver, section 5.4 of Applications of Lie groups to differential equations, Springer Graduate Texts in Mathematics 107 (1986)

• Ian Anderson, The variational bicomplex, Utah State University 1989 (pdf)

• Joseph Krasil'shchik, Alexander Verbovetsky, Homological Methods in Equations of Mathematical Physics (arXiv:math/9808130)

• Joseph Krasil'shchik, Alexandre Vinogradov et al. (eds.) Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, AMS 1999

Other surveys include

• Juha Pohjanpelto, Symmetries, Conservation Laws, and Variational Principles for Differential Equations (2014) (pdf slides)

An early discussion with application to covariant phase spaces and their presymplectic structure is in

• Gregg Zuckerman, Action principles and global geometry, in: Shing-Tung Yau (ed.) Mathematical Aspects of String Theory, World Scientific (1987) 259-284 [pdf, doi:10.1142/0383]

An invariant version (under group action) is in

• Irina Kogan, Peter Olver, The invariant variational bicomplex, pdf

A more detailed version of this is in

• Irina Kogan, Peter Olver, Invariant Euler-Lagrange Equations and the Invariant Variational Bicomplex, pdf

See also

• Victor Kac, An explicit construction of the complex of variational calculus and Lie conformal algebra cohomology, talk at Algebraic Lie Theory, Newton Institute 2009, video

An application to multisymplectic geometry is discussed in

• Thomas Bridges, Peter Hydon, Jeffrey Lawson, Multisymplectic structures and the variational bicomplex (pdf)

Discussion in the context of supergeometry is in

• Gennadi Sardanashvily, Grassmann-graded Lagrangian theory of even and odd variables (arXiv:1206.2508)

Discussion in the convenient context of smooth sets:

• Grigorios Giotopoulos, Hisham Sati, §5.1 & §7.1 in: Field Theory via Higher Geometry I: Smooth Sets of Fields [arXiv:2312.16301]