The homotopy momentum map of
general relativity

The diffeomorphism symmetry of general relativity, a mathematical implementation of the Einstein equivalence principle, is one of its defining features. In contrast to the internal symmetry of gauge theories, diffeomorphisms are external symmetries since they act not only on the fields (i.e. lorentzian metrics), but also on spacetime. The initial value problem, which yields the hamiltonian formulation of the field dynamics, and the presymplectic structure on the space of fields, which yields the Poisson bracket of observables, both depend on the choice of a codimension 1 submanifold as initial time-slice. But such a submanifold is not invariant under diffeomorphisms. In physics terminology: it breaks the symmetry. The consequence is that the basic ingredients of quantization, the hamiltonian and the Poisson bracket, are not compatible with the diffeomorphism symmetry. This issue lies at the heart of some of the fundamental open problems in general relativity and has captivated the interest of many authors since the 1960s.

One of its mathematical symptoms is that the action of the group of diffeomorphisms and the action of the Lie algebra of vector fields are not hamiltonian. More precisely, Noether’s first theorem, which associates to a symmetry a conserved momentum does not define a homomorphism of Lie algebras. (The Noether momenta are the components of the Einstein tensor integrated over the codimension 1 submanifold.) Worse, the space of Noether momenta is not even closed under the Poisson bracket.

In an earlier paper we could show that there is a natural diffeological groupoid describing the choices of initial submanifolds, which exhibits the Poisson brackets as the bracket of its Lie algebroid [BFW13]. Next, we have developed a notion of hamiltonian Lie algebroids, which generalizes the notion of hamiltonian Lie algebra action to the setting of Lie algebroids [BW]. We have conjectured that the Noether charges of general relativity are the components of the momentum section of a hamiltonian Lie algebroid, which would give a conceptual explanation of some of the intriguing features of the constraint functions. Finally, in [BSW] we have interpreted the momenta as elements of a generalized Lie-Rinehart algebra, which is connected to the BV-BFV approach to boundary conditions in classical field theories.

In this paper, we sidestep the choice of initial submanifolds altogether by using higher algebraic structures. We show that the map from vector fields to their Noether currents is part of a homotopy momentum map in the sense of multisymplectic geometry.

In Sec. 2 we study the premultisymplectic form $\omega=EL+\delta\gamma$ of a lagrangian field theory (LFT), where $EL$ is the Euler-Lagrange form and $\gamma$ a boundary form. We prove in Prop. 2.4 that the obstruction of a premultisymplectic vector field $X$ to be hamiltonian lies in bidegree $(0,n-1)$ of the variational bicomplex, where $n$ is the dimension of the spacetime manifold. This shows that the $L_{\infty}$-algebra of hamiltonian forms can be interpreted as generalized current algebra. We introduce the notion of manifest diffeomorphism symmetry (Def. 2.12) and observe that every such symmetry has a hamiltonian momentum map that is given explicitly in terms of the lagrangian and the boundary form (Prop. 2.15).

In Sec. 3 we consider the case of general relativity. Generalizing the concept of tensor fields, we introduce the notion of covariant and contravariant families of forms in the variational bicomplex (Def. 3.3). We then show that the product of families, the contraction of indices, the raising and lowering of indices, etc. have properties that are analogous to tensor fields. In Sec. 3.5 we introduce the notion of covariant derivative of families of forms. In Props. 3.13 and 3.14 we derive divergence formulas that show that horizontally exact forms can be expressed as the contraction of families of forms with the covariant derivative.

Sec. 4 contains the main results. In Thm. 4.1 we prove that the Lepage form $L+\gamma$, which is the primitive of the premultisymplectic form $\omega$, is invariant under the action of spacetime vector fields. This implies that the action has a homotopy momentum map, which is described explicitly in Thm. 4.2.

The study of multisymplectic forms in classical field theory goes back to at least the 1970s. Best known is perhaps the highly influential, but never finished GiMmsy project (named by the initials of the collaborators involved, with the main protagonists Gotay and Marsden capitalized). Their goal was “to explore some of the connections between initial value constraints and gauge transformations” in classical field theories with constraints, such as general relativity [GIMM04, p. 1]. Towards this end they introduced the notion of multimomentum maps [GIMM04, p. 46] (see also [CnCI91, Sec. 4.2]). Given the action $\rho:\mathfrak{g}\to\mathcal{X}(M)$ of a Lie algebra on a manifold $M$ with a closed $(n+1)$-form $\omega$, a multimomentum map was defined as a smooth map $M\to\mathfrak{g}^{*}\otimes\wedge^{n-1}T^{*}M$ or, equivalently, a linear map $f:\mathfrak{g}\to\Omega^{n-1}(M)$, such that $d\,f(a)=-\iota_{\rho(a)}\omega$ for all $a\in\mathfrak{g}$.

Missing from this definition was a requirement of compatibility with the Lie bracket of $\mathfrak{g}$, analogous to hamiltonian momentum maps in symplectic geometry. It seems natural to require $f$ to be $\mathfrak{g}$-equivariant. Alternatively, the hamiltonian forms in $\Omega^{n-1}(M)$ can be equipped with a “Poisson bracket” and $f$ required to commute with the brackets. However, both conditions turn out to be too strong and rarely satisfied. Moreover, the “Poisson bracket” does not satisfy the Jacobi identity, so that it is not immediately clear for what kind of algebraic structure $f$ should be a homomorphism.

This lack of compatibility of algebraic structures leads to issues in the study of the constraints of classical field theories with diffeomorphism symmetries, of which general relativity is the theory “par excellence” [GIMM04, Interlude I, p. 52]. The constraint functions of general relativity are given by the values of the multimomentum map integrated over the Cauchy surface. The resulting map is called the energy-momentum map [GIM04, Sec. 7B]. While the energy-momentum map shows that the constraint functions arise from the multimomentum map, it does not explain the relation between the Lie brackets of the symmetry algebra of vector fields and the Poisson brackets of the constraints. (For the history of this often studied but elusive problem see Sec. 4 of [BFW13].) From the viewpoint of homotopical algebra, this was to be expected: Lie brackets that satisfy the Jacobi identity up to exact terms and maps that preserve the brackets up to exact terms are generally not compatible with homotopies of the underlying complexes. Instead we have to use the homotopy algebraic structure, i.e. the minimal extension of Lie algebras to differential complexes that is stable under quasi-isomorphisms. For a better behaved theory of multimomentum maps, we are thus led to $L_{\infty}$-algebras.

In [Rog12, Thm. 5.2] it was shown that the bracket on hamiltonian forms in $\Omega^{n-1}(M)$ has a natural extension to an $L_{\infty}$-algebra structure on a graded subspace of the de Rham complex, with the 1-bracket given by the de Rham differential.¹¹1In [BFLS98] it was shown that the bracket of integrated local functions on the jet bundle has extensions to alternative $L_{\infty}$-algebra structures on cohomological resolutions. However, these $L_{\infty}$-structures were not given by an explicit construction, depended on choices, and did not suggest a stronger notion of multimomentum maps. It was realized in [CFRZ16, Def./Prop. 5.1] that this is the natural setting for the generalization of hamiltonian momentum maps to the multisymplectic setting, defined as morphisms $\mu:\mathfrak{g}\to L_{\infty}(M,\omega)$ of $L_{\infty}$-algebras. The $\mu_{1}$ component of every homotopy momentum map is a multimomentum map [CFRZ16, Sec. 12.1]. For the obstructions to lifting a multimomentum map to a homotopy momentum map see [CFRZ16, Sec. 9.2].

In local lagrangian field theories, a multimomentum map is given by Noether’s theorem [BHL10, Sec. 4.1]. Finding a homotopy momentum map, however, is a much more difficult problem, even more so in general relativity, where the Hilbert-Einstein lagrangian is non-polynomial in the fields and of second jet order. The situation simplifies greatly if the multi(pre)symplectic form has a primitive, $\omega=d\lambda$, and if the action leaves $\lambda$ invariant. Then a homotopy momentum map can be defined by inserting the fundamental vector fields in $\lambda$ [CFRZ16, Sec. 8.1]. If we want to check whether this applies to a classical field theory, we have to identify the correct $\lambda$ as well as the correct action of the diffeomorphism group. Many authors use for $\lambda$ the boundary 1-form, so that $\omega$ is the universal current in the sense of [Zuc87], and assume that the action is vertical (e.g. [BHL10]). We will show that instead we have to take the sum of the lagrangian and the boundary 1-form for $\lambda$ and the diagonal action (12) on fields and spacetime by vertical and horizontal vector fields.

The spacetime manifold $M$ is assumed to be smooth, finite-dimensional, and second countable. The infinite jet manifold $J^{\infty}F$ of a smooth fibre bundle $F\to M$ is viewed as pro-manifold, so that its de Rham complex, i.e. the variational bicomplex, is an ind-differential complex. For the computations and proofs in this paper, however, this will not play much of a role. The same goes for the diffeological structure on the space of fields $\mathcal{F}=\Gamma(M,F)$, of which we will only use the fact that the diffeological tangent bundle is given by the space of sections of the vertical tangent bundle $T\mathcal{F}=\Gamma(M,VF)$. For the grading and differentials of the variational bicomplex we use the notation of [DF99]: A form in $\Omega^{p,q}(J^{\infty}F)$ has vertical degree $p$ and horizontal degree $q$. The vertical differential is denoted by $\delta$ and the horizontal differential by $d$. Otherwise, we follow [And89], with specific references given in the text. We use the summation convention throughout the paper, so that all repeated indices are being summed over.

For the reader’s convenience, we give a brief review of the main notions of multisymplectic geometry used in this paper. This is also necessary to fix the notation, the choice of gradings, and the signs.

Let $M$ be a manifold with a closed $(n+1)$-form $\omega$. A pair $(X,\alpha)$ consisting of a vector field $X\in\mathcal{X}(M)$ and a form $\alpha\in\Omega^{n-1}(M)$ is called hamiltonian if

A vector field or a form is called hamiltonian if it is part of a hamiltonian pair. We denote the space of hamiltonian vector fields by $\mathcal{X}_{\mathrm{ham}}(M)$ and the space of hamiltonian forms by $\Omega^{n-1}_{\mathrm{ham}}(M)$.

For the pair $(M,\omega)$ we can construct an $L_{\infty}$-algebra $L_{\infty}(M,\omega)$ defined as follows [Rog12, Thm. 5.2]. The $\mathbb{Z}$-graded vector space is

The brackets $l_{k}:\wedge^{k}L_{\infty}(M,\omega)\to L_{\infty}(M,\omega)$ are defined by

for $\deg\alpha_{1}<0$, by

for $k>1$ and $\deg\alpha_{1}=\ldots=\deg\alpha_{k}=0$ where $(X_{i},\alpha_{i})$ are hamiltonian pairs, and by zero in all other cases. With this degree convention, the degree of $l_{k}$ is $2-k$.

We recall that a morphism $\mu:L^{\prime}\to L$ of $L_{\infty}$-algebras is given by a family of linear maps $\mu_{k}:\wedge^{k}L^{\prime}\to L$, $k\geq 1$ of degree $1-k$, subject to relations that are best expressed either in terms of the $L_{\infty}$-operad or in the language of formal pointed manifolds. If the domain $L^{\prime}=\mathfrak{g}$ is a Lie algebra, as is the case for a homotopy momentum map, the conditions for $\mu$ to be a morphism simplify greatly. They are best expressed in terms of the boundary operator $\delta:\wedge^{\bullet}\mathfrak{g}\to\wedge^{\bullet-1}\mathfrak{g}$ of the Chevalley-Eilenberg complex for Lie homology,

A collection of linear maps $\mu_{k}:\wedge^{k}\mathfrak{g}\to L_{\infty}(M,\omega)$ is a homotopy momentum map if and only if [CFRZ16, Prop. 3.8]

for all $1\leq k\leq n$, where we set $\mu_{0}:=0$ and $\mu_{n+1}:=0$. This relation can be interpreted homotopically as follows. Shifting the degree of $\mathfrak{g}$ by $1$ and shifting the degree of the de Rham complex by $n-1$, the right hand side can be expressed in terms of the degree 0 map

where we have used that $S(\mathfrak{g}[1])_{-k}\cong\wedge^{k}\mathfrak{g}$. The maps $\mu_{k}$ have degree $-1$. The condition for $\mu$ to be a morphism of $L_{\infty}$-algebras can be written succinctly as [CFRZ16, Sec. 6.2]

that is, a homotopy momentum map $\mu$ is a null-homotopy of the map of cochain complexes $\nu$.

In degree $k=1$ the condition (1) reads $d\mu_{1}(a_{1})=-\iota_{\rho(a_{1})}\omega$, that is $\bigl{(}\rho(a_{1}),\mu_{1}(a_{1})\bigr{)}$ is a hamiltonian pair. With this relation, $\nu$ can be expressed in terms of the $L_{\infty}$-brackets as

For $k=2$, Eq. (1) is spelled out as

which shows that the failure of $\mu_{1}$ to be a homomorphism of Lie algebras is a $d$-exact term. For $k=3$ we obtain

A variety of ingredients can play a constitutive role in the mathematical study of classical field theories. For the purpose of this paper the following minimal definition will suffice:

Let $\alpha\in\Omega^{p,q}(J^{\infty}F)$, where $p$ denotes the vertical and $q$ the horizontal degree. The vertical differential will be denoted by $\delta$, the horizontal differential by $d$. The form $\alpha$ is represented by a form on a finite dimensional jet manifold $J^{k}F$, which is given by a map $\tilde{\alpha}:J^{k}F\to\wedge^{p+q}T^{*}J^{k}F$. In this way $\alpha$ gives rise to a $k$-th order differential operator

where $j^{k}\varphi:F\to J^{k}F$ is the $k$-th jet prolongation of $\varphi$. In this notation, the integrand of the action can be written as $\tilde{L}(\varphi)=D_{L}(\varphi)$.

The target of the differential operator $D_{\alpha}$ is not a vector space, so it does not make sense to consider the equation “$D_{\alpha}(\varphi)=0$”, even though this is how the corresponding PDE is often written. And even if $F\to M$ is a vector bundle so that $\wedge^{p+q}T^{*}J^{k}F\to F$ is a vector bundle, the right hand side cannot be required to be the zero section, as this would imply that $\varphi$ is the zero section of $F\to M$. Instead, we have to use that there is a zero form $0\in\Omega^{p,q}(J^{k}F)$ in every bidegree. The PDE can then be written more carefully as $D_{\alpha}(\varphi)=D_{0}(\varphi)$. If this equation holds, we will say that $\alpha$ vanishes at $\varphi\in\mathcal{F}$.

If a form $\alpha$ is of top horizontal degree $q=n$, then there is a unique representative $P\alpha$ of its $d$-cohomology class, which has the following property: The form $\alpha$ is $d$-exact at $\varphi$ if and only if $P\alpha$ vanishes at $\varphi$. The map $P:\Omega^{p,n}(J^{\infty}F)\to\Omega^{p,n}(J^{\infty}F)$ is the cohomological version of partial integration and straightforward to compute. It is called the interior Euler operator. Forms in the image of $P$ are called “source” for $p=1$ and “functional” for $p>1$ [Tak79], [And89, Def. 2.5 and Ch. 3].

The map $E:=P\delta:\Omega^{p,n}(J^{\infty}F)\to\Omega^{p+1,n}(J^{\infty}F)$ is called the Euler operator. The source form $EL\in\Omega^{1,n}(J^{\infty}F)$ is called the Euler-Lagrange form. Since the interior Euler operator does not change the $d$-cohomology class, $EL=P\delta L$ and $\delta L$ are in the same $d$-cohomology class, i.e.

for some $\gamma\in\Omega^{1,n-1}(J^{\infty}F)$. The form $\gamma$ is called a boundary form.

From the properties of source forms it follows that a field satisfies the cohomological action principle if and only if it satisfies the Euler-Lagrange equation

In physics terminology, such a field is called on shell.

Let $\gamma$ be a boundary form. The form

of total degree $n$ will be called the Lepage form²²2For the terminology see [Kru83] or [And89, p. 199]. Deligne and Freed call $\lambda$ the “total Lagrangian” [DF99, p. 161].. Let the total differential of $J^{\infty}F$ be denoted by $\mathbf{d}=\delta+d$. The total differential of the Lepage form is

which is the premultisymplectic structure we are interested in.

On $J^{\infty}F$ we have the splitting of vector fields into a vertical and horizontal component which leads to the bigrading on the de Rham complex. Moreover, we have the acyclicity theorem of the variational bicomplex. This leads to the following description of hamiltonian vector fields.

In the proof, we will use the following lemma [Del18, Thm. 11.1.6].

A form $j\in\Omega^{0,n-1}(J^{\infty}F)$ is also called a current. A current is called conserved if it is $d$-closed on shell, i.e. if $dj$ vanishes at every solution of the Euler-Lagrange equation. Prop. 2.4 shows that the degree $(0,n-1)$ component of a hamiltonian form is a conserved current. In this sense, $L_{\infty}(J^{\infty}F,\omega)$ can be viewed as higher current algebra.

The $k$-bracket of hamiltonian forms $\alpha_{1},\ldots,\alpha_{k}$ is given by

where $X_{1},\ldots,X_{k}$ are the hamiltonian vector fields. The 2-bracket is given by

where $(X,\alpha)$ and $(Y,\beta)$ are hamiltonian pairs.

The diffeological tangent space of $\mathcal{F}$ is given by the space of sections of the vertical tangent bundle, $T\mathcal{F}\cong\Gamma(M,VF)$, so that a vector field on $\mathcal{F}$ is given by a map $\Xi:\Gamma(M,F)=\mathcal{F}\to T\mathcal{F}=\Gamma(M,VF)$. This map is called local if it is a differential operator, i.e. if there is a commutative diagram

where $VF=\ker(TF\to TM)$ is the vertical vector bundle. The map $\xi_{0}$ is often called an evolutionary “vector field”.

The map $\xi_{0}$ can be prolonged to a vector field $\xi$ on $J^{\infty}F$, which is the unique vector field such that

commutes. If such a commutative diagram exists, we will say that $\xi\in\mathcal{X}(J^{\infty}F)$ lifts to the vector field $\Xi\in\mathcal{X}(\mathcal{F})$. The following proposition is a purely algebraic characterization of such vector fields.

The kernel of $\Omega^{1,0}(J^{\infty}F)$ defines an integrable distribution on $TJ^{\infty}F$, called the Cartan distribution, which can be interpreted as a flat Ehresmann connection on $TJ^{\infty}F\to M$. The horizontal lift of a vector field $v\in\mathcal{X}(M)$ is denoted by $\hat{v}\in\mathcal{X}(J^{\infty}F)$. Since the connection is flat, the map $\mathcal{X}(M)\to\mathcal{X}(J^{\infty}F)$, $v\mapsto\hat{v}$ is a homomorphism of Lie algebras. The following is a purely algebraic characterization of such lifts.

The last two propositions can be understood geometrically as follows. Assume for the sake of argument that $\mathcal{F}$ is a finite dimensional manifold. The de Rham complex of $\Omega(\mathcal{F}\times M)$ has a bigrading with vertical differential $\delta$ in the direction of $\mathcal{F}$ and horizontal differential in the direction of $M$. A vector field $\xi$ on $\mathcal{F}\times M$ is the lift of a vector field on $\mathcal{F}$ if and only $[\iota_{\xi},d]=0$ and a lift of a vector field on $M$ if and only if $[\iota_{\xi},\delta]=0$. Props. 2.7 and 2.8 show that this characterization is valid also in the variational bicomplex. In order to emphasize this geometric interpretation, we will use for the purpose of this paper the following terminology:

The Lie derivatives of a strictly vertical vector field $\xi$ and of a strictly horizontal vector field $\hat{v}$ are given by

Recall that a form $j\in\Omega^{0,n-1}(J^{\infty}F)$ is also called a current. If there is a strictly vertical vector field $\xi$ such that

then $j$ is called a Noether current and $(\xi,j)$ a Noether pair [DF99, Def. 2.97]. Noether currents are conserved. Noether’s first theorem states that if $\xi$ is a Noether symmetry, then

is a Noether current. The proof is a half-line calculation,

which highlights the advantage of working in the variational bicomplex.

In [DF99, p. 169], a manifest symmetry was defined to be a vector field $X\in\mathcal{X}(J^{\infty}F)$ such that:

• (i)

  $X=\xi+\hat{v}$ is the sum of a strictly vertical vector field $\xi$ and a strictly horizontal vector field $\hat{v}$.

• (ii)

  $\mathcal{L}_{\xi+\hat{v}}(L+\gamma)=0$.

This suggests the following terminology:

The homotopy momentum map of a single vector field is split into a bidegree $(0,n-1)$ and a bidegree $(1,n-2)$ summand as

where

is the Noether current of $\xi_{v}$. In general, the map $\mu_{k}$ splits into a $(0,n-k)$ and a $(1,n-k-1)$ component given by the two lines of the right hand side of the equation

For example, we have

Using Eq. (6), we can write the $l_{2}$-bracket of the momenta as