Multisymplectic Constraint Analysis of Scalar Field Theories, Chern-Simons Gravity, and Bosonic String Theory

The (pre)multisymplectic geometry of the De Donder--Weyl formalism for field theories is further developed for a variety of field theories including a scalar field theory from the canonical Klein-Gordon action, the electric and magnetic Carrollian scalar field theories, bosonic string theory from the Nambu-Goto action, and $2+1$ gravity as a Chern-Simons theory. The Lagrangians for the scalar field theories and for $2+1$ Chern-Simons gravity are found to be singular in the De Donder--Weyl sense while the Nambu-Goto Lagrangian is found to be regular. Furthermore, the constraint structure of the premultisymplectic phase spaces of singular field theories is explained and applied to these theories. Finally, it is studied how symmetries are developed on the multisymplectic phase spaces in the presence of constraints.


Introduction
Covariant phase space formalisms for field theories have been relevant in theoretical physics since the development of the variational calculus for classical field theories. When a field theory under investigation is invariant under Lorentz transformations, it is possible to develop a manifestly Lorentz covariant construction of the variational principle and the symmetries of the Lagrangian. This is evident in the Lagrangian formulation of field theories in which the fields are, in general, tensors on spacetime and the spacetime coordinates are treated as indistinguishable independent variables. In the case of string theory, the fields are scalar fields on the string worldsheet and covariance is similarly maintained. One appropriate geometrical setting for constructing covariant phase spaces for field theories in the Lagrangian formalism is given by jet bundles which are fiber bundle over spacetime whose fiber coordinates consist of the fields and the spacetime derivatives of the fields. The covariant phase space manifold in the Lagrangian setting for first-order field theories ("first-order" in the sense that only first derivatives appear in the Lagrangian) is called the first-order jet bundle which will be referred to colloquially here as the multivelocity phase space, in analogy with the velocity phase space where the Lagrangian formulation of mechanical systems takes place. Thus, the coordinates on the first-order jet bundle, whose fibers over spacetime are the spacetime first derivatives of the fields, are called the multivelocities. In this paper only first-order field theories will be investigated.
Lorentz covariance in Hamiltonian formulations of Lorentz invariant field theories is much more subtle. The first attempt to develop a covariant Hamiltonian formalism was made in the 1930s by Théophile De Donder and Hermann Weyl [1,2] who generalized the variational calculus for the Hamiltonian formalism in mechanics to the multivariable context in field theory, giving rise to what is today referred to as the De Donder-Weyl approach in which the variational principle leads to the so-called (Hamilton)-De Donder-Weyl equations. The phase space manifold on which the De Donder-Weyl formalism for first-order field theories is constructed is called the dual first-order jet bundle and is referred to as the multimomentum phase space. The coordinates on this phase space are called multimomenta and are obtained from the Legendre map given as the partial derivatives of the Lagrangian with respect to all multivelocities. The De Donder-Weyl formulation of field theories has become better understood by learning about the underlying (pre)multisymplectic geometry [3,4] of the multivelocity and multimomentum phase spaces. (Pre)multisymplectic geometry in classical field theory is the geometric multivariable extension of (pre)symplectic geometry in classical mechanics where the physical observables of interest (position, velocity, momentum) depend on a single variable (time). Similarly to how to the simple (single-variable) variational calculus can be performed on (pre)symplectic phase spaces as done in classical mechanics, the multiple variational calculus can be posed geometrically on the (pre)multisymplectic phase spaces.
The De Donder-Weyl formalism, however, was hindered at the time of its development due to a lack of an understanding of constraints. A constraint analysis is needed to obtain the full dynamics of a field theory with a Lagrangian whose Legendre map is singular. In the De Donder-Weyl approach, the Legendre map is singular when the multi-Hessian constructed from the multivelocities is singular. The appropriate constraint analysis that is needed for such singular field theories was not understood at the time when the De Donder-Weyl approach was first developed. Instead, field theories were studied using the canonical Hamiltonian formalism which arises from performing a space+time splitting where, given an initial value Cauchy data surface, one solves a set of partial differential equations for dynamical functions which describe the time evolution of the classical fields of interest. This approach sacrifices the manifest Lorentz covariance exhibited by Lorentz invariant field theories in the Lagrangian formalism. However, the advantage at the time was that the (noncovariant) Dirac-Bergmann constraint analysis [5,6] gave the full dynamics of singular field theories ("singular" with respect to the Legendre map that gives the canonical momenta rather than the multimomenta used in the De Donder-Weyl approach). Furthermore, the canonical formalism gave a straightforward method for performing quantization, making the canonical formalism for field theories exceptionally useful.
The success of the canonical formalism in quantum field theory made it the standard Hamiltonian field theory formalism in theoretical physics. However, understanding how to develop the canonical formulation of field theories in a manifestly Lorentz covariant manner remained a mystery for many years until Crnković and Witten discovered how to construct a classical covariant phase space in which the canonical formulation (on-shell) can be carried out without having to perform a space + time splitting [7]. Soon after, Lee and Wald generalized this new covariant phase space formalism to also work off-shell in [8]. The Dirac-Bergmann constraint analysis makes use of Poisson brackets which are defined on symplectic phase space manifolds; the covariant phase space manifolds presented by Crnković, Witten, Lee, and Wald are infinite-dimensional (pre)symplectic manifolds it is therefore possible to carry out the geometric Gotay-Nester-Hinds constraint algorithm [9] which is equivalent to the Dirac-Bergmann constraint analysis. In fact, Lee and Wald specify in [8] how the covariant Poisson brackets of constraints corresponding to a local symmetry are related to the Lie algebra of the local symmetry under inspection. For a treatment of boundary terms in the covariant phase space formalism presented by Lee and Wald, see [10,11].
The difficulty with performing an analogous constraint analysis a la Dirac on the multivelocity and multimomentum phase spaces on which the De Donder-Weyl formalism takes place arises fundamentally from the fact that Poisson brackets on jet and dual jet bundles are not yet fully understood. A geometric algorithm for performing the desired covariant constraint analysis in the De Donder-Weyl formalism was developed in [12] where the premultisymplectic geometry of the multivelocity and multimomentum phase spaces is used to describe the constraint submanifolds on which field theories are physically relevant. This analysis is geometrically analogous to how the presymplectic geometry of the phase spaces in classical mechanics is used to perform the geometric constraint analysis of mechanical systems equivalent to the Dirac-Bergmann constraint procedure for mechanics developed in [9,13,14,15,16,17]. For classical field theories constructed on mdimensional spacetimes, the jet and dual jet bundles, i.e. the multivelocity and multimomentum phase spaces, are (pre)multisymplectic manifolds equipped with (pre)multisymplectic (m+1)-forms.
These differential forms multisymplectic for regular field theories and are premultisymplectic for singular field theories; the precise mathematical definitions of these forms will be given later. Furthermore, the symmetries of a field theory are Lie groups which act on the phase spaces in such a way that their (pre)multisymplectic forms are preserved and Noether's theorem yields conserved currents called covariant momentum maps which are (m − 1)-forms on the (pre)multisymplectic phase spaces. A general overview of (pre)multisymplectic geometry, including new geometry developed in this work, is provided in Section 2 of this paper. For reviews of the previously known aspects of the multisymplectic formulation of field theories see also, for example, [18,19,20]. The symplectic form on the covariant phase space introduced by Crnković and Witten is derived from the multisymplectic form on the multimomentum phase space in [21]. Furthermore, in [22,23] it is shown how performing a space + time splitting of the multimomentum phase space gives the standard canonical formalism for field theories along with a canonical constraint analysis (equivalent to the constraint analysis a la Dirac) in which covariant momentum maps play a crucial role.
In this paper, the (pre)multisymplectic geometry of the multivelocity and multimomentum phase spaces is used in a variety of field theories to develop: the field equations both in the Lagrangian and in the De Donder-Weyl Hamiltonian formulations, the geometric constraint analysis [12], and the covariant momentum maps [18] which give the conserved currents associated with Noether symmetries. The premultisymplectic constraint algorithm originally formalised in [12] is performed here using a local-coordinate approach which significantly simplifies the entire constraint algorithm; this exposition of the constraint algorithm along with some new properties of the constraints are presented in Section 2.4.
The specific field theories developed in this paper were chosen to point out certain subtle features of the geometric constraint analysis and the investigation of symmetries. This paper presents the first premultisymplectic treatment of Carrollian field theories [24,25,26,27]. It should be noted that the De Donder-Weyl formalism and its (pre)multisymplectic interpretation are not restricted to work only for Lorentz invariant theories; in fact, the Carrollian scalar field theories investigated in the premultisymplectic formalism in this paper are not Lorentz invariant. Carrollian field theories arise from taking the limit c → 0 for the speed of light as first introduced in [28,29]. Furthermore, the multisymplectic formulation of string theory and p-branes on the multimomentum phase space found in [30] is extended in this work by additionally providing the multisymplectic formulation on the multivelocity phase space and discussing the relevant symmetries on the multisymplectic phase spaces. The De Donder-Weyl formulation of string theory (and p-branes in general) can be found in [31] without mention of the underlying multisymplectic structure. Finally, this work reproduces the results found in the premultisymplectic treatment of Chern-Simons gravity done in [32] where the primary constraint submanifold of the multimomentum phase space arising from the De Donder-Weyl Legendre map is presented. This work additionally provides the full premultisymplectic constraint analysis of Chern-Simons gravity, on both the multimomentum and multivelocity phase spaces, where it is found that the field equations define the constraint submanifolds of the socalled Second Order Partial Differential Equation (sopde) type; these constraint submanifolds arise from imposing the sopde condition which guarantees that the final field equations are second order partial differential equations. The treatment of the symmetries of Chern-Simons gravity in the premultisymplectic context is also given. For additional references on the multisymplectic treatment of General Relativity, see [33,34,35,36]; for an alternative geometric description of the De Donder-Weyl formalism applied to gravity see, for example, [40] in which vector-valued forms called polysymplectic forms are used instead of the (pre)multisymplectic forms. The existence of boundaries on the multisymplectic phase spaces will not be considered in this work. For a detailed treatment of boundaries in the multisymplectic and other formalisms of field theories see, for instance, [37,38,39].
The organization of the paper is as follows: Section 2 provides a general overview of the (pre)multisymplectic setting used to formulate first-order classical field theories, including the lo-cal description of the constraint algorithm. Additionally, symmetries and conservation laws in the (pre)multisymplectic framework are reviewed in this section. Section 3 contains the premultisymplectic analysis of the canonical Klein Gordon Lagrangian and of the electric and magnetic Carrollian scalar field theories. All three scalar field theories exhibit different constraint and rigid symmetry structures. In section 4, the Nambu-Goto Lagrangian for bosonic strings and p-branes is found to be regular in the De Donder-Weyl sense and the symmetries of the Nambu-Goto action are presented in multisymplectic framework. Finally, the premultisymplectic treatment of gravity in 2 + 1 dimensions as a Chern-Simons theory [41] is presented in Section 5.
All the manifolds are real, second countable, and of class C ∞ . Manifolds and mappings are assumed to be smooth. Sum over crossed repeated indices is understood.

Preliminary Differential Geometry
This section reviews the foundations of the multisymplectic formulation of first-order field theories and basic concepts of symmetries and conservation laws; the constraint algorithm is also described in the premultisymplectic context.

Multivector Fields
Let E be an n-dimensional manifold and m < n; sections of Λ m (TE) := (m) TE ∧ . . . ∧ TE (the mth exterior product of TE) are called m-multivector fields on E and they are the contravariant skew-symmetric tensor fields of order m on E. (For more details on multivector fields and all the related topics presented here see, for instance, [42,43,44,45]). The set of m-multivector fields on E is denoted X m (E).
For every m-multivector field X ∈ X m (E) and p ∈ E, there exists an open neighborhood U p ⊂ Y and X 1 , ..., X r ∈ X(E) such that there is an open neighborhood U p ⊂ Y and X 1 , ...X m ∈ X(U p ) such that E| Up = X 1 ∧ ... ∧ X m . All multivector fields used in this paper are assumed to be locally decomposable. The contraction between multivector fields X ∈ X m (E) and differentiable k-forms ξ ∈ Ω k (E) is if k ≥ m, and i(X)ξ| Up = 0, if k < m. A distribution D of rank k on E is a subbundle of TE of rank k (and an m-dimensional distribution on E is an m-dimensional subbundle of TE). For every p ∈ E there is a linear subspace D p ⊆ T p Y such that D = p∈Y D p . Given a distribution D ⊆ TE, a nonempty immersed submanifold W ⊆ E is a an integral manifold of D if T p W = D p . The distribution D ⊆ TE is an integrable distribution if at every p ∈ E there is an integral manifold of D containing p. A multivector field X ∈ X m (E) and an m-dimensional distribution D on E are locally associated if there exists an open set U ⊆ E such that X| U is a section of Λ m D| U .
Locally decomposable m-multivector fields are locally associated to m-dimensional distributions. Multivector fields associated with the same distribution form an equivalence class {X} in X m (E). If X 1 , X 2 ∈ X m (E) are two nonvanishing multivector fields, both locally associated with the same distribution on the open set U ⊆ E, then there exists f ∈ C ∞ (U ) such that X 1 = f X 2 (on U ), and this is precisely the equivalence relation with equivalence class denoted {X} U . A multivector field is integrable if its locally associated distribution is integrable. An m-dimensional submanifold W ֒→ E is an integral manifold of Z ∈ X m (E) if, and only if, for every point p ∈ W , the multivector X p spans Λ m T p W . Now, let π : E → M be a fiber bundle with dim M = m and dim E = m + n. Vector fields on E are called π-vertical vector fields when they are tangent to the fibres of π. The set of such vector fields will be denoted as is the integral manifold of the multivector field X ∈ X m (E) at y, then φ is called an integral section of X. If a multivector field X ∈ X m (E) is integrable, then it is π-transverse if, and only if, its integral manifolds are local sections of π; that is, the local sections of π are integral sections of X. If M is an orientable manifold, the condition that a multivector field X ∈ X m (E) is π−transverse can be written as i(X)d m x = 0, where d m x ≡ dx 1 ∧ . . . ∧ dx m is the local coordinate expression (x µ ) on M (µ = 0, ..., m − 1) of the volume form. In particular, it is possible to take a representative X in the class of π-transverse multivector fields such that f = 1 so that The corresponding equivalence classes are called the 1-jets of φ at x, denoted j 1 x φ. Then, the first-order jet bundle It is a bundle over E and M whose natural projections are denoted π 1 : J 1 E → E andπ 1 : J 1 E → M , and dimJ 1 E = m+n+mn. The set of sections of π andπ 1 are denoted by Γ(π) and Γ(π 1 ) respectively. A section ψ ∈ Γ(π 1 ) is said to be a holonomic section if ψ = j 1 φ; that is, ψ is is a holonomic section if it is the first jet prolongation to J 1 E of a section φ ∈ Γ(π). A multivector field X ∈ X m (J 1 E) is said to be a holonomic multivector field if it is integrable and its integral sections are holonomic sections ofπ 1 . Observe that a multivector field X ∈ X m (J 1 E) is a holonomic multivector field if, and only if, (i) X is integrable, (ii) X isπ 1 −transverse, and (iii) the integral sections of X are holonomic.
Given coordinates (x µ , y A ) (A = 1, ..., n) on E adapted to the bundle structure, the induced coordinates on J 1 E are (x µ , y A , y A µ ). Then holonomic sections are written as which defines equivalence classes of multivector fields on J 1 E.
Integral sections of X satisfy Holonomic multivector fields are also called Second Order Partial Differential Equations (sopde).

Multisymplectic Lagrangian Field Theory
Multisymplectic geometry can be viewed as the field theoretic extension of the symplectic geometrization of classical mechanics (for more details on the multisymplectic Lagrangian formulation of first-order field theories see, for instance, [18,19,20,43,46,47,48,49]). The analysis begins by considering an orientable m-dimensional pseudo-Riemannian manifold M . The configuration manifold is taken to be a fiber bundle E over M with n-dimensional fibers with (surjective) projection map π : E → M : (x µ , y A ) → x µ . The fields under consideration are denoted y A (x) and are given by the local sections φ : M → E : x µ → (x µ , y A (x)) of π. The multivelocity phase space on which the Lagrangian formalism takes place is the first-order jet bundle J 1 E of π which has natural coordinates (x µ , y A , y A µ ) and hence dimJ 1 E = n + m + nm; J 1 E is a fiber bundle over E with projection map π 1 : J 1 E → E : (x µ , y A , y A µ ) → (x µ , y A ) and also a fiber bundle over M with projectionπ 1 : J 1 E → M : (x µ , y A , y A µ ) → x µ . Densities on J 1 E can be obtained by lifting the volume form d m x on M to J 1 E. The Lagrangian density is then written as L = L (x µ , y A , y A µ )d m x and L (x µ , y A , y A µ ) ∈ C ∞ (J 1 E) is referred to as the Lagrangian function. The Lagrangian energy E L ∈ C ∞ (J 1 E) is defined as The Lagrangian is said to be regular when the generalized Hessian matrix is non-singular everywhere. Furthermore, the bundle J 1 E comes equipped with an m-form called the m-Poincaré-Cartan form Θ L ∈ Ω m (J 1 E), given by and a closed (and hence locally exact) (m + 1)-form Ω L ∈ Ω m+1 (J 1 E), which is called the (m + 1)-Poincaré-Cartan form, given by . . ∧ dx µm , and so on.
The form Ω L is j-nondegenerate (1 ≤ j ≤ m) if, for every p ∈ J 1 E and Y ∈ X j (J 1 E), it follows that i(Y)Ω L | p = 0 ⇐⇒ Y| p = 0. When Ω L is 1-nondegenerate, it is referred to as being a multisymplectic form. This occurs when L (x µ , y A , y A µ ) is regular. Otherwise, when L is singular, Ω L is 1-degenerate and is referred to as being a premultisymplectic form (although here it will also be referred to as a degenerate multisymplectic form when this is the case).
The couple (J 1 E, Ω L ) is called a Lagrangian system. The field equations for this system are obtained from a variational action principle posed on J 1 E. The action, denoted Here η s is the flow of a vertical vector field on E compactly supported on Σ. For more details see [43,48,49]. Critical sections which are solutions to this variational problem can be characterized in the following equivalent ways: 1. (j 1 φ) * i(X )Ω L = 0, for every X ∈ X(J 1 E).
2. j 1 φ is an integral section of a class ofπ 1 -transverse, locally decomposable, holonomic, multivector fields {X L } ⊂ X m (J 1 E) which satisfy Furthermore, it is possible to choose a representative of the class {X L } which satisfies the normalized transverse condition (1): 3. Given a natural system of coordinates (U ; x µ , y A , y A µ ) on J 1 E, the first jet prolongations When the Lagrangian is regular, the field equations are compatible (and have solutions) on all of J 1 E. In the singular case they are, in general, not compatible on all of J 1 E and it is therefore necessary to implement a constraint algorithm in order to find a submanifold of J 1 E on which the field equations are compatible where consistent solutions exist. This constraint algorithm is explained in detail in Section 2.4.

Multisymplectic Hamiltonian Field Theory
For the De Donder-Weyl Hamiltonian formulation of first-order field theories (see, for instance, [18,19,44,50,51,52,53,54] for details), let Mπ ≡ Λ m 2 T * E be the bundle of m-forms on E vanishing by the action of two π-vertical vector fields. The manifold Mπ is called the extended multimomentum bundle and has local coordinates (x µ , y A , p µ A , p), hence dimMπ = m + n + mn + 1. Then, consider the quotient bundle J 1 E * := Mπ/Λ m 1 (T * E) (where Λ m 1 (T * E) is the bundle of π-semibasic m-forms on E), which is called the multimomentum bundle of E. The bundle J 1 E * has natural local coordinates (x µ , y A , p µ A ) and natural projections As Mπ is a subbundle of Λ m T * E (the multicotangent bundle of E of order m), Mπ is endowed with canonical forms which are known as the multimomentum Liouville m and (m + 1)-forms, whose local expressions are If L ∈ C ∞ (J 1 E) is a Lagrangian function, then the extended Legendre map, F L : J 1 E → Mπ, and the Legendre map, F L := σ • F L : J 1 E → J 1 E * , are locally given as The Lagrangian L is regular if, and only if, F L is a local diffeomorphism (this definition is equivalent to that given above). As a particular case, L is hyper-regular if F L is a global diffeomorphism. A singular Lagrangian L is almost-regular if P 0 := F L (J 1 E) is a submanifold of J 1 E * (where  0 : P 0 ֒→ J 1 E * denotes the natural embedding), F L is a submersion onto its image, and for every p ∈ J 1 E, the fibres F L −1 (F L (p)) are connected submanifolds of J 1 E.
If L is a hyper-regular Lagrangian, then the following diagram illustrates the full structure: The map constructed as h := F L • F L −1 is called the Hamiltonian section associated with the hyper-regular Lagrangian L . The differential forms defined as , are called the Hamilton-Cartan m and (m + 1) forms of J 1 E * associated with the Hamiltonian section h. If the form Ω H is 1-nondegenerate (and hence multisymplectic), then the Hamiltonian so F L * Θ H = Θ L and F L * Ω H = Ω L . For regular, but not hyper-regular, Lagrangians the construction is the same, but on the open sets F L ( The couple (J 1 E * , Ω H ) is called a regular Hamiltonian system when Ω H is multisymplectic and its corresponding field equations are obtained from a variational action principle on J 1 E * . The action, denoted is a functional on the set of sections Γ(τ ) given by Γ(M, J 1 E * ) → R : ψ → M ψ * Θ H . As in the Lagrangian formalism, the objective is to find sections that are critical (stationary): for variations ψ s = η s • ψ of ψ. Critical sections are characterized in the following equivalent ways: 2. ψ is an integral section of a class of integrable, locally decomposable,τ -transverse multivector fields {X H } ⊂ X n (J 1 E * ) which satisfy Furthermore, we can take as a representative of the class {X L } a normalizedτ 1 -transverse multivector field: 3. If (U ; x µ , y A , p µ A ) is a natural system of coordinates in J 1 E * , then ψ ∈ U ⊂ J 1 E * satisfies the Hamilton-De Donder-Weyl equations: For the almost-regular case, P 0 is a fibre bundle over E and M with natural projections τ 0 : P 0 → E andτ 0 := π • τ 0 : P 0 → M . The σ-transverse submanifoldP 0 = F L (J 1 E) ֒→ Mπ with natural embedding 0 :P 0 ֒→ Mπ is diffeomorphic to P 0 according to the restriction of the projection σ toP 0 given asσ :P 0 → P 0 . Takingh :=σ −1 , the Hamilton-Cartan (m + 1)-form is defined as This case is illustrated in the following diagram: is a non-regular Hamiltonian system. Now, the variational principle and the field equations are stated as in the regular case, but for sections Γ(τ 0 ) and the analog to the field equations (6) read As in the Lagrangian singular case, when (P 0 , Ω 0 H ) is a non-regular Hamiltonian system, in the best situations, the Hamiltonian field equations have consistent solutions only on a submanifold of P 0 which is obtained by using the constraint algorithm described in the next section 2.4.
The general geometric structure on which the field theories is developed is depicted in the diagram below:

Geometric Constraint Algorithm
As stated in the previous sections, it is compulsory to implement a constraint algorithm in the Lagrangian and the Hamiltonian formalisms of singular (almost-regular) field theories in order to find the submanifolds of the multivelocity and multimomentum phase spaces on which the field equations are compatible and where consistent solutions exist. The geometric algorithm for uncovering the intrinsic constraint structure in premultisymplectic systems is developed in detail in [12] and is a generalization of the algorithms stated for non-autonomous singular mechanics [13,14]. The constraint algorithm presented in [12] uses the characterization of the variational principle and the resulting field equations via multivector fields as discussed in the previous sections (i.e. equations (3) and (4) in the Lagrangian case, and (6) and (7) in the Hamiltonian case), as the multivector field characterization is the most suitable for its implementation.
This section presents a procedure for obtaining the constraints which define the submanifolds arising at each step of the constraint algorithm in a more practical manner than what is given in previous literature. It should be noted that the procedure described here is done in local charts as opposed to the intrinsic global description given in [12]. The coordinate description of the constraint algorithm will be carried out explicitly in each example field theory studied in this paper.
The general geometrical setting for the Lagrangian and Hamiltonian formalisms for singular field theories consists of taking a fiber bundle κ : F → M , where dim M = m > 1 and dim F = N + m (N > m), and M is an orientable manifold with a volume form whose pull-back to F is denoted ω ∈ Ω m (F ). Let Ω ∈ Ω m+1 (F ) be a premultisymplectic form on F satisfying the so-called variational condition given as Now, consider the problem of finding a submanifold  C : C ֒→ F and a locally decomposable m-vector field X ∈ X m (F ) along C such that where the second equation is the condition that X is κ-transverse, and can be written simply as i(X)ω|C = 1.
The algorithmic procedure consists of the following steps: • Compatibility conditions: First, look for the conditions called compatibility constraints which define the set C 1 ⊂ F where the field equations (10) have solutions. This set is assumed to be a submanifold C 1 ֒→ F . The compatibility constraints can be obtained from direct inspection of the local expression of the field equations (10).
• Tangency conditions: Now, given locally decomposable multivector fields X which are solutions to the field equations (10) on C 1 , ensure the stability of these solutions by imposing the stability or tangency condition of these multivector fields on C 1 . This is done by requiring that the components of X are tangent to C 1 ; that is, if X = µ X µ , for every compatibility constraints defining C 1 , ζ j = i(Zj)α ∈ C ∞ (F ), the stability condition is given by This procedure may produce a new constraint submanifold C 2 ֒→ C 1 ֒→ F ; if this is the case, then it is necessary to impose the tangency of X to C 2 as well. This procedure is reiterated until, in the most favorable cases (which include most physical field theories), no new constraints are produced and hence, a final constraint submanifold C ֒→ . . . C 1 ֒→ F is found such that the solutions X to the field equations are tangent to C. However, note that this procedure does not in general terminate for all field theories, but these cases will not be studied in this work.
• Remark: Recall that the variational principle in the Lagrangian formalism requires that multivector field solutions to the field equations on J 1 E are holonomic multivector fields. However, multivector fields which solve the field equations are not, in general, holonomic and it is therefore sometimes necessary to impose this condition after finding the compatibility constraint submanifold C 1 ֒→ J 1 E. Imposing the holonomy (sopde) condition may give rise to additional constraints which define a new constraint submanifold S 1 ֒→ C 1 ֒→ J 1 E; such constraints will be referred to here as sopde constraints. Then, as described above, the tangency condition for the sopde multivector fields X L ∈ X m (J 1 E) is imposed on all the constraints defining S 1 and subsequently also on any other resulting constraints until (in well behaved field theories) a final constraint submanifold is found.
It is important to note that the sopde condition does not apply in the Hamiltonian formalism and the algorithm is simply applied as described in the previous points.
Integrability conditions for X are typically imposed after the constraint analysis, however, this analysis will be unnecessary in the field theories studied in this work since they are integrable. Now that the geometric constraint algorithm has been described in detail, it is possible to establish various properties of constraints by means of a local analysis. The first of such properties presented here concerns compatibility constraints. As a consequence of the variational condition (9), the premultisymplectic form can be expressed as the following splitting: where Ω ∈ Ω m+1 (F ) and α ∈ Ω 1 (F ) are closed forms. In order to perform such a splitting globally, it is necessary to use an Ehresmann connection on the bundle F → M (although all the results are independent of this choice) [12,14,47]; however, since the field theories in this work are studied only locally, the connection on each local chart U ⊂ F induced by the natural connection on R N × R m → R m will always be used. It thereby follows that, in the Lagrangian and the Hamiltonian formulations of classical field theories, the (pre)multisymplectic forms are split in the following manner: , where F = J 1 E is the multivelocity phase space for the Lagrangian formulation of any generic field theory and F = P 0 ⊂ J 1 E * is the multimomentum phase space for the De Donder-Weyl Hamiltonian formulation of singular field theories. Then, as a consequence of splitting the (pre)multisymplectic forms as shown above, the following lemma and subsequent propositions hold: is a locally decomposable and κ-transverse multivector field on U , then then, denoting f ≡ i(X)ω and where f, g µ ∈ C ∞ (U ) are non-vanishing functions and the result follows.
In particular, taking i(X)ω = 1, it follows that f = 1 = g µ and i(X) where Ω satisfies the variational condition (9) and is written as the splitting (11). If C 1 ֒→ F is the maximal submanifold on which solutions to the field equations (10) exist (the compatibility constraint submanifold), then the following condition holds at p ∈ U ⊂ C 1 : ( Proof ) As C 1 is the maximal submanifold where the field equations are compatible, there exists a locally decomposable multivector X ∈ X m (U ) such that i(X)Ω = 0 and i(X)ω = 0. Then (12) holds and, as ker Therefore, for every Z ∈ ker Ω ∩ ker ω and for p ∈ U it follows that Thus, (13) is a necessary condition of compatibility of the field equations (10). In the particular case where m = 1 (that is, the non-autonomous mechanics) this is also a sufficient condition and then (13) is a geometrical characterization of all the compatibility constraints. Nevertheless, it remains under investigation to prove the converse of Proposition 1 in the general case where m > 1: the compatibility constraint submanifold C 1 is completely determined by (13).
As a final remark, in the Lagrangian and Hamiltonian formalisms of field theories, it is interesting to show the relation between the vector fields belonging to ker F L * and the primary Hamiltonian constraints arising from the Legendre map. This result is analogous to that verified for singular dynamical systems [55,56].
Let (J 1 E, Ω L ) be an almost-regular Lagrangian system. First observe that in natural coordinates of J 1 E the map F L * is represented by the matrix which annihilate the Hessian,

Finally:
Proposition 2 For every constraint ζ i ∈ C ∞ (J 1 E * ) which locally defines the (primary Hamilto- ( Proof ) As ζ i is a primary Hamiltonian constraint it follows that F L * ζ i = 0. Then, and equation (14) follows from the first equality above.

Multisymplectic Symmetries and Noether Currents
In the physics literature it is typical to work with the sections of the various fiber bundles that have been discussed. As mentioned earlier, the fields y A (x) are given by sections of the configuration bundle π : E → M . Furthermore, the transformations which are used to investigate symmetries are also typically constructed on the sections of the configuration bundle in the physics literature. The Lagrangian function is typically defined on the jet prolongations ) and the field transformations naturally produce transformations of the spacetime derivatives of the fields. The De Donder-Weyl formalism is similarly constructed on sections of J 1 E * (or P 0 ). In this paper, the physics is being studied on the configuration manifold E, the jet manifold J 1 E, and the dual jet manifold J 1 E * (or P 0 ) as opposed to working on the sections of those manifolds as it is usual in most physical literature. That is, one transforms y A which is a coordinate on some fiber bundle whose fibers give the field y A (x).
Noether's theorem can be formulated on the multivelocity and multimomentum phase spaces. Noether symmetries are regarded as transformations on the phase spaces which preserve the (pre)multisymplectic forms [18]. The conserved quantities on the multivelocity and multimomentum phase spaces associated with Noether symmetries are called covariant momentum maps which will be discussed later in 2.5.1. Gauge transformations are transformations on the configuration manifold E which do not induce transformations on M ; that is, the vector fields that generate gauge transformations are vector fields on E which are π-vertical. Gauge transformations whose liftings to J 1 E preserve the form Ω L are called gauge symmetries. Alternatively, lifted base space diffeomorphisms (such as spacetime diffeomorphisms or worldsheet diffeomorphisms in the case of string theory) are not referred to as gauge symmetries in this work.
Since physical symmetries preserve Ω L , and consequently Ω H as well, it is necessary to discuss how Lie groups act on the (pre-)multisymplectic multivelocity and multimomentum phase spaces. Afterwards, it will be shown how gauge transformations and base space diffeomorphisms are lifted to the multivelocity and multimomentum phase spaces.

Covariant Momentum Maps and Noether Currents
Covariant momentum maps are the conserved quantities which result from the group action of a Lie group on a multisymplectic manifold. Consider a multisymplectic phase space F with a multisymplectic (m + 1)-form Ω = −dΘ and a Lie group G with group action Φ on F written as and denote Φ g : F → F the corresponding diffeomorphisms induced by this action. If g is the Lie algebra of G, every ξ ∈ g induces a vector field on F with the use of some parameter λ which is used to write g ∈ G in terms of the exponential map The group G is a symmetry group of the multisymplectic phase space F if every g ∈ G generates a multisymplectomorphism (or a covariant canonical transformation) on F ; that is, a diffeomorphism Φ g : F → F such that Φ * g Ω = Ω, or equivalently: where L X ξ denotes the Lie derivative associated with X ξ . Furthermore, exact multisymplectomorphisms (or special covariant canonical transformations) are those which also preserve the form Θ ∈ Ω m (F ): Working with the infinitesimal generators of every ξ ∈ g amounts to J ξ to be linear in ξ as will be the case in the theories that are investigated in this paper. This makes it possible to define a linear map, J| p , by considering a point p ∈ F so that This structure makes it possible to construct another map, J, using J(p) ≡ J| p so that J : F → g * ⊗ Ω m−1 (F ) : p → J(p), for every p ∈ F . The map J is called the covariant momentum map, although this terminology is used to refer to both J and J| ξ . This setting can be concisely specified through the natural pairing ·, · between g * and g written as Working directly with equation (16) and Cartan's identity gives but also 0 = L X ξ dΘ = dL X ξ Θ , so (at least locally) L X ξ Θ = dα ξ for some α ξ ∈ Ω m−1 (F ). Then, equation (18) yields However, equation (16) only determines J ξ up to some exact form dβ ξ with β ξ ∈ Ω m−2 (F ). That is, equation (16) is still satisfied with the redefinition J ξ → J ξ − dβ ξ ; so, in order to keep full generality, the momentum map J ξ takes the form It is worth noting that, for symmetries generated by a vector field X ξ which is a lift from M to F , it follows that L X ξ Θ = 0 ⇒ α ξ = dγ ξ for some γ ξ ∈ Ω m−2 (F ). Exact forms such as dβ ξ and dγ ξ will be ignored in this paper as boundary terms will not be considered.
Furthermore, in the Lagrangian formulation of field theories where F = J 1 E, the covariant momentum maps on J 1 E are said to be in the Lagrangian representation and the Noether current j = j µ d m−1 x µ ∈ Ω m−1 (M ) of a symmetry is obtained by pulling back the associated momentum map J ξ ∈ Ω m−1 (J 1 E) to M , using the first jet prolongations j 1 φ as follows: j = j 1 φ * J ξ .
2.5.2 Lifting Transformations to J 1 E and J 1 E * Consider infinitesimal spacetime diffeomorphisms M → M produced by the coordinate transformation x ′µ = x µ + ξ µ (x). Such spacetime diffeomorphisms are generated by the vector field . Then, the π-projectable vector field ξ E ∈ X(E), which generates the corresponding transformations (x µ , y A ) → (x µ + ξ µ (x), y A + ξ A (x, y)) on the configuration manifold E with ξ A (x, y) ∈ C ∞ (E) can be written as Furthermore, the canonical lift of ξ E ∈ X(E) to J 1 E is given by the following expression [18,20]: The field transformations δy A (x), which are sometimes referred to as the local variation of the fields y A (x), are defined as the Lie derivatives of the fields y A (x) by the vector field ξ on M and take the form, where ξ A (x) = ξ A • φ. The first term in (22), −ξ µ (x) ∂y A ∂x µ (x), is sometimes referred to as the transport term while the second term, ξ A (x), is referred to as the global variation of the fields y A (x) and is given as Since the fields y A (x) are given by the local sections φ : M → E, the field transformations δy A (x) in (22) can be interpreted geometrically as the Lie derivative of the local sections φ defined as [18]: The variation of the spacetime derivatives of the fields, δy A µ (x), can also be characterized by the Lie derivative of local sections as follows: given holonomic local sections ψ = j 1 φ : M → J 1 E, the Lie derivative of ψ with respect to the vector field ξ = −ξ µ (x) ∂ ∂x µ ∈ X(M ) is defined as where X ξ ∈ X(J 1 E) is the canonical lift of ξ ∈ X(M ) to J 1 E given by (21). It thereby follows that It is important to point out that, sometimes, it is only possible to interpret the field variations δy A (x) as Lie derivatives of the jet prolongations j 1 φ; this may occur when the vector fields which generate Noether symmetries on the (pre)multisymplectic phase spaces are not projectable onto E and are not produced from canonical lifts from E to J 1 E. Such examples will come up in sections 3.1.3 and 3.3.3.
In the case of a gauge transformation, the vector field ξ E only has the component along ∂/∂y A , so the canonical lift to J 1 E is given by Canonical lifts from E to J 1 E satisfy the property that they preserve the Lagrangian density if, and only if, they preserve the Poincaré-Cartan form, i.e. L X ξ L = 0 ⇔ L X ξ Θ L = 0 [47,20]. The vector fields on J 1 E used to construct the covariant momentum maps on J 1 E are precisely canonical lifts and, for regular Lagrangians (i.e. when F L is a diffeomorphism), the vector fields used to construct the covariant momentum maps on J 1 E * are the push-forward by the Legendre map of the canonical lifts: Y ξ = F L * X ξ ∈ X(J 1 E * ). When the Lagrangian is singular (in particular, almost-regular), it is sometimes the case that the vector field X ξ ∈ X(J 1 E) is only projectable onto the multimomentum phase space on some constraint submanifold of the multivelocity phase space: Y ξ = F L 0 * X ξ | S⊂J 1 E ∈ X(P 0 ). This scenario occurs in both the Klein-Gordon field theory in Section 3.1 and the magnetic Carrollian scalar field theory in Section 3.3.

Scalar Field Theories
The scalar field theories presented in this section are formulated on an m-dimensional spacetime M and are constructed from two scalar fields, φ(x) and π(x), given by the sections of the configuration bundle E → M : (φ, π) → x µ with local sections denoted as φ : M → E : x µ → (x µ , φ(x), π(x)). Three different scalar field theories will be constructed on this geometric setup while the second and third field theories come from Carroll contractions of the canonical Klein-Gordon Lagrangian. For more details on Carrollian field theories, see for example [24,25,26,27].

Klein-Gordon From Geometric Constraints
In this section, spacetime, M , is taken to be m−dimensional Minkowski space with the Minkowski metric denoted as η µν with signature (− + . . . +).

Lagrangian Formulation
The scalar field theory investigated in this section is given by the canonical Klein-Gordon Lagrangian which originates from the canonical construction of the Klein-Gordon field on the canonical momentum phase space where the canonical momentum, π(x), is given by the Legendre map π(x) = ∂L (x)/∂φ(x). Conversely, in the construction of the canonical Klein-Gordon Lagrangian given here, π(x) is proposed simply as a second variational field while the notion of the canonical Legendre map is abandoned and the relation that is typically obtained from the canonical Legendre map instead shows up as a sopde constraint. The jet bundle which serves as the multivelocity phase space for this specific field theory has coordinates (x µ , φ, π, φ µ , π µ ) and it thereby follows that the first-order jet prolongations of sections φ : M → E have the expression Now, instead of writing the Lagrangian in terms of the fields themselves, the Lagrangian is written as a function on the multivelocity phase space, This Lagrangian is singular as it is directly evident from the Hessian matrix (with respect to the multivelocities), whose components are The null vectors associated with ∂ 2 L ∂φ µ ∂φ ν and respectively. The Lagrangian energy is and the Poincaré-Cartan forms are As L is a singular Lagrangian, Ω L is a premultisymplectic form. Now, take a locally decomposable multivector field which satisfies i(X L )d m x = 1. Then, the field equation (3) gives which, by setting differential forms separately equal to zero, produces the following equations Equation (29d) is a combination of the first three equations and is thereby and identity. Equation (29b) determines the coefficient A 0 , equation (29a) is a relation among the coefficients B 0 and C i i , and equation (29c) is the spatial part of the holonomic or sopde condition for the ∂/∂φ piece of the multivector field. As these equations show, no compatibility constraints appear. As it has been proven in Proposition 1, this can also be analyzed geometrically by enforcing (28). In this case and it follows that which agrees with the fact that no compatibility constraints are produced in this system. Now, imposing the sopde condition A µ = φ µ , B µ = π µ on X L (i.e. demanding that the multivector fields X L are holonomic), it follows that the relevant field equations left over are At this stage, the first equation (30a) gives another relation for the coefficients C i i . Equation (30b) is a sopde constraint which defines the constraint submanifold S 1 ֒→ J 1 E; this is a constraint giving a relation between the fields and multivelocities. Notice that the Lagrangian function (25) on the submanifold S 1 takes the form of the standard Klein-Gordon Lagrangian: As a final step, it is necessary to impose the tangency condition of the multivector fields X L to the submanifold S 1 : These are new relations among the coefficients of X L but are not constraints, therefore S 1 is the final constraint submanifold on which the constraint algorithm terminates. Now, upon taking (x µ , φ(x), π(x), φ ν (x), π ν (x)) to be integral sections of the multivector field X L by setting the field equations on S 1 become The combination of equations (33a) and (33b) gives the Klein-Gordon equation: Furthermore, the integral sections (32) which satisfy the field equations (33a) and (33b) satisfy the tangency condition on S 1 given by (31) particularly as a consequence of (33b).

Hamiltonian Formulation
The De Donder-Weyl Hamiltonian formulation takes place on the bundle J 1 E * which serves as the multimomentum phase space and has coordinates (x µ , φ, π, p µ φ , p µ π ), where the multimomenta p µ φ , p µ π are obtained from the Legendre map F L : The first equation above is invertible and gives the relation among the multimomenta π i and the multivelocities φ i and is therefore not a constraint. The second and third equations give the primary constraints p 0 φ − π = 0, p µ π = 0, which define the submanifold  0 : P 0 ֒→ J 1 E * with local coordinates (x µ , φ, π, p i φ ). Note that, by Proposition 2, the null vectors (26) are given as Furthermore, F L maps onto the primary constraint submanifold P 0 ⊂ J 1 E * as a consequence of the fact that the Lagrangian function L is almost-regular and the restricted Legendre map F L 0 is given by The De Donder-Weyl Hamiltonian H 0 ∈ C ∞ (P 0 ) obtained from E L = F L * 0 H 0 is given as, The Hamilton-Cartan forms obtained from (5) are Now, in order to produce the Hamiltonian field equations, take a locally decomposable multivector field X 0 H = X 0 ∧ X 1 ∧ · · · ∧ X m−1 ∈ X m (P 0 ) satisfying the normalized transverse condition i(X 0 H )d m x = 1, whose components are written as Then, the field equation i(X 0 H )Ω 0 H = 0 yields Similarly to the Lagrangian formalism, equation (35d) is a combination of the first three equations and is thereby an identity. Equations (35b) and (35c) determine the coefficients A 0 and A i respectively and equation (35a) is a relation among the coefficients B 0 and C i i . As before, no compatibility constraints appear; this can also be shown geometrically as in the Lagrangian formulation by searching for compatibility constraints via the application of Proposition 1: set i(Z)dH 0 = 0 for every Z ∈ ker Ω 0 H ∩ ker d m x. However, as it follows that condition (13) holds in agreement with the fact that there are no compatibility constraints (as one would expect from the Lagrangian formulation). Now, taking (x µ , φ(x), π(x), p i φ (x)) to be integral sections of X 0 H given by, and hence Plugging (37b) and (37c) into (37a) yieldsφ which is again the Klein-Gordon equation as expected and thereby displaying the equivalence between the Lagrangian and the Hamiltonian formalisms.

Symmetries
The Lagrangian density associated with the Lagrangian function (25), is invariant under Lorentz transformations Λ µ ν : . Let the Lorentz transformations on M be infinitesimal so that the Lorentz matrices take the form Λ µ ν = δ µ ν + ǫ µ ν , where ǫ µν is the infinitesimal antisymmetric Lorentz matrix. It follows that the spacetime coordinates transform as x ′µ = x µ + ξ µ (x) with ξ µ (x) = ǫ µ ν x ν and the resulting infinitesimal Lorentz transformations of the fields and their spacetime derivatives are given by the Lie derivatives of the first-jet prolongations j 1 φ : M → J 1 E with respect to the vector field ξ = −ξ µ (x) ∂ ∂x µ ∈ X(M ) as defined in (24): which give δL (x) = − ∂ ∂x α ǫ α β x β L (x) . The vector field which generates the Lorentz transformations above on J 1 E is It follows that L X Λ L = 0 holds everywhere on J 1 E while Also, observe that X Λ is tangent to the constraint submanifold S 1 ⊂ J 1 E given by the constraint φ 0 − π = 0 as desired since L X Λ (φ 0 − π) = 0. However, X Λ is not projectable onto E as a result of the component attached to ∂ ∂π ; it is for this reason that the transformations (38) can be interpreted as the Lie derivatives of j 1 φ only (as the sections φ do not provide the term associated with the second term of X Λ that is not projectable onto E). Now L X Λ Θ L | S 1 = 0, so the exact Noether symmetry exists only on the constraint submanifold S 1 . Then, the corresponding momentum map J L (X Λ ) = −i(X Λ )Θ L also exists only on the points of S 1 and is given by In the Hamiltonian formalism on P 0 ⊂ J 1 E * , the symmetry transformations are generated by Y Λ = F L 0 * X Λ | S 1 ∈ X(P 0 ) which is given as

Carrollian Electric Scalar Field Theory
The electric Carrollian contraction [26] of the canonical Klein-Gordon Lagrangian is performed by making the field redefinition given by φ(x) → cφ(x), π(x) → 1 c π(x), and taking the limit c → 0 for the speed of light. It follows that the Minkowski metric becomes degenerate as

Lagrangian Formulation
The Lagrangian function L ∈ C ∞ (J 1 E) obtained from the electric Carrollian contraction of the canonical Klein-Gordon Lagrangian is which is clearly singular since The null vectors of the Hessians ∂ 2 L ∂φ µ ∂φ ν and ∂ 2 L ∂π µ ∂π ν are respectively. The Lagrangian energy function and the Poincaré-Cartan forms are , Since the Lagrangian function (40) is singular, Ω L is a premultisymplectic form. As before, the field equations are obtained by using locally decomposable multivector fields X L = X 0 ∧ X 1 ∧ · · · ∧ X m−1 ∈ X m (J 1 E) such that i(X L )d m x = 1, so it follows that their components are given by Then the field equation is which leads to Equations (42a) and (42b) determine the coefficients B 0 and A 0 ; equation (42c) is an identity as a result of the previous equations as usual. There are no compatibility constraints, which is corroborated by applying Proposition 1: ker and i ∂ ∂φ µ dE L = 0 = i ∂ ∂π µ dE L is satisfied a priori without producing any constraints.
Imposing the sopde condition, A µ = φ µ , B µ = π µ , the field equations (42a) and (42b) produce two sopde constraints which define the constraint submanifold S 1 ֒→ J 1 E. The tangency condition for X L on S 1 is imposec by It is evident above that no new constraints arise, hence, S 1 is the final constraint submanifold in the Lagrangian formalism.
Consider now (x µ , φ(x), π(x), φ ν (x), π ν (x)) which are integral sections of the multivector fields X L and therefore satisfy (32). Then, the sopde constraints (43) yielḋ which can be combined givingφ = 0 , as in [24]. The tangency conditions (44) lead to These equations are fulfilled by the solutions to the field equations, in particular, as a consequence of the second equation in (45).

Hamiltonian Formulation
The multimomenta obtained from the Legendre map are and give the primary constraints p 0 φ −π = 0, p i φ = 0, and p µ π = 0, which define the primary constraint submanifold  0 : P 0 ֒→ J 1 E * with local coordinates (x µ , φ, π) and the restricted Legendre map F L 0 is given by F L =  0 • F L 0 as before. Once again, the primary constraints can be used to compute the null vectors (41) by using equation (14) given in Proposition 2. The Lagrangian function L is almost-regular and the De Donder-Weyl Hamiltonian H 0 ∈ C ∞ (P 0 ) obtained from and the Hamilton-Cartan forms are , The locally decomposable multivector fields X 0 H = X 0 ∧ X 1 ∧ · · · ∧ X m−1 ∈ X m (P 0 ) satisfying the normalized transverse condition i(X 0 H )d m x = 1 used to produce the field equations have the following components: The field equations obtained from i(X 0 H )Ω 0 H = 0 are Once again, using (46a) and (46b), the third equation (46c) is an identity. These equations do not produce any compatibility constraints. This again is in agreement with Proposition 1 which dictates that compatibility constraints may be produced by i(Z)dH 0 = 0, for every Z ∈ ker Ω 0 H ∩ ker d m x, and as ker Ω 0 is obtained computationally, it is confirmed that no compatibility constraints exist.
Upon working on (x µ , φ(x), π(x)) which are integral sections of the multivector fields X 0 H , and hence A µ = ∂φ ∂x µ , B µ = ∂π ∂x µ , it follows that the field equations (46a) and (46b) take the forṁ Notice that plugging in the second equation into the first above gives The equivalence between the Lagrangian and the Hamiltonian formalisms again is evident.

Symmetries
The spacetime symmetry of the Carroll spacetime geometry are the Carroll transformations [26]: where . These transformations can be represented infinitesimally by so that now b k = ǫ 0 k is infinitesimal. Up to linear terms in φ ′ (x ′ ) = φ(x) and π ′ (x ′ ) = π(x), the infinitesimal transformations of the fields, δφ( and similarly for the components of δπ µ (x). Under the transformations presented above, the Lagrangian transforms as δL (x) = − ∂ ∂x α ǫ α β x β L (x) . These transformations are produced by the Lie derivatives of the local sections φ : M → E and j 1 φ : M → J 1 E in the direction of a vector field ξ = −ξ µ (x) ∂ ∂x µ ∈ X(M ) which generates the Carroll transformations, where now ξ µ (x) = ǫ µ ν x ν is given by (48) and (49). The vector field which generates the Carroll transformations on the configuration bundle E, obtained from (23), is while the canonical lift of ξ E to J 1 E which generates the Carroll transformations on the multivelocity phase space is given by Evidently X is tangent to the constraint submanifold S 1 since L X (π − φ 0 ) = 0 and L X π 0 = 0. Finally, since L X L = 0 ⇒ L X Θ L = 0, the momentum map J L (X) corresponding to the Carroll spacetime transformations in the Lagrangian setting is given by In the Hamiltonian setting, which takes place on P 0 ⊂ J 1 E * , the momentum map J H 0 (Y ) ∈ Ω m−1 (P 0 ) is constructed using the vector field Y ∈ X(P 0 ) given by As in the Lagrangian setting, L Y Θ 0 H = 0, it follows that J H 0 (Y ) is given by,

Carrollian Magnetic Scalar Field Theory
The magnetic Carrollian contraction [26] of the canonical Klein-Gordon Lagrangian is performed by reinserting the factors of c (speed of light) into the Lagrangian (25) and taking the limit c → 0. The Minkowski metric (39) becomes degenerate as in the electric Carrollian scalar field theory.

Lagrangian Formulation
The Lagrangian function L ∈ C ∞ (J 1 E) obtained from taking the magnetic Carrollian contraction of the canonical Klein-Gordon Lagragian (25) is This Lagrangian is singular since the components of the Hessian matrix (with respect to the multivelocities) are The null vectors of the Hessians ∂ 2 L ∂φ µ ∂φ ν and ∂ 2 L ∂π µ ∂π ν are respectively. The Lagrangian energy function and the Poincaré-Cartan forms are now , and Ω L is a premultisymplectic form. Taking locally decomposable multivector fields X L = X 0 ∧ X 1 ∧ · · · ∧ X m−1 ∈ X m (J 1 E) satisfying i(X L )d m x = 1, their components are given as and the field equations obtained from i(X L )Ω L = 0 are Equations (51a) and (51b) are relations for the coefficients of the multivector fields while equation (51c) is an identity following from (51a) and (51b) as usual. There are no compatibility constraints in agreement with the geometric analysis presented in Proposition 1, which shows that dE L is satisfied without producing any constraints.
Then, upon imposing the sopde condition A µ = φ µ , B µ = π µ , equations (51a) and (51b) become C i i − π 0 = 0 , φ 0 = 0 . The first equation gives a relation for the coefficients C i i , and the second one is a sopde-constraint which defines the constraint submanifold S 1 ֒→ J 1 E. Now impose the tangency condition of the multivector fields X L to S 1 , These are not constraints, but new relations among the coefficient of X L and hence the final constraint submanifold in the Lagrangian formalism is S 1 . Then, on (x µ , φ(x), π(x), φ ν (x), π ν (x)), the integral sections of the multivector fields X L satisfy equations (32) and the field equations take the form For further details on these equations see [24,26]. Furthermore, from (52) it follows that, on S 1 , The equations above are fulfilled by the solutions to the field equations, in particular, as a consequence of the second equation in (53).

Hamiltonian Formulation
The Legendre map F L : J 1 E → J 1 E * gives the multimomenta It follows that p 0 φ − π = 0 and p µ π = 0 are primary constraints which define the primary constraint submanifold P 0 ֒→ J 1 E * with local coordinates (x µ , φ, π, p i φ ); consequently, the Lagrangian is again almost-regular. As above, the null vectors (50) are given by the partial derivatives of the primary constraints with respect to the multimomenta according to (14) in Proposition 2. The restricted Legendre map F L 0 is given by F L =  0 • F L 0 as usual.
The De Donder-Weyl Hamiltonian H 0 ∈ C ∞ (P 0 ) is given by and the Hamilton-Cartan forms on P 0 are Now take a locally decomposable multivector field X 0 H = X 0 ∧ X 1 ∧ · · · ∧ X m−1 ∈ X m (P 0 ) satisfying i(X 0 H )d m x = 1, so their components are given as The field equations obtained from i(X 0 H )Ω 0 H = 0 are: Equation (54d) is a combination of the equations (54a), (54b), and (54c) which give relations or determine some coefficients. Then, no compatibility constraints appear. Again, this is in agreement with the procedure given in Proposition 1: set i(Z)dH 0 = 0 for every Z ∈ ker Ω 0 H ∩ ker d m x; however, ker Ω 0 H ∩ ker d m x = {0} , and condition (13) from Proposition 1 holds.
Working on (x µ , φ(x), π(x), p i 0 (x)) taken to be integral sections of X 0 H , then (54a), (54b), and (54c) givė Plugging in the third equation into the first equation giveṡ Once again, the equivalence between the Lagrangian and the Hamiltonian formalisms is evident as the resulting field equations are shown to be equivalent.

Symmetries
The Carroll transformations for the magnetic scalar field theory are written slightly differently than how they are written for the electric scalar field 3.2. The Carroll transformations which are symmetries of the magnetic scalar field theory are Under these transformations the Lagrangian transforms as δL (x) = − ∂ ∂x α ǫ α β x β L (x) . Notice that the field π(x) no longer transforms as a Carrollian scalar; instead, π(x) transforms under a Carroll spacetime transformation −ǫ α β x β ∂π ∂x α plus a term −b i φ i (x) which cannot be represented on E. The field transformations behave geometrically as the Lie derivatives of the local sections j 1 φ : M → J 1 E. The vector field generating the Carroll transformations on J 1 E is which is tangent to the constraint submanifold S 1 ⊂ J 1 E, given by the constraint φ 0 = 0, since L X φ 0 = 0. Furthermore, although the vector field X leaves the Lagrangian density invariant (L X L = 0), it does not produce an exact Noether symmetry as Moreover, the vector field X as written above on all of J 1 E is not F L -projectable onto P 0 ⊂ J 1 E * . Instead, X on S 1 given by is F L 0 -projectable onto P 0 , giving Now, letting X be the local extension of X| S 1 to J 1 E whose coordinate expression is given by (55); then, it follows that L X Θ L = 0 on J 1 E. It is thereby possible to define the momentum maps as J L ( X) = −i( X)Θ L ∈ Ω m−1 (J 1 E) and J H 0 (Y ) = −i(Y )Θ 0 H ∈ Ω m−1 (P 0 ), giving:

Bosonic Strings and p-Branes
In the next sections, the multisymplectic formalism for bosonic p-branes is given by working out explicitly the case for string theory (p = 1) and then discussing the generalization to p > 1. The De Donder-Weyl Hamiltonian treatment for p-branes can be found in [31].
The fields of interest are the embeddings of a brane worldvolume in spacetime. Spacetime M is a smooth D = (d + 1)-dimensional manifold with local coordinates x µ (µ = 0, 1, ..., d) and spacetime metric G µν with signature (− + · · · +). The p-brane worldvolume Σ is a smooth manifold with dimΣ = m = p + 1 and local coordinates σ a (with a = 0, 1, ..., p). Given the embedding X : Σ → M : σ a → x µ (σ), the embedding maps x µ (σ) are taken to be fields on Σ. The configuration bundle E over Σ is taken to be the trivial bundle E = Σ×M and comes with a surjective projection map π : E → Σ as usual and sections of E are given by φ : Σ → Σ × M : σ a → (σ a , x µ (σ)). The coordinates on E are (σ a , x µ ), while on J 1 E and J 1 E * the local coordinates are (σ a , x µ , x µ a ) and (σ a , x µ , p a µ ) respectively. It follows that dimE = m + D and dimJ It is also worth noting that the first-order jet prolongations are given as j 1 φ : Σ → J 1 E : σ a → σ a , x µ (σ), ∂x µ ∂σ a (σ) . The Lagrangian density is given by where T p is the p-brane tension which has units of [T p ] = (mass)/(length) p and g = 1 2 g ab dσ a ∧ dσ b is a 2-form on Σ on J 1 E whose pullback by jet prolongations give the induced metric on Σ: The 2-form g can also be constructed on J 1 E * by pushforward of the Legendre map, that is the Hessian matrix is non-singular and the Lagrangian is regular thusly.
The full geometric setting is illustrated in the following figure: R L H

The String Lagrangian Formulation
The Lagrangian for the Nambu-Goto string (p = 1) is from which it follows that where g ba ≡ (g −1 ) ba = 1 detg ǫ bc ǫ ad g dc and so it follows that the Hessian can be written as The Lagrangian energy is where the last equality follows as g ab is a 2 × 2 matrix in the case of the string. Using (59) and (60), the Poincaré-Cartan 2-from and multisymplectic Poincaré-Cartan 3-form on J 1 E are The field equations, i(X L )Ω L = 0, can be computed using a representative of a class ofπ 1transverse and locally decomposable multivector fields X L ∈ X 2 (J 1 E), written as X L = X 0 ∧ X 1 , with components given by the following local expression from which it follows that i(X L )d 2 σ = 1. Then, i(X L )Ω L = 0 is given as Setting differential forms separately equal to zero produces the following equations: Notice that plugging (62) into (63) implies (61), which means that (63) is an identity as expected.
Equation (62) is the sopde condition for X L , i.e. B µ a = x µ a . Furthermore, (61) becomes the Euler-Lagrange equations when working with j 1 φ = σ a , x µ (σ), ∂x µ ∂σ a that are holonomic sections of X L which satisfy Working with such sections that satisfy (64) it follows that (61) becomes which are the Euler-Lagrange equations for this field theory.

The String Hamiltonian Formulation
The De Donder-Weyl Hamiltonian formalism is developed starting from the Legendre map F L : This Legendre map is invertible as expected from the regularity of the generalized Hessian (58). The De Donder-Weyl Hamiltonian is defined as This Hamiltonian can be expressed in terms of the variables on J 1 E * as [31] Π ab ≡ G µν p a µ p b ν ⇒ F L * Π ab = −T 2 detg g ba ⇐⇒ F L * detΠ = (−T 2 detg) 2 det(g −1 ) = T 4 detg , from which it follows that The Hamilton-Cartan 2-form and the multisymplectic Hamilton-Cartan 3-form on J 1 E * are The field equations, i(X H )Ω H = 0, can be computed using a representative of a class ofπ 1transverse locally decomposable multivector fields X H = X 0 ∧ X 1 ∈ X 2 (J 1 E * ), with components given by the following local expression Then, Setting differential forms separately equal to zero produces the following field equations: Note that plugging (68b) into (68c) implies (68a), which means that (68c) is an identity as expected. Furthermore, the variational problem on J 1 E * is solved by integral sections ψ(σ) = (σ a , x µ (σ), p a µ (σ)) of X H for which D µ a = ∂x µ ∂σ a , H c aµ = ∂p c µ ∂σ a , and so the field equations become which are the corresponding Hamilton-De Donder-Weyl equations for the bosonic string; these equations can be plugged into another one to give the Euler-Lagrange equations as usual.

Worldsheet Diffeomorphisms
Consider worldsheet diffeomorphisms produced bỹ It follows that The vector field ξ ∈ X(Σ) which generates the worldsheet diffeomorphisms (69) is given by The lift of the vector field ξ ∈ X(Σ) to E = Σ × M , obtained from (23), is and the canonical lift of ξ E to J 1 E gives The resulting field variation given by the Lie derivative (23) of the local sections φ : Σ → E by the vector field ξ is

It follows that
so the covariant momentum map is given as In the Hamiltonian formalism, the vector field Y ξ = F L * X ξ ∈ X(J 1 E * ) which takes the form generates the worldsheet diffeomorphisms on the multimomentum phase space J 1 E * and is used to construct the covariant momentum map

Spacetime Isometries
Spacetime symmetries, which arise from performing transformations on the coordinates of M , are gauge symmetries in string theory since they are generated by vector fields on the configuration manifold E which are vertical with respect to the projection onto the base space Σ.
Consider infinitesimal spacetime diffeomorphisms x µ → x µ + ζ µ (x) generated by the vector field which is a gauge transformation as the corresponding vector field ζ E ∈ X(E) is π-vertical: The canonical lift to J 1 E, given by the jet prolongation of ζ E , is where L ζ G µν is the Lie derivative of G µν on M with respect to ζ = −ζ µ ∂ ∂x µ ∈ X(M ). Then, when ζ is a Killing vector field, and it therefore follows that the covariant momentum map is given by On the Hamiltonian side, so the covariant momentum map is given by

Generalization to p-Branes (p >1)
The generalization to p-branes (m = p + 1) is straightforward to perform by using the general Lagrangian density (56) for which the corresponding Lagrangian energy E L ∈ C ∞ (J 1 E) is where g ab is now an m × m matrix, so g ba g ab = m = p + 1. Then Proceeding similarly as above, take an m-multivector field X L = X 0 ∧ X 1 ∧ · · · ∧ X m−1 , with components to obtain the field equations, one of which is an identity, another gives the sopde condition B µ a = x µ a , and the third equation upon plugging in the sopde condition is which becomes the Euler-Lagrange equations when working on integral sections of X L : The Hamiltonian formulation is also carried out as in the string case with the Legendre map F L which, for p-branes, is Next, define the matrix from which it follows that The inverse matrix Π ab is given by the following general version of the identity (66): Then the De Donder-Weyl Hamiltonian for the p-brane theory is The De Donder-Weyl field equations are: however, the Hamiltonian is computationally more tedious to deal with than in the case of the string.
The multisymplectic symmetries are now worldvolume diffeomorphisms and spacetime isometires. The worldvolume diffeomorphsims are generated on J 1 E by the vector field and the corresponding momentum map Furthermore, as in the string theory case, the spacetime isometries which produce symmetries of the multisymplectic form Ω L are generated on J 1 E by the following vector field: The corresponding momentum map J L = −i(X ζ )Θ L ∈ Ω m−1 (J 1 E) is given as,

Lagrangian Formulation
The configuration bundle E has coordinates (x µ , e a µ , ω c ρ ); the induced coordinates on J 1 E are (x µ , e a µ , ω c ρ , e a σµ , ω c σρ ) and j 1 φ : M → J 1 E : x µ → x µ , e a µ (x), ω c ρ (x), ∂ σ e a µ (x), ∂ σ ω c ρ (x) are the first-order jet prolongations. The Lagrangian density is given by L = L (x µ , e a µ , ω c ρ , e a σµ , ω c σρ )d 3 x = ǫ µνρ 2η ac e a µ ω c νρ + ǫ abc e a µ ω b ν ω c ρ + It is well-known that this Lagrangian density can be written as a Chern-Simons theory [41], with gauge field A = e a µ P a + ω a µ J a dx µ , where J a = 1 2 ǫ abc J bc are the Lorentz generators in the dual form and P a are the translation generators which satisfy The invariant bilinear form on the Lie algebra of the gauge group is, in general, given as This field theory is singular which can be seen directly from the following components of the Hessian matrix: The null vectors of the Hessian matrix above are given as respectively. The null vector above is expressed as a column vector in the latin (locally Minkowski tangent space) upper index while, at each row of the column vector, the 3 × 3 matrices are indexed by the roman lower indices.
The corresponding Lagrangian energy E L ∈ C ∞ (J 1 E), the Poincaré-Cartan 3-form Θ L ∈ Ω 3 (J 1 E), and the premultisymplectic Poincaré-Cartan 4-form Ω L ∈ Ω 4 (J 1 E) are given by The field equations are obtained using multivector fields written as X L = X 0 ∧ X 1 ∧ X 2 ∈ X 3 (J 1 E) with components given by The geometric field equation (3) gives, Setting differential forms separately equal to zero produces the following two independent field equations: ǫ µνρ η ac B a νµ − ǫ abc e a µ ω b ν = 0 . Nevertheless, solutions to the field equations must be holonomic multivector fields X L which thereby have components (72) for which B a νµ = e a νµ and C c νρ = ω c νρ ; it follows that the field equations (73a) and (73b) become: Both of these equations are sopde constraints and define the submanifold S 1 ֒→ J 1 E. As usual, it is necessary to ensure the tangency of the multivector field X L to the submanifold S 1 . Recall that this is done by taking the Lie derivative of the constraints with respect to the multivector field components (72), where B a νµ = e a νµ and C c νρ = ω c νρ , giving: These equations are relations for the coefficients D a σνµ and H c σνρ and are not constraints; it follows that there are no tangency constraints in this field theory.
Furthermore, the integral sections (x µ , e a µ (x), ω c ρ (x), e a σµ (x), ω c σρ (x)) of these holonomic multivector fields satisfy that B a νµ = ∂ ν e a µ and C c νρ = ∂ ν ω c ρ , and therefore the equations (73a) and (73b) on S 1 become: These are the well-known field equations for gravity in 2 + 1 dimensions; the first equation is Einstein's equation with cosmological constant while the second guarantees that the spin connection is torsion free, thereby fully specifying the spin connection in terms of the dreibein e a µ .

Hamiltonian Formulation
The Legendre map F L : J 1 E → J 1 E * associated to L gives the multimomenta The relations above give the constraints p µν a = 0 and π µν c − 2ǫ µνρ η ab e b ρ = 0, which define the primary constraint submanifold  0 : P 0 ֒→ J 1 E * on which the Hamiltonian formulation takes place. As expected from Proposition 2, the null vectors (71) are given by the partial derivatives of the primary constraints with respect to the multimomenta according to (14): where τ a (x) are the infinitesimal generators of local Lorentz transformations and ρ a (x) are the infinitesimal generators of local translations. It follows from the Lie algebra (70) that δe a µ (x) = − ∂ρ a ∂x µ − ǫ abc e b µ τ c − ǫ abc ω b µ ρ c , δω a µ (x) = − ∂τ a ∂x µ − ǫ abc ω b µ τ c − λǫ abc e b µ ρ c .
The vector field on the configuration bundle E which generates the local Poincaré transformations is The canonical lift of ζ E to J 1 E is The momentum map J L (X ζ ) ∈ Ω 2 (J 1 E) on the multivelocity phase space is given by

Conclusions and Outlook
In this paper, a geometric approach has been used for performing a full constraint analysis of field theories on both the multivelocity and multimomentum phase spaces where the Lagrangian and De Donder-Weyl Hamiltonian formalisms take place respectively. The multivelocity and multimomentum phase spaces are, respectively, the jet and dual jet bundles of the field configuration manifold E which is regarded as a fiber bundle over spacetime (or over the string worldsheet in the case of string theory). These phase spaces are equipped with multisymplectic or premultisymplectic forms which are then used to produce the field equations of the theory under investigation, reveal the symmetries of the field theory, and carry out a full constraint analysis of the singular field theories.
The approach for carrying out the premultisymplectic constraint analysis of field theories involves the use of multivector fields to represent geometrically the solutions of field equations. So, an easy procedural geometric technique for finding the constraints locally has been described, and some new properties of the constraints are also exposed (Propositions 1 and 2). The converse of Proposition 1, which states that the compatibility constraint submanifold C 1 is completely determined by (13), is left for further research along with other results and features regarding the geometric constraint algorithm.
Furthermore, it is shown how further constraint submanifolds (defined by the so-called sopde constraints) may be found in the Lagrangian formalism when imposing the holonomic condition which guarantees that the Lagrangian field equations are second-order partial differential equations. The field equations, i.e. the well-known Euler-Lagrange equations in the Lagrangian case and the Hamilton-De Donder-Weyl equations in the Hamiltonian case, are obtained from the (pre)multisymplectic variational principle in which multivector fields of a particular type serve as solutions to the field equations. By example, in the field theories analyzed in Sections 3.1 and 3.3, it is shown that when sopde constraints exist, the vector fields which generate symmetries of the field theory under investigation may be projectable via the Legendre map only on the sopde constraint submanifold. The geometric constraint analysis finalizes by imposing stability of the solutions to the field equations; this is done by demanding tangency of the solution multivector fields to all constraint submanifolds present in the system until no new constraint submanifolds are produced. It is also worth noting that this work presents a new proven proposition, Proposition 2, which states that the multi-Hessian (2) (which characterizes the Legendre map to the multimomentum phase space) has null vectors which are given by the partial derivative of the primary constraints with respect to the multimomenta.
The technique used in this paper for performing the geometric constraint analysis described above is illustrated in various field theory examples, each of which highlight different aspects of the geometric constraint structure. The scalar field theories presented in this work include a new approach to the study of the canonical Klein-Gordon Lagrangian as well as the first premultisymplectic analysis of Carrollian scalar field theories. The multisymplectic treatment of bosonic string theory from the Nambu-Goto action is given along with the generalization to the theory of bosonic p-branes. The regularity of the Nambu-Goto action with respect to the De Donder-Weyl Legendre map is shown geometrically by inspection of the multisymplectic forms that characterize the theory's multivelocity and multimomentum phase spaces. The connection between the regularity of bosonic string theory in the De Donder-Weyl formalism and its singular behavior in the canonical formulation remains unexplored as, in general, no direct relationship has yet been established between the premultisymplectic constraint analysis presented in this work and the canonical constraint structure arising from the canonical Legendre map. Understanding the connection between the constraint structures in these two formalisms is left for future research. Finally, the premultisymplectic construction of Chern-Simons gravity in 2 + 1 dimensions is given and it is shown that in the Lagrangian formalism there are only sopde constraints present, while in the De Donder-Weyl Hamiltonian formalism there are only primary constraints which are produced by the singular Legendre map.
It is interesting to note that General Relativity is singular (in the De Donder-Weyl sense) in any spacetime dimension. This was previously shown in 3 + 1 dimensions in [33,34,35,36] and here it was shown in 2 + 1 dimensions. The analysis of General Relativity given here and in [33,36] involves treating the connection as an independent variational field; in this setting, the Einstein-Hilbert Lagrangian is singular as it is linear in the multivelocities, thereby leading to a singular multi-Hessian (2). When only the metric is taken to be the fundamental field of the theory (as the connection is assumed to be Levi-Civita), the Einstein-Hilbert Lagrangian is singular for similar reasons as shown in the premultisymplectic setting in [34,35]. No complications arise in generalizations of higher dimension as the singular structure of the relevant Hessians is independent of spacetime dimension. Similarly, theories of massive gravity, multi-gravity, and all theories of the Chern-Simons type (including higher-spin gravity in (2 + 1) dimensions) are De Donder-Weyl singular due to linear dependence of the corresponding Lagrangians on the multivelocities.
Given that the constraint structure of field theories in the De Donder-Weyl formalism is only understood geometrically at this point in time, it would be interesting to develop an algebraic understanding of the constraints a la Dirac. Such a treatment of constraints requires the construction of Poisson brackets on some phase space. In the canonical formalism for field theories, Poisson brackets are well understood in the context of symmetries and constraints on the covariant phase space by Lee and Wald [8] for example; for further insight into such covariant phase spaces (and related ones) see [58]. In the De Donder-Weyl formalism however, the construction of Poisson brackets it still a topic under investigation with many papers published on the matter (see, for example, [21,59,60,61,62,63] and references therein). Covariant Poisson brackets on jet bundles (where the Lagrangian formulation of field theories takes place) has also been discussed extensively in the literature (see for example [64] and references therein) along with the development of the BRST-BV formalism [65]. However, the understanding of the BRST-BV formalism in terms of the (pre)multisympletic structures on jet bundles is not yet fully understood and is left for further research; we also leave gauge fixing in the De Donder-Weyl formulation of the field theories studied in this paper for future work.