A hydrodynamical homotopy co-momentum map and a multisymplectic interpretation of higher order linking numbers

In this article a homotopy co-momentum map (\`a la Callies-Fr\'egier-Rogers-Zambon) trangressing to the standard hydrodynamical co-momentum map of Arnol'd, Marsden and Weinstein and others is constructed and then generalized to a special class of Riemannian manifolds. Also, a covariant phase space interpretation of the coadjoint orbits associated to the Euler evolution for perfect fluids and in particular of Brylinski's manifold of smooth oriented knots is discussed. As an application of the above homotopy co-momentum map, a reinterpretation of the (Massey) higher order linking numbers in terms of conserved quantities within the multisymplectic framework is provided and knot theoretic analogues of first integrals in involution are determined.


Introduction
In this paper we discuss some applications of multisymplectic techniques in a hydrodynamical context. The possibility of applying symplectic techniques therein ultimately comes from Arnol'd's pioneering work culminating in the geometrization of fluid mechanics ( [2,1,3,30]). In particular, in this connection we may mention the paper [38], with its symplectic reinterpretation [34,35,36], and the general portrait depicted in [6]. Here we wish to apply some recently emerged concepts in multisymplectic geometry (mostly building on [7,44,43]) and construct an explicit homotopy co-momentum map ([7]) in a hydrodynamical setting, leading to a multisymplectic interpretation of the so-called higher order linking numbers, viewed à la Massey ( [37,46,23]). The costruction is generalized to cover connected compact oriented Riemannian manifolds having vanishing intermediate de Rham groups. Moreover, a covariant phase space intepretation of the multisymplectic setting is outlined.
We make clear from the outset that our constructions, together with the covariant phase space portrait, will not adhere to the standard multisymplectic approach to continuum mechanics set forth e.g. in [19,31] but they will be based instead on the peculiar structure of an ideal fluid, whose configuration space is the "Lie group" of diffeomorphisms preserving a volume form, which will be directly taken as a multisymplectic form ( [8]).
The layout of the paper is the following. First, in Section 2, we give an example of homotopy co-momentum map in fluid mechanics -in the sense of Callies-Frégier-Rogers-Zambon (CFRZ) ( [7]) -transgressing to Brylinski's symplectic structure on loop spaces and descending, in turn, to the manifold of smooth oriented knots, see [6,4] and below for precise definitions. We briefly discuss the (non) equivariance of the above construction with respect to the group of volume preserving diffeomorphisms of 3-space (see Section 2) and we outline a generalization thereof in a Riemannian framework, signalling potential topological obstructions. Moreover, covariant phase space aspects will be analyzed. In Section 3 we prepare the ground for the forthcoming applications by depicting a hydrodynamical multisymplectic portrait of basic knot theoretic objects, used, in Section 4, to reinterpret the Massey higher order linking numbers in multisymplectic terms: the 1-forms appearing in the hierarchical Massey construction (viewed, in turn, differential geometrically à la Chen) provide an example of first integrals in involution in a multisymplectic framework. The last section is devoted to gathering together the conclusions and to pointing out possible directions for further research. Appropriate background material is provided within the various sections in order to ease readability. This paper is an improved version of part of the preprint [32].

Multisymplectic geometry and hydrodynamics of perfect fluids
In the present section we freely use basic material on symplectic and multisymplectic geometry tailored to our subsequent needs, prominently referring, for additional details, to [35,46,47] for the former and to [44,43] for the latter. For general background on symplectic geometry and (co)momentum maps we quote, among others [1,21,3].

Tools in multisymplectic geometry
All our objects will be smooth, unless differently specified. A (finite dimensional) multisymplectic manifold (M , ω) is a manifold (connected, for simplicity) equipped with a closed (n+1)-form ω (called multisymplectic form or n-plectic form) such that the map α below sending vector fields to n-forms (via contraction) is injective ( [8]). Dropping the last condition leads to the concept of pre-n-plectic form. The n = 1 case retrieves (pre)symplectic manifolds.
In the multisymplectic context, the generalization of the (co)momentum maps of the symplectic case leads to the more refined concept of homotopy co-momentum map, to be presently succinctly reviewed. One first introduces the so-called Roger's Lie n-algebra of observables L ∞ (M , ω) ( [40]). Referring to [40,44,43,7,16] for a full coverage of the relevant apparatus, not needed to full extent here, we just point out that the latter is a graded vector space L whose degree i pieces read together with suitable multilinear maps denoted collectively by . The suffix "Ham" refers to the Hamiltonian (n-1)-forms, i.e. those forms H such that for a vector field X preserving ω (i.e. L X ω = 0), called, in turn, a Hamiltonian vector field pertaining to H . A form β is said to be strictly (resp. globally, resp. locally) conserved by an ωpreserving vector field X if L X β = 0 (resp. L X β is exact, resp. closed). Cartan's formula immediately shows that closed forms are globally conserved; indeed, for such a form Recall, from [44], that a homotopy co-momentum map is an L ∞ -algebra morphismstemming from what is called an infinitesimal action of g on M (with g being the Lie algebra of a generic Lie group G, acting on M by ω-preserving vector fields) given explicitly by a sequence of linear maps fulfilling f 0 = f n+1 = 0 (we have tacitly set Λ −1 (M ) = 0) and together with (for p ∈ Λ k (g)): 2) (k = 1, . . . n + 1). We explain the notation: (withˆdenoting deletion as usual and with ∂ 0 = 0; one has ∂ 2 = 0). Formula (2.2) tells us that the closed forms must actually be exact, with potential − f k (p). Closure can be quickly ascertained as follows (in view of Lemma 2.16 in [44] and keeping in mind that d ω = 0): Notice that the special case k = n + 1 asserts that the function µ n+1 (·) is constant, and its value is fixed by the condition This can be rephrased, upon resorting to Section 9 in [7] (we use a different notation), by asserting that the following g-(n+1)-cocycle in the Chevalley-Eilenberg ought to be a boundary: for a fixed but generic point x ∈ M (the class [c x ] being in general independent of x ∈ M , [7], Cor. 9.3); the operator δ C E is the CE-differential defined by duality: and γ is a path connecting x to x (recall that M is assumed to be connected). We shall resume the above discussion in Subsection 2.3.

The hydrodynamical Poisson bracket
In the present Subsection we briefly review, for motivation and further applications, the symplectic geometrical portrait underlying the theory of perfect fluids, in its simplest instance. We denote by g the (infinite dimensional) Lie subalgebra of X(R 3 ) consisting of the divergence-free vector fields on R 3 that is the "Lie algebra" of the "Lie group" G = sDiff (R 3 ) of volume preserving diffeomorphisms of R 3 . As it is often done, we shall gloss over analytic subtleties (see e.g. [2,3,13,28] for more information). We just recall here that G is a regular Lie group in the sense of Kriegl-Michor ( [28], 38.4) and that its associated exponential map is not even locally surjective (a quite general phenomenon). We shall also tacitly assume that our fields rapidly vanish at infinity, so that convergence problems are avoided and boundary terms are absent. The "hydrodynamical" Lie bracket, equalling minus the standard one: [ξ 1 , ξ 2 ] = curl(ξ 1 × ξ 2 ) will be employed throughout.
Also, following e.g. [3], we shall consider the so-called regular dual g * of g consisting of all 1-forms modulo exact 1-forms: together with the standard pairing (ω ∈ g * , ξ ∈ g) Nevertheless, we shall feel free to use suitable genuine distributional elements as well (i.e. currents, in the sense of de Rham, [12]) from the full topological dual (without introducing new notation for the latter). Everything will be clear from the context.
The (regular) dual g * is naturally interpreted as a Poisson manifold with respect to the hydrodynamical Poisson bracket (Arnol'd -Marsden-Weinstein Lie-Poisson structure) , see e.g. [3,29,30,34,35,36,47]: with v ∈ g (velocity field), w := curl v, its vorticity, with [v] denoting the "gauge" class of v: The Euler evolution, reading, among others, in the so-called vorticity form ∂w ∂t is volume preserving and it also preserves the symplectic leaves of g * given by the : with the coadjoint action reading, explicitly, up to a gradient (not influencing calculations) The Hamiltonian algebra Λ pertaining to O w consists of the so-called Rasetti-Regge currents originally introduced in [38] and further developed in [34,35,36,47,6]): namely, the map . The preceding portrait carries through to the singular vorticity case, in particular when the vorticity field is δ-like and concentrated on a two-dimensional patch, filament or a loop. Dealing with the latter case, we ultimately retrieve the Brylinski manifold Y consisting of smooth oriented knots (smooth embedded loops modulo orientation-preserving reparametrizations) together with its symplectic structure Ω Y and the original Rasetti-Regge currents (see [6,4] for more details): Indeed, recall that, given a volume form ν on a 3-dimensional M , one gets, by transgression, a 2-form Ω on LM via the formula where ev : LM × S 1 → M given by ev(γ, t ) := γ(t ) is the evaluation map of a loop γ ∈ LM at a point t ∈ S 1 ≡ [0, 1]/ (endpoint identification). More explicitly, given tangent vectors u and v at γ, the symplectic form reads The above construction carries through to Y . In this case, the coadjoint orbits are labelled by the equivalence types of knots (via ambient isotopies), by virtue of a result of Brylinski, see [6].

A hydrodynamical homotopy co-momentum map
In this Subsection we elaborate on the previous discussion by introducing an explicit homotopy co-momentum map (HCMM), departing from the standard setting since our group G = sDiff (R 3 ) is infinite dimensional. We start from the observation ( [8]) that the volume form in R 3 , ν := d x ∧ d y ∧ d z can interpreted as a multisymplectic form: in this case the map α is bijective (in particular, injective). In coordinates, if ξ = (ξ i ), then Upon introducing the Hodge * relative to the standard Euclidean metric and the associated "musical isomorphisms", we have (ξ ∈ X(R 3 ), β ∈ Λ 2 (R 3 )): Then we have, for ξ ∈ g (via Cartan's formula) and thus we have an isomorphism g ∼ = Z 2 (R 3 ) (closed 2-forms on R 3 ). This will be important in the sequel. The above also expresses the fact that ν is a strictly conserved 3-form.
We shall now give the promised example of homotopy co-momentum map emerg- where B = B and B is again a vector potential for b, i.e. curl B = b, chosen e.g. in such a way that div B = 0 (Coulomb gauge).
It is immediately checked that The above formula tells us that f 1 (b) is a Hamiltonian 1-form for b (and, conversely, that the vector field b is a Hamiltonian vector field pertaining to f 1 (b), in accordance with (2.1) in Subsection 2.1. Any f 1 (b) above is also a Noether current in the sense of Gotay et al., [19]. In order to complete the definition of a homotopy co-momentum map, we just have to find f 2 , satisfying formula (2.2) above. Indeed, for k = 1 we retrieve (2.7). The case k = 2 reads Then one checks that (using e.g. [44] Lemma 2.18, or the preceding subsection) (recall that ι ξ 1 ∧ξ 2 ν = ν(ξ 1 , ξ 2 , ·)). Therefore, the 1-form µ 2 (ξ 1 , is closed, hence exact, and (2.8) tells us that f 2 (p) is a potential for it and, as such, it is determined up to a constant c(ξ 1 , ξ 2 ). In order to prove that we have a bona fide co-momentum map, we must have, in particular, for q = ξ 1 ∧ξ 2 ∧ξ 3 , the explicit formula which is a priori true up to a constant c(ξ 1 , ξ 2 , ξ 3 ) by virtue of (2.8) and [7], Lemma 9.2. However, the constant is in fact zero since ν(ξ 1 , ξ 2 , ξ 3 ) vanishes at infinity and the same is true for f 2 (∂q) upon solving the related Poisson equation (obtained via a straightforward computation; notice that we use the Riemannian Laplacian, which is minus the standard one). An alternative derivation uses x-independence of the class [c x ]. Upon taking 3): the expression is meaningful in view of the assumed decay at infinity of our objects). This is equivalent to the previous equation (2.9). The function f 2 is the given by the solution of a Poisson equation . This completes the construction of the sought for homotopy co-momentum map.
We define Poisson brackets (PB) via the expression which will be employed below and in Section 4. We may also naturally ask the question whether the above map ( f ) is (infinitesimally) G-equivariant, in the sense of [44]: in particular, one should check the validity of the formula for all ξ, b ∈ g. However, working out the two sides of the above equation easily yields, in particular, for ξ = b, the equality that is, in vector terms 〈B, b〉 = c = 0 since b is compactly supported. However, if one considers a flux tube with non zero helicity 〈B, b〉 (see [33,4,46] and below for further elucidation of this train of concepts), we get a contradiction. Notice that the argument does not depend on the choice of B . The lack of G-equivariance is not surprising, since our construction involves Riemannian geometric features.
We may now recap the preceding discussion via the following: The map ( f ) previously given through the above f j : fulfilling (2.7),(2.5), (2.6), yields a homotopy co-momentum map; explicitly: (ii) The above HCCM transgresses, via the evaluation map ev : (2.11) (iv) The map ( f ) is not G-equivariant in the sense of RWZ.

Proof.
We have just to address point (ii): observe that the relevant piece of the homotopy co-momentum map is f 1 which, under transgression, becomes i.e., up to sign, the Rasetti-Regge current (RR) λ b pertaining to b ∈ g independent of the choice of B . This is in accordance with the general result in [44] asserting that, roughly speaking, homotopy co-momentum maps transgress to homotopy comomentum maps on loop (and even mapping) spaces. Actually, the ansatz for f 1 term was precisely motivated by this phenomenon.

A generalization to Riemannian manifolds
We ought to notice that a hydrodynamically flavoured homotopy co-momentum map can be similarly construed also for an (n +1)-dimensional connected, compact, orientable Riemannian manifold (M , g ), upon taking its Riemannian volume form ν as a multisymplectic form and again the group G of volume preserving diffeomorphism group as symmetry group. The divergence of a vector field X is defined via div X := * d * X = −δX ; the operator * is the Riemannian Hodge star (e.g. [49,28]). We can indeed prove the following result: Proof. As we have already noticed in general, the defining formula triggers a recursive construction starting from f 1 , up to topological obstructions (we have a sequence of closed forms, which must be actually exact, together with the constraint f n (∂q) = (−1) (n+1)(n+2) 2 ν(ξ 1 , . . . ξ n+1 ), with q = ξ 1 ∧ . . . ξ n+1 , for the constant function µ n+1 (·)). In the present case, a natural candidate for the (n-1)-form f 1 can be readily manufactured via Hodge theory (see e.g. [49]): (the direct generalization of the preceding case) after imposing δ f 1 (ξ) = 0 (the analogue of the Coulomb gauge condition), provided one can safely invert the Hodge Laplacian ∆ = d δ + δd , this being the case if H n−1 d R (M ) = 0. One can of course alter the above definition by addition of an exact form. The topological assumptions made ensure that the entire procedure goes through unimpeded due to the formula d f k (ξ 1 ∧ · · · ∧ ξ k ) = µ k (ξ 1 ∧ · · · ∧ ξ k ), k = 2, 3, . . . n.
One has finally to check that f n (∂(ξ 1 ∧ · · · ∧ ξ n+1 )) = −ς(n + 1)ι(ξ 1 ∧ · · · ∧ ξ n+1 )ν but this is true once we observe that, since c x 0 = 0, the class [c x ] = 0 (cf. Subsection 2.1). Therefore, we have, eventually, the compact formulae Remark. We notice that the above result holds, in particular, for homology spheres such as, for instance, the celebrated Poincaré dodecahedral space. We point out that the case in which the intermediate homology groups are at most torsion (hence not detectable by de Rham techniques) is also encompassed: this is e.g. the case of lens spaces. Notice that G-equivariance cannot be expected a priori. Also notice that one could restrict to the natural symmetry group provided by the isometries of (M , g ). See e.g. [42] for a general discussion of topological constraints to existence and uniqueness of homotopy co-momentum maps.

Covariant phase space aspects
We are now going to propose a multisymplectic interpretation of the hydrodynamical bracket which ties neatly with the topics discussed in previous sections, via covariant phase space ideas ( [25,19,15,50,11], but without literally following the standard recipe, as we shall see shortly. Start with a 4-dimensional space-time M = R 3 × R (x, y, z, t ), define the obvious trivial bundle E = M × R 3 → M and interpret Σ := R 3 (x, y, z) as a Cauchy "submanifold" of M . Any divergence-free vector field can be viewed as an initial condition v(x, 0) for the (volume-preserving) Euler evolution (at least for small times, but as we previously said, we do not insist on refined analytical nuances) v(x, t ), yielding a section of E . Using the 3-volume form ν, orienting fibres (notice that, when viewed on E , it is only pre-2-plectic, namely closed but degenerate), and observing that we can set J 1 v := w (:= curl v) (the natural "covariant" jetification of the section v-to be distinguished from the standard jetification j 1 -if we wish to look at v as the vector space counterpart of a connection 1-form, yielding a section of the jet bundle J 1 E → E ) we can rewrite the hydrodynamical bracket, mimicking [15], as since the variations δF δv and δG δv are vertical and divergence-free: δF /δv = curl (δF /δw). Taking again b = curl B et cetera and setting finally F = λ • (see e.g. in particular [35,47]) we see that the expression ( ) can be manipulated to yield the expressive layout (with slight abuse of language) (in full adherence with the discussion carried out in Section 2.2.). The same portrait can be depicted, mutatis mutandis, for the singular case. Ultimately, we reached the following conclusion: (ii) The above construction reproduces the symplectic structure of Y upon taking singular vorticities, concentrated on a smooth oriented knot: the covariant phase space picture is fully retrieved upon passing to a 2-dimensional space-time S 1 × R (λ, t ), with λ ∈ S 1 ≡ Σ being a knot parameter (and staying of course with the same ν).
Remarks. 1. We stress the fact that we did not literally follow the standard "multisymplectic to covariant" recipe developed in [15]. In fact the multisymplectic manifold we consider is not the one prescribed by [19] since we directly took the standard volume form ν on R 3 as a 2-plectic structure (or pre-2-plectic when pulled back to E ), cf. [8]. This neatly matches Brylinski's theory and fits with the stance long advocated, among others, by Rasetti and Regge and Goldin (see e.g. [38,17,20,18], and [47] as well) pinpointing the special and ubiquitous role played by the group G. Another motivation for considering ν is its pivotal role in the formulation of conservation theorems (see [44]). We shall pursue this aspect in what follows.
2. In line with the preceding remark, notice that the above portrait can, in principle, be generalized to any volume form (on an orientable manifold), with its attached group G. The covariant phase space picture should basically persist in the sense that one might construct, in greater generality, an n-plectic structure out of an (n+1)plectic one via an expression akin to ( ). The (non) G-equivariance issue should be relevant in this context.

A Hamiltonian 1-form for links
We may specialize the considerations in Subsection 2.3 to the case of links. The basic and quite natural idea is to associate to a knot (or link ) a perfect fluid whose vorticity in concentrated thereon (cf. the preceding discussion on the Brylinski manifold). As general references for knot theory we quote, among others, [41], together with [5] for the algebraic-topological tools employed here. First recall that in general the Poincaré dual [η S ] ∈ H n−k (M ) of a k-dimensional closed oriented submanifold S of an n-dimensional manifold M is characterized by the property: for any closed, compactly supported k-form ω on M (i : S → M being the inclusion map). We shall indifferently view Poincaré duals as genuine forms or currents in the sense of de Rham ( [12]).
Building on [37,4,46], let L = ∪ n i =1 L i be an oriented link in R 3 with components L i , i = 1, . . . , n -required to be trivial knots and let ω L i denote the Poincaré (or Thom) dual (class) associated to L i : they are 2-forms localized in a cross-section of a suitable tubular neighbourhood T i around L i -with total fibre integral equal to onesee [5], or, as currents, 2-forms which are δ-like on L i , see Figure 1.  Figure 2. We list, for the sake of clarity, (de Rham) cohomology and relative ho-  Proof. The proof is straightforward: indeed for each component L i , the Hamiltonian vector field ξ L i for v L i ≡ v i is minus the vector field associated to the closed 2-form ω L i (via the map α of Section 2). Explicitly, one has (setting ξ L = n i =1 ξ L i ) Remark. Inspection of the very geometry of Poincaré duality shows that the velocity 1-forms v i correspond (upon approximation of the associated Euler equation) to the so-called LIA (Linear Induction Approximation) or binormal evolution of the "vortex ring" L i ("orthogonal" to the discs a i -an easy depiction, cf. Figure 2), see [24] for more information. Formula (3.1) will be the prototype for the calculations in Section 4.
Let us define the Chern-Simons (helicity) 3-form: Integration of C S(L) over R 3 or S 3 yields an integer H (L), the helicity of L: with (i , j ) = ( j , i ) being the Gauss linking number of components L i and L j if i = j and ( j , j ) is the framing of L j , equal to (L j , L j ) with L j being a section of the normal bundle of L j , see e.g. [41,33,39,46] and below. A regular projection of a link onto a plane produces a natural framing called the blackboard framing.
On a Riemannian 3-manifold M , the helicity H pertaining to a perfect fluid with velocity v and vorticity w = curl v is given by (notation as in the previous sections) (the last expression being the differential form counterpart). Concretely, the helicity can be viewed as a measure of the mutual knotting of two generic flow lines, see [33,3,34,35,46] for a more extensive discussion. We used this concept in Subsection 2.3 to prove the non-equivariance of the hydrodynamical HCCM.

A multisymplectic interpretation of Massey products
In this section we resort to the techniques developed in Sections 2 and 3 above and propose a reformulation of the so-called higher order linking numbers in multisymplectic terms. Ordinary and higher order linking numbers provide, among others, a quite useful tool for the investigation of Brunnian phenomena in knot theory: recall that a link is almost trivial or Brunnian if upon removing any component therefrom one gets a trivial link. They can be defined recursively in terms of Massey products, or equivalently, Milnor invariants, by the celebrated Turaev-Porter theorem (see [14,37,46,23]). We are going to review, briefly and quite concretely, the basic steps of the Massey procedure, read differential geometrically as in [37,46,23], presenting at the same time our novel multisymplectic interpretation thereof.
Let L be an oriented link with three or more components L j . The cohomological reinterpretation of the ordinary linking number (1, 2) of two components L 1 and L 2 , say, starts from consideration of the closed 2-form yielding the (integral) de Rham class The linking number (1, 2) is non zero precisely when 〈L 1 , L 2 〉, which, in H 1 (S 3 , L) equals (1, 2)[γ 12 ], is non trivial. If the latter class vanishes (i.e. Ω 12 is exact), we have for some 1-form v 12 . Now, assuming that all the ordinary mutual linking numbers of the components under consideration vanish, one can manufacture the (closed, direct check) 2-form (Massey product) yielding a third order linking number (as a class): If the latter class vanishes, we find a 1-form v 123 such that It is then easy to devise a general pattern, giving rise to forms v I , Ω I (I being a general multiindex). Actually, everything can be organised -via Chen's calculus of iterated path integrals [9,10]-in terms of sequences of nilpotent connections v (k) , k = 1, 2... on a trivial vector bundle over S 3 \L and their attached curvature forms w (k) (ultimately, the Ω I , [37,46,48,22]), everything stemming from the Cartan structure equation together with the ensuing Bianchi identity (the latter implying closure of the forms Ω I ). In order to give a flavour of the general argument, start from the nilpotent connection v (1) with its corresponding curvature w (1) : (we made use of d v 12 + Ω 12 = d v 23 + Ω 23 = 0), and so on.
Also recall that all forms Ω I can be neatly interpreted, via Poincaré duality, as auxiliary (trivial) knots L I , and v I as discs bounded by L I , in adherence to the considerations in Section 3, see [37,46] for more details and worked out examples, including the Whitehead link (involving fourth order linking numbers -with repeated indices) and the Borromean rings (exhibiting a third order linking number). Just notice here that, for instance, formula (4.1) becomes, intersection theoretically ∂a 12 + a 1 ∩ a 2 = 0, see Figure 3. Formula (4.2) can be rewritten as where ξ ≡ ξ 123 = α −1 (Ω 123 ). The above ("vorticity") vector field ξ 123 can be thought of as being concentrated on the knot corresponding to ξ 123 , or, alternatively, in a thin tube around it, when considering a bona fide Poincaré dual, cf. (3.1).
This tells us that v 123 is a Hamiltonian 1-form in the sense of [44] and the formula expresses the fact that Ω 123 is a globally conserved 2-form, and the same holds for Ω 12 and, in general, for Ω I , with their corresponding vector fields ξ I . Specifically, we have the following: Proof. Ad (i). This is clear since the mentioned forms are closed. Ad (ii). The previous discussion can be carried out verbatim for a general multiindex I : (an extension of (3.1)), this yielding the second conclusion.
The following is the main result of this section.  ν(ξ, η, ·) and of the peculiar structure of the vector fields involved (they either partially coincide or have disjoint supports). By the same argument, one gets ι ξ L v I = 0, in view of the Poincaré dual interpretation of v I (cf. Section 3), together with the second assertion; a crucial point to notice is that the auxiliary links obtained via Chen's procedure may be suitably split from their ascendants, this leading to ι ξ L v I = 0, the consequent strict conservation of the v I 's being then immediate.
Notice that, in particular, from ι ξ L v L = 0 (Poincaré dual interpretation again) we also get L ξ L v L = 0 (this is not to be expected a priori in multisymplectic geometry, cf. [44]).
We ought to remark that, upon altering the v I 's by an exact form, we may lose strict conservation, but in any case global conservation is assured (the PB is an exact form, by (2.11) in Section 2 and in view of commutativity of the vector fields ξ I and ξ J ).
Ultimately, we can draw the conclusion that the Massey invariant route to ascertain the Brunnian character of a link can be mechanically understood as a recursive test of a kind of knot theoretic integrability: the Massey linking numbers provide obstructions to the latter.
Thus, somewhat curiously, higher order linking phenomena receive an interpretation in terms of multisymplectic geometry, which is a sort of higher order symplectic geometry. Also, integrability comes in with a twofold meaning: first, higher order linking numbers emerge from the construction of a sequence of flat, i.e. integrable nilpotent connections; second, this very process yields first integrals in involution in a mechanical sense.

Conclusions and outlook
In this note we constructed a homotopy co-momentum map in a hydrodynamical context, whereby we gave, as an application, a multisymplectic reinterpretation of the Massey higher order linking numbers, together with an extension thereof in a Riemannian geometric framework. We have also exhibited a covariant phase space interpretation of the geometrical framework involved.
The multisymplectic approach appears to be very promising for further advancement in this area. Also, the notion of integrability cropping up in our analysis of Massey products may deserve further scrutiny in a general multisymplectic context. We hope to tackle (at least some of ) the open questions raised in this paper elsewhere.