A Hydrodynamical Homotopy Co-Momentum Map and a Multisymplectic Interpretation of Higher-Order Linking Numbers

Abstract

In this paper a homotopy co-momentum map (à la Callies, Frégier, Rogers and Zambon) transgressing to the standard hydrodynamical co-momentum map of Arnol’d, Marsden, 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.

Homotopy Comomentum Maps in Multisymplectic Geometry

“…As explained in the previous section, most of the original results presented here also appeared in the following four preprints [MS21], [MR20b], [MS20] and [MZ21]. To date, three of them have been accepted for publication.…”

“…The contents of this chapter are based on two papers co-authored by the author of thesis and Mauro Spera [MS21,MS20].…”

“…Remark 5.0.1 (Hydrodynamical brackets). In this chapter, we are slightly departing from the hydrodynamical bracket convention employed in [MS21]. In that convention, the infinitesimal action pertaining to a left action is an anti-homomorphism, and the Lie bracket on the space of vector fields X(M ) = Γ(M ) is given by minus the standard one (see [AK98,pag.…”