Classification of Arnold-Beltrami Flows and their Hidden Symmetries

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In the context of mathematical hydrodynamics, we consider the group theory structure which underlies the ABC-flow introduced by Beltrami, Arnold and Childress. Beltrami equation is the eigenstate equation for the first order Laplace-Beltrami operator *d, which we solve by using harmonic analysis. Taking torus T³ constructed as R³/L, where L is a crystallographic lattice, we present a general algorithm to construct solutions of Beltrami equation which utilizes as main ingredient the orbits under the action of the point group Pₗ of three-vectors in the momentum lattice L*. We introduce the new notion of a Universal Classifying Group GUₗ which contains all crystallographic space groups as proper subgroups. We show that the *d-eigenfunctions are naturally arranged into irreducible representations of GUₗ and by means of a systematic use of the branching rules with respect to various possible subgroups H of GUₗ we search and find Beltrami fields with non trivial hidden symmetries. In the case of the cubic lattice the point group Pₗ is the proper octahedral group O₂4 and the Universal Classifying Group is finite group G₁536 of order 1536 which we study in full detail deriving all of its 37 irreducible representations and the associated character table. We show that the O₂4 orbits in the cubic lattice are arranged into 48 equivalence classes, the parameters of the corresponding Beltrami vector fields filling all the 37 irreducible representations of G₁536. In this way we obtain an exhaustive classification of all generalized ABC-flows and of their hidden symmetries. We make several conceptual comments about the possible relation of Arnold-Beltrami flows with (supersymmetric) Chern-Simons gauge theories. We also suggest linear generalizations of Beltrami equation to higher odd-dimensions that possibly make contact with M-theory and the geometry of flux-compactifications.