Global dynamics of electric and magnetic membranes on the complex, quaternionic and octonionic hopf bundles

As a prelude to chiral field soliton realizations, we present a unified path space formulation of global dynamics for systems of relativistic point particles, 2-sphere, 6-sphere membranes and their rank-1, -3 and -7 tensor U(1) gauge fields in D=3, 7 and 15 spacetime dimensions. The path spaces are realized by the complex (C), quaternionic (H) and octonionic (Ω) Hopf bundles. These systems have an electric-magnetic duality in D=4, 8 and 16 dimensions. Their hypercomplex symplectic Kähler structures generalize the geometric prequantization condition for the point charge-monopole complex. In a subsequent analysis of Aharonov-Bohm effects induced by the above tensor U(1) gauge fields in three D-dimensional KP¹ σ-models with a Hopf-Chern-Simons term where (D, K)=(3, C), (7, H) and (15, Ω), these global Wess-Zumino-Novikov-Witten actions play a key role in the London-Nielsen-Olesen limit of very thin membrane solitons.