Topological phase entanglements of membrane solitons in division algebra sigma models with a Hopf term

Abstract

Exploiting the unique connection between the division algebras of the complex numbers ( C), quaternions ( H), octonions ( Ω) and the essential Hopf maps S 2 n − 1 → S n with n = 2, 4, 8, we study S n − 2 -membrane solitons in three D-dimensional KP(1) σ-models with a Hopf term, (D, K) = (3, C), (7, H), and (15, Ω). We present a comprehensive analysis of their topological phase entanglements. Extending Polyakov's approach to Fermi-Bose transmutations to higher dimensions, we detail a geometric regularization of Gauss' linking coefficient, its connections to the self-linking, twisting, writhing numbers of the Feynman paths of the solitons in their thin membrane limit. Alternative forms of the Hopf invariant show the latter as an Aharonov-Bohm-Berry phase of topologically massive, rank ( n − 1) antisymmetric tensor U(1) gauge fields coupled to the S n − 2 -membranes. Via a K-bundle formulation of the dynamics of electrically and magnetically charged extended objects these phases are shown to induce a dyon-like structure on these membranes. We briefly discuss the connections to harmonic mappings, higher dimensional monopoles and instantons. We point out the relevance of the Gauss-Bonnet-Chern theorem on the connection between spin and statistics. By way of the topology of the infinite groups of sphere mappings S n → S n , n = 2, 4, 8, we also analyze the implications of the Hopf phases on the fractional spin and statistics of the membranes.