Chern-Simons Field Theory and Completely Integrable Systems

We show that the classical non-abelian pure Chern-Simons action is related in a natural way to completely integrable systems of the Davey-Stewartson hyerarchy, via reductions of the gauge connection in Hermitian spaces and by performing certain gauge choices. The B\"acklund Transformations are interpreted in terms of Chern-Simons equations of motion or, on the other hand, as a consistency condition on the gauge. A mapping with a nonlinear $\sigma$-model is discussed.Comment: 11 pages, Late

Dynamics in the magnetic/dual magnetic monopole

Inspired by the geometrical methods allowing the introduction of mechanical systems confined in the plane and endowed with exotic galilean symmetry, we resort to the Lagrange-Souriau 2-form formalism, in order to look for a wide class of 3D systems, involving not commuting and/or not canonical variables, but possessing geometric as well gauge symmetries in position and momenta space too. As a paradigmatic example, a charged particle simultaneously interacting with a magnetic monopole and a dual monopole in momenta space is considered. The main features of the motions, conservation laws and the analogies with the planar case are discussed. Possible physical realizations of the model are proposed.Comment: 15 pages, 1 figure, based on a talk for the QTS7 conference, Prague, August 7-13 (2011), accepted in Journal of Physics: Conference Serie

A looping-delooping adjunction for topological spaces

Every principal G-bundle is classified up to equivalence by a homotopy class of maps into the classifying space of G. On the other hand, for every nice topological space Milnor constructed a strict model of loop space, that is a group. Moreover the morphisms of topological groups defined on the loop space of X generate all the bundles over X up to equivalence. In this paper, we show that the relationship between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles with a fixed structure group. Such a resul clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space construction, which are very important in topological K-theory, group cohomology and homotopy theory.Comment: v1: 24 pages; v2: 18 pages; Corrected typos; Revised structure in Introduction, and Sections 1 and 2; Results unchange