Classical Chern-Simons theory. I
This paper is to a non-negligible extent a review but foremost an excellent and coherent formulation of Chern-Simons theory as a standard classical field theory. To achieve this goal, quite a few complex steps have to be taken and some careful considerations are necessary. A basic motivation for carrying out this program is to be found in the fact that the study of this classical field theory in fact yields important insight into some of the corresponding quantized features. The paper is very rich and comprehensive, so we mention here -- in a rather impressionistic mode -- just some of the basic issues; along the road an axiomatisation of general classical, topological field theories is given where both the role of locality (for the Lagrangian) and the basic nature of the Maupertuis' principle are clearly spelled out. The Lagrangian is here the Chern-Simons 3-form of a connection and the attached action in fact quite a rich geometric object. The corresponding Euler-Lagrange equation turns out to be of first order, since the Lagrangian is in fact linear in field derivatives. This equation, in turn, guarantees the existence of a momentum map naturally leading to a groupoid of symmetries. A special section is devoted to the highly relevant case of a 2-manifold with boundary.
scientific article; zbMATH DE number 812544