Chern-Simons Theory with Sources and Dynamical Quantum Groups I: Canonical Analysis and Algebraic Structures

We study the quantization of Chern-Simons theory with group \(G\) coupled to dynamical sources. We first study the dynamics of Chern-Simons sources in the Hamiltonian framework. The gauge group of this system is reduced to the Cartan subgroup of \(G.\) We show that the Dirac bracket between the basic dynamical variables can be expressed in term of dynamical \(r-\)matrix of rational type. We then couple minimally these sources to Chern-Simons theory with the use of a regularisation at the location of the sources. In this case, the gauge symmetries of this theory split in two classes, the bulk gauge transformation associated to the group \(G\) and world lines gauge transformations associated to the Cartan subgroup of \(G\). We give a complete hamiltonian analysis of this system and analyze in detail the Poisson algebras of functions invariant under the action of bulk gauge transformations. This algebra is larger than the algebra of Dirac observables because it contains in particular functions which are not invariant under reparametrization of the world line of the sources. We show that the elements of this Poisson algebra have Poisson brackets expressed in term of dynamical \(r-\)matrix of trigonometric type. This algebra is a dynamical generalization of Fock-Rosly structure. We analyze the quantization of these structures and describe different star structures on these algebras, with a special care to the case where \(G=SL(2,{\mathbb R})\) and \(G=SL(2,{\mathbb C})_{\mathbb R},\) having in mind to apply these results to the study of the quantization of massive spinning point particles coupled to gravity with a cosmological constant in 2+1 dimensions.