The boundary field theory induced by the Chern–Simons theory

Abstract

The Chern–Simons theory defined on a three-dimensional manifold with boundary is written as a two-dimensional field theory defined only on the boundary of the three-manifold. The resulting theory is, essentially, the pull-back to the boundary of a symplectic structure defined on the space of auxiliary fields in terms of which the connection one-form of the Chern–Simons theory is expressed when solving the condition of vanishing curvature. The counting of the physical degrees of freedom living in the boundary associated with the model is performed using Dirac's canonical analysis for the particular case of the gauge group SU(2). The result is that the specific model has one physical local degree of freedom. Moreover, the role of the boundary conditions on the original Chern–Simons theory is displayed and clarified in an example, which shows how the gauge content as well as the structure of the constraints of the induced boundary theory is affected.