The boundary field theory induced by the Chern-Simons theory

TL;DR

This paper explores how Chern-Simons theory on a 3D manifold induces a boundary field theory, analyzing its degrees of freedom and the impact of boundary conditions, with a detailed example for SU(2).

Contribution

It provides a detailed derivation of the boundary theory from Chern-Simons, including the counting of physical degrees of freedom and the influence of boundary conditions.

Findings

The boundary theory has one physical local degree of freedom.

Boundary conditions significantly affect the gauge structure and constraints.

The analysis is exemplified with the SU(2) gauge group.

Abstract

The Chern-Simons theory defined on a 3-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 pullback 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 to 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.