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In this thesis I studied the Symanzik's method for the introduction of theboundary in a field theory and, specifically, I applied this method to threeTopological Field Theories of the Shwartz type: the non-abelian Chern-Simonsmodel, the non-abelian three-dimensional BF theory and its abelianfour-dimensional version. This thesis is organized into three chapters. InChapter 1 the introduction of the boundary in the non-abelian CS model isillustrated. The purpose of this chapter is to describe some techniques knownin literature, which are largely used in the following chapters. In Chapter 2the three-dimensional non-abelian BF theory with a boundary is analyzed. Themost interesting result of this chapter is the interpretation of the boundaryterm of the Ward Identities as the gauge-fixing for the residual gaugeinvariance of the theory on the boundary. In Chapter 3 the techniques developedin the previous chapters are applied to the abelian BF theory in fourspace-time dimensions with a boundary. The most interesting result of thischapter, (and of the entire thesis), is the characterization of the boundarydynamics in terms of canonical commutation relations generated by the algebraof local boundary observables, which exists due to the residual gaugeinvariance of the bulk theory. The results obtained suggest that this model canbe considered as an effective theory for the (3+1)D Topological Insulators.

The dynamics on the three-dimensional boundary of the 4D Topological BF model