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

TL;DR

This thesis explores boundary dynamics in 3D and 4D topological BF theories using Symanzik's method, revealing boundary gauge fixing and algebraic structures that relate to topological insulators.

Contribution

It applies Symanzik's method to analyze boundary effects in 3D and 4D BF theories, characterizing boundary dynamics and residual gauge invariance.

Findings

Boundary term interpreted as gauge-fixing for residual gauge invariance.

Boundary algebra of local observables characterized by canonical commutation relations.

Model proposed as an effective theory for 3+1D topological insulators.

Abstract

In this thesis I studied the Symanzik's method for the introduction of the boundary in a field theory and, specifically, I applied this method to three Topological Field Theories of the Shwartz type: the non-abelian Chern-Simons model, the non-abelian three-dimensional BF theory and its abelian four-dimensional version. This thesis is organized into three chapters. In Chapter 1 the introduction of the boundary in the non-abelian CS model is illustrated. The purpose of this chapter is to describe some techniques known in literature, which are largely used in the following chapters. In Chapter 2 the three-dimensional non-abelian BF theory with a boundary is analyzed. The most interesting result of this chapter is the interpretation of the boundary term of the Ward Identities as the gauge-fixing for the residual gauge invariance of the theory on the boundary. In Chapter 3 the techniques developed in the previous chapters are applied to the abelian BF theory in four space-time dimensions with a boundary. The most interesting result of this chapter, (and of the entire thesis), is the characterization of the boundary dynamics in terms of canonical commutation relations generated by the algebra of local boundary observables, which exists due to the residual gauge invariance of the bulk theory. The results obtained suggest that this model can be considered as an effective theory for the (3+1)D Topological Insulators.