Topologically Protected Duality on The Boundary of Maxwell-BF Theory

Nicola Maggiore,Alberto Blasi

doi:10.3390/sym11070921

Abstract

The Maxwell-BF theory with a single-sided planar boundary is considered in Euclidean four-dimensional spacetime. The presence of a boundary breaks the Ward identities, which describe the gauge symmetries of the theory, and, using standard methods of quantum field theory, the most general boundary conditions and a nontrivial current algebra on the boundary are derived. The electromagnetic structure, which characterizes the boundary, is used to identify the three-dimensional degrees of freedom, which turn out to be formed by a scalar field and a vector field, related by a duality relation. The induced three-dimensional theory shows a strong–weak coupling duality, which separates different regimes described by different covariant actions. The role of the Maxwell term in the bulk action is discussed, together with the relevance of the topological nature of the bulk action for the boundary physics.

Highlights

Topological field theories have been the subject of a thorough investigation in theoretical physics [1,2,3]
The method proposed by Symanzik concerns a space divided into a left and a right hand side, and it has been applied to topological field theories of different types [8], obtaining results relevant for the theory of the fractional quantum Hall effect [9] and of the topological insulators in three and four spacetime dimensions [10]
Since the aim of this paper is to study if and how the non-topological Maxwell term has an impact on the physics on the boundary, we proceed disregarding the solution in Equation (26)

Summary

Introduction

Topological field theories have been the subject of a thorough investigation in theoretical physics [1,2,3]. The Maxwell coupling is expected to be quite relevant in whatever physics may arise on the boundary and that is why we are studying models where the Maxwell term is included in the bulk action This has been done for Chern–Simons theory with both double- [20] and single-sided [21] boundary, with significantly different results. The resulting equation is recognized to be the duality relation which characterizes the existence of fermionic degrees of freedom on the boundary and turns out not to be peculiar of purely topological bulk field theories only. The fact that the physical properties are the same in the holographic theory, whether the Maxwell term is present or not, clarifies the meaning of topological quantum field theories when a boundary is introduced

The Model

The Boundary Algebra

The Action Induced on the 3D Boundary

Holographic Constraint and Duality

Summary of Results and Discussion