3D Dynamics of 4D Topological BF Theory With Boundary

TL;DR

This paper analyzes the boundary effects in 4D abelian topological BF theory, deriving boundary conditions, gauge invariances, and a duality relation, leading to a 3D Lagrangian description.

Contribution

It introduces the most general boundary conditions compatible with 4D BF theory and interprets the boundary algebra as a duality relation between 3D variables.

Findings

Boundary conditions compatible with field equations are identified.

Residual gauge invariance is characterized by Ward identities.

A duality relation between 3D dynamical variables is established.

Abstract

We consider the four dimensional abelian topological BF theory with a planar boundary introduced following the Symanzik's method. We find the most general boundary conditions compatible with the fields equations broken by the boundary. The residual gauge invariance is described by means of two Ward identities which generate an algebra of conserved currents. We interpret this algebra as canonical commutation relations of fields, which we use to construct a three dimensional Lagrangian. As a remarkable by-product, the (unique) boundary condition which we found, can be read as a duality relation between 3D dynamical variables.