Constructing the theory at the boundary, its dynamics and degrees of freedom

TL;DR

This paper introduces a method to derive boundary theories from bulk theories with boundaries, identifying boundary degrees of freedom without gauge fixing, exemplified by the 3D Abelian Chern-Simons theory and its edge modes.

Contribution

It presents a novel approach to determine boundary theories and degrees of freedom directly from the bulk action without prior gauge fixing or assumptions.

Findings

Boundary theory has one degree of freedom characterized by a chiral boson.

The boundary degrees of freedom correspond to Quantum Hall edge modes.

A gauge fixing in the bulk relates to boundary conditions, affecting the action principle.

Abstract

Given a theory in a region of space-time with boundaries, defined by a Lagrangian, we propose a method to find the corresponding theory at the boundary (either space-like or time-like), that allows us to identify the degrees of freedom at the boundary without any gauge fixing nor further assumptions a priori. We start by finding the corresponding action principle at the boundary. From this, we can find the corresponding equations of motion, the symmetries and degrees of freedom at the boundary. Then we characterise the degrees of freedom at the boundary. We exemplify this method by analysing the broadly studied 3-dimensional Abelian Chern-Simons theory. We found, through the Hamiltonian framework, that the boundary theory has one degree of freedom that can be characterised by the chiral boson: the Quantum Hall edge modes. We also find a correspondence between a gauge fixing in the bulk with a boundary condition and discuss whether this is enough to have a well-posed action principle and the possible scenarios.