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

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.

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