  The proper definition of subsystems in gauge theory and gravity requires an
extension of the local phase space by including edge mode fields. Their role is
on the one hand to restore gauge invariance with respect to gauge
transformations supported on the boundary, and on the other hand to parametrize
the largest set of boundary symmetries which can arise if both the gauge
parameters and the dynamical fields are unconstrained at the boundary. In this
work we construct the extended phase space for three-dimensional gravity in
first order connection and triad variables. There, the edge mode fields consist
of a choice of coordinate frame on the boundary and a choice of Lorentz frame
on the bundle, which together constitute the Lorentz-diffeomorphism edge modes.
After constructing the extended symplectic structure and proving its gauge
invariance, we study the boundary symmetries and the integrability of their
generators. We find that the infinite-dimensional algebra of boundary
symmetries with first order variables is the same as that with metric
variables, and explain how this can be traced back to the expressions for the
diffeomorphism Noether charge in both formulations. This concludes the study of
extended phase spaces and edge modes in three-dimensional gravity, which was
done previously by the author in the BF and Chern-Simons formulations.
