  We discuss the effect of different choices in partial gauge fixing of bulk
local Lorentz invariance, on the description of the horizon degrees of freedom
of a Schwarzschild black hole as an SU(2) Chern-Simons theory with specific
sources. A classically equivalent description in terms of an ISO(2)
Chern-Simons theory is also discussed. Further, we demonstrate that both these
descriptions can be partially gauge fixed to a horizon theory with U(1) local
gauge invariance, with the solder form sources being subject to extra
constraints in directions orthogonal to an internal vector field left invariant
by U(1) transformations. Seemingly disparate approaches on characterization of
the horizon theory for the Schwarzschild black hole (as well as spherical
Isolated Horizons in general) are thus shown to be equivalent physically.
