  In this work we discuss the formation of zero energy vortex and chiral edge
modes in a fermionic representation of the Kitaev honeycomb model. We introduce
the representation and show how the associated Jordan-Wigner procedure
naturally defines the so called branch cuts that connect the topological vortex
excitations. Using this notion of the branch cuts we show how to, in the
non-Abelian phase of the model, describe the Majorana zero mode structure
associated with vortex excitations. Furthermore we show how, by intersecting
the edges between Abelian and non-Abelian domains, the branch cuts dictate the
character of the chiral edge modes. In particular we will see in what
situations the exact zero energy Majorana edge modes exist. On a cylinder, and
for the particular instances where the Abelian phase of the model is the full
vacuum, we have been able to exactly solve for the systems edge energy
eigensolutions and derive a recursive formula that exactly describes the edge
mode structure. Penetration depth is also calculated and shown to be dependent
on the momentum of the edge mode. These solutions also describe the overall
character of the fully open non-Abelian domain and are excellent approximations
at moderate distances from the corners.
