  We investigate the exact solution of the honeycomb model proposed by Kitaev
and derive an explicit formula for the projector onto the physical subspace.
The physical states are simply characterized by the parity of the total
occupation of the fermionic eigenmodes. We consider a general lattice on a
torus and show that the physical fermion parity depends in a nontrivial way on
the vortex configuration and the choice of boundary conditions. In the
vortex-free case with a constant gauge field we are able to obtain an
analytical expression of the parity. For a general configuration of the gauge
field the parity can be easily evaluated numerically, which allows the exact
diagonalization of large spin models. We consider physically relevant
quantities, as in particular the vortex energies, and show that their true
value and associated states can be substantially different from the one
calculated in the unprojected space, even in the thermodynamic limit.
