RVB gauge theory and the topological degeneracy in the honeycomb Kitaev model

We relate the Z(2) gauge theory formalism of the Kitaev model to the SU (2) gauge theory of the resonating valence bond physics. Furthermore, we reformulate a known (Feng et al (2007 Phys. Rev. Lett. 98 087204), Chen and Hu (2007 Phys. Rev. B 76 193101), Chen and Nussinov (2008 J. Phys. A: Math. Theor. 41 075001) and Mandal et al (2006 Int. Conf. on Physics Near the Mott Transition)) Jordan-Wigner transformation of the Kitaev model on a torus in a general way that shows that it can be thought of as a Z(2) gauge-fixing procedure. We give an explicit construction of the generators of large gauge transformations on a torus in terms of the spin operators. Using these and the non-trivial loop operators, we construct four mutually anti-commuting operators which commute with the Hamiltonian enabling us to prove that all eigenstates of this model, for the time-reversal symmetric case, are fourfold degenerate in the thermodynamic limit.