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We explore the properties of the SO(3) Majorana representation of spin. Basedon its non-local nature, it is shown that there is an equivalence between theSO(3) Majorana representation and the Jordan-Wigner transformation in one andtwo dimensions. From the relation between the SO(3) Majorana representation andone-dimensional Jordan-Wigner transformation, we show that application of theSO(3) Majorana representation usually results in $Z_{2}$ gauge structure. Basedon lattice Chern-Simons gauge theory, it is shown that the anti-commuting linkvariables in the SO(3) Majorana representation make it equivalent to anoperator form of compact $\text{U(1)}_{1}$ Chern-Simons Jordan-Wignertransformation in 2d. As examples of its application, we discuss two spinmodels, namely the quantum XY model on honeycomb lattice and the $90^{\circ}$compass model on square lattice. It is shown that under the SO(3) Majoranarepresentation both spin models can be exactly mapped into $Z_{2}$ gauge theoryof spinons, with the standard form of $Z_{2}$ Gauss law constraint.

physical review. b./physical review. b

Properties and application of the SO(3) Majorana representation of spin:  equivalence with the Jordan-Wigner transformation and exact $Z_{2}$ gauge  theories for spin models