Harmonic Spinors and $Z_2$ Vortex

Hodge theorem and harmonic spinors are studied in a physics-oriented approach in the present paper. New mathematical results on the harmonic spinors are as follows. Harmonic spinors defined by partial differential operators could be of two types: trivial without topological defects, and having nontrivial topological structures, for example, phase singularities or phase vortices. There could exist a nontrivial harmonic vector field associated with nontrivial harmonic spinor, for example, ${\bf v}_{vortex}$ associated with Weyl 2-spinor. The $Z_2$-vortex is re-visited in the perspective of harmonic spinors leading to a remarkable result that the gauge potential is exactly the same as the nontrivial harmonic vector field associated with the 2-spinor. It is proposed that a discrete symmetry group $SL(2, Z)$ has a role in connection with the continuous group $SU(2)$ similar to the discrete group $Z_2$ in $U(1)$.