Harmonic Spinors and $Z_2$ Vortex

TL;DR

This paper explores harmonic spinors and $Z_2$ vortices from a physics perspective, revealing new mathematical insights into their topological structures and gauge potentials, with implications for symmetry groups in gauge theories.

Contribution

It introduces novel mathematical results on harmonic spinors, linking nontrivial topological structures with gauge potentials and proposing a role for $SL(2, Z)$ in symmetry group relations.

Findings

Harmonic spinors can be trivial or topologically nontrivial.

The gauge potential coincides with the harmonic vector field of a 2-spinor.

A proposed connection between $SL(2, Z)$ and $SU(2)$ symmetry groups.

Abstract

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 $U(1)$.