A ${\bf Z}_2$ Classification for 2D Fermion Level Crossing

We demonstrate that the number of fermionic zero modes of the static $2$-dimensional Dirac operator in the background of $SU(2)$ static gauge-Higgs field configurations is a topological invariant modulo four. Static configurations which are everywhere odd under parity with even-parity pure gauge behaviour at infinity admit $4n$, $n\in {\bf Z},$ zero modes of the Jackiw-Rebbi (JR) type. Odd-parity configurations with odd-parity pure gauge behaviour at infinity are topologically disconnected from the vacuum and admit $4 n + 2$ fermionic zero energy solutions. The classification implies the collapse of half of the fermion zero modes upon embedding a $2$-dimensional gauge-Higgs configuration (string) with odd-parity pure gauge behaviour at infinity into the $3$-dimensional Minkowski space.