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

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

1996-08-28

arxiv.org/abs/hep-ph/9608467v1

Phys.Lett. B396 (1997) 58-63

Physics

High Energy Physics

High Energy Physics - Phenomenology

latex, 8 pages

Scientific paper

10.1016/S0370-2693(97)00106-8

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.

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