A Z2 classification for 2D fermion level crossing

Electroweak vortices may contribute to unsuppressed baryon violating processes in the hot early universe. We explore a possible origin of this phenomenon for a general 2-dimensional embedded defect. 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 ϵ 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 4n + 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.