The $SU(2)$ Global Anomaly Through Level Circling

Abstract

We discuss a novel manifestation of the $SU(2)$ global anomaly in an $SU(2)$ gauge theory with an odd number of chiral quark doublets and arbitrary Yukawa couplings. We argue that the massive 4-dim.($D=4$) Euclidean Dirac operator is nonhermitean with its spectrum of eigenvalues $(\lambda,-\lambda)$ lying in pairs in the complex plane. Consequently the existence of an odd number of normalizable zero modes of the 5-dim.($D=5$) massive Dirac operator is equivalent to a fermionic level exchange phenomenon, level ``circling'', under continuous topologically nontrivial deformations of the external gauge field. More generally global anomalies are a manifestation of fermionic level ``circling'' in any $SP(2n)$ gauge theory with an odd number of massive fermions in the spinor representation and arbitrary Yukawa couplings.