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Some part of the local gauge symmetries in the low energy region, say, lowerthan GUT or the Planck energy can be an induced symmetry describable with theholonomy fields associated with a topologically non-trivial structure ofpartially compactified space. In the case where a six dimensional space iscompactified by the Kaluza-Klein mechanism into a product of the fourdimensional Minkowski space $M_{4}$ and a two dimensional Riemann surface withthe genus $g$, $\Sigma_{g}$, we show that, in a limit where thecompactification mass scale is sent to infinity, a model lagrangian with a U(1)gauge symmetry produces the dynamical gauge fields in $M_{4}$ with a product of$g$ U(1)'s symmetry, i.e., U(1)$\times \cdots\times$U(1). These fields areinduced by a Berry phase mechanism, not by the Kaluza-Klein. The dynamicaldegrees of freedom of the induced fields are shown to come from the holonomies,or the solenoid potentials, associated with the cycles of $\Sigma_{g}$. Theproduction mechanism of kinetic energy terms for the induced fields arediscussed in detail.Comment: 18 pages and 2 figures(available upon request), OU-HET 192, LaTeX  fil

DYNAMICAL GAUGE FIELD INDUCED BY THE BERRY PHASE MECHANISM