Induced fermionic charge in background gauge theories in odd space-time dimensions

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Abstract
We have shown by using gauge- and parity-invariant regularization schemes such as dimensional and higher-covariant-derivative methods that the induced fermionic charge in a U(1) gauge theory in odd space-time dimensions is zero when the fermion mass is strictly equal to zero and recedes towards the boundary when one starts with a nonzero fermion mass and allows it to approach zero. In the latter case the induced charge is exactly the Chern-Simons index.