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The effective gauge field action due to fermions coupled to SU(N) gauge fields in three dimensions is found to change by ±π|n| under a homotopically nontrivial gauge transformation with winding number n. This gauge noninvariance can be eliminated by adding a parity-violating topological term to the action, or by regulating the theory in a way which produces this term automatically in the effective action. The Euler-Heisenberg effective action is calculated in the SU(2) theory and in QED.
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Amanda Redlich (1984) studied this question.
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