# Chiral anomaly in odd spacetime dimensions

In odd number of space-time dimensions, the Fermions are not reducible (i.e. do not have left-chiral and right-chiral counterparts).

Does this mean that there is no such thing as 'chiral' anomalies in odd number of space-time dimensions, when these fermions are coupled to gauge fields?

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There is no chiral anomaly/gauge anomaly if the spacetime dimension $2\ell+1$ is odd, partly because $SO(2\ell+1)$ has real or pseudo-real representations, but no complex representations.
$$\text{Abelian chiral anomaly in}~ 2\ell+2~ \text{dimensions}$$ $$\downarrow$$ $$\text{Parity anomaly in}~ 2\ell+1~ \text{dimensions}$$ $$\downarrow$$ $$\text{Non-Abelian anomaly in}~ 2\ell~ \text{dimensions}.$$