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?


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.

There may instead be parity anomalies in odd spacetime dimensions. In fact, there is a dimensional ladder of related anomalies

$$\text{Abelian chiral anomaly in}~ 2\ell+2~ \text{dimensions}$$ $$ \downarrow$$ $$\text{Parity anomaly in}~ 2\ell+1~ \text{dimensions}$$ $$ \downarrow$$ $$\text{Non-Abelian anomaly in}~ 2\ell~ \text{dimensions}.$$

See e.g. M. Nakahara, Geometry, Topology and Physics, Section 13.6.

  • $\begingroup$ awesome answer, was trying to figure out why myself! $\endgroup$ Nov 3 '12 at 16:52
  • $\begingroup$ Awesome!! What's a non-abelian anomaly? Is there an example of it in the Standard Model? $\endgroup$
    – QuantumDot
    Nov 4 '12 at 0:39

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