Chiral anomaly in odd spacetime dimensions

Question

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?

score 16 · Accepted Answer · answered Nov 03 '12 at 16:17

16

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.

answered Nov 03 '12 at 16:17

Qmechanic

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awesome answer, was trying to figure out why myself! – Dylan O. Sabulsky Nov 03 '12 at 16:52
Awesome!! What's a non-abelian anomaly? Is there an example of it in the Standard Model? – QuantumDot Nov 04 '12 at 00:39

Chiral anomaly in odd spacetime dimensions

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