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I'm currently reading Physics and Geometry by Witten, which I really liked up to the point where he claimed that we exchange representations $R$ and $\tilde R$ under parity transformations, where $R$ is some rep of the gauge group $G$ and $\tilde R$ its dual (page 12 in the pdf).

The spinor rep of the Lorentz group decomposes as $$ S = S_- \oplus S_+ $$

Where $-$ indicates negative and $+$ positive chirality.

Together with the gauge group reps $R$, our fermion representations read

$$ W_+ = S_+ \otimes V_R $$

$$ W_- = S_- \otimes V_{\tilde R},$$

where $V_R$ is the vector bundle associated with the rep $R$. The fact that we must multiply $S_-$ with $V_{\tilde R}$, follows because our fermion representation $W = W_- \oplus W_+$ must be real.

Now, under parity we have clearly $ S_- \leftrightarrow S_+$. (Parity transformations are part of the Lorentz group and it can be nicely seen that these transformation exchange negative and positive chirality reps of the proper orthochronous Lorentz group. ) Al this makes sense to me. But then he argues:

In the basic decomposition $ S = S_- \oplus S_+ $ of the spinor representation S into spinors S± of positive and negative chirality, the distinction between $S_+$ and $S_-$ is a matter of convention. Under a change of the orientation of space-time, called a parity transformation by physicists, $S_+$ and $S_-$ are exchanged. The representations $R$ and $\tilde R$ are therefore exchanged by parity. If we assume that the laws of nature are invariant under parity, then $R$ and $\tilde R$ must be isomorphic.

Uhm what? Why should this be the case?

EDIT: He obviously needs this for his line of arguments, but I can't think of any reason why our gauge group should care about a spacetime transformation. He basically argues that parity is violated because fermions transform according to a complex representations of the gauge group. This argument does not make sense to me either, because $SU(2)$ breaks parity and $SU(2)$ has no complex reps.

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  • $\begingroup$ It is my understanding that he his building a Dirac spinor in even dimensions. In that case, inside a given Dirac spinor, the different parity components have to have isomorphic representations of the Gauge group (otherwise Lorentz transformations and Gauge transformations would not commute). Would that make sense? And remember that "charge conjugation" refers to a Lorentz discrete symmetry that interchanges negative and positive energy solutions, i.e. particles and anti-particle. So, R and tilde R refer to particle and anti-particle gauge irreps. $\endgroup$ – romanovzky May 14 '15 at 21:12

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