# $\mathcal{N}=2$ spin $1/2$ supermultiplet

In Freedman and Van Proeyen's Supergravity, in the footnote on pg. 128, they say

There is a subtle hermiticity requirement for $\mathcal{N}=2$, which requires the multiplet $(-1/2,0,0,1/2)$ must be doubled although it is self-conjugate.

What exactly are they referring to here?

• My guess is that they are referring to the CPT theorem: if the theory contains a multiplet (-1/2,0,0,1/2), it must contain its CPT conjugate too; together these form a full hypermultiplet. The exception is that when the multiplet (-1/2,0,0,1/2) is already CPT invariant, then it is consistent on its own and is called a half hypermultiplet. That happens if it transforms in a pseudo-real representation of gauge/flavour symmetry groups (there is a discussion of this for instance in arxiv.org/abs/1107.0973 ). – Bruno Le Floch Mar 15 '15 at 17:51