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?

  • $\begingroup$ 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 ). $\endgroup$ – Bruno Le Floch Mar 15 '15 at 17:51

Indeed, N=2 is much more stringent than N=1, and, significantly, it contains a central charge. An N=1 scalar multiplet is CPT self-conjugate, and consists of one Majorana spinor, a scalar, and a pseudoscalar, so (-1/2,0,0,1/2). You might, very naively, think that with two supersymmetry generators available, you could still transition from the spin 1/2 state to either the scalar or the pseudo scalar, each with a different susy generator, and then back to the spin -1/2 fermion. But it turns out this is impossible, so you need two such scalar multiplets, rotating into each other by the peculiar central charge of that algebra (N=2).

Fayet (1976), "Fermi-bose hypersymmetry." NuclPhys B113 135-155 in his hyper symmetry paper, saw that, so it would lead to a complex structure instead of relying on it; but if his language were unclear, to see the doubling is actually inevitable in real variables, without assuming anything, you may try the two-liner brute-force proof on pp 15-16 of Zachos' 1979 dissertation. In essence, CPT and hermiticity conspire to require "more room" for the relevant Pascal triangle.

The requirement of two Majorana spinors, two scalars, and two pseudoscalars to realize N=2 is best evident in the two supercurrents (2.1-8) of that second reference: it is manifest that just one spinor and one scalar can only specify one, not two, distinct supercurrents. You must tack together two N=1 scalar supermultiplets to get the N=2 variant.


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