While reading a paper I ran into this particular way of writing a $\cal{N}=3$ fields (in a theory with $N_f$ hypermultiplets) that I couldn't relate to anything I had seen before in the text-books (typically Weinberg's) The claim is that $q^{Aa}$ and $\psi_{\alpha}^{Aa}$ are scalar and fermions such that $a,b$ are $SU(R)_R$ indices, $A,B$ are denoting them to transform in the fundamental representation of $USp(2N_f)$ (why?) and $\alpha, \beta$ are $SO(2,1)$ spinor indices (theory is in $2+1$)

Apparently these fields satisfy some reality conditions given by, the following equations,

$(q^\dagger)_{Aa} = \omega_{AB}\epsilon_{ab}q^{Bb}$


$\bar{\psi} ^\alpha _{Aa} = \omega_{AB} \epsilon_{ab} \epsilon ^{\alpha \beta} \psi _{\beta} ^{Bb}$

(where $\omega$ is some symplectic form, which I have no clue about)

I find the above equations very unfamiliar. For one thing why do the scalar and the fermion have $R$ charge indices? (I am only familiar with the supercharges having a R-symmetry when the central charges are absent) Also I don't understand the $USp(2N_f)$ index.

Apparently this can be related to usual $\cal{N}=2$ fields $(Q,\tilde{Q})$ as,

$q^{A1} = (Q,\tilde{Q})$


$q^{A2} = (-\bar{\tilde{Q}}, \bar{Q})$

I am completely unfamiliar with such a relationship or this thinking of $\cal{N}=2$ fields as being thought of as a pair of $Qs$.

Once I understand the meaning of the above I will post another about its curious way of writing the supersymmetry transformation and the lagrangian with this field content.


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