# Some questions about flavour and R-symmetry in $2+1$ ${\cal N}=3$ theory

• I have heard this fact that for ${\cal N}=3$ theories in $2+1$ with $N_f$ ${\cal N}=3$ matter fields the flavour symmetry group is $USp(N_f)$, $U(N_f)$ or $SO(2N_f)$ depending on whether the gauge representation in which it is lying is real (like adjoint) or complex (like fundamental) or pseudoreal.

I would like to know of the proof (or reference) for the above.

The following questions stem from, this paper.

• If I use the notation of $\phi_1$ and $\phi_2$ for the N=2 chiral multiplets in which the N=3 splits into (why? how?) then at least for the fundamental case why is $\phi_1$ and $\bar{\phi_2}$ in the same flavour representation?

I guess I can legitimately use the notation of $\phi_1$ and $\phi_2$ to denote what is called $\tilde{Q}$ and $Q$ on page $7$ section $2.2$ of the above. Their notation seems to denote that there is some relationship between the fields $\tilde{Q}$ and $Q$ which I think is not true.

• If $\phi_1$ and $\phi_2$ ${\cal N}=2$ multiplets are in the same ${\cal N}=3$ multiplet then why are $\phi_1$ and $\bar{\phi_2}$ in the same flavour representation and their respective conjugates in the conjugate flavour representation?

Apparently this has got to do with two facts,

• that the coupling is $(\phi_1 T^a \phi_2)^2$ in the deforming potential to ${\cal N}=2$ (..equation 2.9 in the paper linked above..)

• and that the SU(2) R-symmetry of ${\cal N}=3$ rotates in a spin-1/2 representation the 2-tuples $(\phi_2, \phi_1)$, and $(-\bar{\phi_1}, \bar{\phi_2})$ (..equation A.2 page 31 of the above linked paper..)

And what is the explicit action of the $SU(2)$ R-symmetry which rotates the above 2-tuples? And around the above equation A.2 why is are these two 2-tuples called "scalrs"? Aren't they full ${\cal N}=2$ chiral multiplets?

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