# $USp$ global symmetry in $d=3$, ${\cal N}=4$ supersymmetric QFT

Define a 3-dimensional QFT with $N=4$ supersymmetry (4 supercharges), and the field content is $g$ $N=4$ hyper-multiplets (that are in a representation $R$ of some group $G$).

Each hyper-multiplet is composed of 2 $N=2$ chiral-multiplets $Q,\tilde{Q}$ with complex conjugate representations $R,\bar R$. Therefore, the global (flavor) symmetry in the system would be $SU(g) \times SU(g) \times U(1)$, where each set of chirals transforms as a fundamental of $SU(g)$, and the $U(1)$ is a rotation between the 2 sets of chirals.

However, if $R$ is real (for example adjoint) then $R,\bar R$ are the same, the 2 sets of chirals are indistinguishable so the global symmetry is the larger $SU(2g)$.

Now, add an $N=4$ vector-multiplet (that is also adjoint of $G$) to the system, the superpotential gets a contribution of interaction terms between the chiral fields in the hyper-multiplet $Q,\tilde{Q}$ and the chiral field in the vector-multiplet $\Phi$ of the form $\sim Q\Phi\tilde{Q}$.

The statement is that this breaks the global symmetry $SU\left(2g\right)$ (of dimension $\left(2g\right)^{2}-1$) to the smaller $USp\left(2g\right)$ (of dimension $g\left(2g+1\right)$).

Why is that? How to see this from the Lagrangian?

• Statement from where? – Qmechanic Aug 15 '18 at 10:15