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I am particularly thinking of the theory described in section $6$ (starting page 31) of this paper, https://arxiv.org/abs/1104.0680.

The exact Lagrangian has never been explicitly stated here but only indicated to be $(a)$ ${\cal N}=2$ supersymmetric Chern-Simons theory on $S^2 \times S^1$ "real" space-time and $(b)$ has a single super charge $Q$ and a single adjoint chiral field $\Phi$ and $(c)$ seems to have a potential term of the form, $\frac{\lambda}{N}\textrm{Tr}[\Phi^4]$ where the gauge group is $SU(N)$.

Can someone help construct the explicit Lagrangian for this theory on the superspace and from that derive the on-shell action of the supersymmetry operator on the classical fields?

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