supersymmetry - Argument for quantum theoretic conformality of $cal{N}=2$ super-Chern-Simon's theory in $2+1$ dimensions

I am using the standard symbols of $V_\mu$ for the gauge field, $\lambda$ for its fermionic superpartner and $F$ and $D$ be scalar fields which make the whole thing a $\cal{N}=2$ vector/gauge superfield in $2+1$ dimensions.

Then the non-Abelian super-Chern-Simon's lagrangian density would be,

$$Tr[\epsilon^{\mu \nu \rho}(V_\mu \partial_\nu V_\rho - \frac{2}{3}V_\mu V_\nu V_\rho) +i\bar{\lambda_a}\lambda_a - 2FD]$$

Clearly this is classically scale invariant.

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  I would like to know of the argument as to why this is also quantum theoretically conformal (..may be there is some obvious symmetry argument which I am missing..)

  
  


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  Also is it true or obvious that if the above is perturbed by a $\lambda Tr[\Phi ^4]$ potential then this might flow to a fixed point which is $\cal{N}=3$ ? And then will it still be superconformal ?

  
  

I would like to know of a way of understanding this phenomenon of supersymmetry enhancement by renormalization flow. If someone could point me to some beginner friendly expository reference regarding this.

Answer

The usual argument for why the Chern-Simons action is exactly conformal is that the action is gauge invariant only if the coupling constant, or Chern-Simons level $k$ (which you have not included in your Lagrangian) is integer valued. If the theory was not conformal there would be a non-zero beta-function which would make the coupling depend continuously on the renormalization scale $\Lambda$. But the integer $k$ can not be a continuos function of $\Lambda$, and hence the beta-function has to vanish. This argument doesn't depend on supersymmetry and hence holds equally well for the $\mathcal{N}=2$ case in your question.

As for the extension to $\mathcal{N}=3$: If I remember correctly you need to add two chiral multiplets $Q$ and $\tilde{Q}$ in conjugate representations of the gauge group and with a kinetic term plus a superpotential of the form you give (something like $W = \frac{1}{k} (\tilde{Q}T^aQ) (\tilde{Q}T^aQ)$). By $\mathcal{N}=3$ supersymmetry the coupling is the same as the Chern-Simons level, and hence the theory is again conformal. Unfortunately I don't know of any good reference which gives a general introduction to this theory. For some recent applications of it see eg papers by Gaiotto and Yin (arXiv:0704.3740) and Aharony, Bergman, Jafferis and Maldacena (arXiv:0806.1218). These references also contain discussions about how the $\mathcal{N}=3$ Chern-Simons theory appears as a conformal fixed point starting from $\mathcal{N}=2$ or $3$ Chern-Simons-matter theory or Yang-Mills-Chern-Simons.