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

This is in continuation to what I was asking here earlier -

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

Or one can look at this paper - http://arxiv.org/pdf/0704.3740

I am mainly having in mind the $\cal{N}=2$ lagrangian that would be obtained by combining the equations 2.4, 2.5 and 2.7 of that paper.

• In that light I would like to understand the meaning of the comment made in that paper just below equation 2.9

• I have heard that one way of arguing the existence of the $\cal{N}=3$ fixed point in the flow of the matter couplings of the $\cal{N}=2$ theory is to see that for large values of the coupling the theory is effectively behaving like the Wess-Zumino model and for small values it is like pure-super-Chern-Simons theory.

What is known about the signs and the derivatives of the beat functions of these two limiting theories?

Like if it known that the sign of the beta function is different for these two cases then it might suffice to argue by continuity that the beta function does go to zero somewhere in between and then that is a strong candidate to be the $\cal{N}=3$ theory.

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