# Central charge at the fixed point of the ${\cal N}=2$ Landau-Ginzburg theory in $1+1$ dimensions

Let me first believe that the ${\cal N}=2$ Landau-Ginzburg theory does in the IR flow to a non-trivial fixed point and that if the potential is of the form $\Phi ^k$ then the central charge of the CFT at that fixed point is indeed $3(1-\frac{2}{k})$.

• Given the above, how does it somehow obviously follow (by Zamlodchikov's c-theorem?) that if I had the potential as any homogeneous function of degree $k$ in say $n$ scalar chiral fields then the central charge at the similar fixed point will be $3n(1-\frac{2}{k})$?

• I have seen some quite terse arguments but I want to know if there is a simple intuitive way to see that the above theory does have a non-trivial IR fixed point (the claim that I initially started off believing).

• Now its also true that the CFT at this fixed point is infact a minimal model and that apprently follows from some representation theory of the ${\cal N}=2$ superconformal algebra. Can someone kindly give me a reference to that?

• Finally are there known generalizations of this construction to higher supersymmetries or higher diemensions?

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One may calculate the exact central charge by arguments rooted in conformal symmetry and the methods underlying CFTs, see e.g. Polchinski's String Theory vol 2 for a sketch. It obviously can't be from Zamolodchikov's c-theorem because that's just an inequality, not an equality able to predict specific numbers such as $c$. SUSY is conserved, one may prove that the primary fields survive the IR limit, and by counting them, one may identify the IR limiting theory with a CFT whose existence and $c$ and dimensions maybe determined independently of the LG starting point. – Luboš Motl May 17 '12 at 9:19
@Lubos Thanks for the reply. Can you kindly specify as to which chapter of Polchinski's book do you you have in mind? Is it his chapter 15 "Advanced CFT"? – user6818 May 17 '12 at 19:35
Yes, advanced CFT. Looking for Landau-Ginzburg in the indices could also help. – Luboš Motl May 19 '12 at 8:43
@Lubos Motl Thanks! About why I mentioned Zamolodchikov's c-theorem - Witten in his lectures when explaining this point seems to say that this theorem allows one to somehow replace a homogeneous potential function of degree k in n variables by one of the "Fermat Type" i.e $\sum _{i=1} ^{n} \Phi_i ^k$ - and then in some sense this theory is locally a "product" and then the central charge is additive. I did not understand from where this line of reasoning comes from. – user6818 May 20 '12 at 19:16