# Conformal invariance of $N = 4$ Supersymmetric Yang-Mills theory

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I will quote the following from the Wikipedia article on Supersymmetry Nonrenormalization theorems [1],

"In $N=4$ super Yang–Mills the $\beta$-function is zero for all couplings, meaning that the theory is conformal."

When we say that the $\beta$-function vanishes for a QFT, we conclude that scale invariance remains preserved at the quantum level.

However scale invariance doesn't necessarily imply conformal invariance, e.g. this paper has two such examples [2]. Hereby I am getting confused, this paper by Sohnius and West [3] originally shows why the $\beta$- function for $N=4$ super Yang–Mills vanishes. But then why is the claim for conformal invariance justified?

recategorized May 21, 2018

I had posted the same question on PSE, and got a good answer. https://physics.stackexchange.com/q/407265/50770

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Isn't the claim of (super)conformal invariance justified by the fact that up to the trace anomaly the trace of the stress energy tensor is zero? This is exactly what you would expect for a CFT and this is exactly what happens at the origin of the moduli space of the $\mathcal{N}=4$ SYM theory.

P.S. In general, if specific conditions are satisfied, for a scale invariant theory one can make $T_{\mu \nu}$ such that $T_{\mu}^{\mu} =0$ which implies the conformal invariance. To understand what happens with the trace anomaly is a long story that need to be carefully studied.

answered May 24, 2018 by (3,625 points)
edited May 24, 2018

Sorry, misprint. Look again.

Is there a proof that under natural conditions, a vanishing trace implies conformal symmetry, or is it just something observed in typical examples?

No, I don't think there is a proof for that.

@conformal_gk I had a misunderstanding earlier that $T_{\mu}^{\mu} = 0$ implies only that the charge associated with dilatations is conserved, but it turns out that the charge associated with special conformal transformations is conserved too. So yeah, my problem is solved, for earlier I had the misconception that only scale invariance is preserved with this this condition.

The relation between scale invariance and conformal invariance is discussed  a bit in chapter 1 of https://arxiv.org/pdf/1601.05000.pdf.

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