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Seiberg traced the nonrenormalization of supersymmetric theories to holomorphy of the superpotential in chiral superspace. However, this overlooks the fact that with a different number of supersymmetric generators, supersymmetry can be real or symplectic, instead of complex. But yet, even for those cases, we still have nonrenormalization. If holomorphy isn't the reason, what is?

If the reason for nonrenormalizability is totally different for different dimensions and number of supersymmetry generators and choice of superfields, is it then ad hoc that nonrenormalizability always holds in each case?

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@QGR Such things are better as comments. – mbq Feb 26 '11 at 21:58

The real reason is, of course, supersymmetry. You can see it in various ways: some quantities don't renormalize since potential divergences cancel between bosonic and fermionic contributions. Or, another way to say it is that potential counterterms are not supersymmetric. This goes beyond renormalizing the action, for example with extended SUSY BPS states get no mass corrections for reasons very similar to non-renormalization theorems.

For certain quantities (superpotential, gauge coupling function, prepotential in N=2) holomorphy is an efficient way to summarize the various restrictions coming from supersymmetry, but it is not the underlying reason for them. Certainly for extended SUSY theories there are more restrictions due to SUSY, not less, but since there may not be a superspace formulation holomorphy is typically not part of the story there.

As for your edited question: with different type of supersymmetry (different number, dimensions etc.) different quantities are simplified in different ways. There is no "unified theory of non-renormalization" in all supersymmetric theories, regardless of context. But, each non-renormalization theorem holds for a whole class of theories and quantities, so I would not describe them as "ad hoc".

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Crisp words, Moshe, +1. – Luboš Motl Feb 26 '11 at 17:41
Thanks @Lubos, we were typing at the same time, saying pretty much the same thing. – user566 Feb 26 '11 at 17:44

First of all, when Seiberg's papers say that a non-renormalization theorem may be proved by holomorphy, be sure that Seiberg is right. But it doesn't mean that holomorphy is the only argument that may become crucial in a proof of such theorems.

For example, a complex spinor may decompose into real or pseudoreal spinors in a lower dimension but some of the non-renormalization theorems may still apply. Also, the BPS states (annihilated by a subset of supercharges) have masses that are totally dictated by the "central charges" and don't receive any quantum corrections; this argument is independent of holomorphy. It simply follows from the vanishing of some $QQ$ bilinears in the state, which - by SUSY algebra - may be rewritten as a difference between the energy/mass and some central charges.

Other proofs of non-renormalization theorems may involve perturbative arguments that things cancel at each order because SUSY prohibits the "wrong terms" in each case, because of some dimensional analysis. In principle, those arguments - although they are familiar in the holomorphic context - don't depend on holomorphy, either.

I think it is misleading to ask about all non-renormalization theorems simultaneously and expect that there is a single word, such as holomorphy, that contains proofs to all of them. Different non-renormalization theorems have different proofs that use different ideas: the required ideas depend not only on the dimension but also on the precise SUSY algebra, precise theory, and even the precise quantity whose non-renormalization we're proving. And in each case, there may exist several highly inequivalent proofs of the same proposition. You would have to ask more specific questions to get more specific answers.

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Here's my stab at answering your question. We have a supersymmetric quantum field theory (any number of dimensions or supersymmetry generators). In general, it's not a superconformal theory, but we can still analyze it using the AdS/CFT correspondence. To be sure, the geometry deviates from pure anti de Sitter. The couplings characterizing the superpotential coefficients form bulk short supermultiplets. The couplings characterizing the Käaut;hler potential coefficients form bulk long supermultiplets. Being supersymmetric, the dual AdS theory has to be in a BPS configuration. This constrains the VEV of bulk short supermultiplets to vary as a power law in the AdS coordinate $z$ with the exponent fixed. Zero beta function here according to the holographic renormalization group. BPS doesn't constrain the $z$-dependence of the VEV of long multiplets. Nonzero beta functions are permissible.

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