Renormalization theorems are common in theories with a sufficient amount of supersymmetry, usually at least 4 supercharges.
Perhaps the first nonrenormalization theorem was introduced by Marcus T. Grisaru, Martin Rocek and Warren Siegel in their 1979 paper Improved methods for supergraphs.
Nonrenormalization theorems in supersymmetric theories are often consequences of the fact that certain objects must have a holomorphic dependence on the quantum fields and coupling constants.
4D SUSY theory involving only chiral superfields, the superpotential is immune from renormalization.
4D SUSY theory the moduli space of the hypermultiplets, called the Higgs branch, has a hyper-Kähler metric and is not renormalized.
In the article Lagrangians of N=2 Supergravity - Matter Systems it was further shown that this metric is independent of the scalars in the vector multiplets.
They also proved that the metric of the Coulomb branch, which is a rigid special Kähler manifold parametrized by the scalars in
Therefore the vacuum manifold is locally a product of a Coulomb and Higgs branch.
Also there are no perturbative corrections to the β-function beyond one-loop, as was shown in 1983 in the article Superspace Or One Thousand and One Lessons in Supersymmetry by Sylvester James Gates, Marcus Grisaru, Martin Rocek and Warren Siegel.
This was demonstrated perturbatively by Martin Sohnius and Peter West in the 1981 article Conformal Invariance in N=4 Supersymmetric Yang-Mills Theory under certain symmetry assumptions on the theory, and then with no assumptions by Stanley Mandelstam in the 1983 article Light Cone Superspace and the Ultraviolet Finiteness of the N=4 Model.
On the other hand the central charge is independent of the chiral multiplets, and so is a linear combination of the FI and Majorana mass terms.
These two theorems were stated and proven in Aspects of N=2 Supersymmetric Gauge Theories in Three Dimensions.
, the R-symmetry is the nonabelian group SU(2) and so the representation of each field is not renormalized.
[clarification needed] linear sigma models, which are superrenormalizable abelian gauge theories with matter in chiral supermultiplets, Edward Witten has argued in Phases of N=2 theories in two-dimensions that the only divergent quantum correction is the logarithmic one-loop correction to the FI term.
In the 1994 article Nonrenormalization Theorem for Gauge Coupling in 2+1D the authors find the renormalization of the level can only be a finite shift, independent of the energy scale, and extended this result to topologically massive theories in which one includes a kinetic term for the gluons.
In Notes on Superconformal Chern-Simons-Matter Theories the authors then showed that this shift needs to occur at one loop, because any renormalization at higher loops would introduce inverse powers of the level, which are nonintegral and so would be in conflict with the quantization condition.