meaning that the theory is conformal. 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. The full nonperturbative proof by Nathan Seiberg appeared in the 1988 article Supersymmetry and Nonperturbative beta Functions. Examples in 3-dimensional theories In 3 dimensions the number counts the number of 2-component Majorana spinors of supercharges. When there is no holomorphicity and few exact results are known. When the superpotential cannot depend on the linear multiplets and in particular is independent of the Fayet–Iliopoulos terms (FI) and Majorana mass terms. 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. When , unlike , the R-symmetry is the nonabelian group SU(2) and so the representation of each field is not renormalized. In a super conformal field theory the conformal dimension of a chiral multiplet is entirely determined by its R-charge, and so these conformal dimensions are not renormalized. Therefore matter fields have no wave function renormalization in superconformal field theories, as was shown in On Mirror Symmetry in Three Dimensional Abelian Gauge Theories. These theories consist of vector multiplets and hypermultiplets. The hypermultiplet metric is hyperkähler and may not be lifted by quantum corrections, but its metric may be modified. No renormalizable interaction between hyper and abelian vector multiplets is possible except for Chern–Simons terms. When , unlike the hypermultiplet metric may no longer be modified by quantum corrections. Examples in 2-dimensional theories In 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. Nonrenormalization from a quantization condition In supersymmetric and nonsupersymmetric theories, the nonrenormalization of a quantity subject to the Dirac quantization condition is often a consequence of the fact that possible renormalizations would be inconsistent with the quantization condition, for example the quantization of the level of a Chern–Simons theory implies that it may only be renormalized at