Abstract
The properties that make massless versions of N = 4 super Yang-Mills theory and a class of N = 2 supersymmetric theories finite are: (i) a universal coupling for the gauge and matter interactions, (ii) anomaly-free representations to which the bosonic and fermionic matter belong, and, (iii) no charge renormalisation, i.e. β( g) = 0. It was conjectured that field theories constructed out of N = 1 matter multiplets are also finite if they too share the above properties. Explicit calculations have verified these theories to be finite up to two loops. The implications of the finiteness conditions for N = 1 finite field theories with SU( M) gauge symmetry are discussed.
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