In the last chapter we saw the customary divergences of quantum field theories getting alleviated by supersymmetry. Over the last few years it has been established that in the case of extended supersymmetry there exist theories for which the divergences are not only alleviated but outright eliminated: the theories are finite, no divergences whatsoever!

The first theory where finiteness was noticed, first to two loops (Jones 1977, Poggio & Pendleton 1977), then to three loops (Grisaru, Roček & Siegel 1980, Avdeev, Tarasov & Vladimirov 1980, Caswell & Zanon 1981), and then established to all orders in perturbation theory (Mandelstam 1983, Howe, Stelle & Townsend 1984, Brink, Lindgren & Nilsson 1983, West 1983, Grisaru & Siegel 1982) was the extended N = 4 supersymmetric Yang–Mills theory with arbitrary compact gauge group G, in four space–time dimensions (Brink, Scherk & Schwarz 1977, Gliozzi, Olive & Scherk 1977). The renormalization-group β-function vanishes identically for this theory and to all orders in perturbation theory one finds finite results for all Green's functions. Individual Feynman graphs may diverge but in each order the divergences cancel among the various graphs. This theory may develop spontaneous G-symmetry breaking and monopoles and may be electric–magnetic self-dual (Osborn 1979, Montonen & Olive 1978). Other finite quantum field theories in four space–time dimensions having only N = 2 supersymmetry have since been found (Howe, Stelle & West 1983) and even N = 1 supersymmetric candidates are being explored (Parkes & West 1984, Jones & Mezincescu 1984, Hamidi & Schwarz 1984), with potential phenomenological aims.