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An affine field theory may be constructed by gauging an affine algebra. The momentum integrals of the affine N = 4 Yang–Mills theory are ultraviolet finite but diverge because the sum over states is infinite. If the affine symmetry is broken by postulating a nonvanishing vacuum expectation value for that component of the scalar field lying in the L0 direction, then the theory acquires a linear mass spectrum. This broken theory is ultraviolet finite too, but the mass spectrum is unbounded. If it is also postulated that the mass spectrum has an upper bound (say, the Planck mass), then the resulting theory appears to be altogether finite. The influence of the exotic states has been estimated and, according to the proposed scenario, is negligible below energies at which gravitational interactions become important. The final effective theory has the symmetry of a compact Lie algebra augmented by the operator L0.

The affine extension of supergravity is investigated. We discuss only the loop algebra since the affine extension of the super-Poincaré algebra appears inconsistent. The construction of the affine supergravity theory has been carried out by the group manifold method and leads to an action describing infinite towers of gravitons and gravitinos that interact subject to the symmetries of the loop algebra. The equations of motion satisfy the usual consistency check.