Gravitational coupling of affine field theories

The coupling of a N=4 affine gauge theory to N=1 supergravity is investigated. For the unbroken Kac-Moody theory the known sum rule for finite Lie multiplets is extended to cover the infinite dimensional Kac-Moody multiplet. For the broken Kac-Moody theory, the infinite Kac-Moody multiplet splits into an infinite set of finite multiplets representing a finite Lie symmetry * the U(1) generated by L₀; in this case there is a separate sum rule for each Lie multiplet. The gravitationally coupled theory is more restricted but at the level of this investigation is not qualitatively different from the corresponding global theory.