Gauge Theories With Infinite Multiplets of Fermions

We study the coupling constant renormalization of gauge theories with an infinite multiplet of fermions, using the zeta function method to make sense of the infinite sums over fermions. If the gauge group K is the maximal compact subgroup of a simple non-compact group G, such infinite multiplets can arise naturally, as reductions of discrete series unitary representations of G. The example K=U(1) and G=SU(1,1) will be studied in detail. Surprisingly, there are abelian gauge theories which are asymptotically free; and others that are UV finite.