The asymptotically-free gauge theories

Abstract

We show how to classify the asymptotically-free gauge theories in four spacetime dimensions, focussing here on the case of purely fermionic matter. The classification depends on the fact (which we prove) that the dimension and Dynkin index of irreducible representations of a simple Lie algebra are both strictly-increasing, integer-valued functions of each Dynkin label. This implies not only that the number of asymptotically-free representations of any one semisimple Lie algebra is finite, but also that they can be written down in a systematic fashion using tables for the asymptotically-free irreducible representations of simple Lie algebras, which we supply. These tables show that at most two out of a possible ten Dynkin labels can be non-zero and that no Dynkin label can exceed four. We also discuss further constraints coming from the need to cancel anomalies, both local and global.