On the structure of the fusion ideal (Q1048046)

Two-dimensional rational conformal field theory has applications to the study of critical phenomena in statistical mechanics and condensed matter physics. Its basic invariant is the associated fusion ring, whose generators are the primary fields of the theory and whose multiplication is determined by dimensions of the spaces of conformal blocks. For a class of conformal field theories associated to loop groups, the fusion ring can be described as a Grothendieck group of positive energy representations, with product structure determined by Connes fusion. In this paper, the author investigates algebraic properties of these fusion rings, and particularly describes a certain bound complexity of the fusion ring of a loop group. He proves that there is a finite level-independent bound on the number of relations defining the fusion ring of positive energy representations of the loop group of a simple, simply connected Lie group. As an illustration, the author gives an explicit computation of the \(G_2\) fusion ring.