Status of the conjecture on the commutative algebra of Verlinde rings used in Braun’s computation

Determine whether the conjecture concerning the commutative algebra structure of Verlinde rings (the fusion rings of positive-energy representations of loop groups at a fixed level) that was used in Braun’s approach to compute the twisted K-theory K_h^•(G) of compact, simply connected, simple Lie groups G at positive integer level h holds in all remaining cases; specifically, establish the conjecture for each such G and h where it is not yet proven, or provide explicit counterexamples.

Background

In the discussion of D-brane charges in the WZW model, the paper reviews methods to compute the non-equivariant twisted K-theory of a compact, simply connected simple Lie group G, which classifies D-branes at a given level. Early computations were followed by a more comprehensive approach due to Braun, which depended on a conjecture about the commutative algebra of Verlinde rings.

While later work by Douglas provided unconditional results using different topological methods, the status of the specific commutative-algebra conjecture invoked by Braun remains unresolved in general, being known in some cases but potentially open in others. Clarifying its validity across all groups and levels would solidify the algebraic foundations underlying those computations.