Nahm's conjecture and Y-systems

Abstract

Nahm's conjecture relates q-hypergeometric modular functions to torsion elements in the Bloch group. An interesting class of such functions can be (conjecturally) obtained from a pair (X,X′) of diagrams, each of which is either a Dynkin diagram of type ADE or a diagram of type T. Using properties of Y-systems, we prove that for a matrix of the form A = C(X) ⊗ C(X')-1 where C(X) and C(X') are the corresponding Cartan matrices, every solution of the equation x =(1 - x)A gives rise to a torsion element of the Bloch group.