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 $(X,X^{\prime })$ 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 $A=\mathcal{C}(X)\otimes \mathcal{C}(X^{\prime }{)}^{-1}$ where $\mathcal{C}(X)$ and $\mathcal{C}(X^{\prime })$ are the corresponding Cartan matrices, every solution of the equation $\mathbf{x}=(1-\mathbf{x}{)}^A$ gives rise to a torsion element of the Bloch group.

11 pages, v4. published version, minor update (typos corrected, references added)

Cited by in corpus (2)

• Positivity and periodicity of $Q$-systems in the WZW fusion ring
• Three Dimensional Topological Field Theories and Nahm Sum Formulas