Nahm's conjecture and Y-systems

TL;DR

This paper explores Nahm's conjecture by linking solutions of specific algebraic equations derived from Dynkin and T-type diagrams to torsion elements in the Bloch group, using properties of Y-systems.

Contribution

It proves that solutions of certain equations associated with Cartan matrices from Dynkin and T diagrams produce torsion elements in the Bloch group, advancing understanding of Nahm's conjecture.

Findings

Solutions of the equation $ extbf{x}=(1- extbf{x})^A$ correspond to torsion elements in the Bloch group.

The paper establishes a connection between Y-systems and torsion elements in the context of Nahm's conjecture.

It provides a proof for a class of functions related to pairs of diagrams of types $ADE$ and $T$.

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 $A=\mathcal{C}(X)\otimes \mathcal{C}(X')^{-1}$ where $\mathcal{C}(X)$ and $\mathcal{C}(X')$ 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.