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# Communications in Number Theory and Physics

## Volume 7 (2013)

### Number 1

### Nahm’s conjecture and $Y$-systems

Pages: 1 – 14

DOI: https://dx.doi.org/10.4310/CNTP.2013.v7.n1.a1

#### Author

#### 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.

Published 11 September 2013