Difference equations arising from cluster algebras

TL;DR

This paper characterizes difference equations from cluster algebras using T-data matrices, generalizes known solutions, and explores their positivity and modular properties, linking to Nahm's conjecture and hypergeometric q-series.

Contribution

It introduces T-data matrices to characterize Y/T-system difference equations and extends the classification of mutation loops beyond period 1 quivers.

Findings

All mutation loops are derived from T-data.

Periodic T-datum systems exhibit positivity.

Proposes a cluster algebra perspective on Nahm's conjecture.

Abstract

We characterize Y/T-system type difference equations arising from cluster algebras by triples of matrices, which we call T-data, that have a certain symplectic property. We show that all mutation loops are essentially obtained from T-data, which generalizes the general solution for period 1 quivers given by Fordy and Marsh. We also show that any T-datum associated with a periodic Y/T-system has the simultaneous positivity. As an application, we propose a version of Nahm's conjecture from a viewpoint of cluster algebras. We conjecture that given a periodic T/Y-system of a certain type, we have a family of hypergeometric q-series that are also modular functions.