Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 094, 17 pages      arXiv:2301.13239      https://doi.org/10.3842/SIGMA.2025.094

Periodic $Y$-Systems and Nahm Sums: The Rank 2 Case

Yuma Mizuno
School of Mathematical Sciences, University College Cork, Western Gateway Building, Western Road, Cork, Ireland

Received June 04, 2025, in final form October 26, 2025; Published online November 03, 2025

Abstract
We classify periodic $Y$-systems of rank 2 satisfying the symplectic property. We find that there are six such $Y$-systems. In all cases, the periodicity follows from the existence of two reddening sequences associated with the time evolution of the $Y$-systems in positive and negative directions, which gives rise to quantum dilogarithm identities associated with Donaldson-Thomas invariants. We also consider $q$-series called the Nahm sums associated with these $Y$-systems. We see that they are included in Zagier's list of rank 2 Nahm sums that are likely to be modular functions. It was recently shown by Wang that they are indeed modular functions.

Key words: cluster algebras; $Y$-systems; Nahm sums.

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