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

TL;DR

This work resolves the classification of periodic $Y$-systems of rank $2$ satisfying a symplectic constraint, proving that exactly six indecomposable systems occur and that their periodicity follows from reddening sequences in both time directions, which also yields quantum dilogarithm identities tied to Donaldson--Thomas invariants. It connects these rank-$2$ systems to Nahm sums, showing that the associated modular Nahm sums arise from $K=A_+(1)^{-1}A_-(1)$ and lie in Zagier's rank-$2$ list, with modularity established by Wang. The paper also delineates how the cluster-algebra framework constructs universal $Y$-solutions from the data $(m r,m n)$, and analyzes the effect of changing slices to classify the six finite-type cases, each associated with a finite-type quiver component (types $A_2$, $A_4$, $E_6$, $D_4$, $A_5$, $A_4$). These results illuminate the intersection of $Y$-systems, cluster algebras, and modularity phenomena in rank $2$, and suggest extensions to higher rank and skew-symmetrizable settings.

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.

Theorems & Definitions (31)

• Definition 1.1
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• Definition 1.3
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• Theorem 1.5
• Remark 1.6
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• Theorem 1.10