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.
Keywords:cluster algebras, $Y$-systems, Nahm sums.
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