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arXiv · 2301.13239

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

Yuma Mizuno
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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.

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Yuma Mizuno. 2023-01-30. Periodic $Y$-Systems and Nahm Sums: The Rank 2 Case. https://doi.org/10.3842/sigma.2025.094

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