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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2301.13239 |
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| _version_ | 1866915593986244608 |
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| author | Mizuno, Yuma |
| author_facet | Mizuno, Yuma |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_13239 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Periodic $Y$-Systems and Nahm Sums: The Rank 2 Case Mizuno, Yuma Quantum Algebra Mathematical Physics Number Theory 11F03, 11P84, 13F60 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. |
| title | Periodic $Y$-Systems and Nahm Sums: The Rank 2 Case |
| topic | Quantum Algebra Mathematical Physics Number Theory 11F03, 11P84, 13F60 |
| url | https://arxiv.org/abs/2301.13239 |