Modularity of tadpole Nahm sums in ranks 4 and 5
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908336796991488 |
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| author | Shi, Changsong Wang, Liuquan |
| author_facet | Shi, Changsong Wang, Liuquan |
| contents | Around 2016, Calinescu, Milas and Penn conjectured that the rank $r$ Nahm sum associated with the $r\times r$ tadpole Cartan matrix is modular, and they provided a proof for $r=2$. The $r=3$ case was recently resolved by Milas and Wang. We prove this conjecture for the next cases $r=4,5$. We also prove the modularity of some companion Nahm sums by establishing the corresponding Rogers--Ramanujan type identities. A key new ingredient in our proofs is some rank reduction formulas which allow us to decompose higher rank tadpole Nahm sums to mixed products of some lower rank Nahm-type sums and theta functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17737 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modularity of tadpole Nahm sums in ranks 4 and 5 Shi, Changsong Wang, Liuquan Number Theory Combinatorics 11P84, 33D15, 33D45, 11F03 Around 2016, Calinescu, Milas and Penn conjectured that the rank $r$ Nahm sum associated with the $r\times r$ tadpole Cartan matrix is modular, and they provided a proof for $r=2$. The $r=3$ case was recently resolved by Milas and Wang. We prove this conjecture for the next cases $r=4,5$. We also prove the modularity of some companion Nahm sums by establishing the corresponding Rogers--Ramanujan type identities. A key new ingredient in our proofs is some rank reduction formulas which allow us to decompose higher rank tadpole Nahm sums to mixed products of some lower rank Nahm-type sums and theta functions. |
| title | Modularity of tadpole Nahm sums in ranks 4 and 5 |
| topic | Number Theory Combinatorics 11P84, 33D15, 33D45, 11F03 |
| url | https://arxiv.org/abs/2504.17737 |