Proofs of Mizuno's Conjectures on Rank Three Nahm Sums of Index $(1,2,2)$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912327573438464 |
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| author | Wang, Boxue Wang, Liuquan |
| author_facet | Wang, Boxue Wang, Liuquan |
| contents | Mizuno provided 15 examples of generalized rank three Nahm sums with symmetrizer $\mathrm{diag}(1,2,2)$ which are conjecturally modular. Using the theory of Bailey pairs and some $q$-series techniques, we establish a number of triple sum Rogers--Ramanujan type identities. These identities confirm the modularity of all of Mizuno's examples except that two Nahm sums are sums of modular forms of weights $0$ and $1$. We also prove Mizuno's conjectural modular transformation formulas for two vector-valued functions consisting of Nahm sums with symmetrizers $\mathrm{diag}(1,1,2)$ and $\mathrm{diag}(1,2,2)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_21725 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Proofs of Mizuno's Conjectures on Rank Three Nahm Sums of Index $(1,2,2)$ Wang, Boxue Wang, Liuquan Number Theory Classical Analysis and ODEs Combinatorics 11P84, 05A30, 33D15, 33D60, 11F03 Mizuno provided 15 examples of generalized rank three Nahm sums with symmetrizer $\mathrm{diag}(1,2,2)$ which are conjecturally modular. Using the theory of Bailey pairs and some $q$-series techniques, we establish a number of triple sum Rogers--Ramanujan type identities. These identities confirm the modularity of all of Mizuno's examples except that two Nahm sums are sums of modular forms of weights $0$ and $1$. We also prove Mizuno's conjectural modular transformation formulas for two vector-valued functions consisting of Nahm sums with symmetrizers $\mathrm{diag}(1,1,2)$ and $\mathrm{diag}(1,2,2)$. |
| title | Proofs of Mizuno's Conjectures on Rank Three Nahm Sums of Index $(1,2,2)$ |
| topic | Number Theory Classical Analysis and ODEs Combinatorics 11P84, 05A30, 33D15, 33D60, 11F03 |
| url | https://arxiv.org/abs/2407.21725 |