Modularity of Some Nahm Sums as Vector-valued Functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909440380239872 |
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| author | Wang, Liuquan Zhang, Huohong |
| author_facet | Wang, Liuquan Zhang, Huohong |
| contents | Zagier observed that modular Nahm sums associated with the same matrix may form a vector-valued modular function on some congruence subgroup. We establish modular transformation formulas for several families of Nahm sums by viewing them as vector-valued functions, and thereby we show that they are indeed modular on the congruence subgroup $Γ_0(N)$ with $N=1,2,3,4$. In particular, we prove two transformation formulas discovered by Mizuno related to the Kanade--Russell mod 9 conjecture and Capparelli's identities. We also establish vector-valued transformation formulas for some theta series. As applications, we give modular transformation formulas for various families of Nahm sums involving those in the Andrews--Gordon identities and Bressoud's identities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_18590 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modularity of Some Nahm Sums as Vector-valued Functions Wang, Liuquan Zhang, Huohong Number Theory Combinatorics 11F03, 11P84, 33D15, 33D60 Zagier observed that modular Nahm sums associated with the same matrix may form a vector-valued modular function on some congruence subgroup. We establish modular transformation formulas for several families of Nahm sums by viewing them as vector-valued functions, and thereby we show that they are indeed modular on the congruence subgroup $Γ_0(N)$ with $N=1,2,3,4$. In particular, we prove two transformation formulas discovered by Mizuno related to the Kanade--Russell mod 9 conjecture and Capparelli's identities. We also establish vector-valued transformation formulas for some theta series. As applications, we give modular transformation formulas for various families of Nahm sums involving those in the Andrews--Gordon identities and Bressoud's identities. |
| title | Modularity of Some Nahm Sums as Vector-valued Functions |
| topic | Number Theory Combinatorics 11F03, 11P84, 33D15, 33D60 |
| url | https://arxiv.org/abs/2412.18590 |