Counterexamples to Zagier's Duality Conjecture on Nahm Sums
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909456334323712 |
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| author | Wang, Liuquan |
| author_facet | Wang, Liuquan |
| contents | Given any positive integer $r$, Nahm's problem is to determine all $r\times r$ rational positive definite matrix $A$, $r$-dimensional rational vector $B$ and rational scalar $C$ such that the rank $r$ Nahm sum associated with $(A,B,C)$ is modular. Around 2007, Zagier conjectured that if the rank $r$ Nahm sum for $(A,B,C)$ is modular, then so is the dual Nahm sum associated with $(A^{-1},A^{-1}B,B^\mathrm{T} A^{-1}B/2-{r}/{24}-C)$. We construct some explicit rank four Nahm sums which are modular while their duals are not modular. This provides counterexamples to Zagier's duality conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09701 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counterexamples to Zagier's Duality Conjecture on Nahm Sums Wang, Liuquan Number Theory Combinatorics 11P84, 33D15, 33D60, 11F03 Given any positive integer $r$, Nahm's problem is to determine all $r\times r$ rational positive definite matrix $A$, $r$-dimensional rational vector $B$ and rational scalar $C$ such that the rank $r$ Nahm sum associated with $(A,B,C)$ is modular. Around 2007, Zagier conjectured that if the rank $r$ Nahm sum for $(A,B,C)$ is modular, then so is the dual Nahm sum associated with $(A^{-1},A^{-1}B,B^\mathrm{T} A^{-1}B/2-{r}/{24}-C)$. We construct some explicit rank four Nahm sums which are modular while their duals are not modular. This provides counterexamples to Zagier's duality conjecture. |
| title | Counterexamples to Zagier's Duality Conjecture on Nahm Sums |
| topic | Number Theory Combinatorics 11P84, 33D15, 33D60, 11F03 |
| url | https://arxiv.org/abs/2411.09701 |