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arXiv · 2411.09701

Counterexamples to Zagier's Duality Conjecture on Nahm Sums

Liuquan Wang
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Abstract

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.

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Liuquan Wang. 2024-11-14. Counterexamples to Zagier's Duality Conjecture on Nahm Sums. https://arxiv.org/abs/2411.09701

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