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arXiv · 2504.17737

Modularity of tadpole Nahm sums in ranks 4 and 5

Changsong Shi·Liuquan Wang
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Abstract

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.

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BibTeXRIS

Changsong Shi, Liuquan Wang. 2025-04-24. Modularity of tadpole Nahm sums in ranks 4 and 5. https://arxiv.org/abs/2504.17737

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