Some New Modular Rank Three Nahm Sums from a Lift-Dual Operation

2024/12/20 by Zhineng Cao, Cao, Zhineng, Liuquan Wang +1 · 3 citations
Mathematics · #11F03 #11P84 #33D15 #33D60 #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2412.15767

openalex publication_date 2024/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Around 2007, Zagier discovered some rank two and rank three Nahm sums, and their modularity have now all been confirmed. Zagier also observed that the dual of a modular Nahm sum is likely to be modular. This duality observation motivates us to discover some new modular rank three Nahm sums by a lift-dual operation. We first lift Zagier's rank two Nahm sums to rank three and then calculate their dual, and we show that these dual Nahm sums are indeed modular. We achieve this by establishing the corresponding Rogers--Ramanujan type identities, which express these Nahm sums as modular infinite products.

Cited by

• Modularity of Nahm Sums Dual to Zagier's Rank-Three Examples
• Nahm sum identities for Cartan matrices of type D_k
• Some New Modular Rank Four Nahm Sums as Lift-dual of Rank Three Examples

Related

• Modularity of Nahm Sums Dual to Zagier's Rank-Three Examples
• Some New Modular Rank Four Nahm Sums as Lift-dual of Rank Three Examples
• Counterexamples to Zagier's Duality Conjecture on Nahm Sums
• Modularity of tadpole Nahm sums in ranks 4 and 5
• Proofs of Two Conjectural Identities on Partial Nahm Sums