Nahm sum identities for Cartan matrices of type D_k

2025/12/08 by Wang, Liuquan, Wang, Shangwen · 1 citation
Mathematics · #05A30 #11F03 #11P84 #33D15 #33D60 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Inequalities and Applications #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.07790

openalex publication_date 2025/12/08 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28

Abstract

Around 2007, Warnaar proved four identities related to Nahm sums associated with twice the inverse of the Cartan matrix of type D_k. Three of these had been conjectured by Flohr, Grabow, and Koehn, while special cases of two of the identities were first conjectured in 1993 by Kedem, Klassen, McCoy, and Melzer. Warnaar's proof relies on a multi-sum identity from Andrews' proof of the Andrews-Gordon identities. We give a new proof of all four identities using the theory of Bailey pairs. Furthermore, we establish a parametric generalization of two of the identities and provide two distinct proofs of this generalization.

Citations

• Some New Modular Rank Four Nahm Sums as Lift-dual of Rank Three Examples
• Modularity of tadpole Nahm sums in ranks 4 and 5
• Some New Modular Rank Three Nahm Sums from a Lift-Dual Operation
• On Some Double Nahm Sums of Zagier
• Modularity of Nahm Sums for the Tadpole Diagram
• Explicit Forms and Proofs of Zagier's Rank Three Examples for Nahm's Problem
• Identities on Zagier's rank two examples for Nahm's problem
• Rogers-Ramanujan type identities and Nil-DAHA
• Conformal Field Theory and Torsion Elements of the Bloch Group
• Change of base in Bailey pairs
• Supersymmetric Analogs of the Gordon-Andrews Identities, and Related TBA Systems

Cited by

• Modularity of Nahm Sums Dual to Zagier's Rank-Three Examples

Related

• Proofs of Two Conjectural Identities on Partial Nahm Sums
• Some New Modular Rank Four Nahm Sums as Lift-dual of Rank Three Examples
• On certain identities involving Nahm-type sums with double poles
• Modularity of tadpole Nahm sums in ranks 4 and 5
• On Some Double Nahm Sums of Zagier