Explicit Forms and Proofs of Zagier's Rank Three Examples for Nahm's Problem

2022/11/08 by Liuquan Wang, Wang, Liuquan · 6 citations
Mathematics · #Advanced Mathematical Identities #Mathematical functions and polynomials #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2211.04375

Abstract

Let r≥ 1 be a positive integer, A a real positive semi-definite symmetric r× r rational matrix, B a rational vector of length r, and C a rational scalar. Nahm's problem is to find all triples (A,B,C) such that the r-fold q-hypergeometric series f_A,B,C(q):=∑_n=(n₁,…,n_r)^T∈ (ℤ_{≥ 0})^r \fracq^{(1)/(2)nT An+nT B+C}(q;q)_n1⋯ (q;q)_nr becomes a modular form, and we call such (A,B,C) a modular triple. When the rank r=3, after extensive computer searches, Zagier provided twelve sets of conjectural modular triples and proved three of them. We prove a number of Rogers-Ramanujan type identities involving triple sums. These identities give modular form representations for and thereby verify all of Zagier's rank three examples. In particular, we prove a conjectural identity of Zagier.

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• Modularity of Nahm Sums Dual to Zagier's Rank-Three Examples
• Tadpole Nahm sum as a Wronskian
• Nahm sum identities for Cartan matrices of type D_k
• On Bilateral Multiple Sums and Rogers-Ramanujan Type Identities
• Some New Modular Rank Four Nahm Sums as Lift-dual of Rank Three Examples
• Proofs of Two Conjectural Identities on Partial Nahm Sums

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