Proofs of Mizuno's Conjectures on Generalized Rank Two Nahm Sums

TL;DR

This paper advances the understanding of generalized rank two Nahm sums by proving modularity for several candidate sets, establishing new identities, and confirming Mizuno's conjectural transformation formulas, thus deepening the connection between Nahm sums and modular forms.

Contribution

It proves modularity for eight new sets of generalized Nahm sums, confirms Mizuno's conjectural transformation formula, and introduces new identities related to these sums.

Findings

Proved modularity for eight candidate Nahm sums.

Confirmed Mizuno's conjectural modular transformation formula.

Discovered new non-modular identities for Nahm sums.

Abstract

Recently, Mizuno studied generalized Nahm sums associated with symmetrizable matrices. He provided 14 sets of candidates of modular Nahm sums in rank two and justified four of them. We prove the modularity for eight other sets of candidates and present conjectural formulas for the remaining two sets of candidates. This is achieved by finding Rogers-Ramanujan type identities associated with these Nahm sums. We also prove Mizuno's conjectural modular transformation formula for a vector-valued function consists of Nahm sums. Meanwhile, we find some new non-modular identities for some other Nahm sums associated with the matrices in Mizuno's candidates.