Proofs of Mizuno's Conjectures on Rank Three Nahm Sums of Index $(1,2,2)$

TL;DR

This work resolves Mizuno's conjectures on rank-three Nahm sums of index $(1,2,2)$ by combining Bailey-pair machinery, $q$-difference equations, and constant-term methods to derive Rogers–Ramanujan type identities that express Nahm sums as sums of eta-quotients. The authors prove modularity for all 15 Mizuno rank-three examples (with two exceptional sums that decompose into modular forms of weights $0$ and $1$) and establish the predicted modular transformation formulas between two vector-valued functions $g$ and $g^{lat}$. They also introduce new Bailey pairs and numerous new theta- and eta-quotient identities, and reveal deep links between the $(1,1,2)$ and $(1,2,2)$ families via a two-vector Nahm-sum framework. Overall, the paper provides a comprehensive modularity verification for Mizuno's families, along with a versatile toolkit for constructing and certifying modular Nahm sums across higher ranks. These results deepen the connection between Nahm sums, modular forms, and vector-valued modular phenomena, with potential impacts on partition theory and conformal field theory.

Abstract

Mizuno provided 15 examples of generalized rank three Nahm sums with symmetrizer $\mathrm{diag}(1,2,2)$ which are conjecturally modular. Using the theory of Bailey pairs and some $q$-series techniques, we establish a number of triple sum Rogers--Ramanujan type identities. These identities confirm the modularity of all of Mizuno's examples except that two Nahm sums are sums of modular forms of weights $0$ and $1$. We also prove Mizuno's conjectural modular transformation formulas for two vector-valued functions consisting of Nahm sums with symmetrizers $\mathrm{diag}(1,1,2)$ and $\mathrm{diag}(1,2,2)$.

Theorems & Definitions (87)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Theorem 1.6
• Lemma 2.1: Bailey's Lemma
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4