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

TL;DR

This paper proves Mizuno's conjectures on the modularity of certain rank three Nahm sums using Bailey pairs and q-series techniques, confirming their modular nature and transformation properties.

Contribution

It establishes the modularity of Mizuno's Nahm sums and proves their conjectural transformation formulas, advancing understanding of their q-series identities.

Findings

Confirmed modularity of Mizuno's Nahm sums.

Proved modular transformation formulas for Nahm sums.

Established new Rogers--Ramanujan type identities.

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 $\mathrm{diag}(1,2,2)$.