Counterexamples to Zagier's Duality Conjecture on Nahm Sums

TL;DR

The paper constructs explicit rank $4$ Nahm sums that are modular while their duals are not, providing counterexamples to Zagier's duality conjecture on Nahm sums. It develops four Bailey pairs and leverages Bailey's transformations to establish new Rogers--Ramanujan-type identities and to re-express Nahm sums and their duals in modular terms. The results demonstrate that dual Nahm sums can sometimes be modular as combinations of eta-quotients or sums of modular forms of different weights, prompting a refined formulation of dualities in Nahm-sum modularity. Overall, the work clarifies the limitations of the naive duality picture and enriches the toolkit for identifying modular Nahm sums. These findings have implications for the structure of modular objects arising from $q$-hypergeometric series and for the broader understanding of Nahm's problem.

Abstract

Given any positive integer $r$, Nahm's problem is to determine all $r\times r$ rational positive definite matrix $A$, $r$-dimensional rational vector $B$ and rational scalar $C$ such that the rank $r$ Nahm sum associated with $(A,B,C)$ is modular. Around 2007, Zagier conjectured that if the rank $r$ Nahm sum for $(A,B,C)$ is modular, then so is the dual Nahm sum associated with $(A^{-1},A^{-1}B,B^\mathrm{T} A^{-1}B/2-{r}/{24}-C)$. We construct some explicit rank four Nahm sums which are modular while their duals are not modular. This provides counterexamples to Zagier's duality conjecture.

Theorems & Definitions (12)

• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• proof : Proof of Theorem \ref{['thm-new-RR']}