Counterexamples to Zagier's Duality Conjecture on Nahm Sums

TL;DR

This paper constructs explicit counterexamples showing that Zagier's duality conjecture for Nahm sums does not hold universally, challenging previous assumptions about the modularity of dual Nahm sums.

Contribution

The paper provides explicit counterexamples of rank four Nahm sums that are modular but whose duals are not, disproving Zagier's duality conjecture.

Findings

Counterexamples of rank four Nahm sums that are modular but have non-modular duals

Disproof of Zagier's duality conjecture for Nahm sums

Implications for the classification of modular Nahm sums

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.