Proofs of Two Conjectural Identities on Partial Nahm Sums

TL;DR

This paper proves two conjectural identities related to partial Nahm sums by transforming them into Hecke-type series and then into modular infinite products, confirming their modularity.

Contribution

It introduces a novel two-step method involving Bailey pairs and Hecke-type series to prove conjectural identities on partial Nahm sums.

Findings

Proved the two conjectural identities on partial Nahm sums.

Established a transformation from Nahm sums to Hecke-type series.

Converted Hecke-type series into modular infinite products.

Abstract

Recently, Wang and Zeng investigated modularity of partial Nahm sums and discovered 14 modular families of such sums. They confirmed modularity for 13 families and proposed a conjecture consisting of two Rogers--Ramanujan type identities for the remaining family. We prove these conjectural identities in two steps. First, employing a transformation formula involving two Bailey pairs, we transform the partial Nahm sums into some specific Hecke-type series. Second, using two distinct approaches, we convert these Hecke-type series to the desired modular infinite products.