Some New Modular Rank Four Nahm Sums as Lift-dual of Rank Three Examples

TL;DR

The paper discovers nine new rank four Nahm sums likely to be modular, proves modularity for four of them via Rogers-Ramanujan type identities, and presents conjectural identities using advanced q-series techniques.

Contribution

It introduces nine new modular rank four Nahm sums obtained through lift-dual operations on rank three sums, with four proven to be modular and several conjectured identities.

Findings

Four Nahm sums proven to be modular via Rogers-Ramanujan identities

Nine new rank four Nahm sums identified using lift-dual operations

Conjectural identities for Nahm sums as modular infinite products

Abstract

We find nine new sets of rank four Nahm sums associated with nine different numeric matrices which are likely to be modular. They are discovered by applying the lift-dual operation to some modular rank three Nahm sums in the works of Zagier and the authors. We prove the modularity of four sets of these Nahm sums by establishing Rogers--Ramanujan type identities which express them as modular infinite products. We use various $q$-series techniques including the constant term method and Bailey pairs to prove these identities. Meanwhile, we present some conjectural identities expressing several Nahm sums as modular infinite products.