Some New Modular Rank Three Nahm Sums from a Lift-Dual Operation

TL;DR

This paper introduces new modular rank three Nahm sums derived from a lift-dual operation, confirming their modularity through Rogers--Ramanujan type identities that express them as infinite products.

Contribution

It develops a novel lift-dual method to generate and verify new modular rank three Nahm sums from rank two sums, expanding the understanding of their modular properties.

Findings

New modular rank three Nahm sums constructed

Dual Nahm sums confirmed to be modular

Established Rogers--Ramanujan type identities for these sums

Abstract

Around 2007, Zagier discovered some rank two and rank three Nahm sums, and their modularity have now all been confirmed. Zagier also observed that the dual of a modular Nahm sum is likely to be modular. This duality observation motivates us to discover some new modular rank three Nahm sums by a lift-dual operation. We first lift Zagier's rank two Nahm sums to rank three and then calculate their dual, and we show that these dual Nahm sums are indeed modular. We achieve this by establishing the corresponding Rogers--Ramanujan type identities, which express these Nahm sums as modular infinite products.