Nahm Sums Dual to Zagier's Rank-Three Examples and Related Identities

Abstract

In 2007, Zagier identified twelve families of candidates for rank-three modular Nahm sums and established the modularity of three of them. The remaining cases were subsequently confirmed by Wang. Zagier's duality observation indicates that the Nahm sums associated with the duals of these examples might still be modular. We confirm that this is indeed true. The modularity of the dual of the sixth example was previously established by Milas and Wang, while that of the dual of the eleventh example was partially established by the present authors. We settle all remaining cases, thereby establishing the modularity of every defined dual in Zagier's rank-three list. We achieve this by proving new Rogers--Ramanujan type identities that express the relevant Nahm sums as finite sums of infinite products. Along the way we discover two new families of rank-three Nahm sums. As applications, we prove some Nahm sum identities conjectured by Cao--Wang, Li and the present authors.