Liuquan Wang

Recently, Mizuno studied generalized Nahm sums associated with symmetrizable matrices. He provided 14 sets of candidates of modular Nahm sums in rank two and justified four of them. We prove the modularity for eight other sets of candidates and present conjectural formulas for the remaining two sets of candidates. This is achieved by finding Rogers–Ramanujan type identities associated with these Nahm sums. We also prove Mizuno’s conjectural modular transformation formula for a vector-valued function consisting of Nahm sums. Meanwhile, we find some new non-modular identities for some other Nahm sums associated with the matrices in Mizuno’s candidates.

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.

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.