Explicit Forms and Proofs of Zagier's Rank Three Examples for Nahm's Problem

Abstract

Let $r\ge 1$ be a positive integer, $A$ a real positive semi-definite symmetric $r\times r$ rational matrix, $B$ a rational vector of length $r$, and $C$ a rational scalar. Nahm's problem is to find all triples $(A,B,C)$ such that the $r$-fold $q$-hypergeometric series $$f_{A,B,C}(q):=\sum _{n=(n_1,\dots ,n_r{)}^{\mathrm{T}}\in ({\mathbb{Z}}_{\ge 0}{)}^r}\frac{q^{\frac{1}{2}{n}^{\mathrm{T}}An+n^{\mathrm{T}}B+C}}{(q;q{)}_{n_1}\cdots (q;q{)}_{n_r}}$$ becomes a modular form, and we call such $(A,B,C)$ a modular triple. When the rank $r=3$, after extensive computer searches, Zagier provided twelve sets of conjectural modular triples and proved three of them. We prove a number of Rogers-Ramanujan type identities involving triple sums. These identities give modular form representations for and thereby verify all of Zagier's rank three examples. In particular, we prove a conjectural identity of Zagier.

37 pages. We made some changes after the first version. Comments are welcome