Explicit forms and proofs of Zagier's rank three examples for Nahm's problem (Q6562852)

Recall the usual \(q\)-series notation, \N\[\N(a;q)_n = \prod_{j=0}^{n-1} (1-aq^j) \N\]\Nand \N\[\N(a;q)_{\infty} = \prod_{j \geq 0} (1-aq^j). \N\]\NA general problem is to determine when the \(q\)-series \N\[\Nf_{A,B,C}(q) = \sum_{n = (n_1,\dots , n_r)^T \in (\mathbb{Z}_{\geq 0})^r} \frac{q^{\frac{1}{2}n^TAn + Bn + C}}{(q;q)_{n_1} \cdots (q;q)_{n_r}} \N\]\Nis a modular form. Here \(A\) is a matrix, \(B\) is a vector, and \(C\) is a scalar. Nahm's problem refers to the case where \(A\) is positive definite.\N\NWhen \(r=3\), \textit{D. Zagier} [Frontiers in number theory, physics, and geometry II. On conformal field theories, discrete groups and renormalization. Papers from the meeting, Les Houches, France, March 9--21, 2003. Berlin: Springer. 3--65 (2007; Zbl 1176.11026)] gave a list of twelve positive-definite or positive semi-definite matrices \(A\) along with several \(B\) and \(C\) for each matrix and conjectured that the corresponding series \(f_{A,B,C}(q)\) are modular. He proved his conjecture for three of the matrices \(A\). In this paper the author establishes the remaining cases. This amounts to proving explicit identities like \N\[\N\sum_{i,j,k \geq 0} \frac{q^{i^2+j^2+k^2 + ij+ik+i+j}}{(q;q)_i(q;q)_j(q;q)_k} = \frac{(q^4;q^4)_{\infty}^2}{(q;q)_{\infty}(q^2;q^2)_{\infty}}\N\]\Nor \N\[\N\sum_{i,j,k \geq 0} \frac{q^{4i^2+2j^2+k^2 +4ij-2ik-2jk + 4i+2j}}{(q^2;q^2)_i(q^2;q^2)_j(q^2;q^2)_k} = \frac{(q^2;q^2)_{\infty}^3(q;q^{12})_{\infty}(q^{11};q^{12})_{\infty}(q^{12};q^{12})_{\infty}}{(q;q)_{\infty}^2(q^4;q^4)_{\infty}},\N\]\Nwhere the modularity of the infinite products is well-known. Most of the identities are proved by reducing the triple sum to a more tractable single or double sum.