In his 2004 paper [Conformal Field Theory and Torsion Elements of the Bloch Group](https://arxiv.org/pdf/hep-th/0404120.pdf), Nahm explains a physical argument due to Kadem, Klassen, McCoy, and Melzer for the following remarkable identity. Let $C \in \operatorname{Mat}\_8(\mathbb{Z})$ denote the inverse of the Cartan matrix of $E\_8$ and $(q)\_n = \prod\_{i=1}^n (1 - q^i)$. Then


$$ \sum\_{n=\mathbb{N}} \frac{q^{2n^2}}{(q)\_{2n}} = \sum\_{v \in \mathbb{N}^8} \frac{q^{vCv}}{\prod\_{j=1}^8 (q)\_{v\_j}} $$


Nahm reports that, as of his writing in 2004, the result identity still not mathematically proven, because the arguments rely on not-yet-proven facts about conformal field theories, but that it has been checked to high orders.


It's been 14 years. Has the above identity been proven in that time, either by firming up the foundations of conformal field theory or through other means?



Has the $E_8$-based generating function for squares numbers been proven?
This identity was actually proven 24 years ago in 



> 
> S.O. Warnaar and P.A. Pearce, ["Exceptional structure of the dilute A 3 model: E8 and E7 Rogers-Ramanujan identities"](https://people.smp.uq.edu.au/OleWarnaar/pubs/Exceptional.pdf) J.Phys. A27 (1994) L891-L898
> 
> 
> 


The proof essentially establishes a finite polynomial identity whose limit under one of the parameters becomes the desired series. I want to remark that Nahm calls this identity a conjecture in the arXiv version of the paper, but in the published version it correctly refers to the article above for a proof.


