Gukov–Pei–Putrov–Vafa conjecture on homological blocks and WRT invariants

Let $\Gamma=(V,E,(w_v)_{v\in V})$ be a finite tree with integral vertex weights, and let $W$ be its negative-definite linking matrix. Let $M(\Gamma)$ be the integral homology sphere obtained by surgery along $\Gamma$, let $\widehat{Z}_\Gamma(q)$ be its homological block for $|q|<1$, and let $\operatorname{WRT}_k(M(\Gamma))$ be the normalized Witten–Reshetikhin–Turaev invariant, with $\operatorname{WRT}_k(S^3)=1$. Set $\zeta_k=e^{2\pi i/k}$. Gukov–Pei–Putrov–Vafa conjecture. For every positive integer $k$,

$$\operatorname{WRT}_k(M(\Gamma))=\frac{1}{2(\zeta_{2k}-\zeta_{2k}^{-1})}\lim_{q\to\zeta_k}\widehat{Z}_\Gamma(q).$$

The conjecture relates quantum WRT invariants to radial limits of the homological blocks introduced by Gukov, Pei, Putrov and Vafa. The supplied text says that the result is proved for Seifert homology spheres and for non-Seifert homology spheres whose surgery diagrams are H-graphs, while the present paper proves it for a wider class of plumbed manifolds.