Gukov–Putrov–Vafa conjecture on the categorification of WRT invariants

Let $M_3$ be a closed oriented $3$-manifold, let $b_1(M_3)$ denote its first Betti number, and let $a,b$ range over $\operatorname{Tor} H_1(M_3,\mathbb Z)/\mathbb Z_2$. Let $\ell k(a,b)$ be the torsion linking pairing, let $\mathcal W_a$ be the stabilizer of $a$ under the Weyl-group action, and let $c\in\mathbb Z_+$ and $\Delta_b\in\mathbb Q$. Define the $S$-matrix by

$$S_{ab}=\frac{e^{4\pi i\ell k(a,b)}+e^{-4\pi i\ell k(a,b)}}{|\mathcal W_a|\sqrt{|\operatorname{Tor} H_1(M_3,\mathbb Z)|}}.$$

Categorification conjecture. There are series $\widehat Z_b(q)$, convergent for $|q|<1$, such that

$$\widehat Z_b(q)\in 2^{-c}q^{\Delta_b}\mathbb Z[[q]],$$

and the WRT invariant decomposes as

$$Z_{SU(2)_k}[M_3]=(i\sqrt{2k})^{b_1(M_3)-1}\sum_{a,b\in\operatorname{Tor} H_1(M_3,\mathbb Z)/\mathbb Z_2}e^{2\pi i k\ell k(a,a)}S_{ab}\,\widehat Z_b(q)\big|_{q\rightarrow e^{\frac{2\pi i}{k}}}.$$

This conjecture proposes homological $q$-series whose radial limits recover the $SU(2)$ Chern–Simons WRT invariant and whose coefficients exhibit an integrality property up to a power of $2$. The source presents it as a conjecture, and no resolution is supplied here.