categorification of q-series In his talk, S. Gukov asked two questions:


*

*What is the categorification of a $q$-serie ?

*How to associate to a 3-manifold a $q$-serie ?
As far as I understand, he was looking for a bigarded cohomology theory $\mathcal{H}^{a,b}$ associated to 3-manifold $X$ which should be given by 
$$\sum_{a,b \in \mathbb{Z}}(-1)^{a} q^{b}dim \mathcal{H}^{a,b}(X)  $$ 


Question


*

*Could someone explain the real motivation of S. Gukov ?

*Is it realistic to ask such questions?

*Is there any progress on that direction ?         

 A: To answer a part of you question, in fact with q-series we can find generating function for a lot of our Characteristic datas in algebraic topology which make sense to use the title "categorification of q-series". For example See C.Soule paper
To answer a part of your question
In general the Hirzebruch genus of a smooth, projective complex $d$-dimensional variety $X$ can be defined by 
$$\chi_y(X)=\sum_{p,q=0}^{\dim_\mathbb C X}(-1)^{p-q}y^{p}\dim H^{p,q}(X,\mathbb Z)$$
The study of such genus is important, since . If we have only nonzero cohomology of type $(p, p)$, then $\chi_y$ is just the Poincaré polynomial which is the generating function for the Betti numbers $b_i(X)$,
$$p(X,y)=\sum_{i=0}^{2\dim_\mathbb CX}b_i(X)y^i$$ 
which are important in calculating the BPS numbers, via Poly-Euler numbers
of with a,b,c parameters . See this thesis and  we can rewrite $\chi_y(X)$ by using Chern numbers see Lemma 2.1.
of this paper
The n-th symmetric product of a space X is defined by
$$X^{(n)}=X\times X\cdots \times X/\Sigma_n$$ the quotient of the product of n copies of $X$ by the natural action of the
symmetric group on n elements, $Σ_n$
Borisov-Libgober and Zhou  proved the following generating series
formula for the Hirzebruch $χ_y$-genus for compact complex algebraic manifold $X$
$$\sum_{n\geq 0}\chi_{-y}(X^{(n)}).t^n=\exp\left(\sum_{r\geq 1}\chi_{-y^r}(X)\right)\frac{t^r}{r}$$
See Eulerian polynomials about Hirzebruch genus written by Friedrich Hirzebruch
A: As far as I know, question 1 is open. 
Let me say a few words about the first part of question 2. One way to associate a q-series to a 3-manifold is via the 3-d index developed by Dimofte, Gaiotto and Gukov:
Dimofte, Tudor; Gaiotto, Davide; Gukov, Sergei, 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17, No. 5, 975-1076 (2013). ZBL1297.81149.
Dimofte, Tudor; Gaiotto, Davide; Gukov, Sergei, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325, No. 2, 367-419 (2014). ZBL1292.57012.
To concretely associate this q-series to an invariant of a 3-manifold, Garoufalidis, Hodgson, Rubinstein, and Segerman were able to show that for any cusped orientable hyperbolic 3-manifold there is a standard 3-d index which can be computed from its canonical cell decomposition. The proof of their main theorem is worth flushing out here. They show that for any triangulation of cusped 3-manifold which is 1-efficient (a property involving normal surfaces) the 3-d index makes sense to compute. Furthermore, if two 1-efficient triangulations are connected by a single 2-3 or 3-2 move (plus 0-2 and 2-0 moves) the 3-d index is the same for both triangulations and if a manifold does not admit a canonical triangulation then all simple refinements of its canonical cell decomposition are 1-efficient and related by simple moves as above. Finally, their paper also includes some computations and shows that the 3-d index of a triangulation converges as a formal q-series (that is it can be written down) if and only if the triangulation is 1-efficient.      
Garoufalidis, Stavros; Hodgson, Craig D.; Rubinstein, J.Hyam; Segerman, Henry, 1-efficient triangulations and the index of a cusped hyperbolic 3-manifold, Geom. Topol. 19, No. 5, 2619-2689 (2015). ZBL1330.57029.
Garoufalidis, Hodgson, Rubinstein, and I did a follow up to this paper, which tries to better understand the connection between the 1-efficiency and convergence of the 3-d index:
https://arxiv.org/pdf/1604.02688.pdf 
An interesting question along these lines, is what is the 3-d index counting? There is some hope that trying to 'categorify' this invariant might shed light on that question as well. In my opinion, this is a relevant interesting question which has already motivated novel approaches to better understanding 3-manifolds. 
