I will provide a brief review focusing on the conjectures proposed by the authors without focusing on too many details.
Firstly let me explain what (advanced) background is needed to understand this paper. One needs to be familiar with the whole technology of producing quantum field theories from M-theory. It is widely known that M5 branes have some QFT living on their worldvolume $M_6$. The famous class $S$-theories (originally due to Gaiotto) are a family of QFTs that live on a 4d manifold $X$ such that $M_6= X \times \Sigma_{g,n}$. The latter is a Riemann surface of genus $g$ with $n$ marked points (in this sense punctured points, for example $\mathbb{R}_n$ is the interval $(-\infty,0)\cup (0,+\infty)$). This gives the famous 4d-2d dualities between QFTs living on $X$ and (Liouville) CFTs living on $\Sigma_{g,n}$. One can re-interpret $M_6$ as $M_3 \times C$ (another 3d manifold involving circles). Then one gets the so-called 3d-3d correspondence. Then compactifying the 6d (2,0) theory on $C$ gives a QFT that goes by the name $T[C]$ living on $X$ whereas compactifying the 6d (2,0) theory on $X$ gives some other theory on $C$. As an example let me mention that the partition function of $T[C]$ on $S^2 \times S^1$ corresponds to the partition function of SL$_K(\mathbb{C})$ Chern-Simons theory at level $k=0$ on $M_3$.
Now, this paper's physics involves exactly such type of dualities. The paper starts of by explaining that in 3d manifolds we have an invariant that goes by the name $WRT$ and is given by
$$ \text{WRT} = \mathcal{Z}_{CS, k}^{\text{SU}(2)}(S^3) $$
while one can define a generalized, in some sense, WRT invariant by replacing $S^3$ with any $M_3$. The conjectures of the authors, that I will now present, involve some sort of categorification of these invariants. I am not sure if I can use the word refinement because it is more than this.
Conjecture 1 The (generalized) WRT invariant can be decomposed in homological blocks as following
$$ \mathcal{Z}_{k}(M_3) \backsim \sum_{a,b \in T } e^{2\pi \, \text{lk}(a,b)} \, S_{ab} \hat{Z_b}(q) \Big|_{q\to e^{2\pi i/k}} $$
where $T = \text{Tor} H_1(M_3,\mathbb{Z})/\mathbb{Z}_2$ and elements of it $a,b$ can be considered as cycles giving boundary conditions in the sense that these cycles are related to the boundary of M2 branes adjacent to an M5. The dual torsion homology gives abelian flat connections. Also, lk($a,b$) denotes the linking pairing on $M_3$ between two 1-cycles (we can think of it as some analogy of the intersection pairing in 4-manifolds and basically it is the number of intersection of $a$ with $b$ with multiplicity). $S$ is a matrix related to the $S$-duality of type IIB theory (since we can reduce the M-theory construction to IIB by a series of standard dualities). Finally, and most importantly, the homological blocks are the quantities $\hat{Z}_{b}(q)$ for a fixed boundary condition $b$.
We can summarize that conjecture 1 is the claim of existence of a new 3-manifold invariant that admits a $q$-series expansion with integer coefficients which is suitable for categorification. This guy, $\hat{Z}_b(q) \in \mathbb{Z}[[q]]$ is related to another new invariant that is predicted by physics.
Without quoting too many details, the whole brane construction and more specifically the internal symmetries we have, dictate the existence of a triply graded invariant, in $\mathbb{Z} \times \mathbb{Z} \times \text{Tor} H_1(M_3,\mathbb{Z})/\mathbb{Z}_2 $, of $M_3$ which looks like
$$ H[M_3] = \oplus_{a \in T} H_a[M] = \oplus_{a \in T} \Big( \oplus_{i,j} H_a^{i,j} \Big) $$
$i \in \mathbb{Z}+\Delta_a$, $j \in \mathbb{Z}$ (for this review it does not matter what $\Delta_a$ is) and $a$ is a boundary condition of course. This invariant categories $\hat{Z}_a(q)$ because
$$ \hat{Z}_a(q) \backsim \sum_{i,j} q^{i}(-1)^j \text{dim} H_a^{i,j}$$
(which suspiciously looks like an index). In the paper the authors describe the physics behind this invariant. Now let me turn to the second conjecture:
Conjecture 2: For the homological block $ \hat{Z}_a(q)$ we saw just above the following holds
$$ \mathcal{I}(q) = \sum_{a \in T} |\text{# of elements in Weyl group}| \, \hat{Z}_a(q) \hat{Z}_a(q^{-1}) $$
where $ \mathcal{I}(q)$ is the superconformal index of the $T[M_3]$ theory. Therefore, indeed, the suspicion of the appearance of an index in the categorification of the WRT invariant is valid (if the conjectures hold of course) .
The authors proceed with further refinements in the case where $M_3$ is a Seifert manifold (a specific type of circle fibration) leading to 4-graded invariants. Then, some discussion about the topologically twisted index (different to the superconformal index), generalization for U$(N)$ theories and the relationship of the above to open Gromov invariants of the CY3 $T^*M_3$ are discussed. Examples follow and these results are applied to quite technical illustrations.
I would say that this paper is fairly technical and it has some new impressive results for people interested in invariants of manifolds. It is certainly worth going through the introduction and section 2 (which is where the main message is delivered). Most of the examples are quite non-trivial and very soon become very technical. I have not been able to go through all of them. Still, this paper is well written, the references within well placed and quoted when needed. This is an impressive study as far as my opinion is concerned with fairly convincing arguments. Related works are 1602.05302 and 1608.02961while for some understanding of the 3d-3d correspondence I recommend the book New Dualities of Supersymmetric Gauge Theories chapter "3d Superconformal Theories form Three-Manifolds" which can also be found on arXiv.
Q&A (4872)
