Consider a knot in a 3-manifold.  Witten tells us to make invariants of this by considering some big integral over connections, with an insertion giving the holonomy around the knot.  Usually you are allowed to color this holonomy on the knot by a representation.

Apparently also you can ask the connection to be singular along the knot; I think the usual story (where?) is that you can take 1/z singularities in the transverse direction to the knot.

It should be also meaningful to consider irregular singularities.  That is, take the transverse directions, and require the connection there to be like a holomorphic connection with irregular singularities.  The point of this post is to discuss this possibility.

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Motivation 1.  There’s an analogy: primes in the ring of integers of a number field, are, from the point of view of etale homotopy, like knots in a 3-manifold.  One difference is that the so-called inertia subgroup of the Galois group corresponds to, but is much bigger than, the subgroup of the fundamental group of the conormal torus generated by the meridian.  They become more comparable after passing to the “tame” quotient of the inertia group.  But one might want instead to do the reverse, and pass from the fundamental group of a 3-manifold to a wild version.

There’s a known analogy for the wild/tame dichotomy in the case of connections on Riemann surfaces; this is allowing for irregular singularities at points.  One can just as well do this in the 3-manifold case; declare that a neighborhood of the knot is $S^1\times (\mathbb{C},0)$ and require that the connections look locally like an irregular connection on a curve in the $\mathbb{C}$ direction.

Actually I don’t want to discuss connections, so I’ll just go across Riemann-Hilbert and allow local systems with Stokes structures along the knot.  That is, the data now is a local system, together with an $S^1$-filtration (i.e. the levels of the filtration can themselves twist around the $S^1$).  One has two options: only allow Stokes structures which actually arise from irregular singularities on Riemann surfaces, or just allow arbitrary ones.

Remark.  Passing through some Tannakian nonsense, we’re replacing ${\pi }_1(M\setminus K)$ with some larger group.  Seems to me it’s a pushout over the peripheral subgroup with some sort of universal Stokes group

$${\pi }_1^{wild}(M\setminus K)={\pi }_1(M\setminus K){\times }_{{\pi }_1(T^2)}\mathfrak{S}$$

Problem.  Determine this universal Stokes group $\mathfrak{S}$, both for the case where only Stokes structures actually arising from irregular singularities are allowed (maybe in some sense this one is known by passing through RH), and in the case where arbitrary Stokes structures are allowed.  In the latter case, it should be somehow built out of positive braids in the annulus vis a vis the dictionary in [STWZ].  Although presently, I’m a little confused about in precisely what happens when tensoring connections with different Stokes structures, although this must in some sense be known.  (I.e., what happens to the knots?  Seems like it must be a cabling, but then the answer would depend on the order of the tensoring.  There’s an outside possibility that one is saved in the case of Stokes structures actually arising from RH by the fact that if the singularities are different, one has higher order than the other, saying something about in which direction the cabling should go; in this case, there’s not after all any choices about this Stokes group.)

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Motivation 2.  In [STZ], we found an extremely mysterious thing.  The cohomology of the moduli space of sheaves microsupported in a legendrian knot had something to do with the HOMFLY polynomial, homology, etc. of the underlying topological knot.  How can this be?

In particular, the Legendrian knot in question lives in the circle bundle over the 2-sphere or the annulus; the geometry of the situation is all wrong for it to have anything to do with Chern-Simons theory.

Speculation: this relation has to do with using the legendrian knot as a prescription for Stokes phenomena for connections on $S^3$ with an irregular singularity along the unknot. 

Let me develop this speculation further.  Henceforth we will have two different knots; for clarity, some $K\subset {\mathbb{R}}^3$ a topological knot, and some Legendrian knot $\mathrm{\Lambda }\subset J^1(S^1)$.

Recall from [AENV] that we can determine Chern-Simons partition functions by quantizing the augmentation polynomial of knot contact homology, i.e., of the relative contact DGA obtained from the conormal torus of the knot in the cosphere bundle over ${\mathbb{R}}^3$.

But the geometry for specifying an irregular singularity for the connection also sits squarely in this framework.  I.e., we use the description of Stokes phenomena by knots e.g. in [STWZ, 3.3] — connections on an annulus with a given irregular singularity type correspond, by RH, to sheaves on a surface microsupported in a given Legendrian which projects to the annulus near the puncture.  E.g., I can take my $\mathrm{\Lambda }$ from above to have been of this type.  Now I just multiply that by $S^1$ (in the tubular neighborhood along the knot) to get a Betti description of the desired irregular connections in ${\mathbb{R}}^3$ — as sheaves microsupported in a certain Legendrian torus $\mathbb{L}$.

This is doing a funny thing: instead cabling so-to-speak in the direction of the knot, I am microlocally cabling transverse to the knot.  That is, to get $\mathbb{L}$, you cut out a neighborhood of the conormal torus to $K$ — this neighborhood being a $J^1(T^2)$ — and replace it by some $\mathrm{\Lambda }\times S^1\subset J^1(S^1\times S^1)$.  Here the original $S^1$ in whose jet bundle $\mathrm{\Lambda }$ sat is identified with the meridian of the conormal torus to $K$.

By contrast, the usual cabling of $K$ by $\mathrm{\Lambda }$ would have this original $S^1$ going along the longitude to $K$.

Anyway, the upshot is: computing the Chern-Simons invariants with this irregular singularity should about to quantizing the augmentation polynomial for the DGA obtained from $\mathbb{L}$.

Problem.  Do this in some examples.

Problem.  Is there a microlocally-transverse-cabling (mt-cabling) formula for the DGA?

There is one very special case: when $K$ is the unknot.  In this case, note that applying Reeb flow to the conormal torus carries it to the conormal torus of another unknot, exchanging the longitude and meridian in the process.

Question.  Does this interchange the usual cabling and the mt-cabling?

Assuming the answer to the above question is close to yes, it follows that the expectation value of the unknot for the irregular singularity specified by $\mathrm{\Lambda }$ agrees with some expectation value of $\mathrm{\Lambda }$ itself.

And the point of all this:

Problem.  “Reduce on a circle” (i.e. take the quotient by the Hopf fibration) to conclude that the expectation value of the $\mathrm{\Lambda }$-singularity along the unknot is equal to the cohomology of the moduli space of connections on $S^2=S^3/hopf$ with an irregular singularity, of type specified by $\mathrm{\Lambda }$, at the north pole $=unknot/hopf$.

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Update — more questions:

Problem: Quantize the wild augmentation varieties.  On the augmentation side, following the prescription in [AENV], we are supposed to count the higher genus curves.

On the sheaf side I can also give a prescription, perhaps closer to mathematical precision and computability.  The microlocalization of sheaves microsupported in $\mathbb{L}$ gives as usual a map

$$Sh{v}_{\mathbb{L}}(S^3)\to Loc(\mathbb{L})$$

The image of this map is identified with the augmentation variety by the usual augmentations-to-sheaves dictionary.  Anyway, by my paper with Takeda this is a (0-shifted) Lagrangian morphism, we want its quantization.  This makes the most sense for rank one local systems, in which case this is giving a Lagrangian morphism to ${\mathbb{C}}^{\ast }\times {\mathbb{C}}^{\ast }$

As mentioned briefly in [STWZ], one can go about doing this as follows.   The relevant Lagrangian skeleton here is type-A arboreal; hence can be quantized by an elaboration of the Ben Zvi – Brochier – Jordan method.  But, maybe there’s a more elementary approach; the Stokes structure is an explicit thing and may have a known quantization (maybe Boalch did it?); this can be glued to the usual quantization of the space of local systems.

 

Problem: For that matter, compute some wild augmentation varieties!

 

Problem: What is Q?  By which I mean: in the above augmentation polynomial description, Q is still visible.  Maybe this can be traced all the way back through the Hopf fibration to tell us what moduli space of bundles we should study to get the rest of the HOMFLY polynomial.  To match what happens by applying nonabelian Hodge theory to [GORS], it should be related to doing something along a different Hopf fibre than the original unknot…