In [STZ], we observed a connection between the sheaf category on an n-strand braid with identically zero Maslov potential, and certain constructions related to the geometric representation theory of gl(n).
Specifically, the relationship was the following. First, consider the $n$-line category, i.e., the braid in question contains no crossings at all, carrying Maslov potential $\mu = 0$. Any object in the category of sheaves with singular support in the braid, acyclic stalks beneath it, and microlocal indices $\mu$, turns out to be just a complete flag in $k^n$; all these objects are isomorphic and the automorphisms are transformations preserving the flag.
That is, the moduli space $M(\equiv, \mu = 0)$ of sheaves is just point/B.
More generally, we showed that if you take any positive braid, the corresponding moduli space is the associated open Bott-Samelson variety, that closing the braid into an annulus amounts to Lusztig’s horocycle transform, and that filling the hole in the annulus amounts to taking the fibre of the Bott-Samelson map. This led to various interpretations of the moduli space in terms of Webster and Williamson’s construction of HOMFLY homology, some of which appeared in [STZ].
I remark in passing that probably one gets some representation-theoretically interesting spaces from cutting holes in the annulus which avoid the braid.
We showed in [STZ] that, closing up the braid $\beta$ into a knot, the number of objects in $M(\beta)$ over a finite field is counted by a certain term of the HOMFLY polynomial of $\beta$. The proof given there is extremely roundabout, but since then I have learned another proof of this result, deriving it directly from the fact that Webster-Williamson computes HOMFLY homology.
On the other hand, it follows from work of Rutherford and Henry that for any Legendrian knot at all, the 2-periodic augmentations are counted by the appropriate term of the HOMFLY polynomial. Via [NRSSZ] and the augmentation counting conjecture (which at least for now remains not only conjectural but without an explicit formulation in the 2-periodic case), this means that objects in the moduli of 2-periodic sheaves are counted by the HOMFLY polynomial.
I would like a representation-theoretic setting for understanding this result. The following records some extremely preliminary remarks regarding what might be involved.
So we study sheaves of 2-periodic complexes on a Legendrian knot.
To specify the discrete data of the problem amounts to choosing a Maslov potential, i.e., an assignment of {0,1} to the strands of the diagram, so it changes at cusps. Equivalently, I make the uniform convention that right-moving is 0 and left-moving is 1, and choose an orientation of the knot.
The line category. Say we have $n|m$ lines, i.e., $n+m$ lines carrying a Maslov potential with $n$ zeroes and $m$ ones. The idea is that this should be related to the theory of gl(n|m).
We study sheaves with singular support in this picture, which are acyclic beneath it.
Note that a line labelled “0” means that, as you move upwards across it, you should either create a rank one summand in degree zero, or destroy a rank one summand in degree one. Likewise, “1” means: create in degree 1, or destroy in degree 0.
So for instance, in the case of two lines where the top one is labelled 1 and the bottom one labelled zero (a slice of the unknot), there are two possible sheaves, whose ranks I record in the usual (even|odd) convention
(1|1)
<——
(1|0)
——>
(0|0)
and
(0|0)
<——
(1|0)
——>
(0|0)
Note that of these, the first looks like a flag in gl(1|1), and the second one is the one relevant for knot theory.
This kind of consideration suggests that maybe the flag varieties for gl(n|m) which have been classically considered are not quite right — they are missing some points, corresponding to the second kind of object above. I speculate wildly that this has something to do with the failure of Borel-Weil-Bott for superalgebras, which might perhaps be repaired by considering these corrected flag varieties.
Crossings.
Unlike for gl(n), there are many different classes of Borel subalgebras in gl(n|m). These correspond to the fact that as you build up a flag, one has to choose at each step whether to pick up an even piece or an odd piece, i.e., exactly to the choice of a Maslov potential.
Similarly, instead of a Weyl group, one has more naturally a Weyl groupoid. That is, the permutation group acts on the set of choices of Maslov potential, and we can take the category which is the quotient of this action. This is generated by transpositions as usual.
In the context of Legendrian knots, note that this means that the crossing category gives a natural correspondence between the flag varieties of different Borel subalgebras related by a single transposition. This sort of consideration should presumably play the same role for gl(n|m) that the (open) Bott-Samelson varieties play for gl(n). In particular, there should be a categorification of the “positive braid” version of the above Weyl groupoid via the cohomology of these spaces.
Cusps.
I have heard many times that gl(n|m) in some sense depends only on the difference $n-m$. This statement should be interpreted as the possibility of having cusps on a Legendrian knot. Indeed, a diagram with a single cusp and lines otherwise traveling straight through gives a correspondence between the flag varieties of gl(n|m) and gl(n-1|m-1).
I think it is possible that a great deal of the above discussion has been developed already in this paper of Brundan and Stroppel, see also this one, they even draw knot diagrams (!) Though, they don’t seem to make any use of crossings, and they draw rounded cusps instead of honest ones. Sadly I am not competent to read this paper…