(trigger warning: M-theory.)

I just learned what a “brane tiling” is, first from a talk of Junya Yagi about this paper, and then some more from this nice survey of Masahiko Yamazaki.

(My attempt to tell) the story is the following.   You make a certain configuration of M5 branes in M theory, of the following sort:

	0	1	2	3	4	5	6	7	8	9	10
S	x	x	x		x		x				x
L	x	x	x	x	x	x
L’	x	x	x	x			x	x

The interesting part of the theory for us is happening in dimensions 4,6.  Usually these are taken to be a torus.  These dimensions are filled by S.  The L, L’ meet S in codimension 1, in straight lines.  Maybe there are many things of each sort.  The fact that they are occupying dimensions 5 and 7 is not so important, what is important is that the intersection of L and L’ is entirely contained in S, and maybe that dimensions 8 and 9 are left free.  I will write s, l, l’ for the restrictions of these branes to dimensions 4,6.

I would like to get a straight answer out of someone about what topological twists are necessary to allow me keep some supersymmetry while putting dimensions 4 and 6 around an arbitrary surface, and letting l and l’ go along arbitrary curves in s, maybe with self-intersection.  If I wanted to let them self-intersect on s, I would also want to let them occupy a line in 5,7 space which varies so that they the self-intersections happen only on S.  E.g., I could take the 5,7 space to be the cotangent to the 4,6 space, and choose the line in 5,7 space to be the conormal.

Anyway, the point is that if you are looking just at s, what you see is a surface with some lines drawn on it.  It might take a bit of optimism to believe that this is related to what we usually discuss on this blog, but just look at Figure 22 on page 29 of Yamazaki’s survey — which explains that the effect of the branes is to create a diagram which looks like what we would call the ranks of the sheaf — and Figure 40 on page 49, where zig-zag paths, i.e. Legendrian fronts, are identified with the kind of branes I called L or L’ above.   (He talks there about D5 and NS5 branes, but the dualities to go to the above discussion are contained in Yagi’s paper.  I prefer the M5 branes, just because it means there is only one thing which I don’t know what it is, instead of two different things.)

That is, now we know what physics we are studying: we make a brane configuration as above.

Here’s what we should do now. In Yamazaki’s survey, it’s pointed out that they only know how to understand the physics when all regions are labeled 0, -1, or +1 — in this case, everything has some interpretation in terms of quivers, although I do not know very specifically either what the questions explicitly are, or what the answers are.

As I understand it, the questions take the following form: dimensionally reduce or compactify the theory taking place in dimensions 0,1,2 (note that all branes occupy these dimensions) on something, and see what you get.

So what we should do now is the following thing.  By searching the literature or asking an appropriate physicist, make a big table of the form:

Question                                Quiver gauge theory answer

(i.e. 3-manifold)                          (fill in this here)

 

In Yamazaki’s survey, it’s stated that while the physicists know exactly what to do in the quiver setting, i.e. when all ranks are -1, 0, 1, they don’t know what to do otherwise.  But, maybe the knotty character varieties will let us give the answer!  In particular, they themselves *will be* the answer to some question:

Question                       Quiver gauge theory answer                       General brane tiling answer

????                                     cluster variety of quiver                                 knotty character variety

 

What is the question?  I don’t know, but probably a reasonable guess is: “what are the line operators in the 3d theory”.  After all, we want the answer to be moduli of objects in a category, and line operators in 3d give a category.  Anyway if that’s the case, the ideal scenario would be that the answer is something like: line operators in the 3d theory are given by some branes which are a line in dimensions 0,1,2, maybe occupy dimension 3, and are an appropriate sheaf (maybe interpreted as a Lagrangian brane) in dimensions 4,5,6,7 = T*s.  [[Something like this should be true because points on the knotty character variety are exactly these Lagrangian branes]].

 

A related we would like to be able to ask is something like:

Question                       Quiver gauge theory answer                       General brane tiling answer

????                                    H*(cl. variety of quiver)                                H*(k char variety)

Because: the answer to this question in the case where s is the 2 sphere and the knot which appears is a positive braid encircling a point, we know the answer H*(k char variety) has something to do with HOMFLY homology.  But if we could ask a physical question to which the answer is H*(k char variety), especially of the sort suggested above in which we have a brane configuration describing a knot, then we would have some path through physics to “explain” *why* H*(k char variety) has something to do with HOMFLY homology — make a bunch of string dualities until we get to the usual place where knot homology appears (!)

 

More generally and maybe most interestingly, any question about a quiver gauge theory whose answer can be computed purely in terms of the geometry of the cluster variety of the quiver, we can probably compute the analogous answer by doing the same thing on the knotty character variety.