Let be a one-dimensional manifold. A thorn on is a ray in , or equivalently a point of .
Thorns are in “general position” if the projection is injective. If is in general position then we define a quiver :
Left, a manifold. Right, some thorns on the manifold.
1. The vertices of are the bounded components of . (We can negotiate whether to include the unbounded components.)
2. The edges of are the thorns, with connecting to if is connected.
3. Orient if takes negative (sorry) values on the vector at the boundary of that points toward .
For example with and as in the figure above, is given by . In general when is homeomorphic to , has type where is the number of thorns. If , then has affine type where is the number of thorns.
When is in general position, the derived category of sheaves on with singular support in is equivalent to the derived category of representations of the quiver . Under this equivalence, the microlocal stalk at the thorn is the cone on the map corresponding to the edge corresponding to .
Bernstein-Gelfand-Ponomaraev construct equivalences between derived representation categories of two different orientations of a quiver. This could partly, not entirely, be deduced from GKS. These equivalences do not respect the -structures.
If is a circle and there are thorns — in one component and in another component of — a Beilinson-Bondal equivalence identifies with the derived category of coherent sheaves on a weighted . Weighted means there is one -orbifold point and one -orbifold point.
For a quiver it is a good idea to fix a dimension vector and study , which parametrizes objects in the heart of the -structure and in a fixed -theory class. The numerator is a vector space and if is of finite ADE type, the denominator acts with finitely many orbits. If has affine ADE type, the denominator acts with one-parameter families of orbits, with a distinguished finite number of them called “nilpotent orbits” by Lusztig. (Under Beilinson-Bondal, these are the ones set-theoretically supported at the two orbifold points.)
The categories of sheaves on (sheaves on sheaves…) are related for different orientations of , and in the finite case Lusztig shows the set of sheaves are preserved by these relations. Lusztig’s argument reminds me of ruling filtrations.
Our experience with Legendrian knots (but these are more like links) says to fix a potential and study those constructible sheaves for which . If is -valued, and obeys a Dyck condition, then these are in the heart of the -structure. The dimension vector is of the form or something like that. The thorny character variety for this potential is the open orbit in .
The cotangent structure on obscures the rotational symmetry of its Fukaya category. Kontsevich has given a description of the stability manifold of representations of the quiver — it is the space of quadratic differentials of the form
modulo the action of st roots of unity on . Note and give the same . The exponent is the unfolding of the type singularity .
Each such quadratic differential gives a flat metric on . This metric is not complete, its completion is obtained by adding points on the boundary (thorns). Some pairs of these points can be joined by geodesics, which are special Lagrangians that generate the Fukaya category. Integrating against them (an oscillatory integral, like the Airy function) gives the central charge.
Last week Xin pointed out that flat coordinates for this metric are given by the real and imaginary parts of .
2 comments
davidtreumann says:
I have the idea that -structures are important for setting up moduli problems.
Let be a Legendrian submanifold. Fix a Maslov potential on .
Proposition: there is at most one -structure on for which the functor is -exact.
Proof: Suppose there is such a -structure. Then must be given by all those with and by all those with in .
Let us call this the “candidate -structure.” The existence of sheaves of microlocal rank one (so they would be in the heart) with negative self-exts shows it is not always a -structure.
In some sense the perverse -structure has this form.
I have the idea that -structures are important for setting up moduli problems.
Let be a Legendrian submanifold. Fix a Maslov potential on .
Proposition: there is at most one -structure on for which the functor is -exact.
Proof: Suppose there is such a -structure. Then must be given by all those with and by all those with in .
Let us call this the “candidate -structure.” The existence of sheaves of microlocal rank one (so they would be in the heart) with negative self-exts shows it is not always a -structure.
In some sense the perverse -structure has this form.