Thorny character varieties

Let M be a one-dimensional manifold.  A thorn on M is a ray in TM, or equivalently a point of TM.

Thorns STM are in “general position” if the projection π:SM is injective.  If S is in general position then we define a quiver QS:

thorns1
Left, a manifold. Right, some thorns on the manifold.

1.  The vertices of QS are the bounded components of Mπ(S).  (We can negotiate whether to include the unbounded components.)

2.  The edges of QS are the thorns, with s connecting v to w if v{π(s)}w is connected.

3.  Orient s:vw if s takes negative (sorry) values on the vector at the boundary of v that points toward w.

For example with M and S as in the figure above, QS is given by .  In general when M is homeomorphic to R, QS has type An1 where n is the number of thorns.  If MS1, then QS has affine type An1 where n is the number of thorns.

When S is in general position, the derived category of sheaves on M with singular support in S is equivalent to the derived category of representations of the quiver QS. Under this equivalence, the microlocal stalk at the thorn s is the cone on the map corresponding to the edge corresponding to s.

 

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 t-structures.

 

If M is a circle and there are n=p+q thorns — p in one component and q in another component of TM — a Beilinson-Bondal equivalence identifies ShS(M) with the derived category of coherent sheaves on a weighted P1.  Weighted means there is one μp-orbifold point and one μq-orbifold point.

 

For a quiver Q it is a good idea to fix a dimension vector d and study Rep(Q,d)/GL(d), which parametrizes objects in the heart of the t-structure and in a fixed K-theory class.  The numerator is a vector space and if Q is of finite ADE type, the denominator acts with finitely many orbits.  If Q 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 Rep(Q) (sheaves on sheaves…) are related for different orientations of Q, and in the finite case Lusztig shows the set of IC 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 p:SZ and study those constructible sheaves for which μs(F)C[p(s)].  If p is {0,1}-valued, and obeys a Dyck condition, then these F are in the heart of the t-structure.  The dimension vector is of the form 1212321 or something like that.  The thorny character variety for this potential is the open orbit in Rep(Q,d).

 

The cotangent structure on TR obscures the rotational symmetry of its Fukaya category. Kontsevich has given a description of the stability manifold of representations of the An quiver — it is the space of quadratic differentials of the form

ef(z)dz2f(z)=zn+1+a1zn1+a2zn2++an

modulo the action of (n+1)st roots of unity on z. Note f(z) and f(z)+2πi give the same ef(z). The exponent is the unfolding of the type A singularity zn+1=0.

Each such quadratic differential gives a flat metric |ef(z)|(dx2+dy2) on C. This metric is not complete, its completion is obtained by adding (n+1) 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 ef(z)dz 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 ef(z)/2dz.

 

2 comments

  1. davidtreumann says:

    I have the idea that t-structures are important for setting up moduli problems.

    Let ΛTM be a Legendrian submanifold. Fix a Maslov potential p on Λ.

    Proposition: there is at most one t-structure on C(Λ) for which the functor μmonp:C(Λ)Loc(Λ) is t-exact.

    Proof: Suppose there is such a t-structure. Then C(Λ)0 must be given by all those F with μmonp(F)Loc(Λ)0 and C(Λ)0 by all those F with μmonp(F) in Loc(Λ)0.

    Let us call this the “candidate t-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 t-structure.

    In some sense the perverse t-structure has this form.

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