A Thurston compactification of the space of stability conditions

• A \(\mathbf{C}\)-linear tringulated category (so has shifts and cones)
• 2-Calabi–Yau, that is: \[ \operatorname{Hom}(x,y) \cong \operatorname{Hom}(y,x[2])^*\]
• For example, suppose \(\Gamma\) is the \(A_n\) chain. Consider the \(A_n\) surface singularity \(Y:x^2+y^2 = z^{n+1}\). Let \(\pi \colon X \to Y\) be the minimal resolution of singularities. Then the exceptional divisor in \(X\) is a chain of \(n\) rational curves. We have \[C_{\Gamma} = \{x \in D^b(\operatorname{Coh} X) \mid R\pi_{*} x = 0\}.\] For example, the bundle \(O(-1)\) supported on the \(i\)-th component of the chain lies in \(C_{\Gamma}\). Call this \(P_i\). It turns out that \(P_1, \dots, P_n\) generate \(C_{\Gamma}\). We have \[ \hom^{0}(P_i, P_i) = \hom^2(P_i,P_i) = 1 \quad \hom^1(P_i, P_{i+1}) = 1,\] and all other homs are 0.
• In general, \(C_{\Gamma}\) is generated by objects \(P_v\), one for each vertex \(v \in \Gamma\), satisfying \[ \hom^0(P_v,P_v) = \hom^2(P_v,P_v) = 1 \quad \hom^1(P_v, P_w) = 1,\] if \(v - w\) is an edge of \(\Gamma\).

• Will take too long to define.
• Gives a metric on the category. That is, for every arrow \(f \colon x \to y\), a non-negative number \(l(f)\) such that the triangle inequality is satisfied: \(l(x \to z) \leq l(x \to y) + l(y \to z)\).
• The set of all stability conditions is a manifold.

On \(T_g\) we have the action of the mapping class group \(\operatorname{Mod}_g\). On \(C_{\Gamma}\), and hence on \(\operatorname{Stab}(C_{\Gamma})\), we have the action of the Artin–Tits group \(A_{\Gamma}\). The action of \(\operatorname{Mod}_g\) is faithful, and the quotient \(T_g \to M_g\) is a covering space with covering group \(\operatorname{Mod}_g\). The action of \(A_{\Gamma}\) on \(\operatorname{Stab}(C_\Gamma)\) is not know to be faithful, but the quotient \(\operatorname{Stab}(C_{\Gamma}) \to \operatorname{Stab}(C_{\Gamma}) / A_{\Gamma} = U_{\Gamma}\) is a covering space and \[\pi_1(\operatorname{Stab}(C_{\Gamma})) = A_{\Gamma}.\]

Conjecture. The action of \(A_{\Gamma}\) on \(\operatorname{Stab}(C_{\Gamma})\) is faithful. Equivalently, \(\operatorname{Stab}(C_{\Gamma})\) is simply connected.

This has major implications. For example, this implies (in particular) that the action of \(A_{\Gamma}\) on \(C_{\Gamma}\) is faithful. And therefore:

Consequence. The word and conjugacy problems for \(A_{\Gamma}\) are solvable.

Another longstanding open conjecture here is the \(K(\pi,1)\)-conjecture.

Conjecture.

1. The space \(U\) is a \(K(\pi,1)\), equivalently (in light of the above conjecture)
2. \(\operatorname{Stab}(C_{\Gamma})\) is contractible.

Question.

1. Fenchel-Nielsen coordinates for \(\operatorname{Stab}(C_{\Gamma})\)?
2. Other approaches to contratibility of \(T_g\)?

Let \(\Gamma\) be any connected quiver.

1. The map \(i\) is injective. Homeomorphism? Not sure.
2. The closure of the image of \(i\) is compact.

3. The boundary contains all “intersection functionals”.

   Intersection functional: fix an \(x \in S\). Define \[ h_x : y \to \dim \operatorname{Hom}^{*}(x,y).\]

   Theorem

   The points \([h_x] \in \mathbf{P}^S\) lie in the closure of the image of \(i\).

   Proof

   Let us recall how the proof goes in Teichmuller theory. Fix a simple closed curve \(\delta\) (draw picture). Choose any metric \(\mu\). Let \(\sigma\) be the Dehn twist in \(\delta\). Consider the pull-back metric \(\sigma^{*}(\mu)\). Then the length of \(\gamma\) under \(\sigma^{*}(\mu)\) is the length of \(\sigma(\gamma)\) under \(\mu\). The curve \(\sigma(\gamma)\) looks like this (draw picture). Its length is \[ l(\sigma \gamma) \leq l(\gamma) + l(\delta)\] This is accurate if there is exactly one intersection point. If there are more, the same local picture holds at each point, and we get \[ l(\sigma \gamma) \leq l(\gamma) + h_{\delta}(\gamma) \cdot l(\delta).\] If we iterate this, we get \[ l_{\sigma^n \mu} (\gamma) \leq l(\gamma) + n h_{\delta}(\gamma) l(\delta).\] The key point is that as \(n \to \infty\), the error in the RHS and LHS is negligible compared to \(n\). And hence the projective functional \(l_{\sigma^n \mu}\) approaches the intersection functional \([h_{\delta}]\).

   The categorical proof is an exact analogue. Fix a spherical object \(x\). Let \(\tau\) be a stability condition. The analogue of the Dehn twist is the spherical twist \(\sigma_x \colon \mathcal{C} \to \mathcal{C}\). It is defined as follows: \[ \sigma_x(y) = \operatorname{Cone} (\operatorname{Hom}(x,y) \otimes x \to y).\] By definition, we have the exact triangle \[ \operatorname{Hom}(x,y) \otimes x \to y \to \sigma_x(y),\] which after a shift becomes \[ y \to \sigma_x(y) \to \operatorname{Hom}(x,y) \otimes x[1].\] By the triangle inequality, this gives \[ l_{\sigma^* \tau} (y) \leq l_\tau y + h_x(y) l_\tau x.\] And by iterating, \[ l_{\sigma^n \tau} (y) \leq l_\tau y + n h_x(y) l_\tau x.\] The key again is to prove that the difference is negligible compared to \(n\), and therefore in \(\mathbf{P}^S\), we have \[ \lim_{n \to \infty} [l_{\sigma^n \tau}] = [h_x].\]

   In the case of the \(A_n\) quiver, we can prove that the closure of the intersection functionals \([h_x]\) is a sphere. The Braid group acts on this sphere by picewise linear maps. We do not know that this is precisely the boundary.

   In the \(n = 2\), case, however, we can prove the best possible outcome and obtain a very nice picture.

   Draw the Farey triangulation.