Contractibility of stability spaces for finite connected acyclic quivers

Establish that for every finite connected acyclic quiver Q, the stability manifold Stab(D^b(Q)) contracts to Toss(D^b(Q)), where Toss(D^b(Q)) denotes the subspace consisting of totally semistable stability conditions on D^b(Q), and prove that Toss(D^b(Q)) is contractible; consequently, deduce that Stab(D^b(Q)) is contractible.

Background

The paper proves that every stability condition on D^b(Q) admits, after a phase rotation, an algebraic heart, and deduces that Stab(D^b(Q)) is connected for any finite connected acyclic quiver Q. As a next step beyond connectedness, the authors focus on the stronger topological property of contractibility for the stability space.

They formulate a conjecture splitting the aim into two parts: first, Stab(D^b(Q)) should contract to the subspace Toss(D^b(Q)) of totally semistable stability conditions; second, Toss(D^b(Q)) itself should be contractible. These two statements together would imply that Stab(D^b(Q)) is contractible.

The conjecture is known in some special cases: it is proved for the affine A_{p,q} quivers, and the contractibility of Toss(D^b(Q)) is established for affine Dynkin (Euclidean) quivers. The general case for arbitrary finite connected acyclic quivers remains unresolved.