Wall-crossing structures on surfaces

TL;DR

This paper explores wall-crossing structures on surfaces induced by Bridgeland stability conditions, demonstrating their transfer to quantum tori and proposing a conjecture linking these structures to quivers with potential, verified for the projective plane.

Contribution

It establishes that geometric stability conditions on certain surfaces have global dimension 2 and formulates a conjecture connecting these to quiver with potential stability data.

Findings

Proves that stability conditions on surfaces with nef anticanonical bundle have global dimension 2.

Formulates a conjecture relating surface stability data to quivers with potential.

Verifies the conjecture for the projective plane.

Abstract

Families of Bridgeland stability conditions induce families of stability data, wall-crossing structures and scattering diagrams on the motivic Hall algebra. These structures can be transferred to the quantum torus if the stability conditions of the family have global dimension at most 2. We show that geometric stability conditions on surfaces with nef anticanonical bundle have global dimension 2 and we study the resulting family of stability data. We formulate a conjecture relating this family to the family of stability data associated with a quiver with potential and we verify this conjecture for the projective plane.