Donaldson-Thomas invariants from tropical disks

Abstract

We prove that the quantum DT invariants associated to quivers with genteel potential can be expressed in terms of certain refined counts of tropical disks. This is based on a quantum version of Bridgeland's description of cluster scattering diagrams in terms of stabilitiy conditions, plus a new version of the description of scattering diagrams in terms of tropical disk counts. The weights with which the tropical disks are counted are expressed in terms of motivic integrals of certain quiver flag varieties. We also show via explicit counterexample that Hall algebra broken lines do not result in consistent Hall algebra theta functions, i.e., they violate the extension of a lemma of Carl-Pumperla-Siebert from the classical setting.

arXiv comment38 pages, 7 figures; published version

References in corpus (8)

• Quantum Dilogarithm
• The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli
• The tropical vertex
• Refined curve counting with tropical geometry
• Donaldson-Thomas theory and cluster algebras
• Tropical refined curve counting from higher genera and lambda classes
• Attractor flow trees, BPS indices and quivers
• Block-Göttsche invariants from wall-crossing

Cited by in corpus (4)

• The canonical wall structure and intrinsic mirror symmetry
• Scattering diagrams, stability conditions, and coherent sheaves on ${\mathbb{P}}^2$
• Scattering diagrams, theta functions, and refined tropical curve counts
• Quivers and curves in higher dimension