Donaldson–Thomas invariants from tropical disks

• Man-Wai Cheung ^[1] ; Travis Mandel ^[2]
  1. [1] Harvard University

     Harvard University

     City of Cambridge, Estados Unidos

     
  2. [2] University of Edinburgh

     University of Edinburgh

     Reino Unido

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 26, Nº. 4, 2020
• Idioma: inglés
• DOI: 10.1007/s00029-020-00580-8
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• Resumen

  • 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 stability 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.