Block–Göttsche invariants from wall-crossing

Abstract

We show how some of the refined tropical counts of Block and Göttsche emerge from the wall-crossing formalism. This leads naturally to a definition of a class of putative<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0010437X14007994_inline1" xlink:type="simple" /><jats:tex-math>$q$</jats:tex-math></jats:alternatives></jats:inline-formula>-deformed Gromov–Witten invariants. We prove that this coincides with another natural<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0010437X14007994_inline2" xlink:type="simple" /><jats:tex-math>$q$</jats:tex-math></jats:alternatives></jats:inline-formula>-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined.