Block–Göttsche invariants from wall-crossing

Filippini, S A; Stoppa, J (2015). Block–Göttsche invariants from wall-crossing. Compositio Mathematica, 151(08):1543-1567.

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.

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.

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