Block-G\"ottsche invariants from wall-crossing

TL;DR

This paper connects tropical counts of Block and G"ottsche with wall-crossing formalism, proposing a new class of q-deformed Gromov-Witten invariants and establishing their equivalence with existing deformations in quiver representation theory.

Contribution

It introduces a novel link between tropical geometry and wall-crossing, defining a new class of q-deformed Gromov-Witten invariants and proving their equivalence with known deformations.

Findings

Refined tropical counts arise from wall-crossing formalism.

A new class of q-deformed Gromov-Witten invariants is defined.

Equivalence established with existing q-deformations in quiver theory.

Abstract

We show how some of the refined tropical counts of Block and G\"ottsche emerge from the wall-crossing formalism. This leads naturally to a definition of a class of putative q-deformed Gromov-Witten invariants. We prove that this coincides with another natural q-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined.