Communications in Number Theory and Physics

Volume 5 (2011)

Number 2

Universal covers and the GW/Kronecker correspondence

Pages: 353 – 395



Jacopo Stoppa (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, United Kingdom)


The tropical vertex is an incarnation of mirror symmetryfound by Gross, Pandharipande and Siebert. It can beapplied to $m$-Kronecker quivers $K(m)$ (together with aresult of Reineke) to compute the Euler characteristics ofthe moduli spaces of their (framed) representations interms of Gromov–Witten invariants (as shown by Gross andPandharipande). In this paper, we study a possible geometricpicture behind this correspondence, in particularconstructing rational tropical curves from subquivers ofthe universal covering quiver $\widetilde{K}(m)$.Additional motivation comes from the physicalinterpretation of $m$-Kronecker quivers in the context ofquiver quantum mechanics (especially, work of Denef).

Published 26 September 2011