Torus fixed points of quiver moduli spaces are given by stable representations of the universal (abelian) covering quiver. As far as the Kronecker quiver is concerned they can be described by stable representations of certain bipartite quivers coming along with a stable colouring. By use of the glueing method it is possible to construct a huge class of such quivers implying a lower bound for the Euler characteristic. For certain roots it is even possible to construct all torus fixed points.

Motivated by string-theoretic arguments Manschot, Pioline and Sen discovered a new remarkable formula for the Poincaré polynomial of a smooth compact moduli space of stable quiver representations which effectively reduces to the abelian case (ie thin dimension vectors). We first prove a motivic generalization of this formula, valid for arbitrary quivers, dimension vectors and stabilities. In the case of complete bipartite quivers we use the refined GW/Kronecker correspondence between Euler characteristics of quiver moduli and Gromov-Witten invariants to identify the MPS formula for Euler characteristics with a standard degeneration formula in Gromov-Witten theory. Finally we combine the MPS formula with localization techniques, obtaining a new formula for quiver Euler characteristics as a sum over trees, and constructing many examples of explicit correspondences between quiver representations and tropical curves. 16G20, 14N35, 14T05

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a b s t r a c tBy a result of Klyachko the Euler characteristic of moduli spaces of stable bundles of rank two on the projective plane is determined. Using similar methods we extend this result to bundles of rank three. The fixed point components correspond to moduli spaces of the subspace quiver. Moreover, the stability condition is given by a certain system of linear inequalities so that the generating function of the Euler characteristic can be determined explicitly.