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.
Keywords
Representations of quivers, Gromov–Witten theory, Quiver
moduli, Tropical curves