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MPS degeneration formula for quiver moduli and refined GW/Kronecker correspondence

Markus Reineke, Jacopo Stoppa and Thorsten Weist

Geometry & Topology 16 (2012) 2097–2134

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.

Representations of quivers, Gromov–Witten theory, Quiver moduli, Tropical curves
Mathematical Subject Classification 2010
Primary: 16G20, 14N35, 14T05
Received: 28 November 2011
Revised: 13 July 2012
Accepted: 29 June 2012
Published: 8 November 2012
Proposed: Richard Thomas
Seconded: Jim Bryan, Yasha Eliashberg
Markus Reineke
Fachbereich C - Mathematik
Bergische Universität Wuppertal
D 42097 Wuppertal
Jacopo Stoppa
Dipartimento di Matematica “F. Casorati”
Università degli Studi di Pavia
Via Ferrata 1
27100 Pavia
Thorsten Weist
Fachbereich C - Mathematik
Bergische Universität Wuppertal
D-D 42097 Wuppertal