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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
MPS degeneration formula for quiver moduli and refined GW/Kronecker correspondence

Markus Reineke, Jacopo Stoppa and Thorsten Weist

Geometry & Topology 16 (2012) 2097–2134
Abstract

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