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MPS
degeneration formula for quiver moduli and refined
GW/Kronecker correspondence
Markus Reineke, Jacopo Stoppa and Thorsten Weist |
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Geometry & Topology 16 (2012) 2097–2134
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 16G20, 14N35, 14T05
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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
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Authors |
| Markus Reineke | |
| Fachbereich C - Mathematik Bergische Universität Wuppertal D 42097 Wuppertal Germany |
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| Jacopo Stoppa | |
| Dipartimento di Matematica “F.
Casorati” Università degli Studi di Pavia Via Ferrata 1 27100 Pavia Italy |
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| Thorsten Weist | |
| Fachbereich C - Mathematik Bergische Universität Wuppertal D-D 42097 Wuppertal Germany |
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