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

Reineke, Markus¹ ; Stoppa, Jacopo² ; Weist, Thorsten³

¹ Fachbereich C - Mathematik, Bergische Universität Wuppertal, D 42097 Wuppertal, Germany
² Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy
³ Fachbereich C - Mathematik, Bergische Universität Wuppertal, D-D 42097 Wuppertal, Germany

Geometry & topology, Tome 16 (2012) no. 4, pp. 2097-2134.

Résumé

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.

Détail

DOI : 10.2140/gt.2012.16.2097

Classification : 16G20, 14N35, 14T05
Keywords: Representations of quivers, Gromov–Witten theory, Quiver moduli, Tropical curves

Bibliographie

[1] T Bridgeland, An introduction to motivic Hall algebras, Adv. Math. 229 (2012) 102

[2] J Engel, M Reineke, Smooth models of quiver moduli, Math. Z. 262 (2009) 817

[3] A Gathmann, H Markwig, The numbers of tropical plane curves through points in general position, J. Reine Angew. Math. 602 (2007) 155

[4] M Gross, R Pandharipande, Quivers, curves, and the tropical vertex, Port. Math. 67 (2010) 211

[5] M Gross, R Pandharipande, B Siebert, The tropical vertex, Duke Math. J. 153 (2010) 297

[6] E N Ionel, T H Parker, Relative Gromov–Witten invariants, Ann. of Math. 157 (2003) 45

[7] A D King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. 45 (1994) 515

[8] M Kontsevich, Y Soibelman, Affine structures and non-Archimedean analytic spaces, from: "The unity of mathematics" (editors P Etingof, V Retakh, I Singer), Progr. Math. 244, Birkhäuser (2006) 321

[9] A M Li, Y Ruan, Symplectic surgery and Gromov–Witten invariants of Calabi–Yau $3$–folds, Invent. Math. 145 (2001) 151

[10] J Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002) 199

[11] I G Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press (1995)

[12] J Manschot, BPS invariants of semi-stable sheaves on rational surfaces

[13] J Manschot, B Pioline, A Sen, Wall crossing from Boltzmann black hole halos, J. High Energy Phys. (2011) 059, 73

[14] G Mikhalkin, Enumerative tropical algebraic geometry in $\mathbb R^2$, J. Amer. Math. Soc. 18 (2005) 313

[15] M Reineke, The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2003) 349

[16] M Reineke, Poisson automorphisms and quiver moduli, J. Inst. Math. Jussieu 9 (2010) 653

[17] M Reineke, T Weist, Refined GW/Kronecker correspondence

[18] J Stoppa, Universal covers and the GW/Kronecker correspondence, Commun. Number Theory Phys. 5 (2011) 353

[19] T Weist, Localization in quiver moduli spaces