Abstract
We prove the equivalence of (a slightly modified version of) the wall-crossing formula of Manschot, Pioline and Sen and the wall-crossing formula of Kontsevich and Soibelman. The former involves abelian analogues of the motivic Donaldson–Thomas type invariants of quivers with stability introduced by Kontsevich and Soibelman, for which we derive positivity and geometricity properties.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Behrend, K., Bryan, J., Szendrői, B.: Motivic degree zero Donaldson–Thomas invariants. Invent. Math. 192 (2013). http://arxiv.org/abs/0909.5088
Bollobás, B.: Modern graph theory. In: Graduate Texts in Mathematics, vol. 184. Springer, New York (1998)
Crawley-Boevey, W., Van den Bergh, M.: Absolutely indecomposable representations and Kac-Moody Lie algebras. Invent. Math. 155(3), 537–559 (2004). http://arxiv.org/abs/math/0106009, with an appendix by Hiraku Nakajima
Dimofte, T., Gukov, S.: Refined, motivic, and quantum. Lett. Math. Phys. 91(1), 1–27 (2010). http://arxiv.org/abs/0904.1420
Engel, J., Reineke, M.: Smooth models of quiver moduli. Math. Z. 262(4), 817–848 (2009). http://arxiv.org/abs/0706.4306
Filippini, S.A., Stoppa, J.: Block-Göttsche invariants from wall-crossing (2012). http://arxiv.org/abs/1212.4976
Gessel I., Wang D.L.: Depth-first search as a combinatorial correspondence. J. Combin. Theory Ser. A 26(3), 308–313 (1979)
Gross, M., Pandharipande, R., Siebert, B.: The tropical vertex, Duke Math. J. 153(2), 297–362 (2010). http://arxiv.org/abs/0902.0779
Harvey J.A., Moore G.: On the algebras of BPS states. Commun. Math. Phys. 197(3), 489–519 (1998)
Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Positivity of Kac polynomials and DT-invariants for quivers (2012). http://arxiv.org/abs/1204.2375
Hausel, T., Rodriguez-Villegas, F.: Mixed Hodge polynomials of character varieties. Invent. Math. 174(3), 555–624 (2008). http://arxiv.org/abs/math/0612668, With an appendix by Nicholas M. Katz
Joyce, D., Song, Y.: A theory of generalized Donaldson–Thomas invariants. Mem. Amer. Math. Soc. 217, (2012) no. 1020. http://arxiv.org/abs/0810.5645
Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson–Thomas invariants and cluster transformations (2008). http://arxiv.org/abs/0811.2435
Kontsevich, M., Soibelman, Y.: Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants. Commun. Num. Theor. Phys. 5, 231–352 (2011). http://arxiv.org/abs/1006.2706
Macdonald, I.G.: Symmetric functions and Hall polynomials, 2nd edn. In: Oxford Mathematical Monographs. Oxford University Press, Oxford (1995). With contributions by A. Zelevinsky
Manschot, J., Pioline, B., Sen, A.: Wall crossing from Boltzmann black hole halos. J. High Energy Phys. 7, 59 (2011). http://arxiv.org/abs/1011.1258
Manschot, J., Pioline, B., Sen, A.: From Black Holes to Quivers (2012). http://arxiv.org/abs/1207.2230
Reineke, M.: The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli. Invent. Math. 152(2), 349–368 (2003). http://arxiv.org/abs/math/0204059
Reineke, M.: Poisson automorphisms and quiver moduli. J. Inst. Math. Jussieu 9(3), 653–667 (2010). http://arxiv.org/abs/0804.3214
Reineke, M.: Cohomology of quiver moduli, functional equations, and integrality of Donaldson–Thomas type invariants. Compost. Math. 147(3), 943–964 (2011). http://arxiv.org/abs/0903.0261
Reineke, M., Stoppa, J., Weist, T.: MPS degeneration formula for quiver moduli and refined GW/Kronecker correspondence. Geom. Topol. 16, 2097–2134 (2012). http://arxiv.org/abs/1110.4847
Reineke M., Weist T.: Refined GW/Kronecker correspondence. Math. Ann. 355(1), 17–56 (2013)
Sen, A.: Equivalence of three wall crossing formulae. Commun. Num. Theor. Phys. 6, 601–659 (2012). http://arxiv.org/abs/1112.2515
Stanley, R.P.: Enumerative combinatorics, vol. 2. In: Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999). With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mozgovoy, S., Reineke, M. Abelian Quiver Invariants and Marginal Wall-Crossing. Lett Math Phys 104, 495–525 (2014). https://doi.org/10.1007/s11005-013-0671-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-013-0671-0