Hostname: page-component-6b989bf9dc-lb7rp Total loading time: 0 Render date: 2024-04-15T01:38:53.946Z Has data issue: false hasContentIssue false
  • English
  • Français

Block–Göttsche invariants from wall-crossing

Part of: Representation theory of rings and algebras Projective and enumerative geometry

Published online by Cambridge University Press:  12 March 2015

S. A. Filippini  and
J. Stoppa
S. A. Filippini
Affiliation:
Institut für Mathematik, Universität Zürich, Wintherthurerstrasse 190, 8057 Zürich, Schweiz email saraangela.filippini@math.uzh.ch
J. Stoppa
Affiliation:
Dipartimento di Matematica ‘F. Casorati’, Università di Pavia, via Ferrata 1, 27100 Pavia, Italia email jacopo.stoppa@unipv.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show how some of the refined tropical counts of Block and Göttsche emerge from the wall-crossing formalism. This leads naturally to a definition of a class of putative $q$-deformed Gromov–Witten invariants. We prove that this coincides with another natural $q$-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined.

Keywords

tropical countswall-crossingGromov–Witten invariants

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 14T05: Tropical geometry 16G20: Representations of quivers and partially ordered sets
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 8 , August 2015 , pp. 1543 - 1567
Copyright
© The Authors 2015 

References

Block, F. and Göttsche, L., Refined curve counting with tropical geometry, Preprint (2014),arXiv:1407.2901 [math.AG].CrossRefGoogle Scholar
Bousseau, P., Around the Gromov–Witten/Kronecker correspondence, talk given at the workshop Mirror symmetry, enumerative geometry and related topics, Pavia, 16–20 May 2014 (report on joint work in progress with R. Thomas).Google Scholar
Dimofte, T. and Gukov, S., Refined, motivic, and quantum, Lett. Math. Phys. 91 (2010), 1–27.CrossRefGoogle Scholar
Gaiotto, D., Moore, G. and Neitzke, A., Four dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys. 299 (2010), 163–224.CrossRefGoogle Scholar
Gathmann, A. and Markwig, H., The numbers of tropical plane curves through points in general position, J. Reine Angew. Math. 602 (2007), 155–177.Google Scholar
Göttsche, L., A conjectural generating function for numbers of curves on surfaces, Comm. Math. Phys. 196 (1998), 523–533.Google Scholar
Göttsche, L. and Shende, V., Refined curve counting on complex surfaces, Geom. Topol. 18 (2014), 2245–2307.CrossRefGoogle Scholar
Gross, M., Pandharipande, R. and Siebert, B., The tropical vertex, Duke Math. J. 153 (2010), 297–362.CrossRefGoogle Scholar
Itenberg, I. and Mikhalkin, G., On Block–Göttsche multiplicities for planar tropical curves, Int. Math. Res. Not. IMRN 2013 (2013), 5289–5320.CrossRefGoogle Scholar
Joyce, D. and Song, Y., A theory of generalized Donaldson–Thomas invariants, Mem. Amer. Math. Soc. 217 (2012), doi:10.1090/S0065-9266-2011-00630-1.Google Scholar
Kontsevich, M. and Soibelman, Y., Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, Preprint (2008), arXiv:0811.2435. See also M. Kontsevich and Y. Soibelman, Motivic Donaldson–Thomas invariants: summary of results, in Mirror symmetry and tropical geometry, Contemporary Mathematics, vol. 527 (American Mathematical Society, Providence, RI, 2010), 55–89.Google Scholar
Kool, M., Shende, V. and Thomas, R., A short proof of the Göttsche conjecture, Geom. Topol. 15 (2011), 397–406.CrossRefGoogle Scholar
Manschot, J., Pioline, B. and Sen, A., Wall-crossing from Boltzmann black hole halos, J. High Energy Phys. 1107 (2011), doi:10.1007/JHEP07(2011)059.Google Scholar
Mikhalkin, G., Enumerative tropical algebraic geometry in ℝ2, J. Amer. Math. Soc. 18 (2005), 313–377.CrossRefGoogle Scholar
Mozgovoy, S. and Reineke, M., Abelian quiver invariants and marginal wall-crossing, Lett. Math. Phys. 104 (2014), 495–525.CrossRefGoogle Scholar
Nagao, K., Wall-crossing of the motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1103.2922 [math.AG].Google Scholar
Pioline, B., Four ways across the wall, in 6th Baltic-Nordic workshop on algebra, geometry, and mathematical physics 2010 (AGMP-6), Journal of Physics: Conference Series, 346 (Institute of Physics Publishing, Bristol, 2012).Google Scholar
Reineke, M., Poisson automorphisms and quiver moduli, J. Inst. Math. Jussieu 9 (2010), 653–667.CrossRefGoogle Scholar
Reineke, M., Stoppa, J. and Weist, T., MPS formula for quiver moduli and refined GW/Kronecker correspondence, Geom. Topol. 16 (2012), 2097–2134.CrossRefGoogle Scholar
Reineke, M. and Weist, T., Refined GW/Kronecker correspondence, Math. Ann. 355 (2013), 17–56.CrossRefGoogle Scholar