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Motivic invariants of quivers via dimensional reduction

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Abstract

We provide a reduction formula for the motivic Donaldson–Thomas invariants associated with a quiver with superpotential. The method is valid provided the superpotential has a linear factor, it allows us to compute virtual motives in terms of ordinary motivic classes of simpler quiver varieties. We outline an application, giving explicit formulas for the motivic Donaldson–Thomas invariants of the orbifolds \({[\mathbb{C} \times \mathbb{C}^2/\mathbb{Z}_n]}\) .

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References

  1. Behrend, K.: Donaldson–Thomas invariants via microlocal geometry. To appear in Annals Math (2005). arXiv:0507523v2

  2. Behrend, K., Bryan, J., Szendrői, B.: Motivic degree zero Donaldson–Thomas invariants (2009). arXiv:0909.5088v1

  3. Bridgeland, T.: An introduction to motivic Hall algebras (2010). arXiv:1002.4372v1

  4. Denef, J., Loeser, F.: Geometry on arc spaces of algebraic varieties. European Congress of Mathematics, vol. 1. (Barcelona, 2000), Prog. Math., vol. 201, pp. 327–348. Birkhauser, Basel (2001)

  5. Ginzburg, V.: Calabi-Yau algebras (2006). arXiv:math/0612139v3

  6. Joyce, D., Song, Y.: A theory of generalized Donaldson–Thomas invariants. arXiv:0810.5645v6

  7. King A.D.: Moduli of representations of finite dimensional algebras. Q. J. Math. 45, 515–530 (1994)

    Article  MATH  Google Scholar 

  8. Kontsevich, M., Soibelman, Y.: Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants (2008). arXiv:1006.2706v1

  9. Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson–Thomas invariants and cluster transformations (2010). arXiv:0811.2435

  10. Kresch A.: Cycle groups for Artin stacks. Invent. Math. 138(3), 495–536 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Maulik D., Nekrasov N., Okounkov A., Pandharipande R.: Gromov witten theory and Donaldson Thomas theory, I. Compositio Mathematica. 142, 1286–1304 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Morrison, A., Nagao, K.: Motivic Donaldson–Thomas invariants of small crepant resolutions (2011). arXiv:1110.5976

  13. Morrison, A., Mozgovoy, S., Nagao, K., Szendrői, B.: Motivic Donaldson–Thomas invariants of the conifold and the refined topological vertex (2011). arXiv:1107.5017v1

  14. Reineke M.: Poisson automorphisms and quiver moduli. J. Inst. Math. Jussieu. 9(3), 653667 (2010)

    Article  MathSciNet  Google Scholar 

  15. Serre, J.P.: Espaces fibrés algébriques Exposé 5, Séminaire C. Chevalley, Anneaux de Chow et applications, 2nd année, IHP. (1958)

  16. Szendrői B.: Non-commutative Donaldson–Thomas theory and the conifold. Geom. Topol. 12, 1171 (2008)

    Article  MathSciNet  Google Scholar 

  17. Toën, B.: Grothendieck rings of Artin n-stacks (2005). arXiv:0509098

  18. Thomas R.: A holomorphic Casson invariant for Calabi-Yau 3-folds and bundles on K3 fibrations. JDG 54, 367–438 (2000)

    MATH  Google Scholar 

  19. Young B.: Generating functions for colored 3D Young diagrams and the Donaldson–Thomas invariants of orbifolds. Duke Math. J. 152(1), 115–153 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

    Andrew Morrison

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  1. Andrew Morrison
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Correspondence to Andrew Morrison.

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Morrison, A. Motivic invariants of quivers via dimensional reduction. Sel. Math. New Ser. 18, 779–797 (2012). https://doi.org/10.1007/s00029-011-0081-z

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