Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 088, 16 pages      arXiv:1210.5181      https://doi.org/10.3842/SIGMA.2012.088
Contribution to the Special Issue “Mirror Symmetry and Related Topics”

Nekrasov's Partition Function and Refined Donaldson-Thomas Theory: the Rank One Case

Balázs Szendrői
Mathematical Institute, University of Oxford, UK

Received June 12, 2012, in final form November 05, 2012; Published online November 17, 2012

Abstract
This paper studies geometric engineering, in the simplest possible case of rank one (Abelian) gauge theory on the affine plane and the resolved conifold. We recall the identification between Nekrasov's partition function and a version of refined Donaldson-Thomas theory, and study the relationship between the underlying vector spaces. Using a purity result, we identify the vector space underlying refined Donaldson-Thomas theory on the conifold geometry as the exterior space of the space of polynomial functions on the affine plane, with the (Lefschetz) SL(2)-action on the threefold side being dual to the geometric SL(2)-action on the affine plane. We suggest that the exterior space should be a module for the (explicitly not yet known) cohomological Hall algebra (algebra of BPS states) of the conifold.

Key words: geometric engineering; Donaldson-Thomas theory; resolved conifold.

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References

  1. Behrend K., Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), 1307-1338, math.AG/0507523.
  2. Behrend K., Bryan J., Szendrői B., Motivic degree zero Donaldson-Thomas invariants, Invent. Math., to appear, arXiv:0909.5088.
  3. Bridgeland T., Hall algebras and curve-counting invariants, J. Amer. Math. Soc. 24 (2011), 969-998, arXiv:1002.4374.
  4. Choi J., Katz S., Klemm A., The refined BPS index from stable pair invariants, arXiv:1210.4403.
  5. Davison B., Maulik D., Schuermann J., Szendrői B., Purity for graded potentials and cluster positivity, unpublished.
  6. Dimca A., Szendrői B., The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on C3, Math. Res. Lett. 16 (2009), 1037-1055, arXiv:0904.2419.
  7. Dimofte T., Gukov S., Refined, motivic, and quantum, Lett. Math. Phys. 91 (2010), 1-27, arXiv:0904.1420.
  8. Efimov A.I., Quantum cluster variables via vanishing cycles, arXiv:1112.3601.
  9. Gopakumar R., Vafa C., M-theory and topological strings - I, hep-th/9809187.
  10. Gukov S., Stosic M., Homological algebra of knots and BPS states, arXiv:1112.0030.
  11. Hollowood T., Iqbal A., Vafa C., Matrix models, geometric engineering and elliptic genera, J. High Energy Phys. 2008 (2008), no. 3, 069, 81 pages, hep-th/0310272.
  12. Iqbal A., Kashani-Poor A.K., SU(N) geometries and topological string amplitudes, Adv. Theor. Math. Phys. 10 (2006), 1-32, hep-th/0306032.
  13. Iqbal A., Kozçaz C., Vafa C., The refined topological vertex, J. High Energy Phys. 2009 (2009), no. 10, 069, 58 pages, hep-th/0701156.
  14. Katz S., Klemm A., Vafa C., Geometric engineering of quantum field theories, Nuclear Phys. B 497 (1997), 173-195, hep-th/9609239.
  15. Klebanov I.R., Witten E., Superconformal field theory on threebranes at a Calabi-Yau singularity, Nuclear Phys. B 536 (1999), 199-218, hep-th/9807080.
  16. Kontsevich M., Soibelman Y., Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Number Theory Phys. 5 (2011), 231-352, arXiv:1006.2706.
  17. Kontsevich M., Soibelman Y., Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
  18. Losev A., Moore G., Nekrasov N., Shatashvili S., Four-dimensional avatars of two-dimensional RCFT, Nuclear Phys. B Proc. Suppl. 46 (1996), 130-145, hep-th/9509151.
  19. Maulik D., Nekrasov N., Okounkov A., Pandharipande R., Gromov-Witten theory and Donaldson-Thomas theory. I, Compos. Math. 142 (2006), 1263-1285, math.AG/0312059.
  20. Morrison A., Mozgovoy S., Nagao K., Szendrői B., Motivic Donaldson-Thomas invariants of the conifold and the refined topological vertex, Adv. Math. 230 (2012), 2065-2093, arXiv:1107.5017.
  21. Nagao K., Nakajima H., Counting invariant of perverse coherent sheaves and its wall-crossing, Int. Math. Res. Not. 2011 (2011), 3885-3938, arXiv:0809.2992.
  22. Nakajima H., Yoshioka K., Instanton counting on blowup. I. 4-dimensional pure gauge theory, Invent. Math. 162 (2005), 313-355, math.AG/0306198.
  23. Nakajima H., Yoshioka K., Instanton counting on blowup. II. K-theoretic partition function, Transform. Groups 10 (2005), 489-519, math.AG/0505553.
  24. Nekrasov N.A., Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003), 831-864, hep-th/0206161.
  25. Nekrasov N.A., Okounkov A., Seiberg-Witten theory and random partitions, in The Unity of Mathematics, Progr. Math., Vol. 244, Birkhäuser Boston, Boston, MA, 2006, 525-596, hep-th/0306238.
  26. Nekrasov N.A., Okounkov A., The index of M-theory, work in progress.
  27. Okounkov A., The index and the vertex, Talk at Brandeis-Harvard-MIT-Northeastern Joint Mathematics Colloquium, December 1, 2011.
  28. Pandharipande R., Thomas R.P., Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009), 407-447.
  29. Saito M., Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), 849-995.
  30. Szendrői B., Non-commutative Donaldson-Thomas invariants and the conifold, Geom. Topol. 12 (2008), 1171-1202, arXiv:0705.3419.
  31. Tachikawa Y., Five-dimensional Chern-Simons terms and Nekrasov's instanton counting, J. High Energy Phys. 2004 (2004), 050, 13 pages, hep-th/0401184.

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