Skip to main content
Log in

Takagi Lectures on Donaldson–Thomas theory

  • Special Feature: The Takagi Lectures
  • Published:
Japanese Journal of Mathematics Aims and scope

Abstract

These are introductory notes on Donaldson–Thomas counts of curves in threefolds and their connections with other branches of mathematics and mathematical physics. They are based on my 2018 Takagi Lectures at The University of Tokyo.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño and C. Vafa, Topological strings and integrable hierarchies, Comm. Math. Phys., 261 (2006), 451–516.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Aganagic, E. Frenkel and A. Okounkov, Quantum q-Langlands correspondence, preprint, arXiv:1701.03146.

  3. M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Comm. Math. Phys., 254 (2005), 425–478.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Aganagic and A. Okounkov, Elliptic stable envelope, preprint, arXiv:1604.00423.

  5. M. Aganagic and A. Okounkov, Quasimap counts and Bethe eigenfunctions, preprint, arXiv:1704.08746.

  6. M. Aganagic and A. Okounkov, in preparation.

  7. M. Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math., 68 (1988), 175–186.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Awata, B. Feigin and J. Shiraishi, Quantum algebraic approach to refined topological vertex, J. High Energy Phys., 2012 (2012), no. 3, 041.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Bagger, N. Lambert, S. Mukhi and C. Papageorgakis, Multiple membranes in M-theory, Phys. Rep., 527 (2013), 1–100.

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Bakalov and A. Kirillov, Jr., Lectures on Tensor Categories and Modular Functors, Univ. Lecture Ser., 21, Amer. Math. Soc., Providence, RI, 2001.

  11. B. Bakker and A. Jorza, Higher rank stable pairs on K3 surfaces, Commun. Number Theory Phys., 6 (2012), 805–847.

    Article  MathSciNet  MATH  Google Scholar 

  12. K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math., 128 (1997), 45–88.

    Article  MathSciNet  MATH  Google Scholar 

  13. T. Braden, A. Licata, N. Proudfoot and B. Webster, Gale duality and Koszul duality, Adv. Math., 225 (2010), 2002–2049.

    Article  MathSciNet  MATH  Google Scholar 

  14. T. Braden, A. Licata, N. Proudfoot and B. Webster, Quantizations of conical symplectic resolutions II: category O and symplectic duality, Astérisque, 384 (2016), 75–179.

    MathSciNet  MATH  Google Scholar 

  15. A. Braverman, M. Finkelberg and H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II, preprint, arXiv:1601.03586.

  16. A. Braverman, M. Finkelberg and H. Nakajima, Coulomb branches of 3d N = 4 quiver gauge theories and slices in the affine Grassmannian, preprint, arXiv:1604.03625.

  17. T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2), 166 (2007), 317–345.

    Article  MathSciNet  MATH  Google Scholar 

  18. T. Bridgeland, Spaces of stability conditions, In: Algebraic Geometry: Seattle 2005. Part 1, Proc. Sympos. Pure Math., 80, Amer. Math. Soc., Providence, RI, 2009, pp. 1–21.

    MathSciNet  MATH  Google Scholar 

  19. T. Bridgeland, Hall algebras and Donaldson–Thomas invariants, preprint, arXiv:1611.03696.

  20. J. Bryan, R. Pandharipande, The local Gromov–Witten theory of curves.With an appendix by Bryan, C. Faber, A. Okounkov and Pandharipande, J. Amer. Math. Soc., 21 (2008), 101–136.

    Article  MathSciNet  Google Scholar 

  21. M. Bullimore, T. Dimofte and D. Gaiotto, The Coulomb branch of 3d N = 4 theories, Comm. Math. Phys., 354 (2017), 671–751.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Bullimore, T. Dimofte, D. Gaiotto and J. Hilburn, Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 gauge theory, J. High Energy Phys., 2016 (2016), no. 10, 108.

    Article  MathSciNet  MATH  Google Scholar 

  23. F. Carlson, Über Potenzreihen mit ganzzahligen Koeffizienten. (German), Math. Z., 9 (1921), 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  24. E. Carlsson and A. Okounkov, Exts and vertex operators, Duke Math. J., 161 (2012), 1797–1815.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Cerf and R. Kenyon, The low-temperature expansion of the Wulff crystal in the 3D Ising model, Comm. Math. Phys., 222 (2001), 147–179.

    Article  MathSciNet  MATH  Google Scholar 

  26. N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Mod. Birkhäuser Class., Birkhäuser Boston, Boston, MA, 2010.

    Book  MATH  Google Scholar 

  27. I. Ciocan-Fontanine, B. Kim and D. Maulik, Stable quasimaps to GIT quotients, J. Geom. Phys., 75 (2014), 17–47.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. de Boer, K. Hori, H. Ooguri and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nuclear Phys. B, 493 (1997), 101–147.

    Article  MathSciNet  MATH  Google Scholar 

  29. S.K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys., 93 (1984), 453–460.

    Article  MathSciNet  MATH  Google Scholar 

  30. S.K. Donaldson and P.B. Kronheimer, The Geometry of Four-Manifolds, Oxford Math. Monogr., Oxford Science Publications, The Clarendon Press, Oxford Univ. Press, New York, 1990.

    Google Scholar 

  31. S.K. Donaldson and R.P. Thomas, Gauge theory in higher dimensions, In: The Geometric Universe, Oxford, 1996, Oxford Univ. Press, Oxford, 1998, pp. 31–47.

    MATH  Google Scholar 

  32. C. Faber and R. Pandharipande, Hodge integrals and Gromov–Witten theory, Invent. Math., 139 (2000), 173–199.

    Article  MathSciNet  MATH  Google Scholar 

  33. B. Fantechi and L. Göttsche, Riemann–Roch theorems and elliptic genus for virtually smooth schemes, Geom. Topol., 14 (2010), 83–115.

    Article  MathSciNet  MATH  Google Scholar 

  34. B. Fantechi, L. Göttsche, L. Illusie, S.L. Kleiman, N. Nitsure and A. Vistoli, Fundamental Algebraic Geometry. Grothendieck’s FGA Explained, Math. Surveys Monogr., 123, Amer. Math. Soc., Providence, RI, 2005.

  35. I.B. Frenkel and N.Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations, Comm. Math. Phys., 146 (1992), 1–60.

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Givental, On the WDVV equation in quantum K-theory, Michigan Math. J., 48 (2000), 295–304.

    Article  MathSciNet  MATH  Google Scholar 

  37. R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys., 3 (1999), 1415–1443.

    Article  MathSciNet  MATH  Google Scholar 

  38. T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math., 135 (1999), 487–518.

    Article  MathSciNet  MATH  Google Scholar 

  39. K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil and E. Zaslow, Mirror Symmetry, Clay Math. Monogr., 1, Amer. Math. Soc., Providence, RI; Clay Math. Inst., Cambridge, MA, 2003.

  40. K. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B, 387 (1996), 513–519.

    Article  MathSciNet  Google Scholar 

  41. A. Iqbal, C. Vafa, N.A. Nekrasov and A. Okounkov, Quantum foam and topological strings, J. High Energy Phys., 2008 (2008), no. 4, 011.

    Article  MathSciNet  MATH  Google Scholar 

  42. G.A. Jones, Characters and surfaces: a survey, In: The Atlas of Finite Groups: Ten Years On, Birmingham, 1995, London Math. Soc. Lecture Note Ser., 249, Cambridge Univ. Press, Cambridge, 1998, pp. 90–118.

    MathSciNet  MATH  Google Scholar 

  43. A. Kapustin, Topological field theory, higher categories, and their applications, In: Proceedings of the International Congress of Mathematicians. Vol. III, Hindustan Book Agency, New Delhi, 2010, pp. 2021–2043.

    Google Scholar 

  44. S. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nuclear Phys. B, 497 (1997), 173–195.

    Article  MathSciNet  MATH  Google Scholar 

  45. B. Khesin and A. Rosly, Polar homology, Canad. J. Math., 55 (2003), 1100–1120.

    Article  MathSciNet  MATH  Google Scholar 

  46. T. Kohno, Conformal Field Theory and Topology. Translated from the 1998 Japanese original by the author, Iwanami Series in Modern Mathematics, Transl. Math. Monogr., 210, Amer. Math. Soc., Providence, RI, 2002.

  47. J. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer-Verlag, 1996.

  48. M. Kontsevich and Y. Soibelman, Motivic Donaldson–Thomas invariants: summary of results, In: Mirror Symmetry and Tropical Geometry, Contemp. Math., 527, Amer. Math. Soc., Providence, RI, 2010, pp. 55–89.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Lehn, Lectures on Hilbert schemes, In: Algebraic Structures and Moduli Spaces, CRM Proc. Lecture Notes, 38, Amer. Math. Soc., Providence, RI, 2004, pp. 1–30.

    Article  MathSciNet  MATH  Google Scholar 

  50. M. Levine and R. Pandharipande, Algebraic cobordism revisited, Invent. Math., 176 (2009), 63–130.

    Article  MathSciNet  MATH  Google Scholar 

  51. J. Li, A degeneration formula of GW-invariants, J. Differential Geom., 60 (2002), 199–293.

    Article  MathSciNet  MATH  Google Scholar 

  52. J. Li and B. Wu, Good degeneration of Quot-schemes and coherent systems, Comm. Anal. Geom., 23 (2015), 841–921.

    Article  MathSciNet  MATH  Google Scholar 

  53. I.G. Macdonald, Symmetric products of an algebraic curve, Topology, 1 (1962), 319–343.

    Article  MathSciNet  MATH  Google Scholar 

  54. D. Maulik, N.A. Nekrasov, A. Okounkov and R. Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory. I, Compos. Math., 142 (2006), 1263–1285.

    Article  MathSciNet  MATH  Google Scholar 

  55. D. Maulik, N.A. Nekrasov, A. Okounkov and R. Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory. II, Compos. Math., 142 (2006), 1286–1304.

    Article  MathSciNet  MATH  Google Scholar 

  56. D. Maulik and A. Oblomkov, Quantum cohomology of the Hilbert scheme of points on An-resolutions, J. Amer. Math. Soc., 22 (2009), 1055–1091.

    Article  MathSciNet  MATH  Google Scholar 

  57. D. Maulik, A. Oblomkov, A. Okounkov and R. Pandharipande, Gromov–Witten/Donaldson–Thomas correspondence for toric 3-folds, Invent. Math., 186 (2011), 435–479.

    Article  MathSciNet  MATH  Google Scholar 

  58. D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, preprint, arXiv:1211.1287.

  59. H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras, Duke Math. J., 76 (1994), 365–416.

    MATH  Google Scholar 

  60. H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2), 145 (1997), 379–388.

    Article  MathSciNet  MATH  Google Scholar 

  61. H. Nakajima, Quiver varieties and Kac–Moody algebras, Duke Math. J., 91 (1998), 515–560.

    Article  MathSciNet  MATH  Google Scholar 

  62. H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, Univ. Lecture Ser., 18, Amer. Math. Soc., Providence, RI, 1999.

  63. H. Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc., 14 (2001), 145–238.

    Article  MathSciNet  MATH  Google Scholar 

  64. H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories. I, Adv. Theor. Math. Phys., 20 (2016), 595–669.

    Article  MathSciNet  MATH  Google Scholar 

  65. H. Nakajima and K. Yoshioka, Lectures on instanton counting, In: Algebraic Structures and Moduli Spaces, CRM Proc. Lecture Notes, 38, Amer. Math. Soc., Providence, RI, 2004, pp. 31–101.

    Article  MathSciNet  MATH  Google Scholar 

  66. H. Nakajima and K. Yoshioka, Instanton counting on blowup. I. 4-dimensional pure gauge theory, Invent. Math., 162 (2005), 313–355.

    Article  MathSciNet  MATH  Google Scholar 

  67. H. Nakajima and K. Yoshioka, Instanton counting on blowup. II. K-theoretic partition function, Transform. Groups, 10 (2005), 489–519.

    Article  MathSciNet  MATH  Google Scholar 

  68. N.A. Nekrasov, Seiberg–Witten prepotential from instanton counting, Adv. Theor. Math. Phys., 7 (2003), 831–864.

    Article  MathSciNet  MATH  Google Scholar 

  69. N.A. Nekrasov, Z-theory: chasing m=f theory. (English, with English and French summaries), In: Strings 04. Part II, C. R. Phys., 6, Elsevier Sci. B. V., Amsterdam, 2005, pp. 261–269.

    MathSciNet  Google Scholar 

  70. N.A. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson–Schwinger equations and qq-characters, preprint, arXiv:1512.05388.

  71. N.A. Nekrasov, BPS/CFT correspondence II: instantons at crossroads, moduli and compactness theorem, Adv. Theor. Math. Phys., 21 (2017), 503–583.

    Article  MathSciNet  MATH  Google Scholar 

  72. N.A. Nekrasov, BPS/CFT correspondence III: gauge origami partition function and qqcharacters, Comm. Math. Phys., 358 (2018), 863–894.

    Article  MathSciNet  MATH  Google Scholar 

  73. N.A. Nekrasov and A. Okounkov, Seiberg–Witten theory and random partitions, In: The Unity of Mathematics, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006, pp. 525–596.

    Article  MathSciNet  MATH  Google Scholar 

  74. N.A. Nekrasov and A. Okounkov, Membranes and sheaves, Algebr. Geom., 3 (2016), 320–369.

    Article  MathSciNet  MATH  Google Scholar 

  75. N.A. Nekrasov and V. Pestun, Seiberg–Witten geometry of four dimensional N=2 quiver gauge theories, preprint, arXiv:1211.2240.

  76. N.A. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, Comm. Math. Phys., 357 (2018), 519–567.

    Article  MathSciNet  MATH  Google Scholar 

  77. N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nuclear Phys. B Proc. Suppl., 192–193 (2009), 91–112.

    Article  MathSciNet  MATH  Google Scholar 

  78. N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, In: XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, (2010), pp. 265–289.

    Chapter  Google Scholar 

  79. A. Oblomkov and A. Okounkov, in preparation.

  80. A. Okounkov, The uses of random partitions, In: XIVth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, (2005), pp. 379–403.

    Google Scholar 

  81. A. Okounkov, Random surfaces enumerating algebraic curves, In: European Congress of Mathematics, Eur. Math. Soc., Zürich, (2005), pp. 751–768.

    Google Scholar 

  82. A. Okounkov, Random partitions and instanton counting, In: International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, (2006), pp. 687–711.

    Google Scholar 

  83. A. Okounkov, Limit shapes, real and imagined, Bull. Amer. Math. Soc. (N.S.), 53 (2016), 187–216.

    Article  MathSciNet  MATH  Google Scholar 

  84. A. Okounkov, Lectures on K-theoretic computations in enumerative geometry, In: Geometry of Moduli Spaces and Representation Theory, IAS/Park City Math. Ser., 24, Amer. Math. Soc., Providence, RI, (2017), pp. 251–380.

    MathSciNet  MATH  Google Scholar 

  85. A. Okounkov, Enumerative geometry and geometric representation theory, In: Algebraic Geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., 97.1, Amer. Math. Soc., Providence, RI, (2018), pp. 419–457.

    MathSciNet  Google Scholar 

  86. A. Okounkov and R. Pandharipande, Gromov–Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2), 163 (2006), 517–560.

    Article  MathSciNet  MATH  Google Scholar 

  87. A. Okounkov and R. Pandharipande, The equivariant Gromov–Witten theory of P1, Ann. of Math. (2), 163 (2006), 561–605.

    Article  MathSciNet  MATH  Google Scholar 

  88. A. Okounkov and R. Pandharipande, Virasoro constraints for target curves, Invent. Math., 163 (2006), 47–108.

    Article  MathSciNet  MATH  Google Scholar 

  89. A. Okounkov and R. Pandharipande, Quantum cohomology of the Hilbert scheme of points in the plane, Invent. Math., 179 (2010), 523–557.

    Article  MathSciNet  MATH  Google Scholar 

  90. A. Okounkov and R. Pandharipande, The local Donaldson–Thomas theory of curves, Geom. Topol., 14 (2010), 1503–1567.

    Article  MathSciNet  MATH  Google Scholar 

  91. A. Okounkov and N.Yu. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc., 16 (2003), 581–603.

    Article  MathSciNet  MATH  Google Scholar 

  92. A. Okounkov, N.Yu. Reshetikhin and C. Vafa, Quantum Calabi–Yau and classical crystals, In: The Unity of Mathematics, Progr. Math., 244, Birkhäuser Boston, Boston, MA, (2006), pp. 597–618.

    MathSciNet  MATH  Google Scholar 

  93. A. Okounkov and A. Smirnov, Quantum difference equation for Nakajima varieties, preprint, arXiv:1602.09007.

  94. H. Ooguri and C. Vafa, Knot invariants and topological strings, Nuclear Phys. B, 577 (2000), 419–438.

    Article  MathSciNet  MATH  Google Scholar 

  95. R. Pandharipande and A. Pixton, Descendents on local curves: stationary theory, In: Geometry and Arithmetic, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, (2012), pp. 283–307.

    Google Scholar 

  96. R. Pandharipande and A. Pixton, Descendents on local curves: rationality, Compos. Math., 149 (2013), 81–124.

    Article  MathSciNet  MATH  Google Scholar 

  97. R. Pandharipande and A. Pixton, Descendent theory for stable pairs on toric 3-folds, J. Math. Soc. Japan, 65 (2013), 1337–1372.

    Article  MathSciNet  MATH  Google Scholar 

  98. R. Pandharipande and A. Pixton, Gromov–Witten/pairs descendent correspondence for toric 3-folds, Geom. Topol., 18 (2014), 2747–2821.

    Article  MathSciNet  MATH  Google Scholar 

  99. R. Pandharipande and A. Pixton, Gromov–Witten/pairs correspondence for the quintic 3-fold, J. Amer. Math. Soc., 30 (2017), 389–449.

    Article  MathSciNet  MATH  Google Scholar 

  100. R. Pandharipande and R.P. Thomas, Curve counting via stable pairs in the derived category, Invent. Math., 178 (2009), 407–447.

    Article  MathSciNet  MATH  Google Scholar 

  101. R. Pandharipande and R.P. Thomas, The 3-fold vertex via stable pairs, Geom. Topol., 13 (2009), 1835–1876.

    Article  MathSciNet  MATH  Google Scholar 

  102. R. Remmert, Classical Topics in Complex Function Theory. Translated from the German by Leslie Kay, Grad. Texts in Math., 172, Springer-Verlag, 1998.

  103. A. Smirnov, Rationality of capped descendent vertex in K-theory, preprint, arXiv:1612.01048.

  104. C.H. Taubes, Casson’s invariant and gauge theory, J. Differential Geom., 31 (1990), 547–599.

    Article  MathSciNet  MATH  Google Scholar 

  105. R.P. Thomas, A holomorphic Casson invariant for Calabi–Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom., 54 (2000), 367–438.

    Article  MATH  Google Scholar 

  106. Y. Toda, Curve counting theories via stable objects I. DT/PT correspondence, J. Amer. Math. Soc., 23 (2010), 1119–1157.

    Article  MathSciNet  MATH  Google Scholar 

  107. Y. Toda, Stability conditions and curve counting invariants on Calabi–Yau 3-folds, Kyoto J. Math., 52 (2012), 1–50.

    Article  MathSciNet  MATH  Google Scholar 

  108. K.K. Uhlenbeck, The Chern classes of Sobolev connections, Comm. Math. Phys., 101 (1985), 449–457.

    Article  MathSciNet  MATH  Google Scholar 

  109. E. Witten, Topological quantum field theory, Comm. Math. Phys., 117 (1988), 353–386.

    Article  MathSciNet  MATH  Google Scholar 

  110. E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys., 121 (1989), 351–399.

    Article  MathSciNet  MATH  Google Scholar 

  111. E. Witten, Two-dimensional gravity and intersection theory on moduli space, In: Surveys in Differential Geometry, Cambridge, MA, 1990, Lehigh Univ., Bethlehem, PA, (1991), pp. 243–310.

    Google Scholar 

  112. E. Witten, Chern–Simons gauge theory as a string theory, In: The Floer Memorial Volume, Progr. Math., 133, Birkhäuser, Basel, (1995), pp. 637–678.

    Article  MathSciNet  MATH  Google Scholar 

  113. E. Witten, Physical methods applied to Donaldson theory, In: Functional Analysis on the Eve of the 21st Century. Vol. 1, New Brunswick, NJ, 1993, Progr. Math., 131, Birkhäuser Boston, Boston, MA, 1995, pp. 283–292.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andrei Okounkov.

Additional information

This article is based on the 20th Takagi Lectures that the author delivered at The University of Tokyo on November 4, 2017.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Okounkov, A. Takagi Lectures on Donaldson–Thomas theory. Jpn. J. Math. 14, 67–133 (2019). https://doi.org/10.1007/s11537-018-1744-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11537-018-1744-8

Keywords and phrases

Mathematics Subject Classification (2010)

Navigation