Communications in Mathematical Physics

, Volume 372, Issue 2, pp 573–597 | Cite as

Magnificent Four with Colors

  • Nikita Nekrasov
  • Nicolò PiazzalungaEmail author


We study the rank N magnificent four theory, which is the supersymmetric localization of U(N) super-Yang–Mills theory with matter (a super-group U(N|N) gauge theory) on a Calabi–Yau fourfold. Our theory contains the higher rank Donaldson–Thomas theory of threefolds. We conjecture an explicit formula for the partition function \({\mathcal {Z}}\), and report on the performed checks. The partition function \({\mathcal {Z}}\) has a free field representation. Surprisingly, it depends on the Coulomb and mass parameters in a simple way. We also clarify the definition of the instanton measure.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



NN thanks A. Okounkov for discussions, and MSRI (Berkeley), Department of Mathematics (UC Berkeley), IHES (Bures-sur-Yvette), and Skoltech (Moscow) for hospitality during the preparation of this work and for opportunity to present its preliminary results there. Research of NN is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. N. 14.641.31.0001. NP thanks A. Grassi, A. Waldron and the participants of the workshops at ICTP Trieste (July 2018) and GGI Florence (April 2018), where this work was presented, for stimulating discussions. The research of NP was supported in part by Istituto Nazionale di Alta Matematica “Francesco Severi” (INdAM) and by Perimeter Institute for Theoretical Physics. (Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research and Innovation.)


  1. 1.
    Nekrasov, N.: Magnificent four. arXiv:1712.08128
  2. 2.
    Witten, E.: BPS Bound states of D0–D6 and D0–D8 systems in a \(B\)-field. JHEP 04, 012 (2002). arXiv:hep-th/0012054 ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Nekrasov, N.: A la recherche de la M-theorie perdue Z theory: Chasing M/f theory. In: Annual International Conference on Strings, Theory and Applications (Strings 2004) Paris, France, 28 June–July 2, 2004 (2004). arXiv:hep-th/0412021
  4. 4.
    Nekrasov, N.: Z-theory: chasing M/F-theory. C. R. Phys. 6, 261–269 (2005)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Nekrasov, N., Okounkov, A.: Membranes and sheaves. arXiv:1404.2323
  6. 6.
    Ooguri, H., Strominger, A., Vafa, C.: Black hole attractors and the topological string. Phys. Rev. D 70, 106007 (2004). arXiv:hep-th/0405146 ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Beasley, C., Gaiotto, D., Guica, M., Huang, L., Strominger, A, Yin, X.: Why \(Z_{BH} = |Z_{top}|^2\). arXiv:hep-th/0608021
  8. 8.
    Vafa, C.: Brane/anti-brane systems and U(N—M) supergroup. arXiv:hep-th/0101218
  9. 9.
    Destainville, N., Govindarajan, S.: Estimating the asymptotics of solid partitions. J. Stat. Phys. 158, 950 (2015). arXiv:1406.5605 ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Knuth, D.E.: A note on solid partitions. Math. Comput. 24, 955–961 (1970)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Douglas, M.R.: Branes within branes. In: Strings, Branes and Dualities. Proceedings, NATO Advanced Study Institute, Cargese, France, May 26–June 14, 1997, pp. 267–275 (1995). arXiv:hep-th/9512077 CrossRefGoogle Scholar
  12. 12.
    Baulieu, L., Kanno, H., Singer, I.M.: Special quantum field theories in eight-dimensions and other dimensions. Commun. Math. Phys. 194, 149–175 (1998). arXiv:hep-th/9704167 ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Cao, Y., Kool, M.: Zero-dimensional Donaldson–Thomas invariants of Calabi–Yau 4-folds. arXiv:1712.07347
  14. 14.
    Witten, E.: Bound states of strings and p-branes. Nucl. Phys. B 460, 335–350 (1996). arXiv:hep-th/9510135 ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Cecotti, S., Girardello, L.: Functional measure, topology and dynamical supersymmetry breaking. Phys. Lett. 110B, 39 (1982)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Sethi, S., Stern, M.: D-brane bound states redux. Commun. Math. Phys. 194, 675–705 (1998). arXiv:hep-th/9705046 ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Moore, G.W., Nekrasov, N., Shatashvili, S.: D particle bound states and generalized instantons. Commun. Math. Phys. 209, 77–95 (2000). arXiv:hep-th/9803265 ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Smilga, A.V.: Quasiclassical expansion for \(Tr (-1)^F e^{- \beta H}\). Commun. Math. Phys. 230, 245–269 (2002). arXiv:hep-th/0110105 ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Hori, K., Kim, H., Yi, P.: Witten index and wall crossing. JHEP 01, 124 (2015). arXiv:1407.2567 ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Szenes, A., Vergne, M.: Mixed toric residues and tropical degenerations. ArXiv Mathematics e-prints (Oct., 2004). arXiv:math/0410064
  21. 21.
    Szenes, A., Vergne, M.: Toric reduction and a conjecture of Batyrev and Materov. Invent. Math. 158, 453–495 (2004). arXiv:math/0306311 ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Mazin, M.: Geometric theory of Parshin residues. PhD thesis, University of Toronto (Canada) (2010)Google Scholar
  23. 23.
    Bachas, C.P., Green, M.B., Schwimmer, A.: (8,0) quantum mechanics and symmetry enhancement in type I’ superstrings. JHEP 01, 006 (1998). arXiv:hep-th/9712086 ADSCrossRefGoogle Scholar
  24. 24.
    Iqbal, A., Nekrasov, N., Okounkov, A., Vafa, C.: Quantum foam and topological strings. JHEP 04, 011 (2008). arXiv:hep-th/0312022 ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Cirafici, M., Sinkovics, A., Szabo, R.J.: Cohomological gauge theory, quiver matrix models and Donaldson–Thomas theory. Nucl. Phys. B 809, 452–518 (2009). arXiv:0803.4188 ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Awata, H., Kanno, H.: Quiver matrix model and topological partition function in six dimensions. JHEP 07, 076 (2009). arXiv:0905.0184 ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Okounkov, A.: Lectures on K-theoretic computations in enumerative geometry. arXiv:1512.07363
  28. 28.
    Cirafici, M., Szabo, R.J.: Curve counting, instantons and McKay correspondences. J. Geom. Phys. 72, 54–109 (2013). arXiv:1209.1486 ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Nekrasov, N., Shadchin, S.: ABCD of instantons. Commun. Math. Phys. 252, 359–391 (2004). arXiv:hep-th/0404225 ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Simons Center for Geometry and PhysicsStony Brook UniversityStony BrookUSA

Personalised recommendations