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Magnificent Four with Colors

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Abstract

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.

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References

  1. Nekrasov, N.: Magnificent four. arXiv:1712.08128

  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

    Article  ADS  MathSciNet  Google Scholar 

  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. Nekrasov, N.: Z-theory: chasing M/F-theory. C. R. Phys. 6, 261–269 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  5. Nekrasov, N., Okounkov, A.: Membranes and sheaves. arXiv:1404.2323

  6. Ooguri, H., Strominger, A., Vafa, C.: Black hole attractors and the topological string. Phys. Rev. D 70, 106007 (2004). arXiv:hep-th/0405146

    Article  ADS  MathSciNet  Google Scholar 

  7. Beasley, C., Gaiotto, D., Guica, M., Huang, L., Strominger, A, Yin, X.: Why \(Z_{BH} = |Z_{top}|^2\). arXiv:hep-th/0608021

  8. Vafa, C.: Brane/anti-brane systems and U(N—M) supergroup. arXiv:hep-th/0101218

  9. Destainville, N., Govindarajan, S.: Estimating the asymptotics of solid partitions. J. Stat. Phys. 158, 950 (2015). arXiv:1406.5605

    Article  ADS  MathSciNet  Google Scholar 

  10. Knuth, D.E.: A note on solid partitions. Math. Comput. 24, 955–961 (1970)

    Article  MathSciNet  Google Scholar 

  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

    Chapter  Google Scholar 

  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

    Article  ADS  MathSciNet  Google Scholar 

  13. Cao, Y., Kool, M.: Zero-dimensional Donaldson–Thomas invariants of Calabi–Yau 4-folds. arXiv:1712.07347

  14. Witten, E.: Bound states of strings and p-branes. Nucl. Phys. B 460, 335–350 (1996). arXiv:hep-th/9510135

    Article  ADS  MathSciNet  Google Scholar 

  15. Cecotti, S., Girardello, L.: Functional measure, topology and dynamical supersymmetry breaking. Phys. Lett. 110B, 39 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  16. Sethi, S., Stern, M.: D-brane bound states redux. Commun. Math. Phys. 194, 675–705 (1998). arXiv:hep-th/9705046

    Article  ADS  MathSciNet  Google Scholar 

  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

    Article  ADS  MathSciNet  Google Scholar 

  18. Smilga, A.V.: Quasiclassical expansion for \(Tr (-1)^F e^{- \beta H}\). Commun. Math. Phys. 230, 245–269 (2002). arXiv:hep-th/0110105

    Article  ADS  MathSciNet  Google Scholar 

  19. Hori, K., Kim, H., Yi, P.: Witten index and wall crossing. JHEP 01, 124 (2015). arXiv:1407.2567

    Article  ADS  MathSciNet  Google Scholar 

  20. Szenes, A., Vergne, M.: Mixed toric residues and tropical degenerations. ArXiv Mathematics e-prints (Oct., 2004). arXiv:math/0410064

  21. Szenes, A., Vergne, M.: Toric reduction and a conjecture of Batyrev and Materov. Invent. Math. 158, 453–495 (2004). arXiv:math/0306311

    Article  ADS  MathSciNet  Google Scholar 

  22. Mazin, M.: Geometric theory of Parshin residues. PhD thesis, University of Toronto (Canada) (2010)

  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

    Article  ADS  Google Scholar 

  24. Iqbal, A., Nekrasov, N., Okounkov, A., Vafa, C.: Quantum foam and topological strings. JHEP 04, 011 (2008). arXiv:hep-th/0312022

    Article  ADS  MathSciNet  Google Scholar 

  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

    Article  ADS  MathSciNet  Google Scholar 

  26. Awata, H., Kanno, H.: Quiver matrix model and topological partition function in six dimensions. JHEP 07, 076 (2009). arXiv:0905.0184

    Article  ADS  MathSciNet  Google Scholar 

  27. Okounkov, A.: Lectures on K-theoretic computations in enumerative geometry. arXiv:1512.07363

  28. Cirafici, M., Szabo, R.J.: Curve counting, instantons and McKay correspondences. J. Geom. Phys. 72, 54–109 (2013). arXiv:1209.1486

    Article  ADS  MathSciNet  Google Scholar 

  29. Nekrasov, N., Shadchin, S.: ABCD of instantons. Commun. Math. Phys. 252, 359–391 (2004). arXiv:hep-th/0404225

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

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.)

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  1. Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY, 11794-3636, USA

    Nikita Nekrasov & Nicolò Piazzalunga

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  1. Nikita Nekrasov
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  2. Nicolò Piazzalunga
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Correspondence to Nicolò Piazzalunga.

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Communicated by X. Yin.

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N. Nekrasov: On leave from Kharkevich IITP (Moscow) and Alikhanov ITEP (Moscow).

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Nekrasov, N., Piazzalunga, N. Magnificent Four with Colors. Commun. Math. Phys. 372, 573–597 (2019). https://doi.org/10.1007/s00220-019-03426-3

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