Letters in Mathematical Physics

, Volume 106, Issue 1, pp 1–27 | Cite as

On Twisted N = 2 5D Super Yang–Mills Theory

  • Jian Qiu
  • Maxim Zabzine


On a five-dimensional simply connected Sasaki–Einstein manifold, one can construct Yang–Mills theories coupled to matter with at least two supersymmetries. The partition function of these theories localises on the contact instantons, however, the contact instanton equations are not elliptic. It turns out that these equations can be embedded into the Haydys–Witten equations (which are elliptic) in the same way the 4D anti-self-dual instanton equations are embedded in the Vafa–Witten equations. We show that under some favourable circumstances, the latter equations will reduce to the former by proving some vanishing theorems. It was also known that the Haydys–Witten equations on product manifolds \({M_5 = M_4 \times \mathbb{R}}\) arise in the context of twisting the 5D maximally supersymmetric Yang–Mills theory. In this paper, we present the construction of twisted N = 2 Yang–Mills theory on Sasaki–Einstein manifolds, and more generally on K-contact manifolds. The localisation locus of this new theory thus provides a covariant version of the Haydys–Witten equation.


five-dimensional gauge theory contact geometry 

Mathematics Subject Classification

81T13 53D10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Pestun, V. : Localization of gauge theory on a four-sphere and supersymmetric Wilson loops. Commun. Math. Phys. 313, 71–129 (2012). arXiv:0712.2824 [hep-th]
  2. 2.
    Källén, J., Zabzine, M.: Twisted supersymmetric 5D Yang-Mills theory and contact geometry. JHEP 1205, 125 (2012). arXiv:1202.1956 [hep-th]
  3. 3.
    Nekrasov, N.: Five dimensional gauge theories and relativistic integrable systems. Nucl. Phys. B 531, 323–344 (1998). arXiv:hep-th/9609219 [hep-th]
  4. 4.
    Baulieu, L., Losev, A., Nekrasov, N.: Chern-Simons and twisted supersymmetry in various dimensions. Nucl. Phys. B 522, 82–104 (1998). arXiv:hep-th/9707174 [hep-th]
  5. 5.
    Harland, D., Nolle, C.: Instantons and killing spinors. JHEP 1203, 082 (2012). arXiv:1109.3552 [hep-th]
  6. 6.
    Corrigan E., Devchand C., Fairlie D.B., Nuyts J.: First order equations for gauge fields in spaces of dimension greater than four. Nucl. Phys. B 214, 452 (1983)MathSciNetCrossRefADSGoogle Scholar
  7. 7.
    Fan H.: Half de Rham complexes and line fields on odd-dimensional manifolds. Trans. Am. Math. Soc. 348, 2947–2982 (1996)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Källén, J., Qiu, J., Zabzine, M.: The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere. JHEP 1208, 157 (2012). arXiv:1206.6008 [hep-th]
  9. 9.
    Qiu, J., Zabzine, M.: 5D Super Yang-Mills on Y p,q Sasaki-Einstein manifolds. arXiv:1307.3149
  10. 10.
    Qiu, J., Zabzine, M.: Factorization of 5D super Yang-Mills on Y p,q spaces. Phys. Rev. D 89, 065040 (2014). arXiv:1403.2945 [hep-th]
  11. 11.
    Qiu, J., Tizzano, L., Winding, J., Zabzine, M.: Gluing Nekrasov partition functions. arXiv:1403.2945 [hep-th]
  12. 12.
    Kim, H.-C., Kim, S.: M5-branes from gauge theories on the 5-sphere. JHEP 1305, 144 (2013). arXiv:1206.6339 [hep-th]
  13. 13.
    Lockhart, G., Vafa, C.: Superconformal partition functions and non-perturbative topological strings. arXiv:1210.5909 [hep-th]
  14. 14.
    Kim, H.-C., Kim, J., Kim, S.: Instantons on the 5-sphere and M5-branes. arXiv:1211.0144 [hep-th]
  15. 15.
    BWolf, M.: Contact manifolds, contact instantons, and twistor geometry. JHEP 1207, 074 (2012). arXiv:1401.5140 [math.DG]
  16. 16.
    Baraglia, D., Hekmati, P.: Moduli spaces of contact instantons. arXiv:1401.5140 [math.DG]
  17. 17.
    Pan, Y.: Note on a cohomological theory of contact-instanton and invariants of contact structures. arXiv:1401.5733 [hep-th]
  18. 18.
    Pan, Y.: 5d Higgs branch localization, seiberg-witten equations and contact geometry. arXiv:1406.5236 [hep-th]
  19. 19.
    Taubes, C.H.: The Seiberg-Witten equations and the Weinstein conjecture. (2006, ArXiv Mathematics e-prints). arXiv:math/0611007
  20. 20.
    Anderson, L.: Five-dimensional topologically twisted maximally supersymmetric Yang-Mills theory. JHEP. 1302, 131 (2013). arXiv:1212.5019 [hep-th]
  21. 21.
    Witten, E.: Monopoles and four-manifolds. Math. Res. Lett 1(6):769–796 (1994). doi: 10.4310/MRL.1994.v1.n6.a13
  22. 22.
    Taubes, C.H.: The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett. (1), 809822 (1994)Google Scholar
  23. 23.
    Witten, E.: Fivebranes and knots. arXiv:1101.3216 [hep-th]
  24. 24.
    Haydys, A.: Fukaya-Seidel category and gauge theory. (2010, ArXiv e-prints). arXiv:1010.2353 [math.SG]
  25. 25.
    Cherkis, S.A.: Octonions, monopoles, and knots. arXiv:1403.6836 [hep-th]
  26. 26.
    Hosomichi, K., Seong, R.-K., Terashima, S.: Supersymmetric gauge theories on the five-sphere. Nucl. Phys. B 865, 376–396 (2012). arXiv:1203.0371 [hep-th]
  27. 27.
    Pan, Y.: Rigid Supersymmetry on 5-dimensional riemannian manifolds and contact geometry. arXiv:1308.1567 [hep-th]
  28. 28.
    Vafa, C., Witten, E.: A strong coupling test of S duality. Nucl. Phys. B 431, 842–77 (1994). arXiv:hep-th/9408074 [hep-th]
  29. 29.
    Martelli, D., Sparks, J., Yau, S.-T.: The Geometric dual of a-maximisation for Toric Sasaki-Einstein manifolds. Commun. Math. Phys. 268, 39–65 (2006). arXiv:hep-th/0503183 [hep-th]
  30. 30.
    Schmude, J.: Localisation on Sasaki-Einstein manifolds from holomophic functions on the cone. arXiv:1401.3266 [hep-th]
  31. 31.
    Berkovits, N.: A ten-dimensional superYang-Mills action with off-shell supersymmetry. Phys. Lett. B 318, 104–106 (1993). arXiv:hep-th/9308128 [hep-th]
  32. 32.
    Alvarez-Gaume, L.: A note on the atiyah-singer index theorem. J. Phys. A Math. Gen. 16(18), 4177 (1983).
  33. 33.
    Alvarez-Gaume, L.: Supersymmetry and the atiyah-singer index theorem. Commun. Math. Phys. 90(2), 161–173 (1983).
  34. 34.
    Blair, D.E.: Riemannian geometry of contact and symplectic manifolds, vol. 203 of Progress in Mathematics, 2nd edn. Birkhäuser Boston, Inc., Boston. (2010). doi: 10.1007/978-0-8176-4959-3
  35. 35.
    Boyer, C.P., Galicki, K.: Sasakian geometry. Oxford mathematical monographs. Oxford University Press, Oxford (2008)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Mathématiques, Université du LuxembourgLuxembourgLuxembourg
  2. 2.Max-Planck-Institut für MathematikBonnGermany
  3. 3.Department of Physics and AstronomyUppsala UniversityUppsalaSweden

Personalised recommendations