Skip to main content
SpringerLink
Log in

5D Super Yang–Mills on Y p,q Sasaki–Einstein Manifolds

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

On any simply connected Sasaki–Einstein five dimensional manifold one can construct a super Yang–Mills theory which preserves at least two supersymmetries. We study the special case of toric Sasaki–Einstein manifolds known as Y p,q manifolds. We use the localisation technique to compute the full perturbative part of the partition function. The full equivariant result is expressed in terms of a certain special function which appears to be a curious generalisation of the triple sine function. As an application of our general result we study the large N behaviour for the case of single hypermultiplet in adjoint representation and we derive the N 3-behaviour in this case.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. 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]

  3. Kim, H.-C., Kim, S.: M5-branes from gauge theories on the 5-sphere. JHEP 1305, 144 (2013). arXiv:1206.6339 [hep-th]

  4. Imamura, Y.: Supersymmetric theories on squashed five-sphere. Prog. Theor. Exp. Phys. 013B04 (2013). arXiv:1209.0561 [hep-th]

  5. Lockhart, G., Vafa, C.: Superconformal partition functions and non-perturbative topological strings. arXiv:1210.5909 [hep-th]

  6. Imamura, Y.: Perturbative partition function for squashed S 5. Prog. Theor. Exp. Phys. 073B01 (2013). arXiv:1210.6308 [hep-th]

  7. Kim, H.-C., Kim, J., Kim, S.: Instantons on the 5-sphere and M5-branes. arXiv:1211.0144 [hep-th]

  8. Kim, H.-C., Kim, S.-S., Lee, K.: 5-Dim superconformal index with enhanced en global symmetry. JHEP 1210, 142 (2012). arXiv:1206.6781 [hep-th]

  9. Terashima, S.: On supersymmetric gauge theories on S 4 × S 1. Phys. Rev. D 89, 125001 (2014). arXiv:1207.2163 [hep-th]

  10. Fujitsuka, M., Honda, M., Yoshida, Y.: Maximal super Yang–Mills theories on curved background with off-shell supercharges. JHEP 1301, 162 (2013). arXiv:1209.4320 [hep-th]

  11. Jafferis, D.L., Pufu, S.S.: Exact results for five-dimensional superconformal field theories with gravity duals. J. High Energy Phys. 2014, 32 (2014). arXiv:1207.4359 [hep-th]

  12. Assel, B., Estes, J., Yamazaki, M.: Wilson loops in 5d N = 1 SCFTs and AdS/CFT. Ann. Henri Poincaré 15, 589 (2014). arXiv:1212.1202 [hep-th]

  13. Minahan, J.A., Nedelin, A., Zabzine, M.: 5D super Yang–Mills theory and the correspondence to AdS 7/CFT 6. J. Phys. A Math. 46, 355401 (2013). arXiv:1304.1016 [hep-th]

  14. Nieri, F., Pasquetti, S., Passerini, F.: 3d & 5d gauge theory partition functions as q-deformed CFT correlators. arXiv:1303.2626 [hep-th]

  15. Hosomichi, K., Seong, R.-K., Terashima, S.: Supersymmetric gauge theories on the five-sphere. arXiv:1203.0371 [hep-th]

  16. Boyer, C.P., Galicki, K.: Sasakian geometry, holonomy, and supersymmetry. Handbook of pseudo-Riemannian geometry and supersymmetry, IRMA Lect. Math. Theor. Phys. Evr. Math. Soc. Zürich 16, 39–83 (2010). arXiv:math/0703231

  17. McDuff, D., Salamon, D.: Introduction to symplectic topology. In: Oxford Mathematical Monographs, 2 edn. Oxford University Press, New York (1999)

  18. Blair, D.E.: Contact manifolds in Riemannian geometry. Lecture notes in mathematics, vol. 509. Springer, New York (1976)

  19. Salamon, D.: Spin geometry and Seiberg–Witten invariants (1996)

  20. Boyer, C.P., Galicki, K.: Sasakian Geometry. Oxford University Press, USA (2008)

  21. Friedrich T., Kath I.: Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator. J. Differ. Geom. 29, 263–279 (1989)

    MATH  MathSciNet  Google Scholar 

  22. Boyer, C.P.: Completely integrable contact Hamiltonian systems and toric contact structures on S 2 × S 3. SIGMA 7, 58 (2011). arXiv:1101.5587 [math.SG]

  23. Martelli, D., Sparks, J.: Toric geometry, Sasaki–Einstein manifolds and a new infinite class of AdS/CFT duals. Commun. Math.Phys. 262, 51–89 (2006). arXiv:hep-th/0411238 [hep-th]

  24. Boyer, C.P., Pati, J.: On the equivalence problem for toric contact structures on S 3-bundles over S 2. Pac. J. Math. 267(2), 277–324 (2014). arXiv:1204.2209 [math.SG]

  25. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: Sasaki–Einstein metrics on S 2 × S 3. Adv. Theor. Math. Phys. 8, 711–734 (2004). arXiv:hep-th/0403002 [hep-th]

  26. 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]

  27. Källén, J., Zabzine, M.: Twisted supersymmetric 5D Yang–Mills theory and contact geometry. JHEP 1205, 125 (2012). arXiv:1202.1956 [hep-th]

  28. Atiyah, M.F.: Elliptic Operators and Compact Groups, vol. 401. Springer, Berlin (1974)

  29. Quine, J.R., Heydari, S.H., Song, R.Y.: Zeta regularized products. Trans. Am. Math. Soc. 338(1), 213–231 (1993). http://www.jstor.org/stable/2154453

  30. Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes: II. Appl. Ann. Math. 88(3), 451–491 (1968). http://www.jstor.org/stable/1970721

  31. Källén, J., Minahan, J., Nedelin, A., Zabzine, M.: N 3-behavior from 5D Yang–Mills theory. JHEP 1210, 184. arXiv:1207.3763 [hep-th]

  32. Moroianu, A.: On the infinitesimal isometries of manifolds with killing spinors. J. Geom. Phys. 35(1), 63–74 (2000). http://www.sciencedirect.com/science/article/pii/S0393044099000790

  33. Figueroa-O’Farrill, J.M.: On the supersymmetries of Anti-de Sitter vacua. Class. Quantum Gravity 16, 2043–2055 (1999). arXiv:hep-th/9902066 [hep-th]

  34. Bär C.: Real Killing spinors and holonomy. Commun. Math. Phys. 154, 509–521 (1993)

    Article  ADS  MATH  Google Scholar 

  35. Boyer, C.P., Galicki, K.: 3-Sasakian manifolds. Surv. Differ. Geom. 7, 123–184 (1999). arXiv:hep-th/9810250 [hep-th]

  36. Martelli, D., Sparks, J.: Toric sasakieinstein metrics on. Phys. Lett. B 621(12), 208–212 (2005). http://www.sciencedirect.com/science/article/pii/S0370269305008609

  37. Cvetic, M., Lu, H., Page, D.N., Pope, C.: New Einstein–Sasaki spaces in five and higher dimensions. Phys. Rev. Lett. 95, 071101 (2005). arXiv:hep-th/0504225 [hep-th]

  38. Kosmann Y.: Dérivées de Lie des spineurs. Ann. di Mat. Pura Appl. (IV) 91, 317–395 (1972)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Unité de Recherche en Mathématiques, Université du Luxembourg, Campus Kirchberg, G 106, 1359, Luxembourg, Luxembourg

    Jian Qiu

  2. Department of Physics and Astronomy, Uppsala University, Box 516, 75120, Uppsala, Sweden

    Maxim Zabzine

Authors
  1. Jian Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maxim Zabzine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maxim Zabzine.

Additional information

Communicated by N. A. Nekrasov

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiu, J., Zabzine, M. 5D Super Yang–Mills on Y p,q Sasaki–Einstein Manifolds. Commun. Math. Phys. 333, 861–904 (2015). https://doi.org/10.1007/s00220-014-2194-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2194-7

Keywords

Search

Navigation