Partition functions for equivariantly twisted \( \mathcal{N}=2 \) gauge theories on toric Kähler manifolds

  • Diego Rodriguez-Gomez
  • Johannes SchmudeEmail author
Open Access
Regular Article - Theoretical Physics


We consider \( \mathcal{N}=2 \) supersymmetric pure gauge theories on toric Kähler manifolds, with particular emphasis on ℂℙ2. By choosing a vector generating a U(1) action inside the torus of the manifold, we construct equivariantly twisted theories. Then, using localization, we compute their supersymmetric partition functions. As expected, these receive contributions from a classical, a one-loop, and an instanton term. It turns out that the one-loop term is trivial and that the instanton contributions are localized at the fixed points of the U(1). In fact the full partition function can be re-written in a factorized form with contributions from each of the fixed points. The full significance of this is yet to be understood.


Supersymmetric gauge theory Nonperturbative Effects Extended Supersymmetry Solitons Monopoles and Instantons 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric field theories on three-manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, The geometry of supersymmetric partition functions, JHEP 01 (2014) 124 [arXiv:1309.5876] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere partition functions and the Zamolodchikov metric, JHEP 11 (2014) 001 [arXiv:1405.7271] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    J. Gomis and N. Ishtiaque, Kähler potential and ambiguities in 4d N = 2 SCFTs, JHEP 04 (2015) 169 [arXiv:1409.5325] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    T.T. Dumitrescu, G. Festuccia and N. Seiberg, Exploring curved superspace, JHEP 08 (2012) 141 [arXiv:1205.1115] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    E. Witten, Topological quantum field theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    A. Karlhede and M. Roček, Topological quantum field theory and N = 2 conformal supergravity, Phys. Lett. B 212 (1988) 51 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    C. Klare and A. Zaffaroni, Extended supersymmetry on curved spaces, JHEP 10 (2013) 218 [arXiv:1308.1102] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    E. Witten, Supersymmetric Yang-Mills theory on a four manifold, J. Math. Phys. 35 (1994) 5101 [hep-th/9403195] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. Johansen, Twisting of N = 1 SUSY gauge theories and heterotic topological theories, Int. J. Mod. Phys. A 10 (1995) 4325 [hep-th/9403017] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    G.W. Moore and E. Witten, Integration over the u-plane in Donaldson theory, Adv. Theor. Math. Phys. 1 (1997) 298 [hep-th/9709193] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    L. Gottsche, H. Nakajima and K. Yoshioka, Instanton counting and Donaldson invariants, math/0606180 [INSPIRE].
  15. [15]
    J. Gomis, T. Okuda and V. Pestun, Exact results fort Hooft loops in gauge theories on S 4, JHEP 05 (2012) 141 [arXiv:1105.2568] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    N. Hama and K. Hosomichi, Seiberg-Witten theories on ellipsoids, JHEP 09 (2012) 033 [arXiv:1206.6359] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    K. Hosomichi, R.-K. Seong and S. Terashima, Supersymmetric gauge theories on the five-sphere, Nucl. Phys. B 865 (2012) 376 [arXiv:1203.0371] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    J. Källén, J. Qiu and M. Zabzine, The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere, JHEP 08 (2012) 157 [arXiv:1206.6008] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    J. Qiu and M. Zabzine, 5D super Yang-Mills on Y p,q Sasaki-Einstein manifolds, Commun. Math. Phys. 333 (2015) 861 [arXiv:1307.3149] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    J. Schmude, Localisation on Sasaki-Einstein manifolds from holomorphic functions on the cone, JHEP 01 (2015) 119 [arXiv:1401.3266] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    J. Qiu and M. Zabzine, On twisted N = 2 5D super Yang-Mills theory, arXiv:1409.1058 [INSPIRE].
  22. [22]
    B. Assel, D. Cassani and D. Martelli, Localization on Hopf surfaces, JHEP 08 (2014) 123 [arXiv:1405.5144] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE].
  25. [25]
    A. Bawane, G. Bonelli, M. Ronzani and A. Tanzini, N = 2 supersymmetric gauge theories on S 2 × S 2 and Liouville gravity, arXiv:1411.2762 [INSPIRE].
  26. [26]
    M. Sinamuli, On N = 2 supersymmetric gauge theories on S 2 × S 2, arXiv:1411.4918 [INSPIRE].
  27. [27]
    A. Van Proeyen, N = 2 supergravity in d = 4, 5, 6 and its matter couplings, extended version of lectures given during the semester Supergravity, superstrings and M-theory,, Institut Henri Poincaré, Paris France November 2000.
  28. [28]
    D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge Univ. Pr., Cambridge U.K. (2012) [INSPIRE].
  29. [29]
    P. van Nieuwenhuizen and A. Waldron, On Euclidean spinors and Wick rotations, Phys. Lett. B 389 (1996) 29 [hep-th/9608174] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    T. Delzant, Hamiltoniens périodiques et images convexes de lapplication moment (in French), Bull. Soc. Math. France 116 (1988) 315.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    V. Guillemin, Kähler structures on toric varieties, J. Diff. Geom. 40 (1994) 285.CrossRefzbMATHGoogle Scholar
  32. [32]
    M. Abreu, Kähler geometry of toric manifolds in symplectic coordinates, in Symplectic and contact topology: interactions and perspectives 35, (2003), pg. 1 [math/0004122].
  33. [33]
    D. Martelli, J. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Commun. Math. Phys. 280 (2008) 611 [hep-th/0603021] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    D. Huybrechts, Complex geometry: an introduction, Universitext, U.S. Government Printing Office, U.S.A. (2005).Google Scholar
  35. [35]
    J. Källén, Cohomological localization of Chern-Simons theory, JHEP 08 (2011) 008 [arXiv:1104.5353] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    J. Källén and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    M. Blau and G. Thompson, Topological gauge theories of antisymmetric tensor fields, Annals Phys. 205 (1991) 130 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    L. Baulieu and M. Schaden, Gauge group TQFT and improved perturbative Yang-Mills theory, Int. J. Mod. Phys. A 13 (1998) 985 [hep-th/9601039] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    R. Goodman and N.R. Wallach, Symmetry, representations and, invariants, Graduate Texts in Mathematics 65, Springer, Germany (2009).Google Scholar
  40. [40]
    H.-C. Kim, J. Kim and S. Kim, Instantons on the 5-sphere and M5-branes, arXiv:1211.0144 [INSPIRE].
  41. [41]
    S. Shadchin, On certain aspects of string theory/gauge theory correspondence, hep-th/0502180 [INSPIRE].
  42. [42]
    S. Pasquetti, Factorisation of N = 2 theories on the squashed 3-sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    F. Nieri, S. Pasquetti and F. Passerini, 3d and 5d gauge theory partition functions as q-deformed CFT correlators, Lett. Math. Phys. 105 (2015) 109 [arXiv:1303.2626] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    F. Nieri, S. Pasquetti, F. Passerini and A. Torrielli, 5D partition functions, q-Virasoro systems and integrable spin-chains, JHEP 12 (2014) 040 [arXiv:1312.1294] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    J. Qiu and M. Zabzine, Factorization of 5D super Yang-Mills theory on Y p,q spaces, Phys. Rev. D 89 (2014) 065040 [arXiv:1312.3475] [INSPIRE].ADSGoogle Scholar
  46. [46]
    J. Qiu, L. Tizzano, J. Winding and M. Zabzine, Gluing Nekrasov partition functions, Commun. Math. Phys. 337 (2015) 785 [arXiv:1403.2945] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of PhysicsUniversidad de OviedoOviedoSpain

Personalised recommendations