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\( \mathcal{N}=2 \) supersymmetric gauge theories on S2 × S2 and Liouville Gravity

Journal of High Energy Physics volume 2015, Article number: 54 (2015) Cite this article

Abstract

We consider \( \mathcal{N}=2 \) supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a U(1) isometry. This is used to explicitly compute the supersymmetric path integral on S2 × S2 via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity.

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References

  1. [1]

    T.T. Dumitrescu, G. Festuccia and N. Seiberg, Exploring curved superspace, JHEP 08 (2012) 141 [arXiv:1205.1115] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  2. [2]

    T.T. Dumitrescu and G. Festuccia, Exploring curved superspace (II), JHEP 01 (2013) 072 [arXiv:1209.5408] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    N. Hama and K. Hosomichi, Seiberg-Witten theories on ellipsoids, JHEP 09 (2012) 033 [Addendum ibid. 10 (2012) 051] [arXiv:1206.6359] [INSPIRE].

  4. [4]

    C. Klare and A. Zaffaroni, Extended supersymmetry on curved spaces, JHEP 10 (2013) 218 [arXiv:1308.1102] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    R.K. Gupta and S. Murthy, All solutions of the localization equations for N = 2 quantum black hole entropy, JHEP 02 (2013) 141 [arXiv:1208.6221] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    E. Witten, Topological quantum field theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    E. Witten, Mirror manifolds and topological field theory, hep-th/9112056 [INSPIRE].

  8. [8]

    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE].

    ADS  Article  Google Scholar 

  10. [10]

    N. Nekrasov, Localizing gauge theories, in XIVth international congress on mathematical physics, Lisbon Portugal July 28 – August 2 2003.

  11. [11]

    G. Bonelli, K. Maruyoshi, A. Tanzini and F. Yagi, N = 2 gauge theories on toric singularities, blow-up formulae and W-algebrae, JHEP 01 (2013) 014 [arXiv:1208.0790] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    U. Bruzzo, M. Pedrini, F. Sala and R.J. Szabo, Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces, arXiv:1312.5554 [INSPIRE].

  14. [14]

    U. Bruzzo, F. Sala and R.J. Szabo, N = 2 quiver gauge theories on A-type ALE spaces, Lett. Math. Phys. 105 (2015) 401 [arXiv:1410.2742] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    H. Samtleben and D. Tsimpis, Rigid supersymmetric theories in 4d Riemannian space, JHEP 05 (2012) 132 [arXiv:1203.3420] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    S.S. Razamat and M. Yamazaki, S-duality and the N = 2 lens space index, JHEP 10 (2013) 048 [arXiv:1306.1543] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    N. Bobev, H. Elvang, D.Z. Freedman and S.S. Pufu, Holography for N = 2* on S4, JHEP 07 (2014) 001 [arXiv:1311.1508] [INSPIRE].

    ADS  Article  Google Scholar 

  18. [18]

    C. Closset and I. Shamir, The N = 1 chiral multiplet on T2 × S2 and supersymmetric localization, JHEP 03 (2014) 040 [arXiv:1311.2430] [INSPIRE].

    ADS  Article  Google Scholar 

  19. [19]

    B. Assel, D. Cassani and D. Martelli, Localization on Hopf surfaces, JHEP 08 (2014) 123 [arXiv:1405.5144] [INSPIRE].

    ADS  Article  Google Scholar 

  20. [20]

    J. Gomis and N. Ishtiaque, Kähler potential and ambiguities in 4d N = 2 SCFTs, JHEP 04 (2015) 169 [arXiv:1409.5325] [INSPIRE].

    ADS  Article  Google Scholar 

  21. [21]

    D. Marmiroli, Phase structure of N = 2* SYM on ellipsoids, arXiv:1410.4715 [INSPIRE].

  22. [22]

    J. Gomis and C. Musema, Partition function of N = 2 gauge theories on S2 × S2, in preparation.

  23. [23]

    G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, Phys. Lett. B 691 (2010) 111 [arXiv:0909.4031] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  24. [24]

    L.F. Alday, F. Benini and Y. Tachikawa, Liouville/Toda central charges from M5-branes, Phys. Rev. Lett. 105 (2010) 141601 [arXiv:0909.4776] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  25. [25]

    C. Klare, A. Tomasiello and A. Zaffaroni, Supersymmetry on curved spaces and holography, JHEP 08 (2012) 061 [arXiv:1205.1062] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  26. [26]

    C. Closset and S. Cremonesi, Comments on N = (2, 2) supersymmetry on two-manifolds, JHEP 07 (2014) 075 [arXiv:1404.2636] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. [27]

    L. Baulieu, G. Bossard and A. Tanzini, Topological vector symmetry of BRSTQFT and construction of maximal supersymmetry, JHEP 08 (2005) 037 [hep-th/0504224] [INSPIRE].

    ADS  Article  Google Scholar 

  28. [28]

    M. Bershtein, B. Feigin and A. Litvinov, Coupling of two conformal field theories and Nakajima-Yoshioka blow-up equations, arXiv:1310.7281 [INSPIRE].

  29. [29]

    A.B. Zamolodchikov, On the three-point function in minimal Liouville gravity, hep-th/0505063 [INSPIRE].

  30. [30]

    A. Belavin and A. Zamolodchikov, Polyakov’s string: twenty five years after. Proceedings, hep-th/0510214 [INSPIRE].

  31. [31]

    S. Ribault and R. Santachiara, Liouville theory with a central charge less than one, arXiv:1503.02067 [INSPIRE].

  32. [32]

    G. Bonelli, K. Maruyoshi and A. Tanzini, Instantons on ALE spaces and super Liouville conformal field theories, JHEP 08 (2011) 056 [arXiv:1106.2505] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. [33]

    G. Bonelli, K. Maruyoshi and A. Tanzini, Gauge theories on ALE space and super Liouville correlation functions, Lett. Math. Phys. 101 (2012) 103 [arXiv:1107.4609] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. [34]

    A.A. Belavin, M.A. Bershtein, B.L. Feigin, A.V. Litvinov and G.M. Tarnopolsky, Instanton moduli spaces and bases in coset conformal field theory, Commun. Math. Phys. 319 (2013) 269 [arXiv:1111.2803] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  35. [35]

    V. Schomerus and P. Suchanek, Liouville’s imaginary shadow, arXiv:1210.1856 [INSPIRE].

  36. [36]

    L. Hadasz and Z. Jaskólski, Super-Liouville-double Liouville correspondence, JHEP 05 (2014) 124 [arXiv:1312.4520] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  37. [37]

    F. Benini and S. Cremonesi, Partition functions of N = (2, 2) gauge theories on S2 and vortices, Commun. Math. Phys. 334 (2015) 1483 [arXiv:1206.2356] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  38. [38]

    N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact results in D = 2 supersymmetric gauge theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  39. [39]

    L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  40. [40]

    N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge theory loop operators and Liouville theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  41. [41]

    H. Awata, H. Fuji, H. Kanno, M. Manabe and Y. Yamada, Localization with a surface operator, irregular conformal blocks and open topological string, Adv. Theor. Math. Phys. 16 (2012) 725 [arXiv:1008.0574] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

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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.

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Affiliations

  1. International School of Advanced Studies (SISSA), via Bonomea 265, 34136, Trieste, Italy

    Aditya Bawane, Giulio Bonelli, Massimiliano Ronzani & Alessandro Tanzini

  2. INFN, Sezione di Trieste, Trieste, Italy

    Aditya Bawane, Giulio Bonelli, Massimiliano Ronzani & Alessandro Tanzini

Authors
  1. Aditya Bawane
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  2. Giulio Bonelli
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  3. Massimiliano Ronzani
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  4. Alessandro Tanzini
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Correspondence to Massimiliano Ronzani.

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ArXiv ePrint: 1411.2762

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Bawane, A., Bonelli, G., Ronzani, M. et al. \( \mathcal{N}=2 \) supersymmetric gauge theories on S2 × S2 and Liouville Gravity. J. High Energ. Phys. 2015, 54 (2015). https://doi.org/10.1007/JHEP07(2015)054

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Keywords

  • Supersymmetric gauge theory
  • Extended Supersymmetry
  • Topological Field Theories