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Localization on round sphere revisited

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Abstract

We consider supersymmetric gauge theories on round 3-sphere with a certain background gauge field. The Lagrangians break the usual symmetry because the background gauge field which we have turned on violates the isometry. In order to maintain the supersymmetry, we choose unfamiliar charged Killing spinors as \( \mathcal{N} \) = 2 SUSY parameters. We perform localization calculous within this setup and find the double sine function as we expected. We comment on more direct relationship between theories on round sphere and squashed sphere via Weyl transformation.

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References

  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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  5. F. Benini and S. Cremonesi, Partition functions of N = (2,2) gauge theories on S 2 and vortices, arXiv:1206.2356 [INSPIRE].

  6. 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].

    Article  ADS  Google Scholar 

  7. N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. Y. Imamura and D. Yokoyama, N = 2 supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].

    ADS  Google Scholar 

  9. D. Gang, Chern-Simons theory on L(p,q) lens spaces and localization, arXiv:0912.4664 [INSPIRE].

  10. J. Kallen, Cohomological localization of Chern-Simons theory, JHEP 08 (2011) 008 [arXiv:1104.5353] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. K. Ohta and Y. Yoshida, Non-Abelian localization for supersymmetric Yang-Mills-Chern-Simons theories on seifert manifold, Phys. Rev. D 86 (2012) 105018 [arXiv:1205.0046] [INSPIRE].

    ADS  Google Scholar 

  12. L.F. Alday, D. Martelli, P. Richmond and J. Sparks, Localization on Three-Manifolds, arXiv:1307.6848 [INSPIRE].

  13. J. Nian, Localization of Supersymmetric Chern-Simons-Matter Theory on a Squashed S 3 with SU(2) × U(1) Isometry, arXiv:1309.3266 [INSPIRE].

  14. O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  16. H. Fuji, S. Hirano and S. Moriyama, Summing up all genus free energy of ABJM matrix model, JHEP 08 (2011) 001 [arXiv:1106.4631] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  17. M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 1203 (2012) P03001 [arXiv:1110.4066] [INSPIRE].

    Article  Google Scholar 

  18. A. Klemm, M. Mariño, M. Schiereck and M. Soroush, ABJM Wilson loops in the Fermi gas approach, arXiv:1207.0611 [INSPIRE].

  19. Y. Imamura and K. Kimura, On the moduli space of elliptic Maxwell-Chern-Simons theories, Prog. Theor. Phys. 120 (2008) 509 [arXiv:0806.3727] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  20. D.L. Jafferis and A. Tomasiello, A simple class of N = 3 gauge/gravity duals, JHEP 10 (2008) 101 [arXiv:0808.0864] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  21. C.P. Herzog, I.R. Klebanov, S.S. Pufu and T. Tesileanu, Multi-Matrix Models and Tri-Sasaki Einstein Spaces, Phys. Rev. D 83 (2011) 046001 [arXiv:1011.5487] [INSPIRE].

    ADS  Google Scholar 

  22. R.C. Santamaria, M. Mariño and P. Putrov, Unquenched flavor and tropical geometry in strongly coupled Chern-Simons-matter theories, JHEP 10 (2011) 139 [arXiv:1011.6281] [INSPIRE].

    Article  ADS  Google Scholar 

  23. D. Martelli and J. Sparks, The large-N limit of quiver matrix models and Sasaki-Einstein manifolds, Phys. Rev. D 84 (2011) 046008 [arXiv:1102.5289] [INSPIRE].

    ADS  Google Scholar 

  24. M. Mariño and P. Putrov, Interacting fermions and N = 2 Chern-Simons-matter theories, arXiv:1206.6346 [INSPIRE].

  25. D. Farquet and J. Sparks, Wilson loops and the geometry of matrix models in AdS 4 /CFT 3, arXiv:1304.0784 [INSPIRE].

  26. S. Cheon, H. Kim and N. Kim, Calculating the partition function of N = 2 Gauge theories on S 3 and AdS/CFT correspondence, JHEP 05 (2011) 134 [arXiv:1102.5565] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. D. Martelli, A. Passias and J. Sparks, The gravity dual of supersymmetric gauge theories on a squashed three-sphere, Nucl. Phys. B 864 (2012) 840 [arXiv:1110.6400] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. A. Tanaka, Comments on knotted 1/2 BPS Wilson loops, JHEP 07 (2012) 097 [arXiv:1204.5975] [INSPIRE].

    Article  ADS  Google Scholar 

  29. K. Nagasaki and S. Yamaguchi, Towards the localization of SUSY gauge theory on a curved space, Int. J. Mod. Phys. A 27 (2012) 1250029 [arXiv:1106.4975] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  30. N. Drukker, T. Okuda and F. Passerini, Exact results for vortex loop operators in 3d supersymmetric theories, arXiv:1211.3409 [INSPIRE].

  31. C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric Field Theories on Three-Manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  32. M. Honda and Y. Yoshida, Localization and Large-N reduction on S 3 for the Planar and M-theory limit, Nucl. Phys. B 865 (2012) 21 [arXiv:1203.1016] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. Y. Asano, G. Ishiki, T. Okada and S. Shimasaki, Large-N reduction for N = 2 quiver Chern-Simons theories on S 3 and localization in matrix models, Phys. Rev. D 85 (2012) 106003 [arXiv:1203.0559] [INSPIRE].

    ADS  Google Scholar 

  34. C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, The Geometry of Supersymmetric Partition Functions, arXiv:1309.5876 [INSPIRE].

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Authors and Affiliations

  1. Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Akinori Tanaka

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  1. Akinori Tanaka
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Correspondence to Akinori Tanaka.

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ArXiv ePrint: 1309.4992

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Tanaka, A. Localization on round sphere revisited. J. High Energ. Phys. 2013, 103 (2013). https://doi.org/10.1007/JHEP11(2013)103

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  • DOI: https://doi.org/10.1007/JHEP11(2013)103

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