Journal of High Energy Physics

, 2018:70 | Cite as

Analytic continuation of dimensions in supersymmetric localization

  • Anastasios Gorantis
  • Joseph A. Minahan
  • Usman Naseer
Open Access
Regular Article - Theoretical Physics
  • 4 Downloads

Abstract

We compute the perturbative partition functions for gauge theories with eight supersymmetries on spheres of dimension d ≤ 5, proving a conjecture by the second author. We apply similar methods to gauge theories with four supersymmetries on spheres with d ≤ 3. The results are valid for non-integer d as well. We further propose an analytic continuation from d = 3 to d = 4 that gives the perturbative partition function for an \( \mathcal{N} \) =1 gauge theory. The results are consistent with the free multiplets and the one-loop β-functions for general \( \mathcal{N} \) = 1 gauge theories. We also consider the analytic continuation of an \( \mathcal{N} \) = 1 preserving mass deformation of the maximally supersymmetric gauge theory and compare to recent holographic results for \( \mathcal{N} \) = 1 super Yang-Mills. We find that the general structure for the real part of the free energy coming from the analytic continuation is consistent with the holographic results.

Keywords

Extended Supersymmetry Matrix Models Supersymmetric Gauge Theory 

Notes

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.

References

  1. [1]
    V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].MathSciNetMATHGoogle Scholar
  2. [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].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    H.-C. Kim and S. Kim, M 5-branes from gauge theories on the 5-sphere, JHEP 05 (2013) 144 [arXiv:1206.6339] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    J. Kallen, Cohomological localization of Chern-Simons theory, JHEP 08 (2011) 008 [arXiv:1104.5353] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    J. Källén and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    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
  8. [8]
    J.A. Minahan and M. Zabzine, Gauge theories with 16 supersymmetries on spheres, JHEP 03 (2015) 155 [arXiv:1502.07154] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  9. [9]
    V. Pestun and M. Zabzine, Introduction to localization in quantum field theory, J. Phys. A 50 (2017) 443001 [arXiv:1608.02953] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  10. [10]
    J.A. Minahan, Localizing gauge theories on S d, JHEP 04 (2016) 152 [arXiv:1512.06924] [INSPIRE].ADSGoogle Scholar
  11. [11]
    J.A. Minahan and U. Naseer, One-loop tests of supersymmetric gauge theories on spheres, JHEP 07 (2017) 074 [arXiv:1703.07435] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    W. Siegel, Inconsistency of supersymmetric dimensional regularization, Phys. Lett. 94B (1980) 37 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    W. Siegel, Supersymmetric dimensional regularization via dimensional reduction, Phys. Lett. 84B (1979) 193 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    L.V. Avdeev, G.A. Chochia and A.A. Vladimirov, On the scope of supersymmetric dimensional regularization, Phys. Lett. 105B (1981) 272 [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    L. Fei, S. Giombi and I.R. Klebanov, Critical O(N ) models in 6 − ϵ dimensions, Phys. Rev. D 90 (2014) 025018 [arXiv:1404.1094] [INSPIRE].ADSGoogle Scholar
  17. [17]
    S. Giombi and I.R. Klebanov, Interpolating between a and F , JHEP 03 (2015) 117 [arXiv:1409.1937] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  18. [18]
    L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Three loop analysis of the critical O(N ) models in 6 − ϵ dimensions, Phys. Rev. D 91 (2015) 045011 [arXiv:1411.1099] [INSPIRE].ADSMathSciNetGoogle Scholar
  19. [19]
    L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Critical Sp(N ) models in 6 − ϵ dimensions and higher spin dS/CFT, JHEP 09 (2015) 076 [arXiv:1502.07271] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  20. [20]
    L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Generalized F -theorem and the ϵ expansion, JHEP 12 (2015) 155 [arXiv:1507.01960] [INSPIRE].ADSMathSciNetGoogle Scholar
  21. [21]
    E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere partition functions and the Zamolodchikov metric, JHEP 11 (2014) 001 [arXiv:1405.7271] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    N. Bobev et al., Holography for \( \mathcal{N} \) = 1 on S 4, JHEP 10 (2016) 095 [arXiv:1605.00656] [INSPIRE].
  23. [23]
    J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 500 (1997) 163 [hep-th/9703100] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    O. Aharony et al., Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    M.A. Rubin and C.R. Ordonez, Symmetric tensor eigen spectrum of the Laplacian on n spheres, J. Math. Phys. 26 (1985) 65 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    R. Camporesi and A. Higuchi, On the eigen functions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20 (1996) 1 [gr-qc/9505009] [INSPIRE].
  27. [27]
    F. Benini and S. Cremonesi, Partition functions of \( \mathcal{N} \) = (2, 2) gauge theories on S 2 and vortices, Commun. Math. Phys. 334 (2015) 1483 [arXiv:1206.2356] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  28. [28]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    J.G. Russo and K. Zarembo, Large-N limit of N = 2 SU(N ) gauge theories from localization, JHEP 10 (2012) 082 [arXiv:1207.3806] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    X. Chen-Lin, J. Gordon and K. Zarembo, \( \mathcal{N} \) = 2 super-Yang-Mills theory at strong coupling, JHEP 11 (2014) 057 [arXiv:1408.6040] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    J.A. Minahan and U. Naseer, in progress.Google Scholar
  32. [32]
    A.D. Kennedy, Clifford algebras in 2ω dimensions, J. Math. Phys. 22 (1981) 1330.ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Anastasios Gorantis
    • 1
  • Joseph A. Minahan
    • 1
  • Usman Naseer
    • 2
  1. 1.Department of Physics and AstronomyUppsala UniversityUppsalaSweden
  2. 2.Center for Theoretical Physics, Massachusetts Institute of TechnologyCambridgeU.S.A.

Personalised recommendations