Advertisement

A one-dimensional theory for Higgs branch operators

  • Mykola Dedushenko
  • Silviu S. Pufu
  • Ran Yacoby
Open Access
Regular Article - Theoretical Physics

Abstract

We use supersymmetric localization to calculate correlation functions of half-BPS local operators in 3d \( \mathcal{N}=4 \) superconformal field theories whose Lagrangian descriptions consist of vectormultiplets coupled to hypermultiplets. The operators we primarily study are certain twisted linear combinations of Higgs branch operators that can be inserted anywhere along a given line. These operators are constructed from the hypermultiplet scalars. They form a one-dimensional non-commutative operator algebra with topological correlation functions. The 2- and 3-point functions of Higgs branch operators in the full 3d \( \mathcal{N}=4 \) theory can be simply inferred from the 1d topological algebra. After conformally mapping the 3d superconformal field theory from flat space to a round three-sphere, we preform supersymmetric localization using a supercharge that does not belong to any 3d \( \mathcal{N}=2 \) subalgebra of the \( \mathcal{N}=4 \) algebra. The result is a simple model that can be used to calculate correlation functions in the 1d topological algebra mentioned above. This model is a 1d Gaussian theory coupled to a matrix model, and it can be viewed as a gauge-fixed version of a topological gauged quantum mechanics. Our results generalize to non-conformal theories on S3 that contain real mass and Fayet-Iliopolous parameters. We also provide partial results in the 1d topological algebra associated with the Coulomb branch, where we calculate correlation functions of local operators built from the vectormultiplet scalars.

Keywords

Extended Supersymmetry Supersymmetric Gauge Theory Conformal Field 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]
    S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large N, Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Extremal correlators in the AdS/CFT correspondence, hep-th/9908160 [INSPIRE].
  3. [3]
    M. Baggio, J. de Boer and K. Papadodimas, A non-renormalization theorem for chiral primary 3-point functions, JHEP 07 (2012) 137 [arXiv:1203.1036] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite Chiral Symmetry in Four Dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    C. Beem, L. Rastelli and B.C. van Rees, \( \mathcal{W} \) symmetry in six dimensions, JHEP 05 (2015) 017 [arXiv:1404.1079] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    C. Beem, W. Peelaers, L. Rastelli and B.C. van Rees, Chiral algebras of class S, JHEP 05 (2015) 020 [arXiv:1408.6522] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    S.M. Chester, J. Lee, S.S. Pufu and R. Yacoby, Exact Correlators of BPS Operators from the 3d Superconformal Bootstrap, JHEP 03 (2015) 130 [arXiv:1412.0334] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    C. Beem, W. Peelaers and L. Rastelli, Deformation quantization and superconformal symmetry in three dimensions, Commun. Math. Phys. 354 (2017) 345 [arXiv:1601.05378] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Y. Tachikawa, A brief review of the 2d/4d correspondences, J. Phys. A 50 (2017) 443012 [arXiv:1608.02964] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  10. [10]
    T.T. Dumitrescu, An introduction to supersymmetric field theories in curved space, J. Phys. A 50 (2017) 443005 [arXiv:1608.02957] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  11. [11]
    D.R. Morrison, Gromov-Witten invariants and localization, J. Phys. A 50 (2017) 443004 [arXiv:1608.02956] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  12. [12]
    S. Pasquetti, Holomorphic blocks and the 5d AGT correspondence, J. Phys. A 50 (2017) 443016 [arXiv:1608.02968] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  13. [13]
    S. Kim and K. Lee, Indices for 6 dimensional superconformal field theories, J. Phys. A 50 (2017) 443017 [arXiv:1608.02969] [INSPIRE].
  14. [14]
    V. Pestun and M. Zabzine, Introduction to localization in quantum field theory, J. Phys. A 50 (2017) 443001 [arXiv:1608.02953] [INSPIRE].
  15. [15]
    K. Zarembo, Localization and AdS/CFT Correspondence, J. Phys. A 50 (2017) 443011 [arXiv:1608.02963] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  16. [16]
    M. Mariño, Localization at large N in Chern-Simons-matter theories, J. Phys. A 50 (2017) 443007 [arXiv:1608.02959] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  17. [17]
    B. Willett, Localization on three-dimensional manifolds, J. Phys. A 50 (2017) 443006 [arXiv:1608.02958] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  18. [18]
    J.A. Minahan, Matrix models for 5d super Yang-Mills, J. Phys. A 50 (2017) 443015 [arXiv:1608.02967] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  19. [19]
    K. Hosomichi, \( \mathcal{N}=2 \) SUSY gauge theories on S 4, J. Phys. A 50 (2017) 443010 [arXiv:1608.02962] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  20. [20]
    T. Dimofte, Perturbative and nonperturbative aspects of complex Chern-Simons theory, J. Phys. A 50 (2017) 443009 [arXiv:1608.02961] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  21. [21]
    J. Qiu and M. Zabzine, Review of localization for 5d supersymmetric gauge theories, J. Phys. A 50 (2017) 443014 [arXiv:1608.02966] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  22. [22]
    V. Pestun, Review of localization in geometry, J. Phys. A 50 (2017) 443002 [arXiv:1608.02954] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  23. [23]
    F. Benini and B. Le Floch, Supersymmetric localization in two dimensions, J. Phys. A 50 (2017) 443003 [arXiv:1608.02955] [INSPIRE].
  24. [24]
    S.S. Pufu, The F-Theorem and F-Maximization, J. Phys. A 50 (2017) 443008 [arXiv:1608.02960] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  25. [25]
    L. Rastelli and S.S. Razamat, The supersymmetric index in four dimensions, J. Phys. A 50 (2017) 443013 [arXiv:1608.02965] [INSPIRE].
  26. [26]
    C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Contact Terms, Unitarity and F-Maximization in Three-Dimensional Superconformal Theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    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
  28. [28]
    E. Gerchkovitz, J. Gomis, N. Ishtiaque, A. Karasik, Z. Komargodski and S.S. Pufu, Correlation Functions of Coulomb Branch Operators, JHEP 01 (2017) 103 [arXiv:1602.05971] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    K. Papadodimas, Topological Anti-Topological Fusion in Four-Dimensional Superconformal Field Theories, JHEP 08 (2010) 118 [arXiv:0910.4963] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M. Baggio, V. Niarchos and K. Papadodimas, Exact correlation functions in SU(2) \( \mathcal{N}=2 \) superconformal QCD, Phys. Rev. Lett. 113 (2014) 251601 [arXiv:1409.4217] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  31. [31]
    M. Baggio, V. Niarchos and K. Papadodimas, tt * equations, localization and exact chiral rings in 4d \( \mathcal{N}=2 \) SCFTs, JHEP 02 (2015) 122 [arXiv:1409.4212] [INSPIRE].
  32. [32]
    M. Baggio, V. Niarchos and K. Papadodimas, On exact correlation functions in SU(N) \( \mathcal{N}=2 \) superconformal QCD, JHEP 11 (2015) 198 [arXiv:1508.03077] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, in The mathematical beauty of physics: A memorial volume for Claude Itzykson. Proceedings, Conference, Saclay, France, June 5-7, 1996, pp. 333–366 (1996) [hep-th/9607163] [INSPIRE].
  34. [34]
    K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    J. de Boer, K. Hori, H. Ooguri and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nucl. Phys. B 493 (1997) 101 [hep-th/9611063] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    J. de Boer, K. Hori, H. Ooguri, Y. Oz and Z. Yin, Mirror symmetry in three-dimensional theories, SL(2, ℤ) and D-brane moduli spaces, Nucl. Phys. B 493 (1997) 148 [hep-th/9612131] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    M. Bullimore, T. Dimofte and D. Gaiotto, The Coulomb Branch of 3d \( \mathcal{N}=4 \) Theories, Commun. Math. Phys. 354 (2017) 671 [arXiv:1503.04817] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    N. Hama, K. Hosomichi and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    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
  44. [44]
    V. Pestun, Localization of the four-dimensional N = 4 SYM to a two-sphere and 1/8 BPS Wilson loops, JHEP 12 (2012) 067 [arXiv:0906.0638] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, in proceedings of Strings, branes and dualities, NATO Advanced Study Institute, Cargese, France, May 26–June 14, 1997, pp. 359–372 [hep-th/9801061] [INSPIRE].
  47. [47]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09), Prague, Czech Republic, August 3–8, 2009, pp. 265–289 [DOI: https://doi.org/10.1142/9789814304634_0015] [arXiv:0908.4052] [INSPIRE].
  48. [48]
    N. Berkovits, A Ten-dimensional superYang-Mills action with off-shell supersymmetry, Phys. Lett. B 318 (1993) 104 [hep-th/9308128] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    B. Assel and J. Gomis, Mirror Symmetry And Loop Operators, JHEP 11 (2015) 055 [arXiv:1506.01718] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    S. Giombi and V. Pestun, Correlators of local operators and 1/8 BPS Wilson loops on S 2 from 2d YM and matrix models, JHEP 10 (2010) 033 [arXiv:0906.1572] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  52. [52]
    S. Giombi and V. Pestun, Correlators of Wilson Loops and Local Operators from Multi-Matrix Models and Strings in AdS, JHEP 01 (2013) 101 [arXiv:1207.7083] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    N. Drukker and J. Plefka, Superprotected n-point correlation functions of local operators in N = 4 super Yang-Mills, JHEP 04 (2009) 052 [arXiv:0901.3653] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    E. Witten, A New Look At The Path Integral Of Quantum Mechanics, arXiv:1009.6032 [INSPIRE].
  55. [55]
    E. Witten, Analytic Continuation Of Chern-Simons Theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001.2933] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    M. Dedushenko, Violation of the phase space general covariance as a diffeomorphism anomaly in quantum mechanics, JHEP 10 (2010) 054 [arXiv:1007.5292] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    A.S. Cattaneo and G. Felder, A path integral approach to the Kontsevich quantization formula, Commun. Math. Phys. 212 (2000) 591 [math/9902090] [INSPIRE].
  58. [58]
    E. Joung and K. Mkrtchyan, Notes on higher-spin algebras: minimal representations and structure constants, JHEP 05 (2014) 103 [arXiv:1401.7977] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  59. [59]
    A. Kapustin, B. Willett and I. Yaakov, Nonperturbative Tests of Three-Dimensional Dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    D.R. Gulotta, C.P. Herzog and S.S. Pufu, From Necklace Quivers to the F-theorem, Operator Counting and T(U(N)), JHEP 12 (2011) 077 [arXiv:1105.2817] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    A. Kapustin and M.J. Strassler, On mirror symmetry in three-dimensional Abelian gauge theories, JHEP 04 (1999) 021 [hep-th/9902033] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Mykola Dedushenko
    • 1
    • 2
  • Silviu S. Pufu
    • 1
  • Ran Yacoby
    • 1
    • 3
  1. 1.Joseph Henry LaboratoriesPrinceton UniversityPrincetonU.S.A.
  2. 2.Walter Burke Institute for Theoretical PhysicsCalifornia Institute of TechnologyPasadenaU.S.A.
  3. 3.Department of Physics and AstrophysicsWeizmann Institute of ScienceRehovotIsrael

Personalised recommendations