Down the rabbit hole with theories of class \( \mathcal{S} \)

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Article

Abstract

We review some of the properties of 3d \( \mathcal{N}=4 \) theories obtained by dimensionally reducing theories of class \( \mathcal{S} \). We study 3d partition functions, and certain limits thereof, for such theories, and the properties implied for these by 3d mirror symmetry.

Keywords

Supersymmetric gauge theory Field Theories in Lower Dimensions Duality in Gauge Field Theories 

Notes

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References

  1. [1]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
  2. [2]
    D. Gaiotto, N=2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4d Superconformal Index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    L.F. Alday, M. Bullimore and M. Fluder, On S-duality of the Superconformal Index on Lens Spaces and 2d TQFT, JHEP 05 (2013) 122 [arXiv:1301.7486] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    S.S. Razamat and M. Yamazaki, S-duality and the N = 2 Lens Space Index, JHEP 10 (2013) 048 [arXiv:1306.1543] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    M. Lemos, W. Peelaers and L. Rastelli, The superconformal index of class S theories of type D, JHEP 05 (2014) 120 [arXiv:1212.1271] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the Twisted D-Series, arXiv:1309.2299 [INSPIRE].
  10. [10]
    O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the E 6 Theory, arXiv:1403.4604 [INSPIRE].
  11. [11]
    A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d Topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [arXiv:1007.0992] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    A. Gadde and S. Gukov, 2d Index and Surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    A. Gorsky and N. Nekrasov, Hamiltonian systems of Calogero type and two-dimensional Yang-Mills theory, Nucl. Phys. B 414 (1994) 213 [hep-th/9304047] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    A. Gorsky and N. Nekrasov, Relativistic Calogero-Moser model as gauged WZW theory, Nucl. Phys. B 436 (1995) 582 [hep-th/9401017] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    A. Gorsky and N. Nekrasov, Elliptic Calogero-Moser system from two-dimensional current algebra, hep-th/9401021 [INSPIRE].
  18. [18]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [INSPIRE].
  19. [19]
    R. Brooks and S.J. Gates Jr., Extended supersymmetry and superBF gauge theories, Nucl. Phys. B 432 (1994) 205 [hep-th/9407147] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  20. [20]
    A. Kapustin and M.J. Strassler, On mirror symmetry in three-dimensional Abelian gauge theories, JHEP 04 (1999) 021 [hep-th/9902033] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    D. Gaiotto and P. Koroteev, On Three Dimensional Quiver Gauge Theories and Integrability, JHEP 05 (2013) 126 [arXiv:1304.0779] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    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
  23. [23]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
  25. [25]
    S. Kim, The complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. B 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
  26. [26]
    Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    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
  28. [28]
    D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    F. Benini, T. Nishioka and M. Yamazaki, 4d Index to 3d Index and 2d TQFT, Phys. Rev. D 86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].ADSGoogle Scholar
  30. [30]
    T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP 07 (2013) 149 [arXiv:1305.3924] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities for orthogonal groups, JHEP 08 (2013) 099 [arXiv:1307.0511] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    F.A.H. Dolan, V.P. Spiridonov and G.S. Vartanov, From 4d superconformal indices to 3d partition functions, Phys. Lett. B 704 (2011) 234 [arXiv:1104.1787] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    A. Gadde and W. Yan, Reducing the 4d Index to the S 3 Partition Function, JHEP 12 (2012) 003 [arXiv:1104.2592] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    Y. Imamura, Relation between the 4d superconformal index and the S 3 partition function, JHEP 09 (2011) 133 [arXiv:1104.4482] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  37. [37]
    S. Pasquetti, Factorisation of N = 2 Theories on the Squashed 3-Sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, arXiv:1211.1986 [INSPIRE].
  39. [39]
    I.G. Macdonald, Symmetric functions and hall polynomials, Oxford University Press, (1995).Google Scholar
  40. [40]
    S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d \( \mathcal{N}=4 \) gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, Coulomb branch Hilbert series and Hall-Littlewood polynomials, arXiv:1403.0585 [INSPIRE].
  42. [42]
    S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, Coulomb branch Hilbert series and Three Dimensional Sicilian Theories, arXiv:1403.2384 [INSPIRE].
  43. [43]
    C. Krattenthaler, V.P. Spiridonov and G.S. Vartanov, Superconformal indices of three-dimensional theories related by mirror symmetry, JHEP 06 (2011) 008 [arXiv:1103.4075] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    F.J. van de Bult, Hyperbolic Hypergeometric Functions, Ph.D. Thesis, University of Amsterdam, Amsterdam, Netherlands (2007), http://www.its.caltech.edu/~vdbult/Thesis.pdf.
  45. [45]
    I. Yaakov, Redeeming Bad Theories, JHEP 11 (2013) 189 [arXiv:1303.2769] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    O. Aharony, IR duality in D = 3 N = 2 supersymmetric USp(2N(c)) and U(N(c)) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    B. Willett and I. Yaakov, N = 2 Dualities and Z Extremization in Three Dimensions, arXiv:1104.0487 [INSPIRE].
  48. [48]
    K. Hosomichi, S. Lee and J. Park, AGT on the S-duality Wall, JHEP 12 (2010) 079 [arXiv:1009.0340] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  49. [49]
    D. Gang, E. Koh, S. Lee and J. Park, Superconformal Index and 3d-3d Correspondence for Mapping Cylinder/Torus, JHEP 01 (2014) 063 [arXiv:1305.0937] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    D. Gang, E. Koh and K. Lee, Superconformal Index with Duality Domain Wall, JHEP 10 (2012) 187 [arXiv:1205.0069] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    S. Benvenuti and S. Pasquetti, 3D-partition functions on the sphere: exact evaluation and mirror symmetry, JHEP 05 (2012) 099 [arXiv:1105.2551] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  53. [53]
    T. Nishioka, Y. Tachikawa and M. Yamazaki, 3d Partition Function as Overlap of Wavefunctions, JHEP 08 (2011) 003 [arXiv:1105.4390] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  54. [54]
    V.P. Spiridonov and G.S. Vartanov, Vanishing superconformal indices and the chiral symmetry breaking, JHEP 06 (2014) 062 [arXiv:1402.2312] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    M. Bullimore, M. Fluder, L. Hollands and P. Richmond, The superconformal index and an elliptic algebra of surface defects, arXiv:1401.3379 [INSPIRE].
  57. [57]
    V. Fock, A. Gorsky, N. Nekrasov and V. Rubtsov, Duality in integrable systems and gauge theories, JHEP 07 (2000) 028 [hep-th/9906235] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  58. [58]
    A. Gorsky and A. Mironov, Integrable many body systems and gauge theories, hep-th/0011197 [INSPIRE].
  59. [59]
    S.S. Razamat, On the N = 2 superconformal index and eigenfunctions of the elliptic RS model, Lett. Math. Phys. 104 (2014) 673 [arXiv:1309.0278] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  60. [60]
    A. Hanany and N. Mekareeya, Tri-vertices and SU(2)’s, JHEP 02 (2011) 069 [arXiv:1012.2119] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  61. [61]
    M. Rahman and A. Verma, A q-integral representation of Rogersq-ultraspherical polynomials and some applications, Constr. Approx. 2 (1986) 1.MathSciNetCrossRefMATHGoogle Scholar
  62. [62]
    H. Awata, S. Odake and J. Shiraishi, Integral representations of the Macdonald symmetric functions, Commun. Math. Phys. 179 (1996) 647 [q-alg/9506006] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  63. [63]
    A. Okounkov, (Shifted) Macdonald Polynomials: q-Integral Representation and Combinatorial Formula, Compos. Math. 112 (1998) 147.MathSciNetCrossRefMATHGoogle Scholar
  64. [64]
    L.F. Alday, D. Martelli, P. Richmond and J. Sparks, Localization on Three-Manifolds, JHEP 10 (2013) 095 [arXiv:1307.6848] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Institute for Advanced StudyPrincetonUnited Kingdom

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