Journal of High Energy Physics

, 2010:79 | Cite as

AGT on the S-duality wall

  • Kazuo Hosomichi
  • Sungjay Lee
  • Jaemo Park


Three-dimensional gauge theory T [G] arises on a domain wall between four-dimensional \( \mathcal{N} = 4 \) SYM theories with the gauge groups G and its S-dual G L. We argue that the \( \mathcal{N} = {2^*} \) mass deformation of the bulk theory induces a mass-deformation of the theory T[G] on the wall. The partition functions of the theory T[SU(2)] and its mass-deformation on the three-sphere are shown to coincide with the transformation coefficient of Liouville one-point conformal block on torus under the S-duality.


Supersymmetric gauge theory Supersymmetry and Duality Conformal and W Symmetry 


  1. [1]
    N.A. Nekrasov, Seiberg-Witten Prepotential From Instanton Counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].MathSciNetGoogle Scholar
  2. [2]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, arXiv:0712.2824 [SPIRES].
  3. [3]
    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] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  4. [4]
    A. Kapustin, B. Willett and I. Yaakov, Nonperturbative Tests of Three-Dimensional Dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [SPIRES].CrossRefADSGoogle Scholar
  5. [5]
    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] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  6. [6]
    N. Wyllard, A N−1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  7. [7]
    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] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  8. [8]
    N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  9. [9]
    F. Passerini, Gauge Theory Wilson Loops and Conformal Toda Field Theory, JHEP 03 (2010) 125 [arXiv:1003.1151] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  10. [10]
    J. Gomis and B. Le Floch, ’t Hooft Operators in Gauge Theory from Toda CFT, arXiv:1008.4139 [SPIRES].
  11. [11]
    K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System and Liouville Field Theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  12. [12]
    H. Awata, H. Fuji, H. Kanno, M. Manabe and Y. Yamada, Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String, arXiv:1008. 0574 [SPIRES].
  13. [13]
    L.F. Alday and Y. Tachikawa, A ffine SL(2) conformal blocks from 4d gauge theories, Lett. Math. Phys. 94 (2010) 87 [arXiv:1005.4469] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  14. [14]
    C. Kozcaz, S. Pasquetti, F. Passerini and N. Wyllard, A ffine sl(N) conformal blocks from N = 2 SU(N) gauge theories, arXiv:1008.1412 [SPIRES].
  15. [15]
    N. Drukker, D. Gaiotto and J. Gomis, The Virtue of Defects in 4D Gauge Theories and 2D CFT s, arXiv:1003.1112 [SPIRES].
  16. [16]
    D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N = 4 Super Yang-Mills Theory, arXiv:0804.2902 [SPIRES].
  17. [17]
    D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, arXiv:0807.3720 [SPIRES].
  18. [18]
    K.A. Intriligator and N. Seiberg, Mirror symmetry in three dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [SPIRES].ADSMathSciNetGoogle Scholar
  19. [19]
    A. Kapustin and M.J. Strassler, On Mirror Symmetry in Three Dimensional Abelian Gauge Theories, JHEP 04 (1999) 021 [hep-th/9902033] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  20. [20]
    D. Gaiotto, N = 2 dualities, arXiv:0904.2715 [SPIRES].
  21. [21]
    J. Teschner, From Liouville theory to the quantum geometry of Riemann surfaces, hep-th/0308031 [SPIRES].
  22. [22]
    J. Teschner, Quantum Liouville theory versus quantized Teichmueller spaces, Fortsch. Phys. 51 (2003) 865 [hep-th/0212243] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  23. [23]
    J. Teschner, On the relation between quantum Liouville theory and the quantized Teichmueller spaces, Int. J. Mod. Phys. A 19S2 (2004) 459 [hep-th/0303149] [SPIRES].MathSciNetGoogle Scholar
  24. [24]
    J. Teschner, An analog of a modular functor from quantized Teichmüller theory, math/0510174.
  25. [25]
    S. Kharchev, D. Lebedev and M. Semenov-Tian-Shansky, Unitary representations of \( {U_q}\left( {\mathfrak{s}\mathfrak{l}\left( {2,\mathbb{R}} \right)} \right) \) , the modular double and the multiparticle q-deformed Toda chains, Commun. Math. Phys. 225 (2002) 573 [hep-th/0102180] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  26. [26]
    A.G. Bytsko and J. Teschner, Quantization of models with non-compact quantum group symmetry: Modular XXZ magnet and lattice sinh-Gordon model, J. Phys. A 39 (2006) 12927 [hep-th/0602093] [SPIRES].MathSciNetGoogle Scholar
  27. [27]
    T. Okuda and V. Pestun, On the instantons and the hypermultiplet mass of N = 2* super Yang-Mills on S 4, arXiv:1004.1222 [SPIRES].
  28. [28]
    D. Tong, Dynamics of N = 2 supersymmetric Chern-Simons theories, JHEP 07 (2000) 019 [hep-th/0005186] [SPIRES].CrossRefADSGoogle Scholar
  29. [29]
    S. Kanno, Y. Matsuo, S. Shiba and Y. Tachikawa, N=2 gauge theories and degenerate fields of Toda theory, Phys. Rev. D 81 (2010) 046004 [arXiv:0911.4787] [SPIRES].ADSMathSciNetGoogle Scholar
  30. [30]
    D.L. Jafferis, The Exact Superconformal R -Symmetry Extremizes Z, arXiv:1012.3210 [SPIRES].
  31. [31]
    N. Hama, K. Hosomichi and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, arXiv:1012.3512 [SPIRES].

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.Yukawa Institute for Theoretical PhysicsKyoto University Kitashirakawa-OiwakechoSakyo, KyotoJapan
  2. 2.Korea Institute for Advanced Study Hoegiro 87(207-43 Cheongnyangni-dong)Dongdaemun-gu, SeoulKorea
  3. 3.DAMTP, Centre for Mathematical SciencesUniversity of CambridgeCambridgeU.K.
  4. 4.Department of Physics & Postech Center for Theoretical Physics, POSTECHPohangKorea

Personalised recommendations