Advertisement

Journal of High Energy Physics

, 2017:50 | Cite as

Generalized Toda theory from six dimensions and the conifold

  • Sam van Leuven
  • Gerben Oling
Open Access
Regular Article - Theoretical Physics
  • 66 Downloads

Abstract

Recently, a physical derivation of the Alday-Gaiotto-Tachikawa correspondence has been put forward. A crucial role is played by the complex Chern-Simons theory arising in the 3d-3d correspondence, whose boundary modes lead to Toda theory on a Riemann surface. We explore several features of this derivation and subsequently argue that it can be extended to a generalization of the AGT correspondence. The latter involves codimension two defects in six dimensions that wrap the Riemann surface. We use a purely geometrical description of these defects and find that the generalized AGT setup can be modeled in a pole region using generalized conifolds. Furthermore, we argue that the ordinary conifold clarifies several features of the derivation of the original AGT correspondence.

Keywords

Conformal and W Symmetry M-Theory Supersymmetric Gauge Theory Supersymmetry and Duality 

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]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    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] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    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
  4. [4]
    D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
  7. [7]
    N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J. Teschner, On the Liouville three point function, Phys. Lett. B 363 (1995) 65 [hep-th/9507109] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    V. Mitev and E. Pomoni, Toda 3-Point Functions From Topological Strings, JHEP 06 (2015) 049 [arXiv:1409.6313] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    M. Isachenkov, V. Mitev and E. Pomoni, Toda 3-Point Functions From Topological Strings II, JHEP 08 (2016) 066 [arXiv:1412.3395] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    L.F. Alday, F. Benini and Y. Tachikawa, Liouville/Toda central charges from M5-branes, Phys. Rev. Lett. 105 (2010) 141601 [arXiv:0909.4776] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    N. Nekrasov and E. Witten, The Omega Deformation, Branes, Integrability and Liouville Theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    A. Mironov, A. Morozov and S. Shakirov, A direct proof of AGT conjecture at β = 1, JHEP 02 (2011) 067 [arXiv:1012.3137] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    R. Dijkgraaf and C. Vafa, Toda Theories, Matrix Models, Topological Strings and N = 2 Gauge Systems, arXiv:0909.2453 [INSPIRE].
  16. [16]
    M. Aganagic, N. Haouzi and S. Shakirov, A n -Triality, arXiv:1403.3657 [INSPIRE].
  17. [17]
    J. Yagi, Compactification on the Ω-background and the AGT correspondence, JHEP 09 (2012) 101 [arXiv:1205.6820] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    M.-C. Tan, M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems, JHEP 07 (2013) 171 [arXiv:1301.1977] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    J. Teschner and G.S. Vartanov, Supersymmetric gauge theories, quantization offlat and conformal field theory, Adv. Theor. Math. Phys. 19 (2015) 1 [arXiv:1302.3778] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    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
  21. [21]
    C. Córdova and D.L. Jafferis, Toda Theory From Six Dimensions, arXiv:1605.03997 [INSPIRE].
  22. [22]
    N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    N. Seiberg, Notes on theories with 16 supercharges, Nucl. Phys. Proc. Suppl. 67 (1998) 158 [hep-th/9705117] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    M.R. Douglas, On D = 5 super Yang-Mills theory and (2, 0) theory, JHEP 02 (2011) 011 [arXiv:1012.2880] [INSPIRE].ADSzbMATHGoogle Scholar
  25. [25]
    D.-E. Diaconescu, D-branes, monopoles and Nahm equations, Nucl. Phys. B 503 (1997) 220 [hep-th/9608163] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N = 4 Super Yang-Mills Theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    F.A. Bais, T. Tjin and P. van Driel, Covariantly coupled chiral algebras, Nucl. Phys. B 357 (1991) 632 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    J. de Boer and T. Tjin, The relation between quantum W algebras and Lie algebras, Commun. Math. Phys. 160 (1994) 317 [hep-th/9302006] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    J. de Boer and J.I. Jottar, Thermodynamics of higher spin black holes in AdS 3, JHEP 01 (2014) 023 [arXiv:1302.0816] [INSPIRE].CrossRefzbMATHGoogle Scholar
  30. [30]
    P. Forgács, A. Wipf, J. Balog, L. Fehér and L. O’Raifeartaigh, Liouville and Toda Theories as Conformally Reduced WZNW Theories, Phys. Lett. B 227 (1989) 214 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    L.F. Alday and Y. Tachikawa, Affine SL(2) conformal blocks from 4d gauge theories, Lett. Math. Phys. 94 (2010) 87 [arXiv:1005.4469] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    C. Kozcaz, S. Pasquetti, F. Passerini and N. Wyllard, Affine sl(N ) conformal blocks from N = 2 SU(N) gauge theories, JHEP 01 (2011) 045 [arXiv:1008.1412] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    N. Wyllard, W-algebras and surface operators in N = 2 gauge theories, J. Phys. A 44 (2011) 155401 [arXiv:1011.0289] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  34. [34]
    N. Wyllard, Instanton partition functions in N = 2 SU(N ) gauge theories with a general surface operator and their W-algebra duals, JHEP 02 (2011) 114 [arXiv:1012.1355] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    Y. Tachikawa, On W-algebras and the symmetries of defects of 6d N = (2, 0) theory, JHEP 03 (2011) 043 [arXiv:1102.0076] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP 06 (2011) 119 [arXiv:1105.0357] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    S. Nawata, Givental J-functions, Quantum integrable systems, AGT relation with surface operator, Adv. Theor. Math. Phys. 19 (2015) 1277 [arXiv:1408.4132] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    C. Córdova and D.L. Jafferis, Complex Chern-Simons from M5-branes on the Squashed Three-Sphere, JHEP 11 (2017) 119 [arXiv:1305.2891] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    N. Hama and K. Hosomichi, Seiberg-Witten Theories on Ellipsoids, JHEP 09 (2012) 033 [arXiv:1206.6359] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    Y. Imamura and D. Yokoyama, N=2 supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].ADSGoogle Scholar
  41. [41]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, The Geometry of Supersymmetric Partition Functions, JHEP 01 (2014) 124 [arXiv:1309.5876] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    D. Gaiotto and J. Maldacena, The gravity duals of N = 2 superconformal field theories, JHEP 10 (2012) 189 [arXiv:0904.4466] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    S. Gukov and D. Pei, Equivariant Verlinde formula from fivebranes and vortices, Commun. Math. Phys. 355 (2017) 1 [arXiv:1501.01310] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    S. Gukov, D. Pei, W. Yan and K. Ye, Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality, arXiv:1605.06528 [INSPIRE].
  45. [45]
    T. Dimofte, Complex Chern-Simons Theory at Level k via the 3d-3d Correspondence, Commun. Math. Phys. 339 (2015) 619 [arXiv:1409.0857] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    C. Córdova and D.L. Jafferis, Five-Dimensional Maximally Supersymmetric Yang-Mills in Supergravity Backgrounds, JHEP 10 (2017) 003 [arXiv:1305.2886] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    R. Dijkgraaf, L. Hollands, P. Sulkowski and C. Vafa, Supersymmetric gauge theories, intersecting branes and free fermions, JHEP 02 (2008) 106 [arXiv:0709.4446] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    N. Itzhaki, D. Kutasov and N. Seiberg, I-brane dynamics, JHEP 01 (2006) 119 [hep-th/0508025] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    J. de Boer and J.I. Jottar, Boundary conditions and partition functions in higher spin AdS 3 /CFT 2, JHEP 04 (2016) 107 [arXiv:1407.3844] [INSPIRE].ADSGoogle Scholar
  53. [53]
    L. Fehér, L. O’Raifeartaigh, P. Ruelle, I. Tsutsui and A. Wipf, On the general structure of Hamiltonian reductions of the WZNW theory, hep-th/9112068 [INSPIRE].
  54. [54]
    P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept. 223 (1993) 183 [hep-th/9210010] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    A. Perez, D. Tempo and R. Troncoso, Higher Spin Black Holes, Lect. Notes Phys. 892 (2015) 265 [arXiv:1402.1465].ADSMathSciNetCrossRefGoogle Scholar
  56. [56]
    L. Donnay, Asymptotic dynamics of three-dimensional gravity, PoS(Modave2015)001 [arXiv:1602.09021] [INSPIRE].
  57. [57]
    M. Bañados, Global charges in Chern-Simons field theory and the (2+1) black hole, Phys. Rev. D 52 (1996) 5816 [hep-th/9405171] [INSPIRE].MathSciNetGoogle Scholar
  58. [58]
    O. Coussaert, M. Henneaux and P. van Driel, The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. 12 (1995) 2961 [gr-qc/9506019] [INSPIRE].
  59. [59]
    M. Bañados, Three-dimensional quantum geometry and black holes, hep-th/9901148 [INSPIRE].
  60. [60]
    A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    A. Campoleoni, S. Fredenhagen and S. Pfenninger, Asymptotic W-symmetries in three-dimensional higher-spin gauge theories, JHEP 09 (2011) 113 [arXiv:1107.0290] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    M. Bershadsky and H. Ooguri, Hidden SL(n) Symmetry in Conformal Field Theories, Commun. Math. Phys. 126 (1989) 49 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    M. Bershadsky, Conformal field theories via Hamiltonian reduction, Commun. Math. Phys. 139 (1991) 71 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    A. LeClair, D. Nemeschansky and N.P. Warner, S matrices for perturbed N = 2 superconformal field theory from quantum groups, Nucl. Phys. B 390 (1993) 653 [hep-th/9206041] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  65. [65]
    M. Bullimore and H.-C. Kim, The Superconformal Index of the (2, 0) Theory with Defects, JHEP 05 (2015) 048 [arXiv:1412.3872] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  66. [66]
    T. Nishioka and Y. Tachikawa, Central charges of para-Liouville and Toda theories from M-5-branes, Phys. Rev. D 84 (2011) 046009 [arXiv:1106.1172] [INSPIRE].ADSGoogle Scholar
  67. [67]
    N. Wyllard, Coset conformal blocks and N = 2 gauge theories, arXiv:1109.4264 [INSPIRE].
  68. [68]
    A.M. Uranga, Brane configurations for branes at conifolds, JHEP 01 (1999) 022 [hep-th/9811004] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    K. Dasgupta and S. Mukhi, Brane constructions, conifolds and M-theory, Nucl. Phys. B 551 (1999) 204 [hep-th/9811139] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    J. McOrist and A.B. Royston, Relating Conifold Geometries to NS5-branes, Nucl. Phys. B 849 (2011) 573 [arXiv:1101.3552] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    P. Candelas and X.C. de la Ossa, Comments on Conifolds, Nucl. Phys. B 342 (1990) 246 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  72. [72]
    P. Ouyang, Holomorphic D7 branes and flavored N = 1 gauge theories, Nucl. Phys. B 699 (2004) 207 [hep-th/0311084] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    R. Minasian, G.W. Moore and D. Tsimpis, Calabi-Yau black holes and (0,4) σ-models, Commun. Math. Phys. 209 (2000) 325 [hep-th/9904217] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    V. Pestun, Localization for \( \mathcal{N} \) = 2 Supersymmetric Gauge Theories in Four Dimensions, arXiv:1412.7134.
  76. [76]
    M. Bershadsky, C. Vafa and V. Sadov, D-branes and topological field theories, Nucl. Phys. B 463 (1996) 420 [hep-th/9511222] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, arXiv:1306.4320 [INSPIRE].

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations