Kähler uniformization from holographic renormalization group flows of M5-branes

  • Martin Fluder
Open Access
Regular Article - Theoretical Physics


In this paper, we initiate the study of holographic renormalization group flows for the metric of four-manifolds. In particular, we derive a set of equations which govern the evolution of a generic Kähler four-manifold along the renormalization group flow in seven-dimensional gauged supergravity. The physical eleven-dimensional M-theory setup is given by a stack of M5-branes wrapping a calibrated Kähler four-cycle inside a Calabi-Yau threefold. By topologically twisting the theory in the ultraviolet, we may choose an arbitrary Kähler metric on the four-cycle as an asymptotic boundary condition. We find that at the infrared fixed point, we reach a Kähler-Einstein metric, which can be interpreted as an indication of “uniformizing” behavior of the flow.


AdS-CFT Correspondence M-Theory Renormalization Group Supergravity Models 


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.


  1. [1]
    M.H. Freedman, The topology of four-dimensional manifolds, J. Diff. Geom. 17 (1982) 357.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    R.S. Hamilton, Three-manifolds with positive ricci curvature, J. Diff. Geom. 17 (1982) 255.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    R.S. Hamilton, The formation of singularities in the Ricci flow, Surveys Diff. Geom. II (1995) 7.Google Scholar
  4. [4]
    W.P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc. 6 (1982) 357.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math/0211159 [INSPIRE].
  6. [6]
    J. Streets and G. Tian, Hermitian Curvature Flow, arXiv:0804.4109.
  7. [7]
    J. Streets and G. Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. 2010 (2010) 3101.MathSciNetMATHGoogle Scholar
  8. [8]
    J. Streets and G. Tian, Regularity results for pluriclosed flow, arXiv:1008.2794.
  9. [9]
    D.H. Friedan, Nonlinear Models in Two + Epsilon Dimensions, Annals Phys. 163 (1985) 318 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  10. [10]
    E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    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].
  12. [12]
    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].
  13. [13]
    E. Witten, Monopoles and four manifolds, Math. Res. Lett. 1 (1994) 769 [hep-th/9411102] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    J.M. Maldacena and C. Núñez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE].
  15. [15]
    J.P. Gauntlett, N. Kim and D. Waldram, M Five-branes wrapped on supersymmetric cycles, Phys. Rev. D 63 (2001) 126001 [hep-th/0012195] [INSPIRE].ADSGoogle Scholar
  16. [16]
    J.P. Gauntlett and N. Kim, M five-branes wrapped on supersymmetric cycles. 2., Phys. Rev. D 65 (2002) 086003 [hep-th/0109039] [INSPIRE].
  17. [17]
    F. Benini and N. Bobev, Exact two-dimensional superconformal R-symmetry and c-extremization, Phys. Rev. Lett. 110 (2013) 061601 [arXiv:1211.4030] [INSPIRE].
  18. [18]
    F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, JHEP 06 (2013) 005 [arXiv:1302.4451] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  19. [19]
    P. Karndumri and E. Ó Colgáin, 3D supergravity from wrapped M5-branes, JHEP 03 (2016) 188 [arXiv:1508.00963] [INSPIRE].
  20. [20]
    I. Bah and V. Stylianou, Gravity duals of N = (0, 2) SCFTs from M5-branes, arXiv:1508.04135 [INSPIRE].
  21. [21]
    M. Bershadsky, C. Vafa and V. Sadov, D-branes and topological field theories, Nucl. Phys. B 463 (1996) 420 [hep-th/9511222] [INSPIRE].
  22. [22]
    M.T. Anderson, C. Beem, N. Bobev and L. Rastelli, Holographic Uniformization, Commun. Math. Phys. 318 (2013) 429 [arXiv:1109.3724] [INSPIRE].
  23. [23]
    D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
  26. [26]
    N. Bobev and P.M. Crichigno, Universal RG Flows Across Dimensions and Holography, JHEP 12 (2017) 065 [arXiv:1708.05052] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    K. Becker, M. Becker and A. Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456 (1995) 130 [hep-th/9507158] [INSPIRE].
  28. [28]
    K. Becker, M. Becker, D.R. Morrison, H. Ooguri, Y. Oz and Z. Yin, Supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau 4 folds, Nucl. Phys. B 480 (1996) 225 [hep-th/9608116] [INSPIRE].
  29. [29]
    J.P. Gauntlett, N.D. Lambert and P.C. West, Branes and calibrated geometries, Commun. Math. Phys. 202 (1999) 571 [hep-th/9803216] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    G.W. Gibbons and G. Papadopoulos, Calibrations and intersecting branes, Commun. Math. Phys. 202 (1999) 593 [hep-th/9803163] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    J.P. Gauntlett, Branes, calibrations and supergravity, in Strings and geometry. Proceedings, Summer School, Cambridge, U.K., March 24 - April 20, 2002, pp. 79-126, hep-th/0305074 [INSPIRE].
  32. [32]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    J.P. Gauntlett, O.A.P. Mac Conamhna, T. Mateos and D. Waldram, AdS spacetimes from wrapped M5 branes, JHEP 11 (2006) 053 [hep-th/0605146] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    P. Figueras, O.A.P. Mac Conamhna and E. Ó Colgáin, Global geometry of the supersymmetric AdS 3 /CF T 2 correspondence in M-theory, Phys. Rev. D 76 (2007) 046007 [hep-th/0703275] [INSPIRE].
  35. [35]
    E. Witten, Some comments on string dynamics, in Future perspectives in string theory. Proceedings, Conference, Strings’95, Los Angeles, U.S.A., March 13-18, 1995, pp. 501-523, hep-th/9507121 [INSPIRE].
  36. [36]
    N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].
  37. [37]
    M. Berkooz, A supergravity dual of a (1,0) field theory in six-dimensions, Phys. Lett. B 437 (1998) 315 [hep-th/9802195] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    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
  40. [40]
    A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, arXiv:1306.4320 [INSPIRE].
  41. [41]
    T. Dimofte, 3d Superconformal Theories from Three-Manifolds, in New Dualities of Supersymmetric Gauge Theories, J. Teschner ed., (2016) pp. 339, [arXiv:1412.7129].
  42. [42]
    M. Pernici, K. Pilch and P. van Nieuwenhuizen, Gauged Maximally Extended Supergravity in Seven-dimensions, Phys. Lett. B 143 (1984) 103 [INSPIRE].
  43. [43]
    H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistent nonlinear K K reduction of 11-d supergravity on AdS 7 × S 4 and selfduality in odd dimensions, Phys. Lett. B 469 (1999) 96 [hep-th/9905075] [INSPIRE].
  44. [44]
    H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistency of the AdS 7 × S 4 reduction and the origin of selfduality in odd dimensions, Nucl. Phys. B 581 (2000) 179 [hep-th/9911238] [INSPIRE].
  45. [45]
    M. Cvetič et al., Embedding AdS black holes in ten-dimensions and eleven-dimensions, Nucl. Phys. B 558 (1999) 96 [hep-th/9903214] [INSPIRE].
  46. [46]
    M. Fluder, work in progress.Google Scholar
  47. [47]
    B.S. Acharya, J.P. Gauntlett and N. Kim, Five-branes wrapped on associative three cycles, Phys. Rev. D 63 (2001) 106003 [hep-th/0011190] [INSPIRE].
  48. [48]
    I. Bah, M. Gabella and N. Halmagyi, BPS M5-branes as Defects for the 3d-3d Correspondence, JHEP 11 (2014) 112 [arXiv:1407.0403] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  49. [49]
    M. Fluder, unpublished notes.Google Scholar
  50. [50]
    J. Song and B. Weinkove, Lecture notes on the Kähler-Ricci flow, arXiv:1212.3653.
  51. [51]
    H.-D. Cao, Deformation of kähler matrics to kähler-einstein metrics on compact kähler manifolds, Invent. Math. 81 (1985) 359.ADSMathSciNetCrossRefMATHGoogle Scholar
  52. [52]
    S.-T. Yau, On the ricci curvature of a compact kähler manifold and the complex monge-ampére equation, I, Commun. Pure Appl. Math. 31 (1978) 339.CrossRefMATHGoogle Scholar
  53. [53]
    A. Futaki, An Obstruction to the Existence of Einstein Kähler Metrics, Invent. Math. 73 (1983) 437.ADSMathSciNetCrossRefMATHGoogle Scholar
  54. [54]
    G. Tian, Kähler-einstein metrics with positive scalar curvature, Invent. Math. 130 (1997) 1.ADSMathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    B.S. Acharya and S. Gukov, M theory and singularities of exceptional holonomy manifolds, Phys. Rept. 392 (2004) 121 [hep-th/0409191] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  56. [56]
    M. Itoh, Moduli of half conformally flat structures, Math. Ann. 296 (1993) 687.MathSciNetCrossRefMATHGoogle Scholar
  57. [57]
    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
  58. [58]
    M. Fluder, \( 4d\mathcal{N}=1/2d \) Yang-Mills Duality in Holography, arXiv:1712.06596 [INSPIRE].
  59. [59]
    I. Bah, C. Beem, N. Bobev and B. Wecht, AdS/CFT Dual Pairs from M5-Branes on Riemann Surfaces, Phys. Rev. D 85 (2012) 121901 [arXiv:1112.5487] [INSPIRE].
  60. [60]
    I. Bah, C. Beem, N. Bobev and B. Wecht, Four-Dimensional SCFTs from M5-Branes, JHEP 06 (2012) 005 [arXiv:1203.0303] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  61. [61]
    M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lond. A 308 (1982) 523 [INSPIRE].
  62. [62]
    C. Beem and A. Gadde, The N = 1 superconformal index for class S fixed points, JHEP 04 (2014) 036 [arXiv:1212.1467] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Walter Burke Institute for Theoretical Physics, California Institute of TechnologyPasadenaU.S.A.

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