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Prog. Theor. Phys. Vol. 109 No. 4 (2003) pp. 489-562

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Invited Papers

Holographic Renormalization Group

Masafumi Fukuma,1 So Matsuura2 and Tadakatsu Sakai3

1Department of Physics, Kyoto University, Kyoto 606-8502, Japan
2Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
3Raymond and Beverly Sackler Faculty of Exact Sciences,
School of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv 69978, Israel

(Received December 20, 2002)

Abstract:

The holographic renormalization group (RG) is reviewed in a self-contained manner. The holographic RG is based on the idea that the radial coordinate of a space-time with asymptotically AdS geometry can be identified with the RG flow parameter of boundary field theory. After briefly discussing basic aspects of the AdS/CFT correspondence, we explain how the concept of the holographic RG emerges from this correspondence. We formulate the holographic RG on the basis of the Hamilton-Jacobi equations for bulk systems of gravity and scalar fields, as introduced by de Boer, Verlinde and Verlinde. We then show that the equations can be solved with a derivative expansion by carefully extracting local counterterms from the generating functional of the boundary field theory. The calculational methods used to obtain the Weyl anomaly and scaling dimensions are presented and applied to the RG flow from the N = 4 SYM to an N = 1 superconformal fixed point discovered by Leigh and Strassler. We further discuss the relation between the holographic RG and the noncritical string theory and show that the structure of the holographic RG should persist beyond the supergravity approximation as a consequence of the renormalizability of the nonlinear σ-model action of noncritical strings. As a check, we investigate the holographic RG structure of higher-derivative gravity systems. We show that such systems can also be analyzed based on the Hamilton-Jacobi equations and that the behavior of bulk fields are determined solely by their boundary values. We also point out that higher-derivative gravity systems give rise to new multicritical points in the parameter space of boundary field theories.


URL : http://ptp.ipap.jp/link?PTP/109/489/
DOI : 10.1143/PTP.109.489

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References:

  1. Y. Nambu, in Symmetries and quark models, ed. R. Chand (Tordon and Breach, 1970), p. 269.
    H. Nielsen, in the 15th International Conference on High Energy Physics (Kiev, 1970).
    L. Susskind, Nuovo Cim. A 69 (1970), 457.
  2. G. 't Hooft, Nucl. Phys. B 72 (1974), 461[CrossRef].
  3. K. G. Wilson, Phys. Rev. D 10 (1974), 2445[APS].
  4. R. Gopakumar and C. Vafa, Adv. Theor. Math. Phys. 3 (1999), 1415, hep-th/9811131[e-print arXiv].
  5. J. Maldacena, Adv. Theor. Math. Phys. 2 (1998), 231, hep-th/9711200[e-print arXiv].
  6. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428 (1998), 105, [CrossRef] hep-th/9802109[e-print arXiv].
  7. E. Witten, Adv. Theor. Math. Phys. 2 (1998), 253, hep-th/9802150[e-print arXiv].
  8. O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, hep-th/9905111[e-print arXiv], and references therein.
  9. G. T. Horowitz and A. Strominger, Nucl. Phys. B 360 (1991), 197[CrossRef].
  10. L. Susskind and E. Witten, hep-th/9805114[e-print arXiv].
  11. E. T. Akhmedov, Phys. Lett. B 442 (1998), 152, [CrossRef] hep-th/9806217[e-print arXiv].
  12. E. Alvarez and C. Gomez, Nucl. Phys. B 541 (1999), 441, [CrossRef] hep-th/9807226[e-print arXiv].
  13. L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, J. High Energy Phys. 12 (1998), 022, [IoP STACKS] hep-th/9810126[e-print arXiv].
  14. M. Porrati and A. Starinets, Phys. Lett. B 454 (1999), 77, [CrossRef] hep-th/9903085[e-print arXiv].
  15. V. Balasubramanian and P. Kraus, Phys. Rev. Lett. 83 (1999), 3605, [APS] hep-th/9903190[e-print arXiv].
  16. D. Z. Freedman, S. S. Gubser, K. Pilch and N. P. Warner, Adv. Theor. Math. Phys. 3 (1999), 363, hep-th/9904017[e-print arXiv].
  17. L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni Nucl. Phys. B 569 (2000), 451, [CrossRef] hep-th/9909047[e-print arXiv].
  18. K. Skenderis and P. K. Townsend, Phys. Lett. B 468 (1999), 46, [CrossRef] hep-th/9909070[e-print arXiv].
  19. O. DeWolfe, D. Z. Freedman, S. S. Gubser and A. Karch, Phys. Rev. D 62 (2000), 046008, [APS] hep-th/9909134[e-print arXiv].
  20. V. Sahakian, Phys. Rev. D 62 (2000), 126011, [APS] hep-th/9910099[e-print arXiv].
  21. E. Alvarez and C. Gomez, Phys. Lett. B 476 (2000), 411, [CrossRef] hep-th/0001016[e-print arXiv].
  22. S. Nojiri, S. D. Odintsov and S. Zerbini, Phys. Rev. D 62 (2000), 064006, [APS] hep-th/0001192[e-print arXiv].
  23. M. Li, Nucl. Phys. B 579 (2000), 525, [CrossRef] hep-th/0001193[e-print arXiv].
  24. V. Sahakian, J. High Energy Phys. 05 (2000), 011, [IoP STACKS] hep-th/0002126[e-print arXiv].
  25. O. DeWolfe and D. Z. Freedman, hep-th/0002226[e-print arXiv].
  26. V. Balasubramanian, E. G. Gimon and D. Minic, J. High Energy Phys. 05 (2000), 014, [IoP STACKS] hep-th/0003147[e-print arXiv].
  27. C. V. Johnson, K. J. Lovis and D. C. Page, J. High Energy Phys. 05 (2001), 036, [IoP STACKS] hep-th/0011166[e-print arXiv].
  28. J. Erdmenger, Phys. Rev. D 64 (2001), 085012, [APS] hep-th/0103219[e-print arXiv].
  29. S. Yamaguchi, J. High Energy Phys. 10 (2002), 002, [IoP STACKS] hep-th/0207171[e-print arXiv].
  30. J. de Boer, E. Verlinde and H. Verlinde, hep-th/9912012[e-print arXiv].
  31. J. de Boer, Fortschr. Phys. 49 (2001), 339, hep-th/0101026[e-print arXiv].
  32. A. M. Polyakov, hep-th/9304146[e-print arXiv].
  33. M. Henningson and K. Skenderis, J. High Energy Phys. 07 (1998), 023, [IoP STACKS] hep-th/9806087[e-print arXiv].
  34. V. Balasubramanian and P. Kraus, Commun. Math. Phys. 208 (1999), 413, [CrossRef] hep-th/9902121[e-print arXiv].
  35. S. de Haro, K. Skenderis and S. Solodukhin, hep-th/0002230[e-print arXiv].
  36. M. J. Duff, Class. Quantum Grav. 11 (1994), 1387, [IoP STACKS] hep-th/9308075[e-print arXiv].
  37. M. Fukuma, S. Matsuura and T. Sakai, Prog. Theor. Phys. 104 (2000), 1089, [IPAP] hep-th/0007062[e-print arXiv].
  38. M. Fukuma and T. Sakai, Mod. Phys. Lett. A 15 (2000), 1703, hep-th/0007200[e-print arXiv].
  39. M. Fukuma, S. Matsuura and T. Sakai, Prog. Theor. Phys. 105 (2001), 1017, [IPAP] hep-th/0103187[e-print arXiv].
  40. M. Fukuma and S. Matsuura, Prog. Theor. Phys. 107 (2002), 1085, [IPAP] hep-th/0112037[e-print arXiv].
  41. A. Fayyazuddin and M. Spalinski, Nucl. Phys. B 535 (1998), 219, [CrossRef] hep-th/9805096[e-print arXiv].
    O. Aharony, A. Fayyazuddin and J. Maldacena, J. High Energy Phys. 07 (1998), 013, [IoP STACKS] hep-th/9806159[e-print arXiv].
  42. M. Blau, K. S. Narain and E. Gava, J. High Energy Phys. 09 (1999), 018, [IoP STACKS] hep-th/9904179[e-print arXiv].
  43. O. Aharony, J. Pawelczyk, S. Theisen and S. Yankielowicz, Phys. Rev. D 60 (1999), 066001, [APS] hep-th/9901134[e-print arXiv].
  44. V. L. Campos, G. Ferretti, H. Larsson, D. Martelli and B. E. Nilsson, J. High Energy Phys. 06 (2000), 023, [IoP STACKS] hep-th/0003151[e-print arXiv].
  45. S. Corley, Phys. Lett. B 484 (2000), 141, [CrossRef] hep-th/0004030[e-print arXiv].
  46. J. Kalkkinen and D. Martelli, Nucl. Phys. B 596 (2001), 415, [CrossRef] hep-th/0007234[e-print arXiv].
  47. S. Nojiri, S. D. Odintsov and S. Ogushi, Phys. Lett. B 494 (2000), 318, [CrossRef] hep-th/0009015[e-print arXiv].
  48. N. Hambli, Phys. Rev. D 64 (2001), 024001, [APS] hep-th/0010054[e-print arXiv].
  49. S. Nojiri, S. D. Odintsov and S. Ogushi, Phys. Lett. B 500 (2001), 199, [CrossRef] hep-th/0011182[e-print arXiv].
  50. J. de Boer, Fortschr. Phys. 49 (2001), 339, hep-th/0101026[e-print arXiv].
  51. J. Kalkkinen, D. Martelli and W. Muck, J. High Energy Phys. 04 (2001), 036, [IoP STACKS] hep-th/0103111[e-print arXiv].
  52. S. Nojiri and S. D. Odintsov, Phys. Lett. B 519 (2001), 145, [CrossRef] hep-th/0106191[e-print arXiv].
  53. S. Nojiri and S. D. Odintsov, Phys. Lett. B 531 (2002), 143, [CrossRef] hep-th/0201210[e-print arXiv].
  54. D. Martelli and W. Muck, hep-th/0205061[e-print arXiv].
  55. R. G. Leigh and M. J. Strassler, Nucl. Phys. B 447 (1995), 95, [CrossRef] hep-th/9503121[e-print arXiv].
  56. S. Ferrara, C. Fronsdal and A. Zaffaroni, Nucl. Phys. B 532 (1998), 153, [CrossRef] hep-th/9802203[e-print arXiv].
  57. L. Andrianopoli and S. Ferrara, Phys. Lett. B 430 (1998), 248, [CrossRef] hep-th/9803171[e-print arXiv].
  58. S. Ferrara, M. A. Lledo and A. Zaffaroni, Phys. Rev. D 58 (1998), 105029, [APS] hep-th/9805082[e-print arXiv].
  59. M. F. Sohnius, Phys. Rep. 128 (1985), 39[CrossRef].
  60. O. Aharony, M. Berkooz and E. Silverstein, J. High Energy Phys. 08 (2001), 006, [IoP STACKS] hep-th/0105309[e-print arXiv].
  61. E. Witten, hep-th/0112258[e-print arXiv].
  62. M. Berkooz, A. Sever and A. Shomer, J. High Energy Phys. 05 (2002), 034, [IoP STACKS] hep-th/0112264[e-print arXiv].
  63. S. Minwalla, Adv. Theor. Math. Phys. 2 (1998), 781, hep-th/9712074[e-print arXiv].
  64. V. K. Dobrev and V. B. Petkova, Phys. Lett. B 162 (1985), 127[CrossRef].
  65. L. Andrianopoli and S. Ferrara, Phys. Lett. B 430 (1998), 248, [CrossRef] hep-th/9803171[e-print arXiv].
  66. M. Gunaydin, D. Minic and M. Zagermann, Nucl. Phys. B 534 (1998), 96, [CrossRef] hep-th/9806042[e-print arXiv].
  67. S. Ferrara and A. Zaffaroni, hep-th/9807090[e-print arXiv].
  68. L. Andrianopoli and S. Ferrara, Lett. Math. Phys. 46 (1998), 265, hep-th/9807150[e-print arXiv].
  69. M. Gunaydin, D. Minic and M. Zagermann, hep-th/9810226[e-print arXiv].
  70. L. Andrianopoli and S. Ferrara, hep-th/9812067[e-print arXiv].
  71. P. S. Howe, K. S. Stelle and P. K. Townsend, Nucl. Phys. B 192 (1981), 332[CrossRef].
  72. P. S. Howe and P. C. West, Phys. Lett. B 389 (1996), 273, [CrossRef] hep-th/9607060[e-print arXiv]; hep-th/9611074[e-print arXiv]; Phys. Lett. B 400 (1997), 307, [CrossRef] hep-th/9611075[e-print arXiv].
  73. H. J. Kim, L. J. Romans and P. van Nieuwenhuizen, Phys. Rev. D 32 (1985), 389[APS].
  74. M. Günaydin and N. Marcus, Class. Quantum Grav. 2 (1985), L11[IoP STACKS].
  75. V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 229 (1983), 381[CrossRef].
  76. D. Anselmi, D. Z. Freedman, M. T. Grisaru and A. A. Johansen, Nucl. Phys. B 526 (1998), 543, [CrossRef] hep-th/9708042[e-print arXiv].
  77. A. Khavaev, K. Pilch and N. P. Warner, Phys. Lett. B 487 (2000), 14, [CrossRef] hep-th/9812035[e-print arXiv].
  78. K. Pilch and N. P. Warner, Adv. Theor. Math. Phys. 4 (2002), 627, hep-th/0006066[e-print arXiv].
  79. M. Berg and H. Samtleben, J. High Energy Phys. 05 (2002), 006, [IoP STACKS] hep-th/0112154[e-print arXiv].
  80. M. Berg and H. Samtleben, J. High Energy Phys. 12 (2002), 070, [IoP STACKS] hep-th/0209191[e-print arXiv].
  81. D. Berenstein, J. M. Maldacena and H. Nastase, J. High Energy Phys. 04 (2002), 013, [IoP STACKS] hep-th/0202021[e-print arXiv].
  82. M. Blau, J. Figueroa-O'Farrill, C. Hull and G. Papadopoulos, J. High Energy Phys. 01 (2002), 047, [IoP STACKS] hep-th/0110242[e-print arXiv]; Class. Quantum Grav. 19 (2002), L87, [IoP STACKS] hep-th/0201081[e-print arXiv].
    M. Blau, J. Figueroa-O'Farrill and G. Papadopoulos, Class. Quantum Grav. 19 (2002), 4753, [IoP STACKS] hep-th/0202111[e-print arXiv].
  83. R. R. Metsaev, Nucl. Phys. B 625 (2002), 70, [CrossRef] hep-th/0112044[e-print arXiv].
  84. R. Corrado, N. Halmagyi, K. D. Kennaway and N. P. Warner, hep-th/0205314[e-print arXiv].
    E. G. Gimon, L. A. Pando Zayas and J. Sonnenschein, hep-th/0206033[e-print arXiv].
    D. Brecher, C. V. Johnson, K. J. Lovis and R. C. Myers, J. High Energy Phys. 10 (2002), 008, [IoP STACKS] hep-th/0206045[e-print arXiv].
  85. Y. Oz and T. Sakai, Phys. Lett. B 544 (2002), 321, [CrossRef] hep-th/0207223[e-print arXiv].
  86. A. B. Zamolodchikov, JETP Lett. 43 (1986), 730 [Pisma Zh. Eksp. Teor. Fiz. 43 (1986), 565].
  87. D. Anselmi, Ann. of Phys. 276 (1999), 361, [CrossRef] hep-th/9903059[e-print arXiv].
  88. G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 (1977), 2752[APS].
  89. C. R. Graham and J. M. Lee, Adv. Math. 87 (1991), 186.
  90. M. Green, J. Schwarz and E. Witten, Superstring Theory (Cambridge University Press, New York, 1987).
  91. S. Nojiri and S. Odintsov, Phys. Lett. B 444 (1998), 92, [CrossRef] hep-th/9810008[e-print arXiv].
    S. Nojiri, S. Odintsov and S. Ogushi, hep-th/9912191[e-print arXiv]; hep-th/0001122[e-print arXiv].
  92. K. G. Wilson and J. B. Kogut, Phys. Rep. 12 (1974), 75[CrossRef].
  93. A. Polyakov, Phys. Lett. B 103 (1981), 207[CrossRef]; Phys. Lett. B 103 (1981), 211[CrossRef].
    V. Knizhnik, A. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A 3 (1988), 819.
  94. F. David, Mod. Phys. Lett. A 3 (1988), 1651.
    J. Distler and H. Kawai, Nucl. Phys. B 321 (1989), 509[CrossRef].
  95. N. Seiberg, Prog. Theor. Phys. Suppl. No. 102 (1990), 319[IPAP].
  96. R. Myer, Phys. Lett. B 199 (1987), 371 [CrossRef]
  97. A. Dhar and S. Wadia, Nucl. Phys. B 590 (2000), 261, [CrossRef] hep-th/0006043[e-print arXiv].
  98. S. Nojiri and S. D. Odintsov, Phys. Lett. B 484 (2000), 119, [CrossRef] hep-th/0004097[e-print arXiv].
  99. R. C. Myers, Phys. Rev. D 36 (1987), 392[APS].
  100. S. Nojiri and S. D. Odintsov, J. High Energy Phys. 07 (2000), 049, [IoP STACKS] hep-th/0006232[e-print arXiv].
    S. Nojiri, S. D. Odintsov and S. Ogushi, hep-th/0010004[e-print arXiv]; hep-th/0105117[e-print arXiv].
  101. S. Nojiri and S. D. Odintsov, Int. J. Mod. Phys. A 15 (2000), 413, hep-th/9903033[e-print arXiv]; Phys. Rev. D 62 (2000), 064018, [APS] hep-th/9911152[e-print arXiv].
  102. J. Polchinski, String Theory, Vol. II (Cambridge University Press, 1998).
  103. M. Fukuma and S. Matsuura, Prog. Theor. Phys. 108 (2002), 375, [IPAP] hep-th/0204257[e-print arXiv].
  104. S. Kamefuchi, L. O'Raifeartaigh and A. Salam, Nucl. Phys. 28 (1961), 529[Elsevier].
  105. D. J. Gross and E. Witten, Nucl. Phys. B 277 (1986), 1[CrossRef].
  106. J. I. Latorre and T. R. Morris, J. High Energy Phys. 11 (2000), 004, [IoP STACKS] hep-th/0008123[e-print arXiv].
  107. I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 550 (2002), 213, [CrossRef] hep-th/0210114[e-print arXiv].
  108. A. M. Polyakov, Nucl. Phys. B 164 (1980), 171[CrossRef].
  109. Y. Makeenko and A. A. Migdal, Nucl. Phys. B 188 (1981), 269[CrossRef].
  110. A. M. Polyakov, Nucl. Phys. B 486 (1997), 23, [CrossRef] hep-th/9607049[e-print arXiv].
  111. A. M. Polyakov, Nucl. Phys. B (Proc. Suppl.) 68 (1998), 1, [CrossRef] hep-th/9711002[e-print arXiv].
  112. A. M. Polyakov, Int. J. Mod. Phys. A 14 (1999), 645, hep-th/9809057[e-print arXiv].
  113. A. M. Polyakov and V. S. Rychkov, Nucl. Phys. B 581 (2000), 116, [CrossRef] hep-th/0002106[e-print arXiv].
  114. A. M. Polyakov and V. S. Rychkov, Nucl. Phys. B 594 (2001), 272, [CrossRef] hep-th/0005173[e-print arXiv].
  115. A. M. Polyakov, Phys. Atom. Nucl. 64 (2001), 540, hep-th/0006132[e-print arXiv].
  116. A. M. Polyakov, Int. J. Mod. Phys. A 17 (Suppl.) (2002), 119, hep-th/0110196[e-print arXiv].
  117. J. B. Kogut and L. Susskind, Phys. Rev. D 11 (1975), 395[APS].
  118. A. Santambrogio and D. Zanon, Phys. Lett. B 545 (2002), 425, [CrossRef] hep-th/0206079[e-print arXiv].
  119. Y. Oz and T. Sakai, hep-th/0208078[e-print arXiv].
  120. S. R. Das, C. Gomez and S. J. Rey, Phys. Rev. D 66 (2002), 046002, [APS] hep-th/0203164[e-print arXiv].
  121. R. G. Leigh, K. Okuyama and M. Rozali, Phys. Rev. D 66 (2002), 046004, [APS] hep-th/0204026[e-print arXiv].
  122. D. Berenstein and H. Nastase, hep-th/0205048[e-print arXiv].

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