Abstract
We propose a method for determining the exact correspondence between the Wilsonian cut-off scale on the boundary and its holographically dual bulk theory. We systematically construct the multi-trace Wilsonian effective action from holographic renormalisation and evolve it by integrating out the asymptotically Anti-de Sitter bulk geometry with scalar probes. The Wilsonian nature of the effective action is shown by proving that it must be either double-trace, closing in on itself under successive integrations, or have an infinite series of multi-trace terms. Focusing on composite scalar operator renormalisation, we relate the Callan-Symanzik equation, the flow of the scalar anomalous dimension and the multi-trace beta functions to their dual RG flows in the bulk. Establishing physical renormalisation conditions on the behaviour of the large-N anomalous dimension then enables us to extract the energy scales. Examples of pure AdS, GPPZ flow, black brane in AdS, M2 and M5 branes are studied before we generalise our results to arbitrary numbers of mass and thermal deformations of an ultra-violet CFT. Relations between the undeformed Wilsonian cut-off, deformation scales and the deformed Wilsonian cut-off are discussed, as is phenomenology of each considered background. We see how a mass gap, the emergent infra-red CFT scaling, etc. arise in different effective theories. We also argue that these results can have alternative interpretations through the flow of the conformal anomaly orthe Ricci scalar curvature of boundary branes. They show consistency with the c-theorem.
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References
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1133 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
L. Susskind and E. Witten, The holographic bound in Anti-de Sitter space, hep-th/9805114 [INSPIRE].
A.W. Peet and J. Polchinski, UV/IR relations in AdS dynamics, Phys. Rev. D 59 (1999) 065011 [hep-th/9809022] [INSPIRE].
E.T. Akhmedov, A remark on the AdS/CFT correspondence and the renormalization group flow, Phys. Lett. B 442 (1998) 152 [hep-th/9806217] [INSPIRE].
L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, Novel local CFT and exact results on perturbations of N = 4 super Yang-Mills from AdS dynamics, JHEP 12 (1998) 022 [hep-th/9810126] [INSPIRE].
J. Distler and F. Zamora, Nonsupersymmetric conformal field theories from stable Anti-de Sitter spaces, Adv. Theor. Math. Phys. 2 (1999) 1405 [hep-th/9810206] [INSPIRE].
V. Balasubramanian and P. Kraus, Space-time and the holographic renormalization group, Phys. Rev. Lett. 83 (1999) 3605 [hep-th/9903190] [INSPIRE].
J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012] [INSPIRE].
V. Balasubramanian, P. Kraus, A.E. Lawrence and S.P. Trivedi, Holographic probes of Anti-de Sitter space-times, Phys. Rev. D 59 (1999) 104021 [hep-th/9808017] [INSPIRE].
D. Freedman, S. Gubser, K. Pilch and N. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].
J. de Boer, The holographic renormalization group, Fortsch. Phys. 49 (2001) 339 [hep-th/0101026] [INSPIRE].
K. Wilson and J.B. Kogut, The renormalization group and the ϵ-expansion, Phys. Rept. 12 (1974) 75 [INSPIRE].
F.J. Wegner and A. Houghton, Renormalization group equation for critical phenomena, Phys. Rev. A 8 (1973) 401 [INSPIRE].
K. Wilson, The renormalization group and critical phenomena, Rev. Mod. Phys. 55 (1983) 583 [INSPIRE].
J. Polchinski, Renormalization and effective lagrangians, Nucl. Phys. B 231 (1984) 269 [INSPIRE].
J. Polonyi, Lectures on the functional renormalization group method, Central Eur. J. Phys. 1 (2003) 1 [hep-th/0110026] [INSPIRE].
T. Faulkner, H. Liu and M. Rangamani, Integrating out geometry: holographic Wilsonian RG and the membrane paradigm, JHEP 08 (2011) 051 [arXiv:1010.4036] [INSPIRE].
I. Heemskerk and J. Polchinski, Holographic and Wilsonian renormalization groups, JHEP 06 (2011) 031 [arXiv:1010.1264] [INSPIRE].
S.-J. Sin and Y. Zhou, Holographic Wilsonian RG flow and sliding membrane paradigm, JHEP 05 (2011) 030 [arXiv:1102.4477] [INSPIRE].
D. Harlow and D. Stanford, Operator dictionaries and wave functions in AdS/CFT and dS/CFT, arXiv:1104.2621 [INSPIRE].
J. Fan, Effective AdS/renormalized CFT, JHEP 09 (2011) 136 [arXiv:1105.0678] [INSPIRE].
E. Akhmedov, I. Gahramanov and E. Musaev, Hints on integrability in the Wilsonian/holographic renormalization group, JETP Lett. 93 (2011) 545 [arXiv:1006.1970] [INSPIRE].
D. Radicevic, Connecting the holographic and Wilsonian renormalization groups, JHEP 12 (2011) 023 [arXiv:1105.5825] [INSPIRE].
D. Elander, H. Isono and G. Mandal, Holographic Wilsonian flows and emergent fermions in extremal charged black holes, JHEP 11 (2011) 155 [arXiv:1109.3366] [INSPIRE].
J.N. Laia and D. Tong, Flowing between fermionic fixed points, JHEP 11 (2011) 131 [arXiv:1108.2216] [INSPIRE].
I. Bredberg, C. Keeler, V. Lysov and A. Strominger, Wilsonian approach to fluid/gravity duality, JHEP 03 (2011) 141 [arXiv:1006.1902] [INSPIRE].
S.-S. Lee, Holographic description of quantum field theory, Nucl. Phys. B 832 (2010) 567 [arXiv:0912.5223] [INSPIRE].
S.-S. Lee, Holographic description of large-N gauge theory, Nucl. Phys. B 851 (2011) 143 [arXiv:1011.1474] [INSPIRE].
M.F. Paulos, Holographic phase space: c-functions and black holes as renormalization group flows, JHEP 05 (2011) 043 [arXiv:1101.5993] [INSPIRE].
S. Kuperstein and A. Mukhopadhyay, The unconditional RG flow of the relativistic holographic fluid, JHEP 11 (2011) 130 [arXiv:1105.4530] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
K. Skenderis and B.C. van Rees, Real-time gauge/gravity duality: prescription, renormalization and examples, JHEP 05 (2009) 085 [arXiv:0812.2909] [INSPIRE].
E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].
I. Papadimitriou, Multi-trace deformations in AdS/CFT: exploring the vacuum structure of the deformed CFT, JHEP 05 (2007) 075 [hep-th/0703152] [INSPIRE].
L. Vecchi, Multitrace deformations, Gamow states and stability of AdS/CFT, JHEP 04 (2011) 056 [arXiv:1005.4921] [INSPIRE].
E.T. Akhmedov, Notes on multitrace operators and holographic renormalization group, hep-th/0202055 [INSPIRE].
W. Mueck, An improved correspondence formula for AdS/CFT with multitrace operators, Phys. Lett. B 531 (2002) 301 [hep-th/0201100] [INSPIRE].
P. Minces, Multitrace operators and the generalized AdS/CFT prescription, Phys. Rev. D 68 (2003) 024027 [hep-th/0201172] [INSPIRE].
T. Hartman and L. Rastelli, Double-trace deformations, mixed boundary conditions and functional determinants in AdS/CFT, JHEP 01 (2008) 019 [hep-th/0602106] [INSPIRE].
S.S. Gubser and I.R. Klebanov, A universal result on central charges in the presence of double trace deformations, Nucl. Phys. B 656 (2003) 23 [hep-th/0212138] [INSPIRE].
P. Mansfield and D. Nolland, One loop conformal anomalies from AdS/CFT in the Schrödinger representation, JHEP 07 (1999) 028 [hep-th/9906054] [INSPIRE].
D. Brattan, J. Camps, R. Loganayagam and M. Rangamani, CFT dual of the AdS Dirichlet problem: fluid/gravity on cut-off surfaces, JHEP 12 (2011) 090 [arXiv:1106.2577] [INSPIRE].
I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, hep-th/0404176 [INSPIRE].
I. Papadimitriou and K. Skenderis, Correlation functions in holographic RG flows, JHEP 10 (2004) 075 [hep-th/0407071] [INSPIRE].
N. Iqbal and H. Liu, Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm, Phys. Rev. D 79 (2009) 025023 [arXiv:0809.3808] [INSPIRE].
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
C. Herzog and D. Son, Schwinger-Keldysh propagators from AdS/CFT correspondence, JHEP 03 (2003) 046 [hep-th/0212072] [INSPIRE].
C. Fefferman and C. Robin Graham, Conformal invariants, in Elie Cartan et les Mathématiques d’aujourd’hui Astérisque (1985) 95.
L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, The supergravity dual of N = 1 super Yang-Mills theory, Nucl. Phys. B 569 (2000) 451 [hep-th/9909047] [INSPIRE].
M. Porrati and A. Starinets, On the canonical c function in 4D field theories possessing supergravity duals, Phys. Lett. B 498 (2001) 285 [hep-th/0009227] [INSPIRE].
M. Porrati and A. Starinets, Holographic duals of 4D field theories, hep-th/0009198 [INSPIRE].
E. Pomoni and L. Rastelli, Large-N field theory and AdS tachyons, JHEP 04 (2009) 020 [arXiv:0805.2261] [INSPIRE].
L. Vecchi, The conformal window of deformed CFT’s in the planar limit, Phys. Rev. D 82 (2010) 045013 [arXiv:1004.2063] [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].
C.G. Callan Jr., S.R. Coleman and R. Jackiw, A new improved energy-momentum tensor, Annals Phys. 59 (1970) 42 [INSPIRE].
K. Higashijima and E. Itou, Unitarity bound of the wave function renormalization constant, Prog. Theor. Phys. 110 (2003) 107 [hep-th/0304047] [INSPIRE].
M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, Addison-Wesley, U.S.A. (1995).
T. Faulkner, H. Liu, J. McGreevy and D. Vegh, Emergent quantum criticality, Fermi surfaces and AdS 2, Phys. Rev. D 83 (2011) 125002 [arXiv:0907.2694] [INSPIRE].
H. Lü, J.-w. Mei, C. Pope and J.F. Vazquez-Poritz, Extremal static AdS black hole/CFT correspondence in gauged supergravities, Phys. Lett. B 673 (2009) 77 [arXiv:0901.1677] [INSPIRE].
S. Carlip, Black hole entropy from conformal field theory in any dimension, Phys. Rev. Lett. 82 (1999) 2828 [hep-th/9812013] [INSPIRE].
S.N. Solodukhin, Conformal description of horizon’s states, Phys. Lett. B 454 (1999) 213 [hep-th/9812056] [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
A. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [Pisma Zh. Eksp. Teor. Fiz. 43 (1986) 565] [INSPIRE].
J.L. Cardy, Is there a c theorem in four-dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].
Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].
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Grozdanov, S. Wilsonian renormalisation and the exact cut-off scale from holographic duality. J. High Energ. Phys. 2012, 79 (2012). https://doi.org/10.1007/JHEP06(2012)079
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DOI: https://doi.org/10.1007/JHEP06(2012)079