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Holographic renormalization as a canonical transformation. (English) Zbl 1294.81227
Summary: The gauge/string dualities have drawn attention to a class of variational problems on a boundary at infinity, which are not well defined unless a certain boundary term is added to the classical action. In the context of supergravity in asymptotically AdS spaces these problems are systematically addressed by the method of holographic renormalization. We argue that this class of a priori ill defined variational problems extends far beyond the realm of holographic dualities. As we show, exactly the same issues arise in gravity in non asymptotically AdS spaces, in point particles with certain unbounded from below potentials, and even fundamental strings in flat or AdS backgrounds. We show that the variational problem in all such cases can be made well defined by the following procedure, which is intrinsic to the system in question and does not rely on the existence of a holographically dual theory: (i) The first step is the construction of the space of the most general asymptotic solutions of the classical equations of motion that inherits a well defined symplectic form from that on phase space. The requirement of a well defined symplectic form is essential and often leads to a necessary repackaging of the degrees of freedom. (ii) Once the space of asymptotic solutions has been constructed in terms of the correct degrees of freedom, then there exists a boundary term that is obtained as a certain solution of the Hamilton-Jacobi equation which simultaneously makes the variational problem well defined and preserves the symplectic form. This procedure is identical to holographic renormalization in the case of asymptotically AdS gravity, but it is applicable to any Hamiltonian system.

##### MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81T13 Yang-Mills and other gauge theories in quantum field theory 81V17 Gravitational interaction in quantum theory 81T20 Quantum field theory on curved space or space-time backgrounds 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 83E50 Supergravity 81T17 Renormalization group methods applied to problems in quantum field theory 53D05 Symplectic manifolds (general theory) 53D22 Canonical transformations in symplectic and contact geometry
##### Keywords:
 [1] Henningson, M.; Skenderis, K., The holographic Weyl anomaly, JHEP, 07, 023, (1998) [2] Balasubramanian, V.; Kraus, P., A stress tensor for anti-de Sitter gravity, Commun. Math. Phys., 208, 413, (1999) [3] Emparan, R.; Johnson, CV; Myers, RC, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev., D 60, 104001, (1999) [4] Kraus, P.; Larsen, F.; Siebelink, R., The gravitational action in asymptotically AdS and flat spacetimes, Nucl. Phys., B 563, 259, (1999) [5] Boer, J.; Verlinde, EP; Verlinde, HL, On the holographic renormalization group, JHEP, 08, 003, (2000) [6] Haro, S.; Solodukhin, SN; Skenderis, K., Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence, Commun. Math. Phys., 217, 595, (2001) [7] Bianchi, M.; Freedman, DZ; Skenderis, K., Holographic renormalization, Nucl. Phys., B 631, 159, (2002) [8] Martelli, D.; Mueck, W., Holographic renormalization and Ward identities with the Hamilton-Jacobi method, Nucl. Phys., B 654, 248, (2003) [9] I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, hep-th/0404176 [SPIRES]. [10] Hartle, JB; Hawking, SW, Wave function of the universe, Phys. Rev., D 28, 2960, (1983) [11] Seiberg, N., Notes on quantum Liouville theory and quantum gravity, Prog. Theor. Phys. Suppl., 102, 319, (1990) [12] I. Papadimitriou, Holographic renormalization for asymptotically flat gravity, in preparation. [13] I. Papadimitriou, Holographic renormalization for Improved Holographic QCD, in preparation. [14] Skenderis, K.; Taylor, M., Kaluza-Klein holography, JHEP, 05, 057, (2006) [15] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Selfadjointness, Academic Press, New York U.S.A (1975). [16] Carreau, M.; Farhi, E.; Gutmann, S.; Mende, PF, The functional integral for quantum systems with Hamiltonians unbounded from below, Ann. Phys., 204, 186, (1990) [17] Papadimitriou, I.; Skenderis, K., Thermodynamics of asymptotically locally AdS spacetimes, JHEP, 08, 004, (2005) [18] Breitenlohner, P.; Freedman, DZ, Stability in gauged extended supergravity, Ann. Phys., 144, 249, (1982) [19] Klebanov, IR; Witten, E., AdS/CFT correspondence and symmetry breaking, Nucl. Phys., B 556, 89, (1999) [20] E. Witten, Multi-trace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [SPIRES]. [21] E. Witten, SL(2$$,$$ Z) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [SPIRES]. [22] Witten, E., Anti-de Sitter space and holography, Adv. Theor. Math. Phys., 2, 253, (1998) [23] Rey, S-J; Theisen, S.; Yee, J-T, Wilson-Polyakov loop at finite temperature in large-N gauge theory and anti-de Sitter supergravity, Nucl. Phys., B 527, 171, (1998) [24] Drukker, N.; Gross, DJ; Ooguri, H., Wilson loops and minimal surfaces, Phys. Rev., D 60, 125006, (1999) [25] Alday, LF; Maldacena, JM, Gluon scattering amplitudes at strong coupling, JHEP, 06, 064, (2007) [26] Skenderis, K., Lecture notes on holographic renormalization, Class. Quant. Grav., 19, 5849, (2002) [27] Lee, J.; Wald, RM, Local symmetries and constraints, J. Math. Phys., 31, 725, (1990) [28] I. Papadimitriou, Holographic renormalization made simple: An example, prepared for International School of Subnuclear Physics. 41st Course: From Quarks to Black Holes - Progress in Understanding the Logic of Nature, Erice, Sicily Italy, August 29 - September 7 2003. [29] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical tables, ninth Dover printing, tenth GPO printing ed., Dover New York (1964). [30] Boer, J.; Solodukhin, SN, A holographic reduction of Minkowski space-time, Nucl. Phys., B 665, 545, (2003) [31] Arnowitt, RL; Deser, S.; Misner, CW, Canonical variables for general relativity, Phys. Rev., 117, 1595, (1960) [32] Gibbons, GW; Hawking, SW, Action integrals and partition functions in quantum gravity, Phys. Rev., D 15, 2752, (1977) [33] Olea, R., Regularization of odd-dimensional AdS gravity: kounterterms, JHEP, 04, 073, (2007) [34] Mansi, DS; Petkou, AC; Tagliabue, G., Gravity in the 3+1-split formalism I: holography as an initial value problem, Class. Quant. Grav., 26, 045008, (2009) [35] C. Fefferman and C. R. Graham, Conformal Invariants, Elie Cartan et les Mathématiques d’aujourd’hui, Astérisque (1985). [36] Imbimbo, C.; Schwimmer, A.; Theisen, S.; Yankielowicz, S., Diffeomorphisms and holographic anomalies, Class. Quant. Grav., 17, 1129, (2000) [37] R.R. Penrose and W. Rindler, Spinors and space-time. Volume II: Spinor and twistor methods in space-time geometry. Cambridge University Press, Cambridge U.K. (1986). [38] Brown, J.; Henneaux, M., Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys., 104, 207, (1986) [39] J. Polchinski, String theory. Volume I: An introduction to the bosonic string Cambridge University Press, Cambridge U.K. (1998).