Abstract
We investigate universal time-dependent exact deformations of Schrödinger geometry. We present 1) scale invariant but non-conformal deformation, 2) non-conformal but scale invariant deformation, and 3) both scale and conformal invariant deformation. All these solutions are universal in the sense that we could embed them in any supergravity constructions of the Schrödinger invariant geometry. We give a field theory interpretation of our time-dependent solutions. In particular, we argue that any time-dependent chemical potential can be treated exactly in our gravity dual approach.
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S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [SPIRES].
J. McGreevy, Holographic duality with a view toward many-body physics, arXiv:0909.0518 [SPIRES].
D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schroedinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [SPIRES].
K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [SPIRES].
C. Duval, G.W. Gibbons and P. Horvathy, Celestial Mechanics, Conformal Structures and Gravitational Waves, Phys. Rev. D 43 (1991) 3907 [hep-th/0512188] [SPIRES].
C. Duval, M. Hassaine and P.A. Horvathy, The geometry of Schródinger symmetry in gravity background/non-relativistic CFT, Annals Phys. 324 (2009) 1158 [arXiv:0809.3128] [SPIRES].
C.P. Herzog, M. Rangamani and S.F. Ross, Heating up Galilean holography, JHEP 11 (2008) 080 [arXiv:0807.1099] [SPIRES].
J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non- relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [SPIRES].
A. Adams, K. Balasubramanian and J. McGreevy, Hot Spacetimes for Cold Atoms, JHEP 11 (2008) 059 [arXiv:0807.1111] [SPIRES].
S.A. Hartnoll and K. Yoshida, Families of IIB duals for nonrelativistic CFTs, JHEP 12 (2008) 071 [arXiv:0810.0298] [SPIRES].
N. Bobev, A. Kundu and K. Pilch, Supersymmetric IIB Solutions with Schródinger Symmetry, JHEP 07 (2009) 107 [arXiv:0905.0673] [SPIRES].
A. Donos and J.P. Gauntlett, Solutions of type IIB and D = 11 supergravity with Schrödinger(z) symmetry, JHEP 07 (2009) 042 [arXiv:0905.1098] [SPIRES].
A. Donos and J.P. Gauntlett, Schrödinger invariant solutions of type IIB with enhanced supersymmetry, JHEP 10 (2009) 073 [arXiv:0907.1761] [SPIRES].
H. Ooguri and C.-S. Park, Supersymmetric non-relativistic geometries in M-theory, Nucl. Phys. B 824 (2010) 136 [arXiv:0905.1954] [SPIRES].
J. Jeong, H.-C. Kim, S. Lee, E.O. Colgain and H. Yavartanoo, Schrödinger invariant solutions of M-theory with Enhanced Supersymmetry, JHEP 03 (2010) 034 [arXiv:0911.5281] [SPIRES].
C.R. Hagen, Scale and conformal transformations in galilean-covariant field theory, Phys. Rev. D 5 (1972) 377 [SPIRES].
U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta 45 (1972) 802 [SPIRES].
S. Schäfer-Nameki, M. Yamazaki and K. Yoshida, Coset Construction for Duals of Non-relativistic CFTs, JHEP 05 (2009) 038 [arXiv:0903.4245] [SPIRES].
Y. Nakayama, Gravity Dual for Reggeon Field Theory and Non-linear Quantum Finance, Int. J. Mod. Phys. A 24 (2009) 6197 [arXiv:0906.4112] [SPIRES].
A. Chamblin and G.W. Gibbons, Nonlinear supergravity on a brane without compactification, Phys. Rev. Lett. 84 (2000) 1090 [hep-th/9909130] [SPIRES].
D. Brecher, A. Chamblin and H.S. Reall, AdS/CFT in the infinite momentum frame, Nucl. Phys. B 607 (2001) 155 [hep-th/0012076] [SPIRES].
A. Kumar and H.K. Kunduri, Gravitational wave solutions in string and M-theory AdS backgrounds, Phys. Rev. D 70 (2004) 104006 [hep-th/0405261] [SPIRES].
K. Sfetsos, Fundamental branes and shock waves, Nucl. Phys. B 726 (2005) 1 [hep-th/0507134] [SPIRES].
C.-S. Chu and P.-M. Ho, Time-dependent AdS/CFT duality and null singularity, JHEP 04 (2006) 013 [hep-th/0602054] [SPIRES].
S.R. Das, J. Michelson, K. Narayan and S.P. Trivedi, Time dependent cosmologies and their duals, Phys. Rev. D 74 (2006) 026002 [hep-th/0602107] [SPIRES].
S.R. Das, J. Michelson, K. Narayan and S.P. Trivedi, Cosmologies with Null Singularities and their Gauge Theory Duals, Phys. Rev. D 75 (2007) 026002 [hep-th/0610053] [SPIRES].
S.R. Coleman and R. Jackiw, Why dilatation generators do not generate dilatations?, Ann. Phys. 67 (1971) 552 [SPIRES].
C.M. Hull and P.K. Townsend, Finiteness and conformal invariance in nonlinear σ-models, Nucl. Phys. B 274 (1986) 349 [SPIRES].
V. Riva and J.L. Cardy, Scale and conformal invariance in field theory: A physical counterexample, Phys. Lett. B 622 (2005) 339 [hep-th/0504197] [SPIRES].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [SPIRES].
J. Polchinski, Scale and conformal invariance in quantum field theory, Nucl. Phys. B 303 (1988) 226 [SPIRES].
Y. Nakayama, Forbidden Landscape from Holography, JHEP 11 (2009) 061 [arXiv:0907.0227] [SPIRES].
Y. Nakayama, No Forbidden Landscape in String/M-theory, JHEP 01 (2010) 030 [arXiv:0909.4297] [SPIRES].
D. Dorigoni and S. Rychkov, Scale Invariance + Unitarity =¿ Conformal Invariance?, arXiv:0910.1087 [SPIRES].
H. Singh, Galilean anti-de-Sitter spacetime in Romans theory, Phys. Lett. B 682 (2009) 225 [arXiv:0909.1692] [SPIRES].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [SPIRES].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [SPIRES].
C.A. Fuertes and S. Moroz, Correlation functions in the non-relativistic AdS/CFT correspondence, Phys. Rev. D 79 (2009) 106004 [arXiv:0903.1844] [SPIRES].
A. Volovich and C. Wen, Correlation Functions in Non-Relativistic Holography, JHEP 05 (2009) 087 [arXiv:0903.2455] [SPIRES].
R.G. Leigh and N.N. Hoang, Fermions and the Sch/nrCFT Correspondence, JHEP 03 (2010) 027 [arXiv:0909.1883] [SPIRES].
Z.F. Ezawa, M. Hotta and A. Iwazaki, Breathing vortex solitons in nonrelativistic Chern-Simons gauge theory, Phys. Rev. Lett. 67 (1991) 411 [Erratum ibid. 67 (1991) 1475] [SPIRES].
R. Jackiw and S.-Y. Pi, Time dependent Chern-Simons solitons and their quantization, Phys. Rev. D 44 (1991) 2524 [SPIRES].
P.A. Horvathy and P. Zhang, Vortices in (abelian) Chern-Simons gauge theory, Phys. Rept. 481 (2009) 83 [arXiv:0811.2094] [SPIRES].
Y. Nakayama and S.-J. Rey, Observables and Correlators in Nonrelativistic ABJM Theory, JHEP 08 (2009) 029 [arXiv:0905.2940] [SPIRES].
V.R. Kaigorodov, Einstein spaces of maximum mobility, Dokl. Akad. Nuk. SSSR 146 (1962) 793 [Sov. Phys. Doklady 7 (1963) 893].
V. Hussin and M. Jacques, On Nonrelativistic Conformal Symmetries And Invariant Tensor Fields, J. Phys. A 19 (1986) 3471.
Y. Nakayama, Index for Non-relativistic Superconformal Field Theories, JHEP 10 (2008) 083 [arXiv:0807.3344] [SPIRES].
Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [SPIRES].
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Nakayama, Y. Universal time-dependent deformations of Schrödinger geometry. J. High Energ. Phys. 2010, 102 (2010). https://doi.org/10.1007/JHEP04(2010)102
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DOI: https://doi.org/10.1007/JHEP04(2010)102