Abstract
Holography for Lifshitz space-times corresponds to dual field theories on a fixed torsional Newton-Cartan (TNC) background. We examine the coupling of non-relativistic field theories to TNC backgrounds and uncover a novel mechanism by which a global U(1) can become local. This involves the TNC vector M μ which sources a particle number current, and which for flat NC space-time satisfies M μ = ∂ μ M with a Schrödinger symmetry realized on M . We discuss various toy model field theories on flat NC space-time for which the new mechanism leads to extra global space-time symmetries beyond the generic Lifshitz symmetry, allowing for an enhancement to Schrödinger symmetry. On the holographic side, the source M also appears in the Lifshitz vacuum with exactly the same properties as for flat NC space-time. In particular, the bulk diffeomorphisms that preserve the boundary conditions realize a Schrödinger algebra on M , allowing for a conserved particle number current. Finally, we present a probe action for a complex scalar field on the Lifshitz vacuum, which exhibits Schrödinger invariance in the same manner as seen in the field theory models.
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References
D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].
K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].
M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
D.T. Son, Newton-Cartan geometry and the quantum Hall effect, arXiv:1306.0638 [INSPIRE].
G. Dautcourt, On the newtonian limit of general relativity, Acta Phys. Pol. B 21 (1990) 755.
L.P. Eisenhart, Dynamical trajectories and geodesics, Ann. Math. 30 (1928) 591.
A. Trautman, Sur la theorie newtonienne de la gravitation, Compt. Rend. Acad. Sci. Paris 247 (1963) 617.
H.P. Kuenzle, Galilei and Lorentz structures on space-time — Comparison of the corresponding geometry and physics, Annales Poincaré Phys. Theor. 17 (1972) 337 [INSPIRE].
C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31 (1985) 1841 [INSPIRE].
C. Duval, G.W. Gibbons and P. Horvathy, Celestial mechanics, conformal structures and gravitational waves, Phys. Rev. D 43 (1991) 3907 [hep-th/0512188] [INSPIRE].
B. Julia and H. Nicolai, Null Killing vector dimensional reduction and Galilean geometrodynamics, Nucl. Phys. B 439 (1995) 291 [hep-th/9412002] [INSPIRE].
M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Torsional Newton-Cartan geometry and Lifshitz holography, Phys. Rev. D 89 (2014) 061901 [arXiv:1311.4794] [INSPIRE].
M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Boundary stress-energy tensor and Newton-Cartan geometry in Lifshitz holography, JHEP 01 (2014) 057 [arXiv:1311.6471] [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Lifshitz space-times for Schrödinger holography, Phys. Lett. B 746 (2015) 318 [arXiv:1409.1519] [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Sources and vevs in Lifshitz holography, in preparation.
E.A. Bergshoeff, J. Hartong and J. Rosseel, Torsional Newton-Cartan geometry and the Schrödinger algebra, Class. Quant. Grav. 32 (2015) 135017 [arXiv:1409.5555] [INSPIRE].
K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime, arXiv:1408.6855 [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Schrödinger invariance from Lifshitz isometries in holography and field theory, arXiv:1409.1522 [INSPIRE].
S.F. Ross and O. Saremi, Holographic stress tensor for non-relativistic theories, JHEP 09 (2009) 009 [arXiv:0907.1846] [INSPIRE].
S.F. Ross, Holography for asymptotically locally Lifshitz spacetimes, Class. Quant. Grav. 28 (2011) 215019 [arXiv:1107.4451] [INSPIRE].
M. Baggio, J. de Boer and K. Holsheimer, Hamilton-Jacobi renormalization for Lifshitz spacetime, JHEP 01 (2012) 058 [arXiv:1107.5562] [INSPIRE].
R.B. Mann and R. McNees, Holographic renormalization for asymptotically Lifshitz spacetimes, JHEP 10 (2011) 129 [arXiv:1107.5792] [INSPIRE].
T. Griffin, P. Hořava and C.M. Melby-Thompson, Conformal Lifshitz gravity from holography, JHEP 05 (2012) 010 [arXiv:1112.5660] [INSPIRE].
Y. Korovin, K. Skenderis and M. Taylor, Lifshitz as a deformation of Anti-de Sitter, JHEP 08 (2013) 026 [arXiv:1304.7776] [INSPIRE].
W. Chemissany and I. Papadimitriou, Generalized dilatation operator method for non-relativistic holography, Phys. Lett. B 737 (2014) 272 [arXiv:1405.3965] [INSPIRE].
W. Chemissany and I. Papadimitriou, Lifshitz holography: the whole shebang, JHEP 01 (2015) 052 [arXiv:1408.0795] [INSPIRE].
M. Guica, K. Skenderis, M. Taylor and B.C. van Rees, Holography for Schrödinger backgrounds, JHEP 02 (2011) 056 [arXiv:1008.1991] [INSPIRE].
J. Hartong and B. Rollier, Asymptotically Schrödinger space-times: tsT transformations and thermodynamics, JHEP 01 (2011) 084 [arXiv:1009.4997] [INSPIRE].
M. Guica, A Fefferman-Graham-like expansion for null warped AdS 3, JHEP 12 (2012) 084 [arXiv:1111.6978] [INSPIRE].
J. Hartong and B. Rollier, Particle number and 3D Schrödinger holography, JHEP 09 (2014) 111 [arXiv:1305.3653] [INSPIRE].
G. Compère, M. Guica and M.J. Rodriguez, Two Virasoro symmetries in stringy warped AdS 3, JHEP 12 (2014) 012 [arXiv:1407.7871] [INSPIRE].
T. Andrade, C. Keeler, A. Peach and S.F. Ross, Schrödinger holography for z < 2, Class. Quant. Grav. 32 (2015) 035015 [arXiv:1408.7103] [INSPIRE].
T. Andrade, C. Keeler, A. Peach and S.F. Ross, Schrödinger holography with z = 2, Class. Quant. Grav. 32 (2015) 085006 [arXiv:1412.0031] [INSPIRE].
D.M. Hofman and B. Rollier, Warped conformal field theory as lower spin gravity, Nucl. Phys. B 897 (2015) 1 [arXiv:1411.0672] [INSPIRE].
A. Gromov and A.G. Abanov, Thermal Hall effect and geometry with torsion, Phys. Rev. Lett. 114 (2015) 016802 [arXiv:1407.2908] [INSPIRE].
M. Geracie, D.T. Son, C. Wu and S.-F. Wu, Spacetime symmetries of the quantum Hall effect, Phys. Rev. D 91 (2015) 045030 [arXiv:1407.1252] [INSPIRE].
M. Geracie and D.T. Son, Hydrodynamics on the lowest Landau level, JHEP 06 (2015) 044 [arXiv:1408.6843] [INSPIRE].
C. Wu and S.-F. Wu, Relativistic gravity and parity-violating nonrelativistic effective field theories, Phys. Rev. D 91 (2015) 126004 [arXiv:1409.8265] [INSPIRE].
M. Geracie, S. Golkar and M.M. Roberts, Hall viscosity, spin density and torsion, arXiv:1410.2574 [INSPIRE].
R. Banerjee, A. Mitra and P. Mukherjee, A new formulation of non-relativistic diffeomorphism invariance, Phys. Lett. B 737 (2014) 369 [arXiv:1404.4491] [INSPIRE].
R. Banerjee, A. Mitra and P. Mukherjee, Localization of the Galilean symmetry and dynamical realization of Newton-Cartan geometry, Class. Quant. Grav. 32 (2015) 045010 [arXiv:1407.3617] [INSPIRE].
T. Brauner, S. Endlich, A. Monin and R. Penco, General coordinate invariance in quantum many-body systems, Phys. Rev. D 90 (2014) 105016 [arXiv:1407.7730] [INSPIRE].
K. Jensen and A. Karch, Revisiting non-relativistic limits, JHEP 04 (2015) 155 [arXiv:1412.2738] [INSPIRE].
R. Andringa, E. Bergshoeff, S. Panda and M. de Roo, Newtonian gravity and the Bargmann algebra, Class. Quant. Grav. 28 (2011) 105011 [arXiv:1011.1145] [INSPIRE].
C. Hoyos, B.S. Kim and Y. Oz, Lifshitz field theories at non-zero temperature, hydrodynamics and gravity, JHEP 03 (2014) 029 [arXiv:1309.6794] [INSPIRE].
R. Penrose and W. Rindler, Spinors and space-time. Volume 2: spinor and twistor methods in space-time geometry, Cambridge University Press, Cambridge U.K. (1986).
J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz, Diffeomorphisms and holographic anomalies, Class. Quant. Grav. 17 (2000) 1129 [hep-th/9910267] [INSPIRE].
K. Skenderis, Asymptotically Anti-de Sitter space-times and their stress energy tensor, Int. J. Mod. Phys. A 16 (2001) 740 [hep-th/0010138] [INSPIRE].
X. Bekaert and K. Morand, Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view, arXiv:1412.8212 [INSPIRE].
S. El-Showk, Y. Nakayama and S. Rychkov, What Maxwell theory in D <> 4 teaches us about scale and conformal invariance, Nucl. Phys. B 848 (2011) 578 [arXiv:1101.5385] [INSPIRE].
R. Andringa, E.A. Bergshoeff, J. Rosseel and E. Sezgin, 3D Newton-Cartan supergravity, Class. Quant. Grav. 30 (2013) 205005 [arXiv:1305.6737] [INSPIRE].
K. Kuchar, Gravitation, geometry, and nonrelativistic quantum theory, Phys. Rev. D 22 (1980) 1285 [INSPIRE].
E. Bergshoeff, J. Gomis, M. Kovacevic, L. Parra, J. Rosseel and T. Zojer, Nonrelativistic superparticle in a curved background, Phys. Rev. D 90 (2014) 065006 [arXiv:1406.7286] [INSPIRE].
E.B. Kiritsis and G. Kofinas, On Hořava-Lifshitz ‘black holes’, JHEP 01 (2010) 122 [arXiv:0910.5487] [INSPIRE].
M. Henkel, Schrödinger invariance in strongly anisotropic critical systems, J. Statist. Phys. 75 (1994) 1023 [hep-th/9310081] [INSPIRE].
M. Henkel and J. Unterberger, Schrödinger invariance and space-time symmetries, Nucl. Phys. B 660 (2003) 407 [hep-th/0302187] [INSPIRE].
U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation., Helv. Phys. Acta 45 (1972) 802 [INSPIRE].
M. Perroud, Projective representations of the Schrödinger group, Helv. Phys. Acta 50 (1977) 233 [INSPIRE].
V. Bargmann, On unitary ray representations of continuous groups, Ann. Math. 59 (1954) 1 [INSPIRE].
C. Keeler, G. Knodel and J.T. Liu, What do non-relativistic CFTs tell us about Lifshitz spacetimes?, JHEP 01 (2014) 062 [arXiv:1308.5689] [INSPIRE].
M. Blau, J. Hartong and B. Rollier, Geometry of Schrödinger space-times, global coordinates and harmonic trapping, JHEP 07 (2009) 027 [arXiv:0904.3304] [INSPIRE].
M. Blau, J. Hartong and B. Rollier, Geometry of Schrödinger space-times II: particle and field probes of the causal structure, JHEP 07 (2010) 069 [arXiv:1005.0760] [INSPIRE].
M. Mulligan, C. Nayak and S. Kachru, An isotropic to anisotropic transition in a fractional quantum Hall state, Phys. Rev. B 82 (2010) 085102 [arXiv:1004.3570] [INSPIRE].
X. Bekaert and K. Morand, Embedding nonrelativistic physics inside a gravitational wave, Phys. Rev. D 88 (2013) 063008 [arXiv:1307.6263] [INSPIRE].
K. Balasubramanian and K. Narayan, Lifshitz spacetimes from AdS null and cosmological solutions, JHEP 08 (2010) 014 [arXiv:1005.3291] [INSPIRE].
A. Donos and J.P. Gauntlett, Lifshitz solutions of D = 10 and D = 11 supergravity, JHEP 12 (2010) 002 [arXiv:1008.2062] [INSPIRE].
R.N. Caldeira Costa and M. Taylor, Holography for chiral scale-invariant models, JHEP 02 (2011) 082 [arXiv:1010.4800] [INSPIRE].
D. Cassani and A.F. Faedo, Constructing Lifshitz solutions from AdS, JHEP 05 (2011) 013 [arXiv:1102.5344] [INSPIRE].
W. Chemissany and J. Hartong, From D3-branes to Lifshitz space-times, Class. Quant. Grav. 28 (2011) 195011 [arXiv:1105.0612] [INSPIRE].
B. Gouteraux and E. Kiritsis, Quantum critical lines in holographic phases with (un)broken symmetry, JHEP 04 (2013) 053 [arXiv:1212.2625] [INSPIRE].
J. Gath, J. Hartong, R. Monteiro and N.A. Obers, Holographic models for theories with hyperscaling violation, JHEP 04 (2013) 159 [arXiv:1212.3263] [INSPIRE].
B. Goutéraux, Universal scaling properties of extremal cohesive holographic phases, JHEP 01 (2014) 080 [arXiv:1308.2084] [INSPIRE].
B. Goutéraux, Charge transport in holography with momentum dissipation, JHEP 04 (2014) 181 [arXiv:1401.5436] [INSPIRE].
A. Donos, B. Goutéraux and E. Kiritsis, Holographic Metals and Insulators with Helical Symmetry, JHEP 09 (2014) 038 [arXiv:1406.6351] [INSPIRE].
D.V. Khveshchenko, Taking a critical look at holographic critical matter, arXiv:1404.7000 [INSPIRE].
A. Karch, Conductivities for hyperscaling violating geometries, JHEP 06 (2014) 140 [arXiv:1405.2926] [INSPIRE].
S.A. Hartnoll and A. Karch, Scaling theory of the cuprate strange metals, Phys. Rev. B 91 (2015) 155126 [arXiv:1501.03165] [INSPIRE].
P. Hořava, Quantum gravity at a Lifshitz point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].
T. Griffin, P. Hořva and C.M. Melby-Thompson, Lifshitz gravity for Lifshitz holography, Phys. Rev. Lett. 110 (2013) 081602 [arXiv:1211.4872] [INSPIRE].
S. Janiszewski and A. Karch, Non-relativistic holography from Hořava gravity, JHEP 02 (2013) 123 [arXiv:1211.0005] [INSPIRE].
U.H. Danielsson and L. Thorlacius, Black holes in asymptotically Lifshitz spacetime, JHEP 03 (2009) 070 [arXiv:0812.5088] [INSPIRE].
M.C.N. Cheng, S.A. Hartnoll and C.A. Keeler, Deformations of Lifshitz holography, JHEP 03 (2010) 062 [arXiv:0912.2784] [INSPIRE].
M. Baggio, Deformations of CFTs and holography, Ph.D. thesis, University of Amsterdam, Institute for Theoretical Physics (ITF), Amsterdam, The Netherlands (2013).
K. Holsheimer, On the marginally relevant operator in ‡ = 2 Lifshitz holography, JHEP 03 (2014) 084 [arXiv:1311.4539] [INSPIRE].
K. Jensen, Aspects of hot Galilean field theory, JHEP 04 (2015) 123 [arXiv:1411.7024] [INSPIRE].
C. Hoyos, B.S. Kim and Y. Oz, Lifshitz hydrodynamics, JHEP 11 (2013) 145 [arXiv:1304.7481] [INSPIRE].
G. Policastro, D.T. Son and A.O. Starinets, The shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 87 (2001) 081601 [hep-th/0104066] [INSPIRE].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
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Hartong, J., Kiritsis, E. & Obers, N.A. Field theory on Newton-Cartan backgrounds and symmetries of the Lifshitz vacuum. J. High Energ. Phys. 2015, 6 (2015). https://doi.org/10.1007/JHEP08(2015)006
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DOI: https://doi.org/10.1007/JHEP08(2015)006