Abstract
We prove a c-theorem for holographic renormalization group flows in a Schrödinger spacetime that demonstrates that the effective radius L(r) monotonically decreases from the UV to the IR, where r is the bulk radial coordinate. This result assumes that the bulk matter satisfies the null energy condition, but holds regardless of the value of the critical exponent z. We also construct several numerical examples in a model where the Schrödinger background is realized by a massive vector coupled to a real scalar. The full Schrödinger group is realized when z = 2, and in this case it is possible to construct solutions with constant effective z(r) = 2 along the entire flow.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
A.B. 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].
L. Di Pietro and Z. Komargodski, Cardy formulae for SUSY theories in d = 4 and d = 6, JHEP 12 (2014) 031 [arXiv:1407.6061] [INSPIRE].
A.A. Ardehali, J.T. Liu and P. Szepietowski, c − a from the \( \mathcal{N}=1 \) superconformal index, JHEP 12 (2014) 145 [arXiv:1407.6024] [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
E. Álvarez and C. Gómez, Geometric holography, the renormalization group and the c-theorem, Nucl. Phys. B 541 (1999) 441 [hep-th/9807226] [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].
D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c-theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].
V. Sahakian, Holography, a covariant c function and the geometry of the renormalization group, Phys. Rev. D 62 (2000) 126011 [hep-th/9910099] [INSPIRE].
M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [INSPIRE].
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].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
J.T. Liu and Z. Zhao, Holographic Lifshitz flows and the null energy condition, arXiv:1206.1047 [INSPIRE].
S. Moroz, Below the Breitenlohner-Freedman bound in the nonrelativistic AdS/CFT correspondence, Phys. Rev. D 81 (2010) 066002 [arXiv:0911.4060] [INSPIRE].
T. Ishii and T. Nishioka, Flows to Schrödinger Geometries, Phys. Rev. D 84 (2011) 125007 [arXiv:1109.6318] [INSPIRE].
H. Braviner, R. Gregory and S.F. Ross, Flows involving Lifshitz solutions, Class. Quant. Grav. 28 (2011) 225028 [arXiv:1108.3067] [INSPIRE].
I. Adam, I.V. Melnikov and S. Theisen, A Non-Relativistic Weyl Anomaly, JHEP 09 (2009) 130 [arXiv:0907.2156] [INSPIRE].
P.R.S. Gomes and M. Gomes, On Ward Identities in Lifshitz-like Field Theories, Phys. Rev. D 85 (2012) 065010 [arXiv:1112.3887] [INSPIRE].
T. Griffin, P. Hořava and C.M. Melby-Thompson, Conformal Lifshitz Gravity from Holography, JHEP 05 (2012) 010 [arXiv:1112.5660] [INSPIRE].
M. Baggio, J. de Boer and K. Holsheimer, Anomalous Breaking of Anisotropic Scaling Symmetry in the Quantum Lifshitz Model, JHEP 07 (2012) 099 [arXiv:1112.6416] [INSPIRE].
I. Arav, S. Chapman and Y. Oz, Lifshitz Scale Anomalies, JHEP 02 (2015) 078 [arXiv:1410.5831] [INSPIRE].
K. Jensen, Anomalies for Galilean fields, arXiv:1412.7750 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1510.06975
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Liu, J.T., Zhong, W. A holographic c-theorem for Schrödinger spacetimes. J. High Energ. Phys. 2015, 1–18 (2015). https://doi.org/10.1007/JHEP12(2015)179
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/JHEP12(2015)179