Abstract
We compute the leading part of the trace anomaly for a free non-relativistic scalar in 2 + 1 dimensions coupled to a background Newton-Cartan metric. The anomaly is proportional to 1/m, where m is the mass of the scalar. We comment on the implications of a conjectured a-theorem for non-relativistic theories with boost invariance.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.J. Duff, Observations on Conformal Anomalies, Nucl. Phys. B 125 (1977) 334 [INSPIRE].
M.J. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hep-th/9308075] [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].
J.L. Cardy, Is There a c-Theorem in Four-Dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].
H. Osborn, Derivation of a Four-dimensional c-Theorem, Phys. Lett. B 222 (1989) 97 [INSPIRE].
I. Jack and H. Osborn, Analogs for the c-Theorem for Four-dimensional Renormalizable Field Theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].
H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
Z. Komargodski, The Constraints of Conformal Symmetry on RG Flows, JHEP 07 (2012) 069 [arXiv:1112.4538] [INSPIRE].
I. Adam, I.V. Melnikov and S. Theisen, A Non-Relativistic Weyl Anomaly, JHEP 09 (2009) 130 [arXiv:0907.2156] [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].
T. Griffin, P. Hořava and C.M. Melby-Thompson, Conformal Lifshitz Gravity from Holography, JHEP 05 (2012) 010 [arXiv:1112.5660] [INSPIRE].
I. Arav, S. Chapman and Y. Oz, Lifshitz Scale Anomalies, JHEP 02 (2015) 078 [arXiv:1410.5831] [INSPIRE].
I. Arav, S. Chapman and Y. Oz, Non-Relativistic Scale Anomalies, JHEP 06 (2016) 158 [arXiv:1601.06795] [INSPIRE].
S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [INSPIRE].
K. Jensen, Anomalies for Galilean fields, arXiv:1412.7750 [INSPIRE].
R. Auzzi, S. Baiguera and G. Nardelli, On Newton-Cartan trace anomalies, JHEP 02 (2016) 003 [Erratum ibid. 02 (2016) 177] [arXiv:1511.08150] [INSPIRE].
L. Bonora, P. Pasti and M. Bregola, Weyl COCYCLES, Class. Quant. Grav. 3 (1986) 635 [INSPIRE].
C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, W.H. Freeman, San Francisco U.S.A. (1973) [ISBN: 978-0-7167-0344-0].
D.T. Son and M. Wingate, General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas, Annals Phys. 321 (2006) 197 [cond-mat/0509786] [INSPIRE].
C. Hoyos and D.T. Son, Hall Viscosity and Electromagnetic Response, Phys. Rev. Lett. 108 (2012) 066805 [arXiv:1109.2651] [INSPIRE].
D.T. Son, Newton-Cartan Geometry and the Quantum Hall Effect, arXiv:1306.0638 [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].
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, On the coupling of Galilean-invariant field theories to curved spacetime, arXiv:1408.6855 [INSPIRE].
K. Jensen and A. Karch, Revisiting non-relativistic limits, JHEP 04 (2015) 155 [arXiv:1412.2738] [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].
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, Schrödinger Invariance from Lifshitz Isometries in Holography and Field Theory, Phys. Rev. D 92 (2015) 066003 [arXiv:1409.1522] [INSPIRE].
S. Moroz, C. Hoyos and L. Radzihovsky, Galilean invariance at quantum Hall edge, Phys. Rev. B 91 (2015) 195409 [Erratum ibid. 91 (2015) 199906] [arXiv:1502.00667] [INSPIRE].
C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann Structures and Newton-cartan Theory, Phys. Rev. D 31 (1985) 1841 [INSPIRE].
L. Bonora, P. Cotta-Ramusino and C. Reina, Conformal Anomaly and Cohomology, Phys. Lett. B 126 (1983) 305 [INSPIRE].
S.M. Christensen and S.A. Fulling, Trace Anomalies and the Hawking Effect, Phys. Rev. D 15 (1977) 2088 [INSPIRE].
L.S. Brown, Stress Tensor Trace Anomaly in a Gravitational Metric: Scalar Fields, Phys. Rev. D 15 (1977) 1469 [INSPIRE].
J.S. Dowker and R. Critchley, The Stress Tensor Conformal Anomaly for Scalar and Spinor Fields, Phys. Rev. D 16 (1977) 3390 [INSPIRE].
S.W. Hawking, Zeta Function Regularization of Path Integrals in Curved Space-Time, Commun. Math. Phys. 55 (1977) 133 [INSPIRE].
N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (1982) [INSPIRE].
D.V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].
V. Mukhanov and S. Winitzki, Introduction to quantum effects in gravity, Cambridge University Press (2007) [INSPIRE].
S.N. Solodukhin, Entanglement Entropy in Non-Relativistic Field Theories, JHEP 04 (2010) 101 [arXiv:0909.0277] [INSPIRE].
A.O. Barvinsky and G.A. Vilkovisky, Covariant perturbation theory. 2: Second order in the curvature. General algorithms, Nucl. Phys. B 333 (1990) 471 [INSPIRE].
D. Anselmi, D.Z. Freedman, M.T. Grisaru and A.A. Johansen, Nonperturbative formulas for central functions of supersymmetric gauge theories, Nucl. Phys. B 526 (1998) 543 [hep-th/9708042] [INSPIRE].
K.A. Intriligator and B. Wecht, The Exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
R. Auzzi and B. Keren-Zur, Superspace formulation of the local RG equation, JHEP 05 (2015) 150 [arXiv:1502.05962] [INSPIRE].
E. Bergshoeff, J. Rosseel and T. Zojer, Newton-Cartan (super)gravity as a non-relativistic limit, Class. Quant. Grav. 32 (2015) 205003 [arXiv:1505.02095] [INSPIRE].
E. Bergshoeff, J. Rosseel and T. Zojer, Newton-Cartan supergravity with torsion and Schrödinger supergravity, JHEP 11 (2015) 180 [arXiv:1509.04527] [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].
J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [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 W. Zhong, A holographic c-theorem for Schrödinger spacetimes, JHEP 12 (2015) 179 [arXiv:1510.06975] [INSPIRE].
M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) 033001 [arXiv:1512.03554] [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: 1605.08684
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
Auzzi, R., Nardelli, G. Heat kernel for Newton-Cartan trace anomalies. J. High Energ. Phys. 2016, 47 (2016). https://doi.org/10.1007/JHEP07(2016)047
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2016)047