Abstract
We derive a torsionfull version of three-dimensional \( \mathcal{N}=2 \) Newton-Cartan supergravity using a non-relativistic notion of the superconformal tensor calculus. The “superconformal” theory that we start with is Schrödinger supergravity which we obtain by gauging the Schrödinger superalgebra. We present two non-relativistic \( \mathcal{N}=2 \) matter multiplets that can be used as compensators in the superconformal calculus. They lead to two different off-shell formulations which, in analogy with the relativistic case, we call “old minimal” and “new minimal” Newton-Cartan supergravity. We find similarities but also point out some differences with respect to the relativistic case.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
A.G. Abanov and A. Gromov, Electromagnetic and gravitational responses of two-dimensional noninteracting electrons in a background magnetic field, Phys. Rev. B 90 (2014) 014435 [arXiv:1401.3703] [INSPIRE].
A. Gromov and A.G. Abanov, Thermal Hall effect and geometry with torsion, Phys. Rev. Lett. 114 (2015) 016802 [arXiv:1407.2908] [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].
K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [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].
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].
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].
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, 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].
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 and A. Karch, Revisiting non-relativistic limits, JHEP 04 (2015) 155 [arXiv:1412.2738] [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Field theory on Newton-Cartan backgrounds and symmetries of the Lifshitz vacuum, JHEP 08 (2015) 006 [arXiv:1502.00228] [INSPIRE].
M. Geracie, K. Prabhu and M.M. Roberts, Curved non-relativistic spacetimes, newtonian gravitation and massive matter, J. Math. Phys. 56 (2015) 103505 [arXiv:1503.02682] [INSPIRE].
M. Geracie, K. Prabhu and M.M. Roberts, Fields and fluids on curved non-relativistic spacetimes, JHEP 08 (2015) 042 [arXiv:1503.02680] [INSPIRE].
E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Ann. École Norm. Sup. 40 (1923) 325.
E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie) (suite), Ann. École Norm. Sup. 41 (1924) 1.
C.W. Misner, K. Thorne and J. Wheeler, Gravitation, W.H. Freeman and company, San Francisco, U.S.A. (1973).
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].
R. Andringa, E.A. Bergshoeff, J. Rosseel and E. Sezgin, 3D Newton-Cartan supergravity, Class. Quant. Grav. 30 (2013) 205005 [arXiv:1305.6737] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
M. Mariño, Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories, J. Phys. A 44 (2011) 463001 [arXiv:1104.0783] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [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].
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).
J.A. de Azcarraga and J. Lukierski, Galilean superconformal symmetries, Phys. Lett. B 678 (2009) 411 [arXiv:0905.0141] [INSPIRE].
M. Sakaguchi, Super Galilean conformal algebra in AdS/CFT, J. Math. Phys. 51 (2010) 042301 [arXiv:0905.0188] [INSPIRE].
A. Bagchi and I. Mandal, Supersymmetric extension of galilean conformal algebras, Phys. Rev. D 80 (2009) 086011 [arXiv:0905.0580] [INSPIRE].
J.P. Gauntlett, J. Gomis and P.K. Townsend, Supersymmetry and the physical phase space formulation of spinning particles, Phys. Lett. B 248 (1990) 288 [INSPIRE].
M. Leblanc, G. Lozano and H. Min, Extended superconformal Galilean symmetry in Chern-Simons matter systems, Annals Phys. 219 (1992) 328 [hep-th/9206039] [INSPIRE].
C. Duval and P.A. Horvathy, On Schrödinger superalgebras, J. Math. Phys. 35 (1994) 2516 [hep-th/0508079] [INSPIRE].
M. Sakaguchi and K. Yoshida, More super Schrödinger algebras from P SU (2, 2|4), JHEP 08 (2008) 049 [arXiv:0806.3612] [INSPIRE].
P.S. Howe, J.M. Izquierdo, G. Papadopoulos and P.K. Townsend, New supergravities with central charges and Killing spinors in (2 + 1)-dimensions, Nucl. Phys. B 467 (1996) 183 [hep-th/9505032] [INSPIRE].
J. Gomis, K. Kamimura and P.K. Townsend, Non-relativistic superbranes, JHEP 11 (2004) 051 [hep-th/0409219] [INSPIRE].
H. Afshar, E.A. Bergshoeff, A. Mehra, P. Parekh and B. Rollier, Hořava-Lifshitz gravity and Schrödinger scalar field theories, work in progress, UG-15-55.
B. de Wit, R. Philippe and A. Van Proeyen, The improved tensor multiplet in N = 2 supergravity, Nucl. Phys. B 219 (1983) 143 [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: 1509.04527
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
Bergshoeff, E., Rosseel, J. & Zojer, T. Newton-Cartan supergravity with torsion and Schrödinger supergravity. J. High Energ. Phys. 2015, 180 (2015). https://doi.org/10.1007/JHEP11(2015)180
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2015)180