Journal of High Energy Physics

, 2015:180 | Cite as

Newton-Cartan supergravity with torsion and Schrödinger supergravity

Open Access
Regular Article - Theoretical Physics

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.

Keywords

Gauge Symmetry Supergravity Models Holography and condensed matter physics (AdS/CMT) Classical Theories of Gravity 

References

  1. [1]
    C. Hoyos and D.T. Son, Hall viscosity and electromagnetic response, Phys. Rev. Lett. 108 (2012) 066805 [arXiv:1109.2651] [INSPIRE].CrossRefADSGoogle Scholar
  2. [2]
    D.T. Son, Newton-Cartan geometry and the quantum Hall effect, arXiv:1306.0638 [INSPIRE].
  3. [3]
    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].CrossRefADSGoogle Scholar
  4. [4]
    A. Gromov and A.G. Abanov, Thermal Hall effect and geometry with torsion, Phys. Rev. Lett. 114 (2015) 016802 [arXiv:1407.2908] [INSPIRE].CrossRefADSGoogle Scholar
  5. [5]
    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008)106005 [arXiv:0808.1725] [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  7. [7]
    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].ADSMathSciNetGoogle Scholar
  8. [8]
    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].ADSGoogle Scholar
  9. [9]
    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].CrossRefADSMathSciNetGoogle Scholar
  10. [10]
    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].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    K. Jensen, On the coupling of galilean-invariant field theories to curved spacetime, arXiv:1408.6855 [INSPIRE].
  12. [12]
    J. Hartong, E. Kiritsis and N.A. Obers, Lifshitz space-times for Schrödinger holography, Phys. Lett. B 746 (2015) 318 [arXiv:1409.1519] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  13. [13]
    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].ADSGoogle Scholar
  14. [14]
    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].CrossRefADSMathSciNetGoogle Scholar
  15. [15]
    K. Jensen and A. Karch, Revisiting non-relativistic limits, JHEP 04 (2015) 155 [arXiv:1412.2738] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  16. [16]
    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].CrossRefADSGoogle Scholar
  17. [17]
    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].CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    M. Geracie, K. Prabhu and M.M. Roberts, Fields and fluids on curved non-relativistic spacetimes, JHEP 08 (2015) 042 [arXiv:1503.02680] [INSPIRE].CrossRefADSGoogle Scholar
  19. [19]
    E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Ann. École Norm. Sup. 40 (1923) 325.MathSciNetGoogle Scholar
  20. [20]
    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.MathSciNetGoogle Scholar
  21. [21]
    C.W. Misner, K. Thorne and J. Wheeler, Gravitation, W.H. Freeman and company, San Francisco, U.S.A. (1973).Google Scholar
  22. [22]
    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].CrossRefADSGoogle Scholar
  23. [23]
    R. Andringa, E.A. Bergshoeff, J. Rosseel and E. Sezgin, 3D Newton-Cartan supergravity, Class. Quant. Grav. 30 (2013) 205005 [arXiv:1305.6737] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  24. [24]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].CrossRefADSMathSciNetMATHGoogle Scholar
  25. [25]
    M. Mariño, Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories, J. Phys. A 44 (2011) 463001 [arXiv:1104.0783] [INSPIRE].ADSGoogle Scholar
  26. [26]
    G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  27. [27]
    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].CrossRefADSGoogle Scholar
  28. [28]
    D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).CrossRefMATHGoogle Scholar
  29. [29]
    J.A. de Azcarraga and J. Lukierski, Galilean superconformal symmetries, Phys. Lett. B 678 (2009) 411 [arXiv:0905.0141] [INSPIRE].CrossRefADSGoogle Scholar
  30. [30]
    M. Sakaguchi, Super Galilean conformal algebra in AdS/CFT, J. Math. Phys. 51 (2010) 042301 [arXiv:0905.0188] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  31. [31]
    A. Bagchi and I. Mandal, Supersymmetric extension of galilean conformal algebras, Phys. Rev. D 80 (2009) 086011 [arXiv:0905.0580] [INSPIRE].ADSMathSciNetGoogle Scholar
  32. [32]
    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].CrossRefADSGoogle Scholar
  33. [33]
    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].CrossRefADSMathSciNetMATHGoogle Scholar
  34. [34]
    C. Duval and P.A. Horvathy, On Schrödinger superalgebras, J. Math. Phys. 35 (1994) 2516 [hep-th/0508079] [INSPIRE].CrossRefADSMathSciNetMATHGoogle Scholar
  35. [35]
    M. Sakaguchi and K. Yoshida, More super Schrödinger algebras from P SU (2, 2|4), JHEP 08 (2008) 049 [arXiv:0806.3612] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  36. [36]
    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].CrossRefADSMathSciNetGoogle Scholar
  37. [37]
    J. Gomis, K. Kamimura and P.K. Townsend, Non-relativistic superbranes, JHEP 11 (2004) 051 [hep-th/0409219] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  38. [38]
    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.Google Scholar
  39. [39]
    B. de Wit, R. Philippe and A. Van Proeyen, The improved tensor multiplet in N = 2 supergravity, Nucl. Phys. B 219 (1983) 143 [INSPIRE].CrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Van Swinderen Institute for Particle Physics and GravityUniversity of GroningenGroningenThe Netherlands
  2. 2.Institute for Theoretical PhysicsVienna University of TechnologyViennaAustria
  3. 3.Albert Einstein Center for Fundamental PhysicsUniversity of BernBernSwitzerland

Personalised recommendations