Abstract
We derive the action and symmetries of the bosonic sector of non-Lorentzian IIB supergravity by taking the non-relativistic string limit. We find that the bosonic field content is extended by a Lagrange multiplier that implements a restriction on the Ramond-Ramond fluxes. We show that the SL(2, ℝ) transformation rules of non-Lorentzian IIB supergravity form a novel, nonlinear polynomial realization. Using classical invariant theory of polynomial equations and binary forms, we will develop a general formalism describing the polynomial realization of SL(2, ℝ) and apply it to the special case of non-Lorentzian IIB supergravity. Using the same formalism, we classify all the relevant SL(2, ℝ) invariants. Invoking other bosonic symmetries, such as the local boost and dilatation symmetry, we show how the bosonic part of the non-Lorentzian IIB supergravity action is formed uniquely from these SL(2, ℝ) invariants. This work also points towards the concept of a non-Lorentzian bootstrap, where bosonic symmetries in non-Lorentzian supergravity are used to bootstrap the bosonic dynamics in Lorentzian supergravity, without considering the fermions.
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K.T. Grosvenor, C. Hoyos, F. Peña-Benitez and P. Surówka, Space-Dependent Symmetries and Fractons, Front. in Phys. 9 (2022) 792621 [arXiv:2112.00531] [INSPIRE].
G. Oling and Z. Yan, Aspects of Nonrelativistic Strings, Front. in Phys. 10 (2022) 832271 [arXiv:2202.12698] [INSPIRE].
E. Bergshoeff, J. Figueroa-O’Farrill and J. Gomis, A non-lorentzian primer, SciPost Phys. Lect. Notes 69 (2023) 1 [arXiv:2206.12177] [INSPIRE].
E.A. Bergshoeff and J. Rosseel, Non-Lorentzian Supergravity, in Handbook of Quantum Gravity, C. Bambi, L. Modesto, I. Shapiro eds., Springer, Singapore (2023) [https://doi.org/10.1007/978-981-19-3079-9_52-1] [arXiv:2211.02604] [INSPIRE].
J. Hartong, N.A. Obers and G. Oling, Review on Non-Relativistic Gravity, arXiv:2212.11309 [https://doi.org/10.3389/fphy.2023.1116888] [INSPIRE].
I.R. Klebanov and J.M. Maldacena, (1 + 1)-dimensional NCOS and its U(N) gauge theory dual, Adv. Theor. Math. Phys. 4 (2000) 283 [hep-th/0006085] [INSPIRE].
J. Gomis and H. Ooguri, Nonrelativistic closed string theory, J. Math. Phys. 42 (2001) 3127 [hep-th/0009181] [INSPIRE].
U.H. Danielsson, A. Guijosa and M. Kruczenski, IIA/B, wound and wrapped, JHEP 10 (2000) 020 [hep-th/0009182] [INSPIRE].
J. Gomis, J. Oh and Z. Yan, Nonrelativistic String Theory in Background Fields, JHEP 10 (2019) 101 [arXiv:1905.07315] [INSPIRE].
A.D. Gallegos, U. Gürsoy and N. Zinnato, Torsional Newton Cartan gravity from non-relativistic strings, JHEP 09 (2020) 172 [arXiv:1906.01607] [INSPIRE].
Z. Yan and M. Yu, Background Field Method for Nonlinear Sigma Models in Nonrelativistic String Theory, JHEP 03 (2020) 181 [arXiv:1912.03181] [INSPIRE].
J. Gomis, Z. Yan and M. Yu, Nonrelativistic Open String and Yang-Mills Theory, JHEP 03 (2021) 269 [arXiv:2007.01886] [INSPIRE].
R. Andringa, E. Bergshoeff, J. Gomis and M. de Roo, ‘Stringy’ Newton-Cartan Gravity, Class. Quant. Grav. 29 (2012) 235020 [arXiv:1206.5176] [INSPIRE].
E.A. Bergshoeff et al., String Theory and String Newton-Cartan Geometry, J. Phys. A 53 (2020) 014001 [arXiv:1907.10668] [INSPIRE].
L. Bidussi et al., Torsional string Newton-Cartan geometry for non-relativistic strings, JHEP 02 (2022) 116 [arXiv:2107.00642] [INSPIRE].
S. Ebert, H.-Y. Sun and Z. Yan, Dual D-brane actions in nonrelativistic string theory, JHEP 04 (2022) 161 [arXiv:2112.09316] [INSPIRE].
E.A. Bergshoeff et al., Non-relativistic ten-dimensional minimal supergravity, JHEP 12 (2021) 123 [arXiv:2107.14636] [INSPIRE].
E.A. Bergshoeff et al., A non-relativistic limit of NS-NS gravity, JHEP 06 (2021) 021 [arXiv:2102.06974] [INSPIRE].
J. Gomis, J. Gomis and K. Kamimura, Non-relativistic superstrings: a new soluble sector of AdS5 × S5, JHEP 12 (2005) 024 [hep-th/0507036] [INSPIRE].
D.T. Son, Newton-Cartan Geometry and the Quantum Hall Effect, arXiv:1306.0638 [INSPIRE].
T. Harmark, J. Hartong and N.A. Obers, Nonrelativistic strings and limits of the AdS/CFT correspondence, Phys. Rev. D 96 (2017) 086019 [arXiv:1705.03535] [INSPIRE].
T. Harmark et al., Strings with Non-Relativistic Conformal Symmetry and Limits of the AdS/CFT Correspondence, JHEP 11 (2018) 190 [arXiv:1810.05560] [INSPIRE].
T. Harmark, J. Hartong, N.A. Obers and G. Oling, Spin Matrix Theory String Backgrounds and Penrose Limits of AdS/CFT, JHEP 03 (2021) 129 [arXiv:2011.02539] [INSPIRE].
N. Lambert, R. Mouland and T. Orchard, Non-Lorentzian SU(1, n) Spacetime Symmetry In Various Dimensions, Front. in Phys. 10 (2022) 864800 [arXiv:2112.14860] [INSPIRE].
N. Lambert, A. Lipstein, R. Mouland and P. Richmond, Five-dimensional path integrals for six-dimensional conformal field theories, JHEP 02 (2022) 151 [arXiv:2109.04829] [INSPIRE].
J.H. Schwarz and P.C. West, Symmetries and Transformations of Chiral N = 2D = 10 Supergravity, Phys. Lett. B 126 (1983) 301 [INSPIRE].
C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [INSPIRE].
J.H. Schwarz, An SL(2, Z) multiplet of type IIB superstrings, Phys. Lett. B 360 (1995) 13 [Erratum ibid. 364 (1995) 252] [hep-th/9508143] [INSPIRE].
E. Bergshoeff, J. Gomis and Z. Yan, Nonrelativistic String Theory and T-Duality, JHEP 11 (2018) 133 [arXiv:1806.06071] [INSPIRE].
J. Gomis, Z. Yan and M. Yu, T-Duality in Nonrelativistic Open String Theory, JHEP 02 (2021) 087 [arXiv:2008.05493] [INSPIRE].
Z. Yan and M. Yu, KLT factorization of nonrelativistic string amplitudes, JHEP 04 (2022) 068 [arXiv:2112.00025] [INSPIRE].
E. Bergshoeff, J. Lahnsteiner, L. Romano and J. Rosseel, The supersymmetric Neveu-Schwarz branes of non-relativistic string theory, JHEP 08 (2022) 218 [arXiv:2204.04089] [INSPIRE].
L. Susskind, Another conjecture about M(atrix) theory, hep-th/9704080 [INSPIRE].
N. Seiberg, Why is the matrix model correct?, Phys. Rev. Lett. 79 (1997) 3577 [hep-th/9710009] [INSPIRE].
A. Sen, D0-branes on Tn and matrix theory, Adv. Theor. Math. Phys. 2 (1998) 51 [hep-th/9709220] [INSPIRE].
W. Taylor, M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory, Rev. Mod. Phys. 73 (2001) 419 [hep-th/0101126] [INSPIRE].
E.A. Bergshoeff et al., Branched SL(2, ℤ) duality, JHEP 10 (2022) 131 [arXiv:2208.13815] [INSPIRE].
P.J. Olver, Classical invariant theory, Cambridge University Press (1999) [https://doi.org/10.1017/cbo9780511623660].
D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, Springer Science & Business Media (1994) [ISBN: 9783540569633].
S. Ebert and Z. Yan, Anisotropic Compactification of Nonrelativistic M-Theory, arXiv:2309.04912 [INSPIRE].
J. Gomis and Z. Yan, Worldsheet formalism for decoupling limits in string theory, arXiv:2311.10565.
C. Blair, J. Lahnsteiner, N. Obers and Z. Yan, Matrix theory reloaded: U-duality, non-Lorentzian backgrounds and decoupling limits of M-theory, to appear.
E. Bergshoeff et al., Generalized Newton-Cartan geometries for particles and strings, Class. Quant. Grav. 40 (2023) 075010 [arXiv:2207.00363] [INSPIRE].
Z. Yan, Torsional deformation of nonrelativistic string theory, JHEP 09 (2021) 035 [arXiv:2106.10021] [INSPIRE].
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge Univ. Press, Cambridge, U.K. (2012). [https://doi.org/10.1017/cbo9781139026833].
T. Banks and N. Seiberg, Strings from matrices, Nucl. Phys. B 497 (1997) 41 [hep-th/9702187] [INSPIRE].
L. Motl, Proposals on nonperturbative superstring interactions, hep-th/9701025 [INSPIRE].
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, Matrix string theory, Nucl. Phys. B 500 (1997) 43 [hep-th/9703030] [INSPIRE].
C. Blair, J. Lahnsteiner, N. Obers and Z. Yan, Unification of decoupling limits in string theory and M-theory, arXiv:2311.10564.
R. Gopakumar, S. Minwalla, N. Seiberg and A. Strominger, (OM) theory in diverse dimensions, JHEP 08 (2000) 008 [hep-th/0006062] [INSPIRE].
J. Hartong and E. Have, Nonrelativistic Expansion of Closed Bosonic Strings, Phys. Rev. Lett. 128 (2022) 021602 [arXiv:2107.00023] [INSPIRE].
J. Hartong and E. Have, Nonrelativistic approximations of closed bosonic string theory, JHEP 02 (2023) 153 [arXiv:2211.01795] [INSPIRE].
D. Hilbert, Ueber die Theorie der algebraischen Formen, in Algebra ∙ Invariantentheorie Geometrie, Springer (1970), p. 199–257 [https://doi.org/10.1007/978-3-662-26737-0_16].
D. Hilbert, Über die vollen Invariantensysteme, in Algebra ∙ Invariantentheorie Geometrie, Springer (1970), p. 287–344 [https://doi.org/10.1007/978-3-662-26737-0_19].
K. Iwasawa, On some types of topological groups, Annals Math. 50 (1949) 507.
M.B. Green and M. Gutperle, Effects of D instantons, Nucl. Phys. B 498 (1997) 195 [hep-th/9701093] [INSPIRE].
L.J. Romans, Massive N = 2a Supergravity in Ten-Dimensions, Phys. Lett. B 169 (1986) 374 [INSPIRE].
I. Jeon, K. Lee, J.-H. Park and Y. Suh, Stringy Unification of Type IIA and IIB Supergravities under N = 2 D = 10 Supersymmetric Double Field Theory, Phys. Lett. B 723 (2013) 245 [arXiv:1210.5078] [INSPIRE].
S.M. Ko, C. Melby-Thompson, R. Meyer and J.-H. Park, Dynamics of Perturbations in Double Field Theory & Non-Relativistic String Theory, JHEP 12 (2015) 144 [arXiv:1508.01121] [INSPIRE].
D.S. Berman, C.D.A. Blair and R. Otsuki, Non-Riemannian geometry of M-theory, JHEP 07 (2019) 175 [arXiv:1902.01867] [INSPIRE].
D. Hansen, J. Hartong and N.A. Obers, Action Principle for Newtonian Gravity, Phys. Rev. Lett. 122 (2019) 061106 [arXiv:1807.04765] [INSPIRE].
A.D. Gallegos, U. Gürsoy, S. Verma and N. Zinnato, Non-Riemannian gravity actions from double field theory, JHEP 06 (2021) 173 [arXiv:2012.07765] [INSPIRE].
J.J. Fernández-Melgarejo and L. Romano, work in progress.
E. Bergshoeff, C.D.A. Blair, J. Lahnsteiner and J. Rosseel, Membrane Newton-Cartan Supergravity in Eleven Dimensions, to appear.
E.A. Bergshoeff, M. de Roo, S.F. Kerstan and F. Riccioni, IIB supergravity revisited, JHEP 08 (2005) 098 [hep-th/0506013] [INSPIRE].
C.D.A. Blair, D. Gallegos and N. Zinnato, A non-relativistic limit of M-theory and 11-dimensional membrane Newton-Cartan geometry, JHEP 10 (2021) 015 [arXiv:2104.07579] [INSPIRE].
Acknowledgments
We would like to thank Chris Blair, Niels Obers, Gerben Oling, and Luca Romano for useful discussions. In particular, we thank Stephen Ebert for valuable discussions and comments on a draft of the paper. We would also like to thank the organizers of the workshop on Beyond Lorentzian Geometry II at ICMS, Edinburgh in February 2023 and the workshop on Non-Relativistic Strings and Beyond at Nordita, Stockholm in May 2023, where this work was presented and completed. K.T.G. has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101024967. Z.Y. is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 31003710. U.Z. is supported by TUBITAK - 2218 National Postdoctoral Research Fellowship Program with grant number 118C512. Nordita is supported in part by NordForsk.
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Bergshoeff, E.A., Grosvenor, K.T., Lahnsteiner, J. et al. Non-Lorentzian IIB supergravity from a polynomial realization of SL(2, ℝ). J. High Energ. Phys. 2023, 22 (2023). https://doi.org/10.1007/JHEP12(2023)022
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DOI: https://doi.org/10.1007/JHEP12(2023)022