Abstract
We study a decoupling limit of M-theory where the three-form gauge potential becomes critical. This limit leads to nonrelativistic M-theory coupled to a non-Lorentzian spacetime geometry. Nonrelativistic M-theory is U-dual to M-theory in the discrete light cone quantization, a non-perturbative approach related to the Matrix theory description of M-theory. We focus on the compactification of nonrelativistic M-theory over a two-torus that exhibits anisotropic behaviors due to the foliation structure of the spacetime geometry. We develop a frame covariant formalism of the toroidal geometry, which provides a geometrical interpretation of the recently discovered polynomial realization of SL(2 , ℤ) duality in nonrelativistic type IIB superstring theory. We will show that the nonrelativistic IIB string background fields transform as polynomials of an effective Galilean “boost velocity” on the two-torus. As an application, we construct an action principle describing a single M5-brane in nonrelativistic M-theory and study its compactification over the anisotropic two-torus. This procedure leads to a D3-brane action in nonrelativistic IIB string theory that makes the SL(2 , ℤ) invariance manifest in the polynomial realization.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].
B. de Wit, J. Hoppe and H. Nicolai, On the quantum mechanics of supermembranes, Nucl. Phys. B 305 (1988) 545 [INSPIRE].
L. Susskind, Another conjecture about M(atrix) theory, hep-th/9704080 [INSPIRE].
J.B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51 (1979) 659 [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].
J. Gomis and H. Ooguri, Nonrelativistic closed string theory, J. Math. Phys. 42 (2001) 3127 [hep-th/0009181] [INSPIRE].
J.A. Garcia, A. Guijosa and J.D. Vergara, A membrane action for OM theory, Nucl. Phys. B 630 (2002) 178 [hep-th/0201140] [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].
S. Ebert, H.-Y. Sun and Z. Yan, Dual D-brane actions in nonrelativistic string theory, JHEP 04 (2022) 161 [arXiv:2112.09316] [INSPIRE].
J. Brugues, J. Gomis and K. Kamimura, Newton-Hooke algebras, non-relativistic branes and generalized pp-wave metrics, Phys. Rev. D 73 (2006) 085011 [hep-th/0603023] [INSPIRE].
D. Pereñiguez, p-brane Newton-Cartan geometry, J. Math. Phys. 60 (2019) 112501 [arXiv:1908.04801] [INSPIRE].
E. Bergshoeff, J.M. Izquierdo and L. Romano, Carroll versus Galilei from a brane perspective, JHEP 10 (2020) 066 [arXiv:2003.03062] [INSPIRE].
E. Bergshoeff et al., p-brane Galilean and Carrollian geometries and gravities, arXiv:2308.12852 [INSPIRE].
C.D.A. Blair, J. Lahnsteiner, N.A.J. Obers and Z. Yan, Unification of decoupling limits in string and M-theory, arXiv:2311.10564 [INSPIRE].
E. Bergshoeff, J. Gomis and Z. Yan, Nonrelativistic string theory and T-duality, JHEP 11 (2018) 133 [arXiv:1806.06071] [INSPIRE].
U.H. Danielsson, A. Guijosa and M. Kruczenski, IIA/B, wound and wrapped, JHEP 10 (2000) 020 [hep-th/0009182] [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].
G. Oling and Z. Yan, Aspects of nonrelativistic strings, Front. in Phys. 10 (2022) 832271 [arXiv:2202.12698] [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].
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].
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].
R. Gopakumar, S. Minwalla, N. Seiberg and A. Strominger, (OM) theory in diverse dimensions, JHEP 08 (2000) 008 [hep-th/0006062] [INSPIRE].
J. Gomis and Z. Yan, Worldsheet formalism for decoupling limits in string theory, arXiv:2311.10565 [INSPIRE].
J.H. Schwarz, Superstring dualities, Nucl. Phys. B Proc. Suppl. 49 (1996) 183 [hep-th/9509148] [INSPIRE].
J.H. Schwarz, The power of M theory, Phys. Lett. B 367 (1996) 97 [hep-th/9510086] [INSPIRE].
P.S. Aspinwall, Some relationships between dualities in string theory, Nucl. Phys. B Proc. Suppl. 46 (1996) 30 [hep-th/9508154] [INSPIRE].
E.A. Bergshoeff et al., Branched SL(2, Z) duality, JHEP 10 (2022) 131 [arXiv:2208.13815] [INSPIRE].
E.A. Bergshoeff et al., Non-Lorentzian IIB supergravity from a polynomial realization of SL(2, R), arXiv:2306.04741 [INSPIRE].
D. Roychowdhury, Nonrelativistic expansion of M5 brane and M theory background, arXiv:2212.03458 [INSPIRE].
E. Bergshoeff, D.S. Berman, J.P. van der Schaar and P. Sundell, Critical fields on the M5-brane and noncommutative open strings, Phys. Lett. B 492 (2000) 193 [hep-th/0006112] [INSPIRE].
D. Berman, M5 on a torus and the three-brane, Nucl. Phys. B 533 (1998) 317 [hep-th/9804115] [INSPIRE].
D. Berman, The M5-brane on a torus, Lect. Notes Phys. 525 (1999) 398 [hep-th/9812053] [INSPIRE].
P. Pasti, D.P. Sorokin and M. Tonin, Covariant action for a D = 11 five-brane with the chiral field, Phys. Lett. B 398 (1997) 41 [hep-th/9701037] [INSPIRE].
P. Pasti, D.P. Sorokin and M. Tonin, Duality symmetric actions with manifest space-time symmetries, Phys. Rev. D 52 (1995) R4277 [hep-th/9506109] [INSPIRE].
P. Pasti, D.P. Sorokin and M. Tonin, On Lorentz invariant actions for chiral p forms, Phys. Rev. D 55 (1997) 6292 [hep-th/9611100] [INSPIRE].
E.A. Bergshoeff et al., String theory and string Newton-Cartan geometry, J. Phys. A 53 (2020) 014001 [arXiv:1907.10668] [INSPIRE].
E.A. Bergshoeff et al., A non-relativistic limit of NS-NS gravity, JHEP 06 (2021) 021 [arXiv:2102.06974] [INSPIRE].
L. Bidussi et al., Torsional string Newton-Cartan geometry for non-relativistic strings, JHEP 02 (2022) 116 [arXiv:2107.00642] [INSPIRE].
E.A. Bergshoeff et al., Non-relativistic ten-dimensional minimal supergravity, JHEP 12 (2021) 123 [arXiv:2107.14636] [INSPIRE].
J. Gomis, J. Oh and Z. Yan, Nonrelativistic string theory in background fields, JHEP 10 (2019) 101 [arXiv:1905.07315] [INSPIRE].
Z. Yan, Torsional deformation of nonrelativistic string theory, JHEP 09 (2021) 035 [arXiv:2106.10021] [INSPIRE].
J. Gomis, Z. Yan and M. Yu, Nonrelativistic open string and Yang-Mills theory, JHEP 03 (2021) 269 [arXiv:2007.01886] [INSPIRE].
J. Klusoň and P. Novosad, Non-relativistic M2-brane, JHEP 06 (2019) 072 [arXiv:1903.12450] [INSPIRE].
D. Roychowdhury, Nonrelativistic expansion of M2 branes and M theory backgrounds, JHEP 11 (2022) 152 [arXiv:2208.05646] [INSPIRE].
A. Candiello and K. Lechner, Duality in supergravity theories, Nucl. Phys. B 412 (1994) 479 [hep-th/9309143] [INSPIRE].
O. Aharony, String theory dualities from M theory, Nucl. Phys. B 476 (1996) 470 [hep-th/9604103] [INSPIRE].
J. Gomis, K. Kamimura and P.K. Townsend, Non-relativistic superbranes, JHEP 11 (2004) 051 [hep-th/0409219] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
M. Perry and J.H. Schwarz, Interacting chiral gauge fields in six-dimensions and Born-Infeld theory, Nucl. Phys. B 489 (1997) 47 [hep-th/9611065] [INSPIRE].
J.H. Schwarz, Coupling a selfdual tensor to gravity in six-dimensions, Phys. Lett. B 395 (1997) 191 [hep-th/9701008] [INSPIRE].
I.A. Bandos et al., Covariant action for the superfive-brane of M theory, Phys. Rev. Lett. 78 (1997) 4332 [hep-th/9701149] [INSPIRE].
E. Bergshoeff, D.P. Sorokin and P.K. Townsend, The M5-brane Hamiltonian, Nucl. Phys. B 533 (1998) 303 [hep-th/9805065] [INSPIRE].
E.A. Bergshoeff et al., SL(2, R)-invariant IIB brane actions, JHEP 02 (2007) 007 [hep-th/0611036] [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].
R. Gopakumar, J.M. Maldacena, S. Minwalla and A. Strominger, S duality and noncommutative gauge theory, JHEP 06 (2000) 036 [hep-th/0005048] [INSPIRE].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
C. Vafa, Evidence for F theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
S.A. Cherkis and J.H. Schwarz, Wrapping the M theory five-brane on K3, Phys. Lett. B 403 (1997) 225 [hep-th/9703062] [INSPIRE].
J. Park and W. Sim, Supersymmetric heterotic action out of M5 brane, JHEP 08 (2009) 047 [arXiv:0905.2393] [INSPIRE].
U.H. Danielsson and G. Ferretti, The heterotic life of the D particle, Int. J. Mod. Phys. A 12 (1997) 4581 [hep-th/9610082] [INSPIRE].
S. Kachru and E. Silverstein, On gauge bosons in the matrix model approach to M theory, Phys. Lett. B 396 (1997) 70 [hep-th/9612162] [INSPIRE].
L. Motl and L. Susskind, Finite N heterotic matrix models and discrete light cone quantization, hep-th/9708083 [INSPIRE].
M. Heydeman, J.H. Schwarz and C. Wen, M5-brane and D-brane scattering amplitudes, JHEP 12 (2017) 003 [arXiv:1710.02170] [INSPIRE].
Z. Yan and M. Yu, KLT factorization of nonrelativistic string amplitudes, JHEP 04 (2022) 068 [arXiv:2112.00025] [INSPIRE].
M. Dine and A. Rajaraman, Multigraviton scattering in the matrix model, Phys. Lett. B 425 (1998) 77 [hep-th/9710174] [INSPIRE].
M.R. Douglas and H. Ooguri, Why matrix theory is hard, Phys. Lett. B 425 (1998) 71 [hep-th/9710178] [INSPIRE].
K. Iwasawa, On some types of topological groups, Ann. Math. 50 (1949) 507.
N.A. Obers and B. Pioline, U duality and M theory, Phys. Rept. 318 (1999) 113 [hep-th/9809039] [INSPIRE].
Acknowledgments
We would like to thank Eric Bergshoeff, Chris Blair, Eric D’Hoker, Thomas Dumitrescu, Kevin Grosvenor, Johannes Lahnsteiner, Per Kraus, Florian Niedermann, Niels Obers, and Utku Zorba for their useful discussions and comments on the paper. The main results of this paper were presented on February 8, 2023, at the Beyond Lorentzian Geometry II workshop held at ICMS in Edinburgh and on May 9, 2023, at the Non-Relativistic Strings and Beyond workshop held at Nordita in Stockholm. S.E. is supported by the Bhaumik Institute. Z.Y. is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 31003710. Nordita is supported in part by NordForsk.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2309.04912
Rights and permissions
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.
About this article
Cite this article
Ebert, S., Yan, Z. Anisotropic compactification of nonrelativistic M-theory. J. High Energ. Phys. 2023, 135 (2023). https://doi.org/10.1007/JHEP11(2023)135
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2023)135