Abstract
We consider the general \( \mathcal{N}=4 \), d = 3 Galilean superalgebra with arbitrary central charges and study its dynamical realizations. Using the nonlinear realization techniques, we introduce a class of actions for \( \mathcal{N}=4 \) three-dimensional non-relativistic superparticle, such that they are linear in the central charge Maurer-Cartan one-forms. As a prerequisite to the quantization, we analyze the phase space constraints structure of our model for various choices of the central charges. The first class constraints generate gauge transformations, involving fermionic κ-gauge transformations. The quantization of the model gives rise to the collection of free \( \mathcal{N}=4 \), d = 3 Galilean superfields, which can be further employed, e.g., for description of three-dimensional non-relativistic \( \mathcal{N}=4 \) supersymmetric theories.
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References
C. Duval and P.A. Horvathy, Non-relativistic conformal symmetries and Newton-Cartan structures, J. Phys. A 42 (2009) 465206 [arXiv:0904.0531] [INSPIRE].
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].
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.A. Bergshoeff and J. Rosseel, Three-Dimensional Extended Bargmann Supergravity, Phys. Rev. Lett. 116 (2016) 251601 [arXiv:1604.08042] [INSPIRE].
J.M. Lévy-Leblond, Galilei group and Galilean invariance, in Group Theory and its Applications, vol. II, E.M. Loebl eds., Acad. Press, N.Y., U.S.A. (1971), p. 221.
D.R. Grigore, The projective unitary irreducible representations of the Galilei group in (1+2)-dimensions, J. Math. Phys. 37 (1996) 460 [hep-th/9312048] [INSPIRE].
J. Lukierski, P.C. Stichel and W.J. Zakrzewski, Galilean-invariant (2+1)-dimensional models with a Chern-Simons-like term and D = 2 noncommutative geometry, Annals Phys. 260 (1997) 224 [hep-th/9612017] [INSPIRE].
R. Jackiw and V.P. Nair, Anyon spin and the exotic central extension of the planar Galilei group, Phys. Lett. B 480 (2000) 237 [hep-th/0003130] [INSPIRE].
J. Lukierski, I. Próchnicka, P.C. Stichel and W.J. Zakrzewski, Galilean exotic planar supersymmetries and nonrelativistic supersymmetric wave equations, Phys. Lett. B 639 (2006) 389 [hep-th/0602198] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
J. Lukierski, D = 4 Extended Galilei Superalgebras with Central Charges, Phys. Lett. B 694 (2011) 478 [arXiv:1009.0182] [INSPIRE].
K. Jensen and A. Karch, Revisiting non-relativistic limits, JHEP 04 (2015) 155 [arXiv:1412.2738] [INSPIRE].
E.A. Bergshoeff, J. Rosseel and P.K. Townsend, On nonrelativistic 3D Spin-1 theories, in 12th International Workshop on Supersymmetries and Quantum Symmetries (SQS’17) Dubna, Russia, July 31 – August 5, 2017, arXiv:1801.02527 [INSPIRE].
R. Casalbuoni, The Classical Mechanics for Bose-Fermi Systems, Nuovo Cim. A 33 (1976) 389 [INSPIRE].
J.H. Schwarz, Quantum Superspace Representation of Superconformal Algebras, Nucl. Phys. B 185 (1981) 221 [INSPIRE].
A. Frydryszak, N Extended Free Superfields (N = 2, N = 4, N = 6, N = 8) From Quantization of Supersymmetric Particle Model, Phys. Rev. D 30 (1984) 2172 [INSPIRE].
A. Frydryszak, Multichiral free superfields with one central charge for N = 2, N = 4, N = 6, N = 8, Phys. Rev. D 35 (1987) 2432 [INSPIRE].
A. Barducci, R. Casalbuoni and L. Lusanna, Supersymmetries and the Pseudoclassical Relativistic electron, Nuovo Cim. A 35 (1976) 377 [INSPIRE].
F.A. Berezin and M.S. Marinov, Particle Spin Dynamics as the Grassmann Variant of Classical Mechanics, Annals Phys. 104 (1977) 336 [INSPIRE].
P.S. Howe, S. Penati, M. Pernici and P.K. Townsend, Wave Equations for Arbitrary Spin From Quantization of the Extended Supersymmetric Spinning Particle, Phys. Lett. B 215 (1988) 555 [INSPIRE].
P.S. Howe, S. Penati, M. Pernici and P.K. Townsend, A Particle Mechanics Description of Antisymmetric Tensor Fields, Class. Quant. Grav. 6 (1989) 1125 [INSPIRE].
A. Barducci, R. Casalbuoni and J. Gomis, Non-relativistic Spinning Particle in a Newton-Cartan Background, JHEP 01 (2018) 002 [arXiv:1710.10970] [INSPIRE].
C. Duval and P.A. Horvathy, On Schrödinger superalgebras, J. Math. Phys. 35 (1994) 2516 [hep-th/0508079] [INSPIRE].
S. Weinberg, Dynamics at infinite momentum, Phys. Rev. 150 (1966) 1313 [INSPIRE].
L. Susskind, Hadronic Currents, Phys. Rev. 165 (1968) 1547 [INSPIRE].
T. Heinzl, Light cone quantization: Foundations and applications, Lect. Notes Phys. 572 (2001) 55 [hep-th/0008096] [INSPIRE].
J. Gomis, K. Kamimura and P.K. Townsend, Non-relativistic superbranes, JHEP 11 (2004) 051 [hep-th/0409219] [INSPIRE].
J. Lukierski, P.C. Stichel and W.J. Zakrzewski, Exotic Galilean conformal symmetry and its dynamical realisations, Phys. Lett. A 357 (2006) 1 [hep-th/0511259] [INSPIRE].
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].
K. Kamimura and S. Onda, Contractions of AdS brane algebra and superGalileon Lagrangians, J. Math. Phys. 54 (2013) 062503 [arXiv:1303.5506] [INSPIRE].
V. Bargmann, On unitary ray representations of continuous groups, Annals Math. 59 (1954) 1.
R. Puzalowski, Galilean supersymmetry, Acta Phys. Austriaca 50 (1978) 45 [INSPIRE].
E. Witten and D.I. Olive, Supersymmetry Algebras That Include Topological Charges, Phys. Lett. B 78 (1978) 97 [INSPIRE].
Z. Hlousek and D. Spector, Topological charges and central charges in (3+1)-dimensional supersymmetry, Phys. Lett. B 283 (1992) 75 [INSPIRE].
O. Bergman and C.B. Thorn, SuperGalilei invariant field theories in (2+1)-dimensions, Phys. Rev. D 52 (1995) 5997 [hep-th/9507007] [INSPIRE].
E. Bergshoeff, J. Gomis, M. Kovačević, L. Parra, J. Rosseel and T. Zojer, Nonrelativistic superparticle in a curved background, Phys. Rev. D 90 (2014) 065006 [arXiv:1406.7286] [INSPIRE].
R. Haag, J.T. Lopuszanski and M. Sohnius, All Possible Generators of Supersymmetries of the S-Matrix, Nucl. Phys. B 88 (1975) 257 [INSPIRE].
B. Zumino, Normal forms of complex matrices, J. Math. Phys. 3 (1962) 1055.
S. Ferrara, C.A. Savoy and B. Zumino, General Massive Multiplets in Extended Supersymmetry, Phys. Lett. B 100 (1981) 393 [INSPIRE].
M. Sohnius, K.S. Stelle and P.C. West, Off Mass Shell Formulation of Extended Supersymmetric Gauge Theories, Phys. Lett. B 92 (1980) 123 [INSPIRE].
M.F. Sohnius, K.S. Stelle and P.C. West, Dimensional reduction by Legendre transformation generates off-shell supersymmetric Yang-Mills theories, Nucl. Phys. B 173 (1980) 127 [INSPIRE].
M. Le Bellac and J.M. Lévy-Leblond, Galilean electromagnetism, Nuovo Cim. B 14 (1973) 217.
A. Bagchi, R. Basu and A. Mehra, Galilean Conformal Electrodynamics, JHEP 11 (2014) 061 [arXiv:1408.0810] [INSPIRE].
A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Galilean Yang-Mills Theory, JHEP 04 (2016) 051 [arXiv:1512.08375] [INSPIRE].
S. Fedoruk and J. Lukierski, The algebraic structure of Galilean superconformal symmetries, Phys. Rev. D 84 (2011) 065002 [arXiv:1105.3444] [INSPIRE].
J. Lukierski, Holomorphic and real Euclidean supersymmetries in three-dimensions and four-dimensions, Czech. J. Phys. B 37 (1987) 359 [INSPIRE].
T. Kugo and P.K. Townsend, Supersymmetry and the Division Algebras, Nucl. Phys. B 221 (1983) 357 [INSPIRE].
J. Lukierski and A. Nowicki, Quaternionic Supergroups and D = 4 Euclidean Extended Supersymmetries, Annals Phys. 166 (1986) 164 [INSPIRE].
B. Zumino, Nonlinear Realization of Supersymmetry in de Sitter Space, Nucl. Phys. B 127 (1977) 189 [INSPIRE].
E.A. Ivanov and V.I. Ogievetsky, The Inverse Higgs Phenomenon in Nonlinear Realizations, Teor. Mat. Fiz. 25 (1975) 164 [INSPIRE].
J.A. de Azcarraga and J. Lukierski, Supersymmetric Particles with Internal Symmetries and Central Charges, Phys. Lett. B 113 (1982) 170 [INSPIRE].
W. Siegel, Hidden Local Supersymmetry in the Supersymmetric Particle Action, Phys. Lett. B 128 (1983) 397 [INSPIRE].
J.P. Gauntlett, G.W. Gibbons, C.M. Hull and P.K. Townsend, BPS states of D = 4 N = 1 supersymmetry, Commun. Math. Phys. 216 (2001) 431 [hep-th/0001024] [INSPIRE].
S. Fedoruk and V.G. Zima, Uniform twistor-like formulation of massive and massless superparticles with tensorial central charges, Nucl. Phys. Proc. Suppl. 102 (2001) 233 [hep-th/0104178] [INSPIRE].
J.A. de Azcarraga and J. Lukierski, Gupta-Bleuler Quantization of Massive Superparticle Models in D = 6, D = 8 and D = 10, Phys. Rev. D 38 (1988) 509 [INSPIRE].
Z. Hasiewicz, J. Kowalski-Glikman, J. Lukierski and J.W. van Holten, BRST Formulation of the Gupta-blEuler Quantization Method, J. Math. Phys. 32 (1991) 2358 [INSPIRE].
J.A. de Azcarraga and J. Lukierski, Supersymmetric Particle Model With Additional Bosonic Coordinates, Z. Phys. C 30 (1986) 221 [INSPIRE].
I.L. Buchbinder, E.A. Ivanov and I.B. Samsonov, The low-energy N = 4 SYM effective action in diverse harmonic superspaces, Phys. Part. Nucl. 48 (2017) 333 [arXiv:1603.02768] [INSPIRE].
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Fedoruk, S., Ivanov, E. & Lukierski, J. From \( \mathcal{N}=4 \) Galilean superparticle to three-dimensional non-relativistic \( \mathcal{N}=4 \) superfields. J. High Energ. Phys. 2018, 19 (2018). https://doi.org/10.1007/JHEP05(2018)019
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DOI: https://doi.org/10.1007/JHEP05(2018)019