Abstract
We formulate the equations which determine a potential function in an \( \mathcal{N} \) = 1 higher derivative supersymmetric mechanics compatible with the osp(2|1)⊕so(d) symmetry and provide a few explicit examples.
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V.P. Akulov and A.I. Pashnev, Quantum superconformal model in (1, 2) space, Theor. Math. Phys. 56 (1983) 862 [INSPIRE].
S. Fubini and E. Rabinovici, Superconformal quantum mechanics, Nucl. Phys. B 245 (1984) 17 [INSPIRE].
E.A. Ivanov, S.O. Krivonos and V.M. Leviant, Geometric superfield approach to superconformal mechanics, J. Phys. A 22 (1989) 4201 [INSPIRE].
D.Z. Freedman and P.F. Mende, An exactly solvable N particle system in supersymmetric quantum mechanics, Nucl. Phys. B 344 (1990) 317 [INSPIRE].
V.K. Dobrev, Non-relativistic holography — A group-theoretical perspective, Int. J. Mod. Phys. A 29 (2014) 1430001 [arXiv:1312.0219] [INSPIRE].
P. Claus et al., Black holes and superconformal mechanics, Phys. Rev. Lett. 81 (1998) 4553 [hep-th/9804177] [INSPIRE].
G.W. Gibbons and P.K. Townsend, Black holes and Calogero models, Phys. Lett. B 454 (1999) 187 [hep-th/9812034] [INSPIRE].
J. Michelson and A. Strominger, Superconformal multiblack hole quantum mechanics, JHEP 09 (1999) 005 [hep-th/9908044] [INSPIRE].
A. Maloney, M. Spradlin and A. Strominger, Superconformal multiblack hole moduli spaces in four-dimensions, JHEP 04 (2002) 003 [hep-th/9911001] [INSPIRE].
A.V. Galajinsky, Comments on N = 4 superconformal extension of the Calogero model, Mod. Phys. Lett. A 18 (2003) 1493 [hep-th/0302156] [INSPIRE].
S. Fedoruk, E. Ivanov and O. Lechtenfeld, Superconformal Mechanics, J. Phys. A 45 (2012) 173001 [arXiv:1112.1947] [INSPIRE].
A. Galajinsky, \( \mathcal{N} \)= 4 superconformal mechanics from the SU(2) perspective, JHEP 02 (2015) 091 [arXiv:1412.4467] [INSPIRE].
E. Ivanov, S. Sidorov and F. Toppan, Superconformal mechanics in SU(2|1) superspace, Phys. Rev. D 91 (2015) 085032 [arXiv:1501.05622] [INSPIRE].
S. Fedoruk and E. Ivanov, New realizations of the supergroup D(2, 1; α) in \( \mathcal{N} \)= 4 superconformal mechanics, JHEP 10 (2015) 087 [arXiv:1507.08584] [INSPIRE].
A. Galajinsky and O. Lechtenfeld, Superconformal SU(1, 1|n) mechanics, JHEP 09 (2016) 114 [arXiv:1606.05230] [INSPIRE].
E. Ivanov, O. Lechtenfeld and S. Sidorov, SU(2|2) supersymmetric mechanics, JHEP 11 (2016) 031 [arXiv:1609.00490] [INSPIRE].
A. Galajinsky, Couplings in D(2, 1; α) superconformal mechanics from the SU(2) perspective, JHEP 03 (2017) 054 [arXiv:1702.01955] [INSPIRE].
D. Chernyavsky, Super 0-brane action on the coset space of D(2, 1; α) supergroup, JHEP 09 (2017) 054 [arXiv:1707.00437] [INSPIRE].
N. Kozyrev et al., Curved Witten-Dijkgraaf-Verlinde-Verlinde equation and \( \mathcal{N} \) = 4 mechanics, Phys. Rev. D 96 (2017) 101702 [arXiv:1710.00884] [INSPIRE].
N. Kozyrev et al., \( \mathcal{N} \) = 4 supersymmetric mechanics on curved spaces, Phys. Rev. D 97 (2018) 085015 [arXiv:1711.08734] [INSPIRE].
S. Fedoruk, E. Ivanov, O. Lechtenfeld and S. Sidorov, Quantum SU(2|1) supersymmetric Calogero-Moser spinning systems, JHEP 04 (2018) 043 [arXiv:1801.00206] [INSPIRE].
D. Chernyavsky, SU(1, 1|N) superconformal mechanics with fermionic gauge symmetry, JHEP 04 (2018) 009 [arXiv:1803.00343] [INSPIRE].
S. Krivonos, O. Lechtenfeld and A. Sutulin, N-extended supersymmetric Calogero models, Phys. Lett. B 784 (2018) 137 [arXiv:1804.10825] [INSPIRE].
J. Mateos Guilarte and M.S. Plyushchay, Nonlinear symmetries of perfectly invisible PT-regularized conformal and superconformal mechanics systems, JHEP 01 (2019) 194 [arXiv:1806.08740] [INSPIRE].
L. Inzunza and M.S. Plyushchay, Hidden symmetries of rationally deformed superconformal mechanics, Phys. Rev. D 99 (2019) 025001 [arXiv:1809.08527] [INSPIRE].
D. Chernyavsky, On OSp(N|2) superconformal mechanics, JHEP 02 (2019) 170 [arXiv:1810.01626] [INSPIRE].
E. Ivanov, O. Lechtenfeld and S. Sidorov, Deformed N = 8 supersymmetric mechanics, Symmetry 11 (2019) 135.
S. Krivonos, O. Lechtenfeld, A. Provorov and A. Sutulin, Extended supersymmetric Calogero model, Phys. Lett. B 791 (2019) 385 [arXiv:1812.10168] [INSPIRE].
S. Krivonos, O. Lechtenfeld and A. Sutulin, Supersymmetric many-body Euler-Calogero-Moser model, Phys. Lett. B 790 (2019) 191 [arXiv:1812.03530] [INSPIRE].
A. Galajinsky and O. Lechtenfeld, Spinning extensions of D(2, 1; α) superconformal mechanics, JHEP 03 (2019) 069 [arXiv:1902.06851] [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].
P.A. Horvathy, Non-relativistic conformal and supersymmetries, Int. J. Mod. Phys. A 3 (1993) 339 [arXiv:0807.0513] [INSPIRE].
C. Duval and P.A. Horvathy, On Schrödinger superalgebras, J. Math. Phys. 35 (1994) 2516 [hep-th/0508079] [INSPIRE].
M. Henkel and J. Unterberger, Supersymmetric extensions of Schrödinger-invariance, Nucl. Phys. B 746 (2006) 155 [math-ph/0512024] [INSPIRE].
S. Fedoruk, E. Ivanov and O. Lechtenfeld, Supersymmetric Calogero models by gauging, Phys. Rev. D 79 (2009) 105015 [arXiv:0812.4276] [INSPIRE].
P.D. Alvarez, J.L. Cortes, P.A. Horvathy and M.S. Plyushchay, Super-extended noncommutative Landau problem and conformal symmetry, JHEP 03 (2009) 034 [arXiv:0901.1021] [INSPIRE].
A. Galajinsky and I. Masterov, Remark on quantum mechanics with N = 2 Schrödinger supersymmetry, Phys. Lett. B 675 (2009) 116 [arXiv:0902.2910] [INSPIRE].
A. Galajinsky, N = 2 superconformal Newton-Hooke algebra and many-body mechanics, Phys. Lett. B 680 (2009) 510 [arXiv:0906.5509] [INSPIRE].
A. Galajinsky and O. Lechtenfeld, Harmonic N = 2 mechanics, Phys. Rev. D 80 (2009) 065012 [arXiv:0907.2242] [INSPIRE].
A. Galajinsky, Conformal mechanics in Newton-Hooke spacetime, Nucl. Phys. B 832 (2010) 586 [arXiv:1002.2290] [INSPIRE].
P.A. Horvathy, M.S. Plyushchay and M. Valenzuela, Supersymmetry of the planar Dirac-Deser-Jackiw-Templeton system and of its non-relativistic limit, J. Math. Phys. 51 (2010) 092108 [arXiv:1002.4729] [INSPIRE].
S. Fedoruk and J. Lukierski, The algebraic structure of Galilean superconformal symmetries, Phys. Rev. D 84 (2011) 065002 [arXiv:1105.3444] [INSPIRE].
N. Kozyrev, S. Krivonos, O. Lechtenfeld and A. Nersessian, Higher-derivative N = 4 superparticle in three-dimensional spacetime, Phys. Rev. D 89 (2014) 045013 [arXiv:1311.4540] [INSPIRE].
I. Masterov, Dynamical realizations of N = 1 l-conformal Galilei superalgebra, J. Math. Phys. 55 (2014) 102901 [arXiv:1407.1438] [INSPIRE].
I. Masterov, Higher-derivative mechanics with N = 2 l-conformal Galilei supersymmetry, J. Math. Phys. 56 (2015) 022902 [arXiv:1410.5335] [INSPIRE].
I. Masterov, New realizations of \( \mathcal{N} \) = 2l-conformal Newton-Hooke superalgebra, Mod. Phys. Lett. A 30 (2015) 1550073 [arXiv:1412.1751] [INSPIRE].
S. Fedoruk, E. Ivanov and J. Lukierski, From \( \mathcal{N} \) = 4 Galilean superparticle to three-dimensional non-relativistic \( \mathcal{N} \) = 4 superfields, JHEP 05 (2018) 019 [arXiv:1803.03159] [INSPIRE].
M. Henkel, Local scale invariance and strongly anisotropic equilibrium critical systems, Phys. Rev. Lett. 78 (1997) 1940 [cond-mat/9610174] [INSPIRE].
J. Negro, M.A. del Olmo and A. Rodriguez-Marco, Nonrelativistic conformal groups, J. Math. Phys. 38 (1997) 3786.
U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta 45 (1972) 802 [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].
S. Fedoruk, E. Ivanov and J. Lukierski, Galilean Conformal mechanics from nonlinear realizations, Phys. Rev. D 83 (2011) 085013 [arXiv:1101.1658] [INSPIRE].
C. Duval and P. Horvathy, Conformal Galilei groups, Veronese curves and Newton-Hooke spacetimes, J. Phys. A 44 (2011) 335203 [arXiv:1104.1502] [INSPIRE].
J. Gomis and K. Kamimura, Schrödinger equations for higher order non-relativistic particles and N-galilean conformal symmetry, Phys. Rev. D 85 (2012) 045023 [arXiv:1109.3773] [INSPIRE].
K. Andrzejewski, J. Gonera and P. Máslanka, Nonrelativistic conformal groups and their dynamical realizations, Phys. Rev. D 86 (2012) 065009 [arXiv:1204.5950] [INSPIRE].
A. Galajinsky and I. Masterov, Dynamical realization of l-conformal Galilei algebra and oscillators, Nucl. Phys. B 866 (2013) 212 [arXiv:1208.1403] [INSPIRE].
K. Andrzejewski, J. Gonera and A. Kijanka-Dec, Nonrelativistic conformal transformations in Lagrangian formalism, Phys. Rev. D 87 (2013) 065012 [arXiv:1301.1531] [INSPIRE].
K. Andrzejewski, J. Gonera, P. Kosiński and P. Máslanka, On dynamical realizations of l-conformal Galilei groups, Nucl. Phys. B 876 (2013) 309 [arXiv:1305.6805] [INSPIRE].
K. Andrzejewski and J. Gonera, Dynamical interpretation of nonrelativistic conformal groups, Phys. Lett. B 721 (2013) 319 [INSPIRE].
A. Galajinsky and I. Masterov, On dynamical realizations of l-conformal Galilei and Newton-Hooke algebras, Nucl. Phys. B 896 (2015) 244 [arXiv:1503.08633] [INSPIRE].
D. Chernyavsky and A. Galajinsky, Ricci-flat spacetimes with l-conformal Galilei symmetry, Phys. Lett. B 754 (2016) 249 [arXiv:1512.06226] [INSPIRE].
D. Chernyavsky, Coset spaces and Einstein manifolds with l-conformal Galilei symmetry, Nucl. Phys. B 911 (2016) 471 [arXiv:1606.08224] [INSPIRE].
I. Masterov, Remark on higher-derivative mechanics with l-conformal Galilei symmetry, J. Math. Phys. 57 (2016) 092901 [arXiv:1607.02693] [INSPIRE].
S. Krivonos, O. Lechtenfeld and A. Sorin, Minimal realization of ℓ-conformal Galilei algebra, Pais-Uhlenbeck oscillators and their deformation, JHEP 10 (2016) 078 [arXiv:1607.03756] [INSPIRE].
O. Baranovsky, Higher-derivative generalization of conformal mechanics, J. Math. Phys. 58 (2017) 082903 [arXiv:1704.04880] [INSPIRE].
D. Chernyavsky and D. Sorokin, Three-dimensional (higher-spin) gravities with extended Schrödinger and l-conformal Galilean symmetries, JHEP 07 (2019) 156 [arXiv:1905.13154] [INSPIRE].
A. Galajinsky and I. Masterov, Dynamical realizations of l-conformal Newton-Hooke group, Phys. Lett. B 723 (2013) 190 [arXiv:1303.3419] [INSPIRE].
K. Andrzejewski, A. Galajinsky, J. Gonera and I. Masterov, Conformal Newton-Hooke symmetry of Pais-Uhlenbeck oscillator, Nucl. Phys. B 885 (2014) 150 [arXiv:1402.1297] [INSPIRE].
K. Andrzejewski, Hamiltonian formalisms and symmetries of the Pais-Uhlenbeck oscillator, Nucl. Phys. B 889 (2014) 333 [arXiv:1410.0479] [INSPIRE].
K. Andrzejewski, Conformal Newton-Hooke algebras, Niederer’s transformation and Pais-Uhlenbeck oscillator, Phys. Lett. B 738 (2014) 405 [arXiv:1409.3926] [INSPIRE].
I. Masterov, N = 2 supersymmetric extension of l-conformal Galilei algebra, J. Math. Phys. 53 (2012) 072904 [arXiv:1112.4924] [INSPIRE].
N. Aizawa, N = 2 Galilean superconformal algebras with central extension, J. Phys. A 45 (2012) 475203 [arXiv:1206.2708] [INSPIRE].
N. Aizawa, Z. Kuznetsova and F. Toppan, Chiral and real N = 2 supersymmetric l-conformal Galilei algebras, J. Math. Phys. 54 (2013) 093506 [arXiv:1307.5259] [INSPIRE].
A. Galajinsky and I. Masterov, N = 4l-conformal Galilei superalgebra, Phys. Lett. B 771 (2017) 401 [arXiv:1705.02814] [INSPIRE].
A. Galajinsky and S. Krivonos, N = 4 ℓ-conformal Galilei superalgebras inspired by D(2, 1; α) supermultiplets, JHEP 09 (2017) 131 [arXiv:1706.08300] [INSPIRE].
N. Aizawa, P.S. Isaac and J. Segar, ℤ2 × ℤ2 generalizations of N = 1 superconformal Galilei algebras and their representations, J. Math. Phys. 60 (2019) 023507 [arXiv:1808.09112] [INSPIRE].
R. de Lima Rodrigues, W. Pires de Almeida and I. Fonseca Neto, Supersymmetric classical mechanics: Free case, hep-th/0201242 [INSPIRE].
A. Galajinsky, O. Lechtenfeld and K. Polovnikov, Calogero models and nonlocal conformal transformations, Phys. Lett. B 643 (2006) 221 [hep-th/0607215] [INSPIRE].
I. Masterov, B. Merzlikin, work in progress.
P.O. Kazinski, S.L. Lyakhovich and A.A. Sharapov, Lagrange structure and quantization, JHEP 07 (2005) 076 [hep-th/0506093] [INSPIRE].
D.S. Kaparulin, S.L. Lyakhovich and A.A. Sharapov, Classical and quantum stability of higher-derivative dynamics, Eur. Phys. J. C 74 (2014) 3072 [arXiv:1407.8481] [INSPIRE].
D.S. Kaparulin and S.L. Lyakhovich, On the stability of a nonlinear oscillator with higher derivatives, Russ. Phys. J. 57 (2015) 1561 [INSPIRE].
D.S. Kaparulin, S.L. Lyakhovich and A.A. Sharapov, Stable interactions via proper deformations, J. Phys. A 49 (2016) 155204 [arXiv:1510.08365] [INSPIRE].
M. Henkel and S. Stoimenov, Infinite-dimensional meta-conformal Lie algebras in one and two spatial dimensions, J. Stat. Mech. (2019) 084009.
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Masterov, I., Merzlikin, B. Superfield approach to higher derivative \( \mathcal{N} \) = 1 superconformal mechanics. J. High Energ. Phys. 2019, 165 (2019). https://doi.org/10.1007/JHEP11(2019)165
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DOI: https://doi.org/10.1007/JHEP11(2019)165