×

Minimal unitary representation of \(\operatorname{SU}(2,2)\) and its deformations as massless conformal fields and their supersymmetric extensions. (English) Zbl 1312.81095

Summary: We study the minimal unitary representation (minrep) of \(\operatorname{SO}(4,2)\) over a Hilbert space of functions of three variables, obtained by quantizing its quasiconformal action on a five dimensional space. The minrep of \(\operatorname{SO}(4,2)\), which coincides with the minrep of \(\operatorname{SU}(2,2)\) similarly constructed, corresponds to a massless conformal scalar in four space-time dimensions. There exists a one-parameter family of deformations of the minrep of \(\operatorname{SU}(2,2)\). For positive (negative) integer values of the deformation parameter \(\zeta\), one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in \((0,\zeta/2)((-\zeta/2,0))\) representation of the \(\operatorname{SL}(2,\mathbb{C})\) subgroup. We construct the supersymmetric extensions of the minrep of \(\operatorname{SU}(2,2)\) and its deformations to those of \(\operatorname{SU}(2,2|N)\). The minimal unitary supermultiplet of \(\operatorname{SU}(2,2|4)\), in the undeformed case, simply corresponds to the massless \(N=4\) Yang-Mills supermultiplet in four dimensions. For each given nonzero integer value of \(\zeta\), one obtains a unique supermultiplet of massless conformal fields of higher spin. For \(\operatorname{SU}(2,2|4)\), these supermultiplets are simply the doubleton supermultiplets studied in the work of M. Gunaydin et al. [Nucl. Phys. B 534, No. 1–2, 96–120 (1998); erratum ibid. 538, No. 1–2, 531 (1999; Zbl 1041.83515)].{
©2010 American Institute of Physics}

MSC:

81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T60 Supersymmetric field theories in quantum mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory
20C35 Applications of group representations to physics and other areas of science
14D15 Formal methods and deformations in algebraic geometry
22E70 Applications of Lie groups to the sciences; explicit representations

Citations:

Zbl 1041.83515
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Joseph, A., Minimal realizations and spectrum generating algebras, Commun. Math. Phys., 36, 325 (1974) · Zbl 0285.17007 · doi:10.1007/BF01646204
[2] Vogan, D. A. Jr., Noncommutative Harmonic Analysis and Lie Groups (Marseille, 1980), 880, 506-535 (1981)
[3] Kostant, B., Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), 42, 85-124 (1990)
[4] Binegar, B.; Zierau, R., Unitarization of a singular representation of SO(p,q), Commun. Math. Phys., 138, 245 (1991) · Zbl 0748.22009 · doi:10.1007/BF02099491
[5] Kazhdan, D.; Savin, G., Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday, Part I (Ramat Aviv, 1989), 2, 209-223 (1990)
[6] Brylinski, R.; Kostant, B., Functional Analysis on the Eve of the 21st Century, Vol. 1 (New Brunswick, NJ, 1993), 131, 13-63 (1995) · Zbl 0851.22017
[7] Brylinski, R.; Kostant, B., Minimal representations, geometric quantization, and unitarity, Proc. Natl. Acad. Sci. U.S.A., 91, 6026 (1994) · Zbl 0803.58023 · doi:10.1073/pnas.91.13.6026
[8] Gross, B. H.; Wallach, N. R., Lie Theory and Geometry, 123, 289-304 (1994)
[9] Li, J.-S., Representation Theory of Lie Groups (Park City, UT, 1998), 8, 293-340 (2000)
[10] Kobayashi, T.; Ørsted, B., Analysis on the minimal representation of O(p,q). I. Realization via conformal geometry, Adv. Math., 180, 486 (2003) · Zbl 1046.22004 · doi:10.1016/S0001-8708(03)00012-4
[11] Kobayashi, T.; Ørsted, B., Analysis on the minimal representation O(p,q). II. Branching laws, Adv. Math., 180, 513 (2003) · Zbl 1049.22006 · doi:10.1016/S0001-8708(03)00013-6
[12] Kobayashi, T.; Ørsted, B., Analysis on the minimal representation of O(p,q). III. Ultrahyperbolic equations on \(<mml:math display=''inline`` overflow=''scroll``>\), Adv. Math., 180, 551 (2003) · Zbl 1039.22005 · doi:10.1016/S0001-8708(03)00014-8
[13] Gover, A. R.; Waldron, A., The SO(d+2,2) minimal representation and ambient tractors: The conformal geometry of momentum space · Zbl 1208.81098
[14] 14.D.Kazhdan, B.Pioline, and A.Waldron, “Minimal representations, spherical vectors and exceptional theta series. I,” Commun. Math. Phys.CMPHAY0010-3616226, 1 (2002);10.1007/s002200200601e-print arXiv:hep-th/0107222. · Zbl 1143.11316
[15] 15.M.Günaydin, K.Koepsell, and H.Nicolai, “Conformal and quasiconformal realizations of exceptional Lie groups,” Commun. Math. Phys.CMPHAY0010-3616221, 57 (2001);10.1007/PL00005574e-print arXiv:hep-th/0008063. · Zbl 0992.22016
[16] 16.M.Günaydin and O.Pavlyk, “Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups,” J. High Energy Phys.JHEPFG1126-670808, (2005) 101;10.1088/1126-6708/2005/08/101e-print arXiv:hep-th/0506010.
[17] 17.M.Günaydin, K.Koepsell, and H.Nicolai, “The minimal unitary representation of <mml:math display=”inline“ overflow=”scroll“>E8(8),” Adv. Theor. Math. Phys.1095-07615, 923 (2002); e-print arXiv:hep-th/0109005. · Zbl 1029.17008
[18] Günaydin, M.; Sierra, G.; Townsend, P. K., Exceptional supergravity theories and the magic square, Phys. Lett., 133B, 72 (1983)
[19] 19.M.Günaydin and O.Pavlyk, “Minimal unitary realizations of exceptional U-duality groups and their subgroups as quasiconformal groups,” J. High Energy Phys.JHEPFG1126-670801, (2005) 019;10.1088/1126-6708/2005/01/019e-print arXiv:hep-th/0409272.
[20] 20.M.Günaydin and O.Pavlyk, “A unified approach to the minimal unitary realizations of noncompact groups and supergroups,” J. High Energy Phys.JHEPFG1126-670809, (2006) 050;10.1088/1126-6708/2006/09/050e-print arXiv:hep-th/0604077.
[21] 21.M.Günaydin, A.Neitzke, O.Pavlyk, and B.Pioline, “Quasi-conformal actions, quaternionic discrete series and twistors: SU(2,1) and <mml:math display=”inline“ overflow=”scroll“>G2(2),” Commun. Math. Phys.CMPHAY0010-3616283, 169 (2008);10.1007/s00220-008-0563-9e-print arXiv:0707.1669. · Zbl 1154.22023
[22] Gross, B. H.; Wallach, N. R., On quaternionic discrete series representations, and their continuations, J. Reine Angew. Math., 481, 73 (1996) · Zbl 0857.22012
[23] 23.M.Gunaydin and O.Pavlyk, “Spectrum generating conformal and quasiconformal U-duality goups, supergravity and spherical vectors,” JHEP070, 1004 (2010); e-print arXiv:0901.1646. · Zbl 1272.83084
[24] 24.M.Gunaydin and O.Pavlyk, “Quasiconformal realizations of <mml:math display=”inline“ overflow=”scroll“>E(6(6), <mml:math display=”inline“ overflow=”scroll“>E(7(7), <mml:math display=”inline“ overflow=”scroll“>E(8(8), and <mml:math display=”inline“ overflow=”scroll“>SO(n+3,m+3), <mml:math display=”inline“ overflow=”scroll“>E=4, and <mml:math display=”inline“ overflow=”scroll“>N>4 supergravity and spherical vectors,” Adv. Theor. Math. Phys.1095-076113, 1 (2009); e-print arXiv:0904.0784. · Zbl 1208.83105
[25] 25.J. M.Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys.1095-07612, 231 (1998); e-print arXiv:hep-th/9711200. · Zbl 0914.53047
[26] Gunaydin, M.; Marcus, N., The spectrum of the \(<mml:math display=''inline`` overflow=''scroll``>\) compactification of the chiral N=2, D=10 supergravity and the unitary supermultiplets of U(2,2/4), Class. Quantum Grav., 2, L11 (1985) · Zbl 0575.53060 · doi:10.1088/0264-9381/2/2/001
[27] 27.M.Gunaydin, D.Minic, and M.Zagermann, “4D doubleton conformal theories, CPT and IIB string on AdS(5) x S(5),” Nucl. Phys. BNUPBBO0550-3213534, 96 (1998);10.1016/S0550-3213(98)00543-4e-print arXiv:hep-th/9806042. · Zbl 1041.83515
[28] 28.M.Gunaydin, D.Minic, and M.Zagermann, “Novel supermultiplets of <mml:math display=”inline“ overflow=”scroll“>SU(2,2<mml:mo form=”infix“>∣4) and the <mml:math display=”inline“ overflow=”scroll“>AdS5/CFT4 duality,” Nucl. Phys. BNUPBBO0550-3213544, 737 (1999);10.1016/S0550-3213(99)00007-3e-print arXiv:hep-th/9810226. · Zbl 0944.81027
[29] Malkin, I. A.; Man’ko, V. I., Symmetry of the hydrogen atom, Soviet J. Nucl. Phys., 3, 267 (1966)
[30] Nambu, Y., Suppl. Prog. Theor. Phys., 37-38, 368 (1966) · doi:10.1143/PTPS.37.368
[31] Nambu, Y., Infinite-Component wave equations with hydrogenlike mass spectra, Phys. Rev., 160, 1171 (1967) · doi:10.1103/PhysRev.160.1171
[32] Barut, A. O.; Kleinert, H., Transition probabilities of the H-atom from noncompact dynamical groups, Phys. Rev., 156, 1541 (1967) · doi:10.1103/PhysRev.156.1541
[33] Barut, A. O.; Kleinert, H., Current operators and majorana equation for the hydrogen atom from dynamical groups, Phys. Rev., 157, 1180 (1967) · doi:10.1103/PhysRev.157.1180
[34] Barut, A. O.; Kleinert, H., Dynamical group O(4,2) for baryons and the behaviour of form factors, Phys. Rev., 161, 1464 (1967) · doi:10.1103/PhysRev.161.1464
[35] Mack, G.; Todorov, I., Irreducibility of the ladder representations of U(2,2) when restricted to the Poincare subgroup, J. Math. Phys., 10, 2078 (1969) · Zbl 0183.29003 · doi:10.1063/1.1664804
[37] Mack, G., All unitary ray represenations of the conformal group SU(2,2) with positive energy, Commun. Math. Phys., 55, 1 (1977) · Zbl 0352.22012 · doi:10.1007/BF01613145
[38] Knapp, A. W.; Speh, B., Irreducible unitary representations of SU(2,2), J. Funct. Anal., 45, 41 (1982) · Zbl 0543.22011 · doi:10.1016/0022-1236(82)90004-0
[39] Kac, V. G., Lie Superalgebras, Adv. Math., 26, 8 (1977) · Zbl 0366.17012 · doi:10.1016/0001-8708(77)90017-2
[40] Kac, V. G., A Sketch of lie superalgebra theory, Commun. Math. Phys., 53, 31 (1977) · Zbl 0359.17009 · doi:10.1007/BF01609166
[41] Dobrev, V. K.; Petkova, V. B., On the group theoretical approach to extended conformal supersymmetry: Classification of multiplets, Lett. Math. Phys., 9, 287 (1985) · Zbl 0585.17003 · doi:10.1007/BF00397755
[42] Dobrev, V. K.; Petkova, V. B., All positive energy unitary irreducible representations of extended conformal supersymmetry, Phys. Lett., 162B, 127 (1985) · Zbl 0585.17003
[43] Dirac, P. A. M., A remarkable representation of the 3+2 de Sitter group, J. Math. Phys., 4, 901 (1963) · Zbl 0125.40704 · doi:10.1063/1.1704016
[44] Flato, M.; Fronsdal, C., Quantum field theory of singletons: The Rac, J. Math. Phys., 22, 1100 (1981) · doi:10.1063/1.524993
[45] Fronsdal, C., The dirac supermultiplet, Phys. Rev. D, 26, 1988 (1982) · doi:10.1103/PhysRevD.26.1988
[46] Angelopoulos, E.; Flato, M.; Fronsdal, C.; Sternheimer, D., Massless particles, conformal group and de sitter universe, Phys. Rev. D, 23, 1278 (1981) · doi:10.1103/PhysRevD.23.1278
[47] Günaydin, M.; Saclioglu, C., Oscillator-like unitary representations of noncompact groups with a Jordan structure and the noncompact groups of supergravity, Commun. Math. Phys., 87, 159 (1982) · Zbl 0498.22019 · doi:10.1007/BF01218560
[48] Bars, I.; Gunaydin, M., Unitary representations of noncompact supergroups, Commun. Math. Phys., 91, 31 (1983) · Zbl 0531.17002 · doi:10.1007/BF01206048
[49] Gunaydin, M.; Warner, N. P., Unitary supermultiplets of OSp(8/4, R) and the Spectrum of the \(<mml:math display=''inline`` overflow=''scroll``>\) compactification of eleven-dimensional supergravity, Nucl. Phys. B, 272, 99 (1986) · doi:10.1016/0550-3213(86)90342-1
[50] Gunaydin, M.; van Nieuwenhuizen, P.; Warner, N. P., General construction of the unitary representations of anti-de sitter superalgebras and the spectrum of the \(<mml:math display=''inline`` overflow=''scroll``>\) compactification of eleven-dimensional supergravity, Nucl. Phys. B, 255, 63 (1985) · doi:10.1016/0550-3213(85)90129-4
[51] de Alfaro, V.; Fubini, S.; Furlan, G., Conformal invariance in quantum mechanics, Nuovo Cimento Soc. Ital. Fis., A, 34, 569 (1976) · Zbl 0449.22024 · doi:10.1007/BF02785666
[52] Casahorran, J., On a novel supersymmetric connection between harmonic and isotonic oscillators, Physica A, 217, 429 (1995) · doi:10.1016/0378-4371(95)00070-N
[53] Carinena, J. F.; Perelomov, A. M.; Ranada, M. F.; Santander, M., A quantum exactly solvable non-linear oscillator related with the isotonic oscillator, J. Phys. A: Math. Theor., 41, 085301 (2008) · Zbl 1138.81380 · doi:10.1088/1751-8113/41/8/085301
[54] Perelomov, A., Generalized Coherent States and Their Applications (1986) · Zbl 0605.22013
[55] Binegar, B., Conformal superalgebras, massless representations, and hidden symmetries, Phys. Rev. D, 34, 525 (1986) · Zbl 1222.81266 · doi:10.1103/PhysRevD.34.525
[56] 56.P.Claus, M.Gunaydin, R.Kallosh, J.Rahmfeld, and Y.Zunger, “Supertwistors as quarks of SU(2,2∣4),” J. High Energy Phys.JHEPFG1126-670805, (1999) 019;10.1088/1126-6708/1999/05/019e-print arXiv:hep-th/9905112. · Zbl 1005.81035
[57] Fernando, S. and Günaydin, M. (work in progress).
[58] Günaydin, M. (1989)
[59] Bekaert, X.; Grigoriev, M., Manifestly conformal descriptions and higher symmetries of bosonic singletons, SIGMA, 6, 038 (2010) · Zbl 1241.70049
[60] 60.M. A.Vasiliev, “Bosonic conformal higher-spin fields of any symmetry,” Nucl. Phys. BNUPBBO0550-3213829, 176 (2010);10.1016/j.nuclphysb.2009.12.010e-print arXiv:0909.5226. · Zbl 1203.81152
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.