Quasiconformal Group Approach to Higher Spin Algebras, their Deformations and Supersymmetric Extensions

The quasiconformal method provides us with a unified approach to the construction of minimal unitary representations (minrep) of noncompact groups, their deformations as well as their supersymmetric extensions. We review the quasiconformal construction of the minrep of SO(d,2), its deformations and their applications to unitary realizations of AdS(d+1)/CFTd higher spin algebras and their deformations for arbitrary d and supersymmetric extensions for d ≤ 6. AdS(d+1)/CFTd higher spin algebras, their deformations and supersymmetric extensions are given by the enveloping algebras of the quasiconformal realizations of the minrep, its deformations and supersymmetric extensions, respectively, and are in one-to-one correspondence with massless conformal fields for arbitrary d and massless conformal supermultiplets for d ≤ 6.

[1]  Sudarshan Fernando,et al.  Massless conformal fields, AdS(d + 1)/CFTd higher spin algebras and their deformations , 2015, 1511.02167.

[2]  M. Gunaydin,et al.  Minimal unitary representation of 5d superconformal algebra F(4) and AdS6/CFT5 higher spin (super)-algebras , 2014, 1409.2185.

[3]  M. Günaydin,et al.  Deformed twistors and higher spin conformal (super-)algebras in six dimensions , 2014 .

[4]  E. Skvortsov,et al.  Elements of Vasiliev theory , 2014, 1401.2975.

[5]  M. Günaydin,et al.  Deformed twistors and higher spin conformal (super-)algebras in four dimensions , 2013, 1312.2907.

[6]  X. Yin,et al.  The higher spin/vector model duality , 2012, 1208.4036.

[7]  A. Sagnotti.  Notes on strings and higher spins , 2011, 1112.4285.

[8]  M. Gunaydin,et al.  SU(2) deformations of the minimal unitary representation of OSp(8*|2N) as massless 6D conformal supermultiplets , 2010, 1008.0702.

[9]  M. Gunaydin,et al.  Minimal unitary representation of SO∗(8)=SO(6,2) and its SU(2) deformations as massless 6D conformal fields and their supersymmetric extensions , 2010, 1005.3580.

[10]  X. Yin,et al.  Higher spins in AdS and twistorial holography , 2010, 1004.3736.

[11]  X. Yin,et al.  Higher spin gauge theory and holography: the three-point functions , 2009, 0912.3462.

[12]  M. Gunaydin,et al.  Minimal unitary representation of SU(2,2) and its deformations as massless conformal fields and their supersymmetric extensions , 2009, 0908.3624.

[13]  A. Gover,et al.  The so(d+2,2) Minimal Representation and Ambient Tractors: the Conformal Geometry of Momentum Space , 2009, 0903.1394.

[14]  C. Iazeolla.  On the Algebraic Structure of Higher-Spin Field Equations and New Exact Solutions , 2008, 0807.0406.

[15]  O. Pavlyk,et al.  A unified approach to the minimal unitary realizations of noncompact groups and supergroups , 2006, hep-th/0604077.

[16]  V. Souček,et al.  The Uniqueness of the Joseph Ideal for the Classical Groups , 2005, math/0512296.

[17]  O. Pavlyk,et al.  Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups , 2005, hep-th/0506010.

[18]  M. Vasiliev,et al.  Nonlinear higher spin theories in various dimensions , 2005, hep-th/0503128.

[19]  M. Eastwood.  The Cartan Product , 2005 .

[20]  M. Vasiliev.  Higher spin superalgebras in any dimension and their representations , 2004 .

[21]  O. Pavlyk,et al.  Minimal Unitary Realizations of Exceptional U-duality Groups and Their Subgroups as Quasiconformal Groups , 2004, hep-th/0409272.

[22]  A. Polyakov,et al.  AdS dual of the critical O(N) vector model , 2002, hep-th/0210114.

[23]  M. Eastwood.  Higher symmetries of the laplacian , 2002, hep-th/0206233.

[24]  E. Sezgin,et al.  Massless higher spins and holography , 2002, hep-th/0205131.

[25]  E. Sezgin,et al.  7D bosonic higher spin gauge theory: symmetry algebra and linearized constraints , 2001, hep-th/0112100.

[26]  Toshiyuki Kobayashi,et al.  Analysis on the minimal representation of O(p,q) II. Branching laws , 2001, math/0111085.

[27]  Toshiyuki Kobayashi,et al.  Analysis on the minimal representation of O(p;q) { I. Realization via conformal geometry , 2001, math/0111083.

[28]  Toshiyuki Kobayashi,et al.  Analysis on the minimal representation of O(p,q) -- III. ultrahyperbolic equations on R^{p-1,q-1} , 2001, math/0111086.

[29]  K. Koepsell,et al.  The Minimal Unitary Representation of E_8(8) , 2001, hep-th/0109005.

[30]  D. Kazhdan,et al.  Minimal Representations, Spherical Vectors¶and Exceptional Theta Series , 2001, hep-th/0107222.

[31]  E. Sezgin,et al.  Towards massless higher spin extension of D=5, N=8 gauged supergravity , 2001, hep-th/0107186.

[32]  M. Gunaydin,et al.  Supercoherent states of OSp(8 ∗ |2N) , conformal superfields and the AdS 7 / CFT 6 duality , 2001, hep-th/0106161.

[33]  E. Sezgin,et al.  Doubletons and 5D higher spin gauge theory , 2001, hep-th/0105001.

[34]  K. Koepsell,et al.  Conformal and Quasiconformal Realizations¶of Exceptional Lie Groups , 2000, hep-th/0008063.

[35]  M. Gunaydin,et al.  Unitary supermultiplets of OSp(8 ∗ |4) and the AdS 7 / CFT 6 duality , 1999, hep-th/9910110.

[36]  M. Vasiliev.  Higher Spin Gauge Theories: Star-Product and AdS Space , 1999, hep-th/9910096.

[37]  D. Minic,et al.  Novel supermultiplets of SU (2, 2|4) and the AdS5/CFT4 duality☆ , 1998, hep-th/9810226.

[38]  M. Laoues.  Some Properties of Massless Particles in Arbitrary Dimensions , 1998, hep-th/9806101.

[39]  D. Minic,et al.  4D doubleton conformal theories, CPT and IIB strings on AdS5 × S5 , 1998, hep-th/9806042.

[40]  M. Laoues,et al.  MASSLESSNESS IN n-DIMENSIONS , 1998, hep-th/9806100.

[41]  B. Kostant,et al.  Minimal representations, geometric quantization, and unitarity. , 1994, Proceedings of the National Academy of Sciences of the United States of America.

[42]  M. Vasiliev.  More on equations of motion for interacting massless fields of all spins in (3+1)-dimensions , 1992 .

[43]  B. Binegar,et al.  Unitarization of a singular representation ofSO(p, q) , 1991 .

[44]  E. Fradkin,et al.  Conformal superalgebras of higher spins , 1989 .

[45]  M. Vasiliev,et al.  Massless representations and admissibility condition for higher spin superalgebras , 1989 .

[46]  M. Günaydin,et al.  Unitary lowest weight representations of the noncompact supergroup OSp(2n/2m,R) , 1988 .

[47]  M. Gunaydin,et al.  The spectrum of the S^5 compactification of the chiral N=2, D=10 supergravity and the unitary supermultiplets of U(2,2/4) , 1985 .

[48]  M. Günaydin,et al.  Exceptional Supergravity Theories and the MAGIC Square , 1983 .

[49]  M. Günaydin,et al.  Unitary representations of non-compact supergroups , 1983 .

[50]  M. Günaydin,et al.  Oscillator-like unitary representations of non-compact groups with a jordan structure and the non-compact groups of supergravity , 1982 .

[51]  M. Flato,et al.  Quantum field theory of singletons. The Rac , 1981 .

[52]  M. Flato,et al.  Massless Particles, Conformal Group and De Sitter Universe , 1981 .

[53]  M. Flato,et al.  One massless particle equals two Dirac singletons , 1978 .

[54]  W. Nahm.  Supersymmetries and their Representations , 1978 .

[55]  V. Alfaro,et al.  Conformal invariance in quantum mechanics , 1976 .

[56]  A. Joseph.  Minimal realizations and spectrum generating algebras , 1974 .

[57]  Francesco Calogero,et al.  Solution of the One‐Dimensional N‐Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials , 1971 .

[58]  C. Marchioro,et al.  Lower Bounds to the Ground‐State Energy of Systems Containing Identical Particles , 1969 .

[59]  Paul Adrien Maurice Dirac,et al.  A Remarkable Representation of the 3 + 2 de Sitter Group , 1963 .

[60]  B. Kostant,et al.  Lagrangian Models of Minimal Representations of E 6 E 7 and E 8 , 1995 .

[61]  B. Gross,et al.  A Distinguished Family of Unitary Representations for the Exceptional Groups of Real Rank = 4 , 1994 .

[62]  M. Gunaydin.  SINGLETON AND DOUBLETON SUPERMULTIPLETS OF SPACE-TIME SUPERGROUPS AND INFINITE SPIN SUPERALGEBRAS , 1989 .

[63]  E. Fradkin,et al.  Cubic interaction in extended theories of massless higher-spin fields , 1987 .

[64]  P. Nieuwenhuizen,et al.  General construction of the unitary representations of anti-de sitter superalgebras and the spectrum of the S4 compactification of 11-dimensional supergravity☆ , 1985 .

[65]  M Gunaydin,et al.  The spectrum of the S5 compactification of the chiral N=2, D=10 supergravity and the unitary supermultiplets of U(2,2/4) , 1985 .

[66]  M. Günaydin.  Oscillator-like unitary representations of non-compact groups and supergroups and extended supergravity theories , 1983 .

[67]  D. Vogan.  Singular unitary representations , 1981 .

[68]  C. Frønsdal.  The Dirac Supermultiplet , 1981 .