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From Coxeter Higher-Spin Theories to Strings and Tensor Models

Authors
M. A. Vasiliev,Richard Milne
Miguel Cazalilla,Juan-Maria Gonzalez-Leal,Emilio J. Márquez Navarro,Rajesh Verma,Jacqueline Daykin,Bryan Webber,Galinovskiy Andrey,Andrew D. Jackson,Sylvester Gates,Patrick Dorey,Feng Han,francesco del galdo,Takeshi Osawa,Olaf Joehren,Jean-Christophe Wallet,Yves RAYNAUD,Federico Zertuche,Alberto Lerda,Ronald Jackson,Vladimir Orlov,Ivan Starokon,Anton Zimmerling,Tonia Moore,Joachim Dengler,Djuna Croon,Michael Kmiec,Keisuke Yano,Kristen Smith,Mahendra Ramachandran,Vladimir Mićić,Alexandru Ciubotaru,Peter Wolfsteiner,Torben Gaedt,Garrett Poe,Andreas Stergiou,Jesse Thaler,Alain Jaccard,Michael Johnson,Marc Torrens,Mikołaj Baranowski,Benjamin Bellenie,Andy Van Brocklin,Joseph Fotsing,Jelena Vasiljevic,Digamber Porob,David Van Wie,Seok Ki Choi,Vishnu Ram OV,Felipe Lumbreras,Antonio Amores Arrocha,Urfat Nuriyev,Steven Juhyung Lee,Shahin Atashbar Tehrani,Matthew Sullivan,Wai-Ching Lam,Martin Köhne,Sandra S. 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arXiv:1804.06520v3 [hep-th] 16 Aug 2018
FIAN/TD/02-18
From Coxeter Higher-Spin Theories to Strings and Tensor Models
M.A. Vasiliev
I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute,
Leninsky prospect 53, 119991, Moscow, Russia
To my father
Abstract
A new class of higher-spin gauge theories associated with various Coxeter groups is
proposed. The emphasize is on the Bp–models. The cases of B1 and its infinite
graded-symmetric product sym (×B1) correspond to the usual higher-spin theory
and its multi-particle extension, respectively. The multi-particle B2–higher-spin theory
is conjectured to be associated with String Theory. Bp–higher-spin models with p > 2
are anticipated to be dual to the rank-p boundary tensor sigma-models. Bp higher-
spin models with p 2 possess two coupling constants responsible for higher-spin
interactions in AdS background and stringy/tensor effects, respectively. The brane-
like idempotent extension of the Coxeter higher-spin theory is proposed allowing to
unify in the same model the fields supported by space-times of different dimensions.
Consistency of the holographic interpretation of the boundary matrix-like model in
the B2-higher-spin model is shown to demand N 4 SUSY, suggesting duality with
the N = 4 SYM upon spontaneous breaking of higher-spin symmetries. The proposed
models are shown to admit unitary truncations.
1
Contents
1 Introduction 3
2 Standard higher-spin equations 7
3 Deformed oscillator algebras 8
3.1 Single oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Many oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Coxeter groups and Cherednik algebras . . . . . . . . . . . . . . . . . . . . . 10
4 Higher-spin modules and framed oscillator algebras 12
5 Framed Cherednik algebras 14
5.1 AN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2 General Coxeter group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Coxeter higher-spin equations 16
6.1 General scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.2 Spinor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.2.1 3d Coxeter higher-spin models . . . . . . . . . . . . . . . . . . . . . . 18
6.2.2 4d Coxeter higher-spin models . . . . . . . . . . . . . . . . . . . . . . 19
6.3 Vector Coxeter higher-spin models in any dimension . . . . . . . . . . . . . . 21
7 Higher forms and invariants 23
8 Color extensions 25
9 Idempotent extension 26
9.1 General scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
9.2 Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
9.2.1 Vector-like models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
9.2.2 N = 4 SUSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10 Unitarity 30
11 Field frames of multi-particle theories 32
11.1 Definition and field equations . . . . . . . . . . . . . . . . . . . . . . . . . . 32
11.2 Frame generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
11.3 Unity idempotent map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
12 Interpretation 38
12.1 Strings and tensor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
12.2 Space-time metamorphoses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
13 Conclusion 43
2
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