## Abstract

We aim at formulating a higher-spin gravity theory around AdS_{2} relevant for holography. As a first step, we investigate its kinematics by identifying the low-dimensional cousins of the standard higher-dimensional structures in higher-spin gravity such as the singleton, the higher-spin symmetry algebra, the higher-rank gauge and matter fields, etc. In particular, the higher-spin algebra is given here by [*λ*] and parameterized by a real parameter *λ*. The singleton is defined to be a Verma module of the AdS_{2} isometry subalgebra so (2*,* 1) *⊂* [*λ*] with conformal weight \( \Delta =\frac{1\pm \lambda }{2}. \) On the one hand, the spectrum of local modes is determined by the Flato-Fronsdal theorem for the tensor product of two such singletons. It is given by an infinite tower of massive scalar fields in AdS_{2} with ascending masses expressed in terms of *λ*. On the other hand, the higher-spin fields arising through the gauging of [*λ*] algebra do not propagate local degrees of freedom. Our analysis of the spectrum suggests that AdS_{2} higher-spin gravity is a theory of an infinite collection of massive scalars with fine-tuned masses, interacting with infinitely many topological gauge fields. Finally, we discuss the holographic CFT_{1} duals of the kinematical structures identified in the bulk.

## References

S. Sachdev and J. Ye,

*Gapless spin fluid ground state in a random, quantum Heisenberg magnet*,*Phys. Rev. Lett.***70**(1993) 3339 [cond-mat/9212030] [INSPIRE].A. Kitaev,

*A simple model of quantum holography (part 1)*, talk at*KITP*, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.A. Kitaev,

*A simple model of quantum holography (part 2)*, talk at*KITP*, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.J. Maldacena and D. Stanford,

*Remarks on the Sachdev-Ye-Kitaev model*,*Phys. Rev.***D 94**(2016) 106002 [arXiv:1604.07818] [INSPIRE].G. Sárosi,

*AdS*_{2}*holography and the SYK model*, PoS(Modave2017)001 [arXiv:1711.08482] [INSPIRE].V. Rosenhaus,

*An introduction to the SYK model*, arXiv:1807.03334 [INSPIRE].C. Teitelboim,

*Gravitation and Hamiltonian Structure in Two Space-Time Dimensions*,*Phys. Lett.***126B**(1983) 41 [INSPIRE].R. Jackiw,

*Lower Dimensional Gravity*,*Nucl. Phys.***B 252**(1985) 343 [INSPIRE].T. Fukuyama and K. Kamimura,

*Gauge Theory of Two-dimensional Gravity*,*Phys. Lett.***160B**(1985) 259 [INSPIRE].K.B. Alkalaev,

*On higher spin extension of the Jackiw-Teitelboim gravity model*,*J. Phys.***A 47**(2014) 365401 [arXiv:1311.5119] [INSPIRE].K.B. Alkalaev,

*Global and local properties of AdS*_{2}*higher spin gravity*,*JHEP***10**(2014) 122 [arXiv:1404.5330] [INSPIRE].D. Grumiller, M. Leston and D. Vassilevich,

*Anti-de Sitter holography for gravity and higher spin theories in two dimensions*,*Phys. Rev.***D 89**(2014) 044001 [arXiv:1311.7413] [INSPIRE].D. Grumiller, J. Salzer and D. Vassilevich,

*AdS*_{2}*holography is (non-)trivial for (non-)constant dilaton*,*JHEP***12**(2015) 015 [arXiv:1509.08486] [INSPIRE].H.A. González, D. Grumiller and J. Salzer,

*Towards a bulk description of higher spin SYK*,*JHEP***05**(2018) 083 [arXiv:1802.01562] [INSPIRE].B.L. Feigin,

*Lie algebras gl*(*λ*)*and cohomologies of Lie algebras of differential operators*,*Russ. Math. Surv.***43**(1988) 169.M.A. Vasiliev,

*Higher Spin Algebras and Quantization on the Sphere and Hyperboloid*,*Int. J. Mod. Phys.***A 6**(1991) 1115 [INSPIRE].A. Jevicki, K. Suzuki and J. Yoon,

*Bi-Local Holography in the SYK Model*,*JHEP***07**(2016) 007 [arXiv:1603.06246] [INSPIRE].A. Jevicki and K. Suzuki,

*Bi-Local Holography in the SYK Model: Perturbations*,*JHEP***11**(2016) 046 [arXiv:1608.07567] [INSPIRE].S.R. Das and A. Jevicki,

*Large N collective fields and holography*,*Phys. Rev.***D 68**(2003) 044011 [hep-th/0304093] [INSPIRE].D.J. Gross and V. Rosenhaus,

*A line of CFTs: from generalized free fields to SYK*,*JHEP***07**(2017) 086 [arXiv:1706.07015] [INSPIRE].C. Peng, \( \mathcal{N} \) = (0

*,*2)*SYK, Chaos and Higher-Spins*,*JHEP***12**(2018) 065 [arXiv:1805.09325] [INSPIRE].X. Bekaert,

*Singletons and their maximal symmetry algebras*, in*Modern Mathematical Physics. Proceedings, 6th Summer School: Belgrade, Serbia, September 14–23, 2010*, pp. 71–89, arXiv:1111.4554 [INSPIRE].E. Angelopoulos and M. Laoues,

*Masslessness in n-dimensions*,*Rev. Math. Phys.***10**(1998) 271 [hep-th/9806100] [INSPIRE].E. Angelopoulos, M. Flato, C. Fronsdal and D. Sternheimer,

*Massless Particles, Conformal Group and de Sitter Universe*,*Phys. Rev.***D 23**(1981) 1278 [INSPIRE].P. Breitenlohner and D.Z. Freedman,

*Stability in Gauged Extended Supergravity*,*Annals Phys.***144**(1982) 249 [INSPIRE].A. Kitaev,

*Notes on*\( \tilde{\mathrm{SL}} \)(2*,*ℝ)*representations*, arXiv:1711.08169 [INSPIRE].I.R. Klebanov and E. Witten,

*AdS/CFT correspondence and symmetry breaking*,*Nucl. Phys.***B 556**(1999) 89 [hep-th/9905104] [INSPIRE].M.R. Gaberdiel and R. Gopakumar,

*An AdS*_{3}*Dual for Minimal Model CFTs*,*Phys. Rev.***D 83**(2011) 066007 [arXiv:1011.2986] [INSPIRE].E. Sezgin and P. Sundell,

*Holography in 4D (super) higher spin theories and a test via cubic scalar couplings*,*JHEP***07**(2005) 044 [hep-th/0305040] [INSPIRE].K. Alkalaev,

*Mixed-symmetry tensor conserved currents and AdS/CFT correspondence*,*J. Phys.***A 46**(2013) 214007 [arXiv:1207.1079] [INSPIRE].T. Basile, X. Bekaert and E. Joung,

*Twisted Flato-Fronsdal Theorem for Higher-Spin Algebras*,*JHEP***07**(2018) 009 [arXiv:1802.03232] [INSPIRE].M. Grigoriev, I. Lovrekovic and E. Skvortsov,

*New Conformal Higher Spin Gravities in 3d*,*JHEP***01**(2020) 059 [arXiv:1909.13305] [INSPIRE].T. Basile, X. Bekaert and N. Boulanger,

*Flato-Fronsdal theorem for higher-order singletons*,*JHEP***11**(2014) 131 [arXiv:1410.7668] [INSPIRE].C. Brust and K. Hinterbichler,

*Free*□^{k}*scalar conformal field theory*,*JHEP***02**(2017) 066 [arXiv:1607.07439] [INSPIRE].M. Flato and C. Fronsdal,

*One Massless Particle Equals Two Dirac Singletons: Elementary Particles in a Curved Space. 6.*,*Lett. Math. Phys.***2**(1978) 421 [INSPIRE].M.A. Vasiliev,

*Higher spin superalgebras in any dimension and their representations*,*JHEP***12**(2004) 046 [hep-th/0404124] [INSPIRE].F.A. Dolan,

*Character formulae and partition functions in higher dimensional conformal field theory*,*J. Math. Phys.***47**(2006) 062303 [hep-th/0508031] [INSPIRE].R.R. Metsaev,

*Massive fields in AdS*_{3}*and compactification in AdS space time*,*Nucl. Phys. Proc. Suppl.***102**(2001) 100 [hep-th/0103088] [INSPIRE].A.Yu. Artsukevich and M.A. Vasiliev,

*On Dimensional Degression in AdS*_{d},*Phys. Rev.***D 79**(2009) 045007 [arXiv:0810.2065] [INSPIRE].D.J. Gross and V. Rosenhaus,

*All point correlation functions in SYK*,*JHEP***12**(2017) 148 [arXiv:1710.08113] [INSPIRE].S.F. Prokushkin and M.A. Vasiliev,

*Higher spin gauge interactions for massive matter fields in 3-D AdS space-time*,*Nucl. Phys.***B 545**(1999) 385 [hep-th/9806236] [INSPIRE].S.E. Konstein and M.A. Vasiliev,

*Extended Higher Spin Superalgebras and Their Massless Representations*,*Nucl. Phys.***B 331**(1990) 475 [INSPIRE].E.S. Fradkin and V.Ya. Linetsky,

*Infinite dimensional generalizations of finite dimensional symmetries*,*J. Math. Phys.***32**(1991) 1218 [INSPIRE].E.S. Fradkin and V.Ya. Linetsky,

*Infinite dimensional generalizations of simple Lie algebras*,*Mod. Phys. Lett.***A 5**(1990) 1967 [INSPIRE].N. Boulanger, D. Ponomarev, E.D. Skvortsov and M. Taronna,

*On the uniqueness of higher-spin symmetries in AdS and CFT*,*Int. J. Mod. Phys.***A 28**(2013) 1350162 [arXiv:1305.5180] [INSPIRE].E. Joung and K. Mkrtchyan,

*Notes on higher-spin algebras: minimal representations and structure constants*,*JHEP***05**(2014) 103 [arXiv:1401.7977] [INSPIRE].S. Fernando and M. Günaydin,

*Massless conformal fields, AdS*_{d+1}*/CFT*_{d}*higher spin algebras and their deformations*,*Nucl. Phys.***B 904**(2016) 494 [arXiv:1511.02167] [INSPIRE].M.A. Vasiliev,

*Nonlinear equations for symmetric massless higher spin fields in (A)dS(d)*,*Phys. Lett.***B 567**(2003) 139 [hep-th/0304049] [INSPIRE].R. Howe,

*Transcending classical invariant theory*,*J. Am. Math. Soc.***2**(1989) 535.M.A. Vasiliev,

*Higher spin gauge theories: Star product and AdS space*, hep-th/9910096 [INSPIRE].D. Arnaudon, M. Bauer and L. Frappat,

*On Casimir’s ghost*,*Commun. Math. Phys.***187**(1997) 429 [q-alg/9605021] [INSPIRE].C. Iazeolla and P. Sundell,

*A Fiber Approach to Harmonic Analysis of Unfolded Higher-Spin Field Equations*,*JHEP***10**(2008) 022 [arXiv:0806.1942] [INSPIRE].K.S. Stelle and P.C. West,

*Spontaneously Broken de Sitter Symmetry and the Gravitational Holonomy Group*,*Phys. Rev.***D 21**(1980) 1466 [INSPIRE].X. Bekaert, S. Cnockaert, C. Iazeolla and M.A. Vasiliev,

*Nonlinear higher spin theories in various dimensions*, in*Higher spin gauge theories: Proceedings, 1st Solvay Workshop: Brussels, Belgium, 12–14 May, 2004*, pp. 132–197, hep-th/0503128 [INSPIRE].X. Bekaert and J. Mourad,

*The Continuous spin limit of higher spin field equations*,*JHEP***01**(2006) 115 [hep-th/0509092] [INSPIRE].K.B. Alkalaev and M.A. Grigoriev,

*Continuous spin fields of mixed-symmetry type*,*JHEP***03**(2018) 030 [arXiv:1712.02317] [INSPIRE].X. Bekaert, J. Mourad and M. Najafizadeh,

*Continuous-spin field propagator and interaction with matter*,*JHEP***11**(2017) 113 [arXiv:1710.05788] [INSPIRE].K. Alkalaev, A. Chekmenev and M. Grigoriev,

*Unified formulation for helicity and continuous spin fermionic fields*,*JHEP***11**(2018) 050 [arXiv:1808.09385] [INSPIRE].M.A. Vasiliev,

*Unfolded representation for relativistic equations in (2+1) anti-de Sitter space*,*Class. Quant. Grav.***11**(1994) 649 [INSPIRE].M.A. Vasiliev,

*Higher spin gauge interactions for matter fields in two-dimensions*,*Phys. Lett.***B 363**(1995) 51 [hep-th/9511063] [INSPIRE].O.V. Shaynkman and M.A. Vasiliev,

*Scalar field in any dimension from the higher spin gauge theory perspective*,*Theor. Math. Phys.***123**(2000) 683 [hep-th/0003123] [INSPIRE].M.A. Vasiliev,

*Properties of equations of motion of interacting gauge fields of all spins in (3+1)-dimensions*,*Class. Quant. Grav.***8**(1991) 1387 [INSPIRE].M.A. Vasiliev,

*More on equations of motion for interacting massless fields of all spins in (3+1)-dimensions*,*Phys. Lett.***B 285**(1992) 225 [INSPIRE].K.B. Alkalaev, M. Grigoriev and E.D. Skvortsov,

*Uniformizing higher-spin equations*,*J. Phys.***A 48**(2015) 015401 [arXiv:1409.6507] [INSPIRE].M.A. Vasiliev,

*From Coxeter Higher-Spin Theories to Strings and Tensor Models*,*JHEP***08**(2018) 051 [arXiv:1804.06520] [INSPIRE].A. Sharapov and E. Skvortsov,

*Formal Higher Spin Gravities*,*Nucl. Phys.***B 941**(2019) 838 [arXiv:1901.01426] [INSPIRE].V. Mazorchuk,

*Lectures on sl*_{2}(*C*)*-modules*, Imperial College Press, (2009).

##
**Open Access**

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

ArXiv ePrint: 1911.13212

## Rights and permissions

This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.

## About this article

### Cite this article

Alkalaev, K., Bekaert, X. Towards higher-spin AdS_{2}/CFT_{1} holography.
*J. High Energ. Phys.* **2020**, 206 (2020). https://doi.org/10.1007/JHEP04(2020)206

Received:

Revised:

Accepted:

Published:

DOI: https://doi.org/10.1007/JHEP04(2020)206

### Keywords

- Higher Spin Gravity
- Higher Spin Symmetry