Abstract
We examine rectangular W-algebras with so(M) or sp(2M) symmetry, which can be realized as the asymptotic symmetry of higher spin gravities with restricted matrix extensions. We compute the central charges of the algebras and the levels of so(M) or sp(2M) affine subalgebras by applying the Hamiltonian reductions of so or sp type Lie algebras. For simple cases with generators of spin up to two, we obtain their operator product expansions by requiring the associativity. We further claim that the W-algebras can be realized as the symmetry algebras of dual coset CFTs and provide several strong supports. The analysis can be regarded as a check of extended higher spin holographies including full quantum corrections. We also extend the analysis by introducing \( \mathcal{N} \) = 1 supersymmetry.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.J. Gross, High-energy symmetries of string theory, Phys. Rev. Lett.60 (1988) 1229 [INSPIRE].
M.A. Vasiliev, From Coxeter higher-spin theories to strings and tensor models, JHEP08 (2018) 051 [arXiv:1804.06520] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Ronne, Extended higher spin holography and Grassmannian models, JHEP11 (2013) 038 [arXiv:1306.0466] [INSPIRE].
S.F. Prokushkin and M.A. Vasiliev, Higher spin gauge interactions for massive matter fields in 3D AdS space-time, Nucl. Phys.B 545 (1999) 385 [hep-th/9806236] [INSPIRE].
C. Candu and C. Vollenweider, On the coset duals of extended higher spin theories, JHEP04 (2014) 145 [arXiv:1312.5240] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and I. Rienacker, Higher spin algebras and large N = 4 holography, JHEP03 (2018) 097 [arXiv:1801.00806] [INSPIRE].
D. Kumar and M. Sharma, Symmetry algebras of stringy cosets, JHEP08 (2019) 179 [arXiv:1812.11920] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS3dual for minimal model CFTs, Phys. Rev.D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Ronne, Higher spin AdS3supergravity and its dual CFT, JHEP02 (2012) 109 [arXiv:1111.2139] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Ronne, Higher spin AdS3holography with extended supersymmetry, JHEP10 (2014) 163 [arXiv:1406.1521] [INSPIRE].
Y. Hikida and P.B. Rønne, Marginal deformations and the Higgs phenomenon in higher spin AdS3holography, JHEP07 (2015) 125 [arXiv:1503.03870] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Large N = 4 holography, JHEP09 (2013) 036 [arXiv:1305.4181] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Higher spins & strings, JHEP11 (2014) 044 [arXiv:1406.6103] [INSPIRE].
T. Creutzig and Y. Hikida, Rectangular W-algebras, extended higher spin gravity and dual coset CFTs, JHEP02 (2019) 147 [arXiv:1812.07149] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Triality in minimal model holography, JHEP07 (2012) 127 [arXiv:1205.2472] [INSPIRE].
C. Candu and M.R. Gaberdiel, Duality in N = 2 minimal model holography, JHEP02 (2013) 070 [arXiv:1207.6646] [INSPIRE].
E. Joung, J. Kim, J. Kim and S.-J. Rey, Asymptotic symmetries of colored gravity in three dimensions, JHEP03 (2018) 104 [arXiv:1712.07744] [INSPIRE].
M. Henneaux and S.-J. Rey, Nonlinear W∞as asymptotic symmetry of three-dimensional higher spin Anti-de Sitter gravity, JHEP12 (2010) 007 [arXiv:1008.4579] [INSPIRE].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP11 (2010) 007 [arXiv:1008.4744] [INSPIRE].
C. Ahn, The large N ’t Hooft limit of coset minimal models, JHEP10 (2011) 125 [arXiv:1106.0351] [INSPIRE].
M.R. Gaberdiel and C. Vollenweider, Minimal model holography for SO(2N), JHEP08 (2011) 104 [arXiv:1106.2634] [INSPIRE].
C. Candu, M.R. Gaberdiel, M. Kelm and C. Vollenweider, Even spin minimal model holography, JHEP01 (2013) 185 [arXiv:1211.3113] [INSPIRE].
B.L. Feigin, The Lie algebras gl(λ) and cohomologies of Lie algebras of differential operators, Russ. Math. Surv.43 (1988) 169.
T. Creutzig, Y. Hikida and P.B. Rønne, N = 1 supersymmetric higher spin holography on AdS3, JHEP02 (2013) 019 [arXiv:1209.5404] [INSPIRE].
C. Candu and C. Vollenweider, The N = 1 algebra \( {\mathcal{W}}_{\infty } \) [μ] and its truncations, JHEP11 (2013) 032 [arXiv:1305.0013] [INSPIRE].
T. Creutzig and Y. Hikida, Rectangular W-(super)algebras and their representations, arXiv:1906.05868 [INSPIRE].
A. Castro, R. Gopakumar, M. Gutperle and J. Raeymaekers, Conical defects in higher spin theories, JHEP02 (2012) 096 [arXiv:1111.3381] [INSPIRE].
E. Perlmutter, T. Prochazka and J. Raeymaekers, The semiclassical limit of WNCFTs and Vasiliev theory, JHEP05 (2013) 007 [arXiv:1210.8452] [INSPIRE].
Y. Hikida, Conical defects and N = 2 higher spin holography, JHEP08 (2013) 127 [arXiv:1212.4124] [INSPIRE].
C. Candu, C. Peng and C. Vollenweider, Extended supersymmetry in AdS3higher spin theories, JHEP12 (2014) 113 [arXiv:1408.5144] [INSPIRE].
A. Achucarro and P.K. Townsend, A Chern-Simons action for three-dimensional Anti-de Sitter supergravity theories, Phys. Lett.B 180 (1986) 89 [INSPIRE].
E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys.B 311 (1988) 46 [INSPIRE].
E. Bergshoeff, B. de Wit and M.A. Vasiliev, The structure of the super-\( {\mathcal{W}}_{\infty } \) (μ) algebra, Nucl. Phys.B 366 (1991) 315 [INSPIRE].
L. Frappat, P. Sorba and A. Sciarrino, Dictionary on Lie superalgebras, hep-th/9607161 [INSPIRE].
K. Thielemans, A Mathematica package for computing operator product expansions, Int. J. Mod. Phys.C 2 (1991) 787 [INSPIRE].
M. Beccaria, C. Candu and M.R. Gaberdiel, The large N = 4 superconformal W∞algebra, JHEP 06 (2014) 117 [arXiv:1404.1694] [INSPIRE].
M.R. Gaberdiel and C. Peng, The symmetry of large N = 4 holography, JHEP05 (2014) 152 [arXiv:1403.2396] [INSPIRE].
T. Creutzig and A.R. Linshaw, The super W1+∞algebra with integral central charge, Trans. Am. Math. Soc.367 (2015) 5521 [arXiv:1209.6032].
T. Creutzig and A.R. Linshaw, Cosets of affine vertex algebras inside larger structures, J. Algebra517 (2019) 396 [arXiv:1407.8512] [INSPIRE].
I. Bakas and E. Kiritsis, Grassmannian coset models and unitary representations of W∞, Mod. Phys. Lett.A 5 (1990) 2039 [INSPIRE].
S. Odake and T. Sano, W1+∞and super W∞algebras with SU(N) symmetry, Phys. Lett.B 258 (1991) 369.
T. Creutzig and A.R. Linshaw, Orbifolds of symplectic fermion algebras, arXiv:1404.2686 [INSPIRE].
V.G. Kac and M. Wakimoto, Integrable highest weight modules over affine superalgebras and Appell’s function, Commun. Math. Phys.215 (2001) 631 [math-ph/0006007].
P. Goddard, A. Kent and D.I. Olive, Virasoro algebras and coset space models, Phys. Lett.B 152 (1985) 88.
K. Hanaki and C. Peng, Symmetries of holographic super-minimal models, JHEP08 (2013) 030 [arXiv:1203.5768] [INSPIRE].
M. Henneaux, G. Lucena Gómez, J. Park and S.-J. Rey, Super-W∞asymptotic symmetry of higher-spin AdS3supergravity, JHEP06 (2012) 037 [arXiv:1203.5152] [INSPIRE].
V.G. Kac and M. Wakimoto, Quantum reduction and representation theory of superconformal algebras, math-ph/0304011 [INSPIRE].
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: 1906.05872
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Creutzig, T., Hikida, Y. & Uetoko, T. Rectangular W-algebras of types so(M) and sp(2M) and dual coset CFTs. J. High Energ. Phys. 2019, 23 (2019). https://doi.org/10.1007/JHEP10(2019)023
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)023