Abstract
This is a long-overdue companion paper to [1]. We study the relation between sl(3|2) Chern-Simons supergravity on AdS3 and two-dimensional CFT’s with \( \mathcal{N} \) = 2 super-\( {\mathcal{W}}_3 \) symmetry. Specifically, we carry out a complete analysis of asymptotic symmetries in a basis that makes the superconformal structure transparent, allowing us to establish the precise dictionary between currents and transformation parameters in the bulk and their boundary counterparts. We also discuss the incorporation of sources and display in full detail the corresponding holographic Ward identities. By imposing suitable hermiticity conditions on the CFT currents, we identify the superalgebra su(2, 1|1, 1) as the appropriate real form of sl(3|2) in Lorentzian signature. We take the opportunity to review some of the properties of the \( \mathcal{N} \) = 2 super-\( {\mathcal{W}}_3 \) conformal algebra, including its multiplet structure, OPE’s and spectral flow invariance, correcting some minor typos present in the literature.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Bañados, A. Castro, A. Faraggi and J.I. Jottar, Extremal Higher Spin Black Holes, JHEP 04 (2016) 077 [arXiv:1512.00073] [INSPIRE].
V.G. Drinfeld and V.V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, J. Sov. Math. 30 (1984) 1975 [INSPIRE].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744] [INSPIRE].
A. Campoleoni, S. Fredenhagen and S. Pfenninger, Asymptotic W-symmetries in three-dimensional higher-spin gauge theories, JHEP 09 (2011) 113 [arXiv:1107.0290] [INSPIRE].
M. Henneaux and S.-J. Rey, Nonlinear W∞ as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity, JHEP 12 (2010) 007 [arXiv:1008.4579] [INSPIRE].
M.R. Gaberdiel and T. Hartman, Symmetries of Holographic Minimal Models, JHEP 05 (2011) 031 [arXiv:1101.2910] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Minimal Model Holography, J. Phys. A 46 (2013) 214002 [arXiv:1207.6697] [INSPIRE].
M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Hypersymmetry bounds and three-dimensional higher-spin black holes, JHEP 08 (2015) 021 [arXiv:1506.01847] [INSPIRE].
H. Lü, C.N. Pope, L.J. Romans, X. Shen and X.J. Wang, Polyakov construction of the N = 2 super-W3 algebra, Phys. Lett. B 264 (1991) 91 [INSPIRE].
H.S. Tan, Exploring Three-dimensional Higher-Spin Supergravity based on sl(N|N − 1) Chern-Simons theories, JHEP 11 (2012) 063 [arXiv:1208.2277] [INSPIRE].
S. Datta and J.R. David, Supersymmetry of classical solutions in Chern-Simons higher spin supergravity, JHEP 01 (2013) 146 [arXiv:1208.3921] [INSPIRE].
C. Peng, Dualities from higher-spin supergravity, JHEP 03 (2013) 054 [arXiv:1211.6748] [INSPIRE].
B. Chen, J. Long and Y.-N. Wang, Conical Defects, Black Holes and Higher Spin (Super-)Symmetry, JHEP 06 (2013) 025 [arXiv:1303.0109] [INSPIRE].
S. Datta and J.R. David, Black holes in higher spin supergravity, JHEP 07 (2013) 110 [arXiv:1303.1946] [INSPIRE].
A. Castro, N. Iqbal and E. Llabrés, Eternal Higher Spin Black Holes: a Thermofield Interpretation, JHEP 08 (2016) 022 [arXiv:1602.09057] [INSPIRE].
M. Henneaux, W. Merbis and A. Ranjbar, Asymptotic dynamics of AdS3 gravity with two asymptotic regions, JHEP 03 (2020) 064 [arXiv:1912.09465] [INSPIRE].
J. Cotler and K. Jensen, AdS3 gravity and random CFT, arXiv:2006.08648 [INSPIRE].
L.J. Romans, The N = 2 super-W3 algebra, Nucl. Phys. B 369 (1992) 403 [INSPIRE].
W. Boucher, D. Friedan and A. Kent, Determinant Formulae and Unitarity for the N = 2 Superconformal Algebras in Two-Dimensions or Exact Results on String Compactification, Phys. Lett. B 172 (1986) 316 [INSPIRE].
S. Mizoguchi, Determinant Formula and Unitarity for the W3 Algebra, Phys. Lett. B 222 (1989) 226 [INSPIRE].
C. Candu and M.R. Gaberdiel, Duality in N = 2 Minimal Model Holography, JHEP 02 (2013) 070 [arXiv:1207.6646] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Ronne, Higher spin AdS3 supergravity and its dual CFT, JHEP 02 (2012) 109 [arXiv:1111.2139] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Triality in Minimal Model Holography, JHEP 07 (2012) 127 [arXiv:1205.2472] [INSPIRE].
K. Thielemans, An Algorithmic approach to operator product expansions, W algebras and W strings, Ph.D. Thesis, Katholieke Universiteit Leuven (1994) hep-th/9506159 [INSPIRE].
C. Candu and C. Vollenweider, The \( \mathcal{N} \) = 1 algebra \( {\mathcal{W}}_{\infty}\left[\mu \right] \) and its truncations, JHEP 11 (2013) 032 [arXiv:1305.0013] [INSPIRE].
M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, Black holes in three dimensional higher spin gravity: A review, J. Phys. A 46 (2013) 214001 [arXiv:1208.5182] [INSPIRE].
A. Castro, Lectures on Higher Spin Black Holes in AdS3 Gravity, Acta Phys. Polon. B 47 (2016) 2479 [INSPIRE].
J. de Boer and J.I. Jottar, Thermodynamics of higher spin black holes in AdS3, JHEP 01 (2014) 023 [arXiv:1302.0816] [INSPIRE].
J. de Boer and J.I. Jottar, Boundary conditions and partition functions in higher spin AdS3/CFT2, JHEP 04 (2016) 107 [arXiv:1407.3844] [INSPIRE].
M. Gutperle and P. Kraus, Higher Spin Black Holes, JHEP 05 (2011) 022 [arXiv:1103.4304] [INSPIRE].
P. Kraus and E. Perlmutter, Partition functions of higher spin black holes and their CFT duals, JHEP 11 (2011) 061 [arXiv:1108.2567] [INSPIRE].
M.R. Gaberdiel, T. Hartman and K. Jin, Higher Spin Black Holes from CFT, JHEP 04 (2012) 103 [arXiv:1203.0015] [INSPIRE].
M. Bañados, R. Canto and S. Theisen, The Action for higher spin black holes in three dimensions, JHEP 07 (2012) 147 [arXiv:1204.5105] [INSPIRE].
C. Bunster, M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Generalized Black Holes in Three-dimensional Spacetime, JHEP 05 (2014) 031 [arXiv:1404.3305] [INSPIRE].
A. Perez, D. Tempo and R. Troncoso, Higher spin black hole entropy in three dimensions, JHEP 04 (2013) 143 [arXiv:1301.0847] [INSPIRE].
M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Chemical potentials in three-dimensional higher spin anti-de Sitter gravity, JHEP 12 (2013) 048 [arXiv:1309.4362] [INSPIRE].
A. Perez, D. Tempo and R. Troncoso, Higher spin gravity in 3D: Black holes, global charges and thermodynamics, Phys. Lett. B 726 (2013) 444 [arXiv:1207.2844] [INSPIRE].
M. Bañados, G. Düring, A. Faraggi and I. Reyes, Phases of higher spin black holes: Hawking-Page, transitions between black holes and a critical point, Phys. Rev. D 96 (2017) 046017 [arXiv:1611.08025] [INSPIRE].
J. Brown and M. Henneaux, On the Poisson Brackets of Differentiable Generators in Classical Field Theory, J. Math. Phys. 27 (1986) 489 [INSPIRE].
A. Castro, R. Gopakumar, M. Gutperle and J. Raeymaekers, Conical Defects in Higher Spin Theories, JHEP 02 (2012) 096 [arXiv:1111.3381] [INSPIRE].
M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, Spacetime Geometry in Higher Spin Gravity, JHEP 10 (2011) 053 [arXiv:1106.4788] [INSPIRE].
L. Frappat, P. Sorba and A. Sciarrino, Dictionary on Lie superalgebras, hep-th/9607161 [INSPIRE].
J. de Boer, A. Castro, E. Hijano, J.I. Jottar and P. Kraus, Higher spin entanglement and \( {\mathcal{W}}_N \) conformal blocks, JHEP 07 (2015) 168 [arXiv:1412.7520] [INSPIRE].
D. Melnikov, A. Mironov and A. Morozov, On skew tau-functions in higher spin theory, JHEP 05 (2016) 027 [arXiv:1602.06233] [INSPIRE].
O. Hulík, J. Raeymaekers and O. Vasilakis, Multi-centered higher spin solutions from \( {\mathcal{W}}_N \) conformal blocks, JHEP 11 (2018) 101 [arXiv:1809.01387] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, Exact Virasoro Blocks from Wilson Lines and Background-Independent Operators, JHEP 07 (2017) 092 [arXiv:1612.06385] [INSPIRE].
M. Besken, A. Hegde and P. Kraus, Anomalous dimensions from quantum Wilson lines, arXiv:1702.06640 [INSPIRE].
Y. Hikida and T. Uetoko, Correlators in higher-spin AdS3 holography from Wilson lines with loop corrections, PTEP 2017 (2017) 113B03 [arXiv:1708.08657] [INSPIRE].
Y. Hikida and T. Uetoko, Conformal blocks from Wilson lines with loop corrections, Phys. Rev. D 97 (2018) 086014 [arXiv:1801.08549] [INSPIRE].
Y. Hikida and T. Uetoko, Superconformal blocks from Wilson lines with loop corrections, JHEP 08 (2018) 101 [arXiv:1806.05836] [INSPIRE].
M. Beşken, E. D’Hoker, A. Hegde and P. Kraus, Renormalization of gravitational Wilson lines, JHEP 06 (2019) 020 [arXiv:1810.00766] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2008.05541
Rights and permissions
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.
About this article
Cite this article
Castro, A., Faraggi, A. & Osorio, I. A note on the \( \mathcal{N} \) = 2 super-\( {\mathcal{W}}_3 \) holographic dictionary. J. High Energ. Phys. 2020, 177 (2020). https://doi.org/10.1007/JHEP12(2020)177
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2020)177