Abstract
We study theories with W-algebra symmetries and their relation to WZNW-type models on (super-)groups generalizing the H +3 WZNW to Liouville correspondence. Correlation functions of the WZNW models are expressed in terms of correlators of CFTs with W-algebra symmetry. The symmetries of the theories involved in these correspondences are related by the Drinfeld-Sokolov reduction of Lie algebras to W-algebras. The W-algebras considered in this paper are the Bershadsky-Polyakov algebra for sl(3) and the quasi-superconformal algebra for generic sl(N|M). The quantum W-algebras obtained from affine sl(N) are constructed using embeddings of sl(2) into sl(N), and these can in turn be characterized by partitions of N. The above cases correspond to N + 2 = 2 + N 1 and its supergroup extension. Finally, sl(2N) and the correspondence corresponding to 2 N = N 2 is also analyzed. These are all W-algebras that are generated by fields of at most dimension two.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept. 223 (1993) 183 [hep-th/9210010] [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].
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].
M.R. Gaberdiel and R. Gopakumar, An AdS 3 dual for minimal model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [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].
T. Creutzig, Y. Hikida and P.B. Rønne, Higher spin AdS 3 supergravity and its dual CFT, JHEP 02 (2012) 109 [arXiv:1111.2139] [INSPIRE].
C. Candu and M.R. Gaberdiel, Supersymmetric holography on AdS 3, JHEP 09 (2013) 071 [arXiv:1203.1939] [INSPIRE].
M. Henneaux, G. Lucena Gómez, J. Park and S.-J. Rey, Super-W ∞ asymptotic symmetry of higher-spin AdS 3 supergravity, JHEP 06 (2012) 037 [arXiv:1203.5152] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Rønne, \( \mathcal{N}=1 \) supersymmetric higher spin holography on AdS 3, JHEP 02 (2013) 019 [arXiv:1209.5404] [INSPIRE].
M. Beccaria, C. Candu, M.R. Gaberdiel and M. Groher, \( \mathcal{N}=1 \) extension of minimal model holography, JHEP 07 (2013) 174 [arXiv:1305.1048] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Large- \( \mathcal{N}=4 \) holography, JHEP 09 (2013) 036 [arXiv:1305.4181] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Rønne, Extended higher spin holography and Grassmannian models, JHEP 11 (2013) 038 [arXiv:1306.0466] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Ronne, Higher spin AdS 3 holography with extended supersymmetry, JHEP 10 (2014) 163 [arXiv:1406.1521] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Higher spins & strings, JHEP 11 (2014) 044 [arXiv:1406.6103] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
N. Wyllard, A N − 1 conformal Toda field theory correlation functions from conformal \( \mathcal{N}=2 \) SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
K. Gawedzki, Noncompact WZW conformal field theories, IHES-P-91-73 (1991).
S. Ribault and J. Teschner, H +3 -WZNW correlators from Liouville theory, JHEP 06 (2005) 014 [hep-th/0502048] [INSPIRE].
Y. Hikida and V. Schomerus, H +3 WZNW model from Liouville field theory, JHEP 10 (2007) 064 [arXiv:0706.1030] [INSPIRE].
Y. Hikida and V. Schomerus, Structure constants of the OSP(1|2) WZNW model, JHEP 12 (2007) 100 [arXiv:0711.0338] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Ronne, Supergroup-extended super Liouville correspondence, JHEP 06 (2011) 063 [arXiv:1103.5753] [INSPIRE].
V.G. Knizhnik, Superconformal algebras in two-dimensions, Theor. Math. Phys. 66 (1986) 68 [INSPIRE].
M.A. Bershadsky, Superconformal algebras in two-dimensions with arbitrary N, Phys. Lett. B 174 (1986) 285 [INSPIRE].
V.G. Kac and M. Wakimoto, Quantum reduction and representation theory of superconformal algebras, math-ph/0304011 [INSPIRE].
J. de Boer and T. Tjin, The relation between quantum W algebras and Lie algebras, Commun. Math. Phys. 160 (1994) 317 [hep-th/9302006] [INSPIRE].
A.M. Polyakov, Gauge transformations and diffeomorphisms, Int. J. Mod. Phys. A 5 (1990) 833 [INSPIRE].
M. Bershadsky, Conformal field theories via Hamiltonian reduction, Commun. Math. Phys. 139 (1991) 71 [INSPIRE].
A. Sevrin and W. Troost, Extensions of the Virasoro algebra and gauged WZW models, Phys. Lett. B 315 (1993) 304 [hep-th/9306033] [INSPIRE].
L. Feher, L. O’Raifeartaigh, P. Ruelle, I. Tsutsui and A. Wipf, On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories, Phys. Rept. 222 (1992) 1.
F.A. Bais, T. Tjin and P. van Driel, Covariantly coupled chiral algebras, Nucl. Phys. B 357 (1991) 632 [INSPIRE].
A.B. Zamolodchikov, Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor. Math. Phys. 65 (1985) 1205 [INSPIRE].
L.J. Romans, Quasisuperconformal algebras in two-dimensions and Hamiltonian reduction, Nucl. Phys. B 357 (1991) 549 [INSPIRE].
T. Creutzig and Y. Hikida, Branes in the OSP(1|2) WZNW model, Nucl. Phys. B 842 (2011) 172 [arXiv:1004.1977] [INSPIRE].
V.A. Fateev, A.B. Zamolodchikov and Al.B. Zamolodchikov, unpublished.
Y. Hikida and V. Schomerus, The FZZ-duality conjecture: a proof, JHEP 03 (2009) 095 [arXiv:0805.3931] [INSPIRE].
T. Creutzig, Y. Hikida and P.B. Rønne, The FZZ duality with boundary, JHEP 09 (2011) 004 [arXiv:1012.4731] [INSPIRE].
K. Hosomichi, \( \mathcal{N}=2 \) Liouville theory with boundary, JHEP 12 (2006) 061 [hep-th/0408172] [INSPIRE].
J. de Boer and T. Tjin, Quantization and representation theory of finite W algebras, Commun. Math. Phys. 158 (1993) 485 [hep-th/9211109] [INSPIRE].
I. Bars, Free fields and new cosets of current algebras, Phys. Lett. B 255 (1991) 353 [INSPIRE].
B.L. Feigin and A.M. Semikhatov, W (2) n algebras, Nucl. Phys. B 698 (2004) 409 [math/0401164] [INSPIRE].
T. Creutzig and A.R. Linshaw, Cosets of affine vertex algebras inside larger structures, arXiv:1407.8512 [INSPIRE].
L.F. Alday and Y. Tachikawa, Affine SL(2) conformal blocks from 4d gauge theories, Lett. Math. Phys. 94 (2010) 87 [arXiv:1005.4469] [INSPIRE].
C. Kozcaz, S. Pasquetti, F. Passerini and N. Wyllard, Affine sl(N) conformal blocks from \( \mathcal{N}=2 \) SU(N) gauge theories, JHEP 01 (2011) 045[arXiv:1008.1412] [INSPIRE].
E. Frenkel, S. Gukov and J. Teschner, Surface operators and separation of variables, arXiv:1506.07508 [INSPIRE].
N. Wyllard, W-algebras and surface operators in \( \mathcal{N}=2 \) gauge theories, J. Phys. A 44 (2011) 155401 [arXiv:1011.0289] [INSPIRE].
N. Wyllard, Instanton partition functions in \( \mathcal{N}=2 \) SU(N) gauge theories with a general surface operator and their W-algebra duals, JHEP 02 (2011) 114 [arXiv:1012.1355] [INSPIRE].
J. Fay, Lecture Notes in Mathematics. Vol. 352: Theta functions on Riemann surfaces, Springer-Verlag, Springer Berlin (1973).
D. Mumford, Progress in Mathematics 43: Tata lectures on theta, Vols. I, II, Birkhäuser, Boston U.S.A. (1984).
L. Álvarez-Gaumé, G.W. Moore and C. Vafa, Theta functions, modular invariance and strings, Commun. Math. Phys. 106 (1986) 1 [INSPIRE].
E.P. Verlinde and H.L. Verlinde, Chiral bosonization, determinants and the string partition function, Nucl. Phys. B 288 (1987) 357 [INSPIRE].
D. Bernard, On the Wess-Zumino-Witten models on Riemann surfaces, Nucl. Phys. B 309 (1988) 145 [INSPIRE].
T. Tjin and P. Van Driel, Coupled WZNW — Toda models and covariant KdV hierarchies, ITFA-91-04.
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: 1509.07516
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Creutzig, T., Hikida, Y. & Rønne, P.B. Correspondences between WZNW models and CFTs with W-algebra symmetry. J. High Energ. Phys. 2016, 48 (2016). https://doi.org/10.1007/JHEP02(2016)048
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2016)048