Abstract
We study the AGT correspondence between four-dimensional supersymmetric gauge field theory and two-dimensional conformal field theories in the context of \( {\mathcal{W}}_N \) minimal models. The origin of the AGT correspondence is in a special integrable structure which appears in the properly extended conformal theory. One of the basic manifestations of this integrability is the special orthogonal basis which arises in the extended theory. We propose modification of the AGT representation for the \( {\mathcal{W}}_N \) conformal blocks in the minimal models. The necessary modification is related to the reduction of the orthogonal basis. This leads to the explicit combinatorial representation for the conformal blocks of minimal models and employs the sum over N-tupels of Young diagrams with additional restrictions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [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.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].
N. Wyllard, A(N − 1) conformal Toda field theory correlation functions from conformal N =2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
A. Mironov and A. Morozov, The power of Nekrasov functions, Phys. Lett. B 680 (2009) 188 [arXiv:0908.2190] [INSPIRE].
A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Integrable structure, W-symmetry and AGT relation, JHEP 01 (2012) 051 [arXiv:1109.4042] [INSPIRE].
V. Belavin and B. Feigin, Super Liouville conformal blocks from N = 2 SU(2) quiver gauge theories, JHEP 07 (2011) 079 [arXiv:1105.5800] [INSPIRE].
V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [arXiv:1012.1312] [INSPIRE].
R. Santachiara and A. Tanzini, Moore-Read fractional quantum Hall wavefunctions and SU(2) quiver gauge theories, Phys. Rev. D 82 (2010) 126006 [arXiv:1002.5017] [INSPIRE].
B. Estienne, V. Pasquier, R. Santachiara and D. Serban, Conformal blocks in Virasoro and W theories: duality and the Calogero-Sutherland model, Nucl. Phys. B 860 (2012) 377 [arXiv:1110.1101] [INSPIRE].
K. Papadodimas and S. Raju, Correlation functions in holographic minimal models, Nucl. Phys. B 856 (2012) 607 [arXiv:1108.3077] [INSPIRE].
C.-M. Chang and X. Yin, Correlators in W N minimal model revisited, JHEP 10 (2012) 050 [arXiv:1112.5459] [INSPIRE].
B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous gl ∞ : tensor products of Fock modules and \( {\mathcal{W}}_N \) characters, arXiv:1002.3113.
B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous gl ∞ : Semi-infinite construction of representations, arXiv:1002.3100.
A. Zamolodchikov, Higher equations of motion in Liouville field theory, Int. J. Mod. Phys. A 19S2 (2004) 510 [hep-th/0312279] [INSPIRE].
M. Bershtein and O. Foda, AGT, Burge pairs and minimal models, arXiv:1404.7075 [INSPIRE].
P. Bowcock and G.M.T. Watts, Null vectors of the W(3) algebra, Phys. Lett. B 297 (1992) 282 [hep-th/9209105] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory. I, JHEP 11 (2007) 002 [arXiv:0709.3806] [INSPIRE].
S. Mironov, A. Morozov and Y. Zenkevich, Generalized Jack polynomials and the AGT relations for the SU(3) group, JETP Lett. 99 (2014) 109 [arXiv:1312.5732] [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: 1404.7094
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
Alkalaev, K.B., Belavin, V.A. Conformal blocks of \( {\mathcal{W}}_N \) minimal models and AGT correspondence. J. High Energ. Phys. 2014, 24 (2014). https://doi.org/10.1007/JHEP07(2014)024
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2014)024