Abstract
In [1] we proposed a formula for the 3-point structure constants of generic primary fields in the Toda field theory, derived using topological strings and the AGT-W correspondence from the partition functions of the non-Lagrangian T N theories on S 4. In this article, we obtain from it the well-known formula by Fateev and Litvinov and show that the degeneration on a first level of one of the three primary fields on the Toda side corresponds to a particular Higgsing of the T N theories.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. Mitev and E. Pomoni, Toda 3-point functions from topological strings, JHEP 06 (2015) 049 [arXiv:1409.6313] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, Germany (1997).
H. Dorn and H.J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].
V.A. Fateev and A.V. Litvinov, On differential equation on four-point correlation function in the conformal Toda field theory, JETP Lett. 81 (2005) 594 [hep-th/0505120] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory. I, JHEP 11 (2007) 002 [arXiv:0709.3806] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory II, JHEP 01 (2009) 033 [arXiv:0810.3020] [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 N =2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
C. Kozcaz, S. Pasquetti and N. Wyllard, A&B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [INSPIRE].
L. Bao, V. Mitev, E. Pomoni, M. Taki and F. Yagi, Non-lagrangian theories from brane junctions, JHEP 01 (2014) 175 [arXiv:1310.3841] [INSPIRE].
H. Hayashi, H.-C. Kim and T. Nishinaka, Topological strings and 5d T N partition functions, JHEP 06 (2014) 014 [arXiv:1310.3854] [INSPIRE].
J. Gomis, T. Okuda and V. Pestun, Exact results for ’t Hooft loops in gauge theories on S 4, JHEP 05 (2012) 141 [arXiv:1105.2568] [INSPIRE].
H. Awata and Y. Yamada, Five-dimensional AGT conjecture and the deformed Virasoro algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [INSPIRE].
H. Awata and Y. Yamada, Five-dimensional AGT relation and the deformed β-ensemble, Prog. Theor. Phys. 124 (2010) 227 [arXiv:1004.5122] [INSPIRE].
R. Schiappa and N. Wyllard, An A(r) threesome: matrix models, 2D CFTs and 4D N = 2 gauge theories, J. Math. Phys. 51 (2010) 082304 [arXiv:0911.5337] [INSPIRE].
A. Mironov, A. Morozov, S. Shakirov and A. Smirnov, Proving AGT conjecture as HS duality: extension to five dimensions, Nucl. Phys. B 855 (2012) 128 [arXiv:1105.0948] [INSPIRE].
H. Itoyama, T. Oota and R. Yoshioka, 2D-4D connection between q-virasoro/W block at root of unity limit and instanton partition function on ALE space, Nucl. Phys. B 877 (2013) 506 [arXiv:1308.2068] [INSPIRE].
L. Bao, E. Pomoni, M. Taki and F. Yagi, M 5-branes, toric diagrams and gauge theory duality, JHEP 04 (2012) 105 [arXiv:1112.5228] [INSPIRE].
F. Nieri, S. Pasquetti and F. Passerini, 3D and 5D gauge theory partition functions as q-deformed CFT correlators, Lett. Math. Phys. 105 (2015) 109 [arXiv:1303.2626] [INSPIRE].
F. Nieri, S. Pasquetti, F. Passerini and A. Torrielli, 5D partition functions, q-Virasoro systems and integrable spin-chains, JHEP 12 (2014) 040 [arXiv:1312.1294] [INSPIRE].
M. Aganagic, N. Haouzi, C. Kozcaz and S. Shakirov, Gauge/Liouville triality, arXiv:1309.1687 [INSPIRE].
M. Aganagic, N. Haouzi and S. Shakirov, A n -triality, arXiv:1403.3657 [INSPIRE].
M. Taki, On AGT-W conjecture and q-deformed W-algebra, arXiv:1403.7016 [INSPIRE].
R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [INSPIRE].
M.C.N. Cheng, R. Dijkgraaf and C. Vafa, Non-perturbative topological strings and conformal blocks, JHEP 09 (2011) 022 [arXiv:1010.4573] [INSPIRE].
M.-C. Tan, An M-theoretic derivation of a 5D and 6D AGT correspondence and relativistic and elliptized integrable systems, JHEP 12 (2013) 031 [arXiv:1309.4775] [INSPIRE].
A. Iqbal and C. Vafa, BPS degeneracies and superconformal index in diverse dimensions, Phys. Rev. D 90 (2014) 105031 [arXiv:1210.3605] [INSPIRE].
H. Awata, H. Kubo, S. Odake and J. Shiraishi, Quantum deformation of the W (N ) algebra, q-alg/9612001 [INSPIRE].
F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].
H. Awata and H. Kanno, Instanton counting, Macdonald functions and the moduli space of D-branes, JHEP 05 (2005) 039 [hep-th/0502061] [INSPIRE].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
H. Hayashi and G. Zoccarato, Exact partition functions of Higgsed 5D T N theories, JHEP 01 (2015) 093 [arXiv:1409.0571] [INSPIRE].
H. Hayashi, Y. Tachikawa and K. Yonekura, Mass-deformed T N as a linear quiver, JHEP 02 (2015) 089 [arXiv:1410.6868] [INSPIRE].
A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].
O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE].
O. Aharony, A. Hanany and B. Kol, Webs of (p, q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto and J. Maldacena, The gravity duals of N = 2 superconformal field theories, JHEP 10 (2012) 189 [arXiv:0904.4466] [INSPIRE].
O. DeWolfe, A. Hanany, A. Iqbal and E. Katz, Five-branes, seven-branes and five-dimensional E(n) field theories, JHEP 03 (1999) 006 [hep-th/9902179] [INSPIRE].
A. Mikhailov, N. Nekrasov and S. Sethi, Geometric realizations of BPS states in N = 2 theories, Nucl. Phys. B 531 (1998) 345 [hep-th/9803142] [INSPIRE].
O. DeWolfe, T. Hauer, A. Iqbal and B. Zwiebach, Constraints on the BPS spectrum of N = 2, D = 4 theories with A-D-E flavor symmetry, Nucl. Phys. B 534 (1998) 261 [hep-th/9805220] [INSPIRE].
O. Bergman and A. Fayyazuddin, String junction transitions in the moduli space of N = 2 SYM, Nucl. Phys. B 535 (1998) 139 [hep-th/9806011] [INSPIRE].
T. Dimofte, S. Gukov and L. Hollands, Vortex counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].
M. Taki, Surface operator, bubbling Calabi-Yau and AGT relation, JHEP 07 (2011) 047 [arXiv:1007.2524] [INSPIRE].
M. Aganagic and S. Shakirov, Knot homology and refined Chern-Simons index, Commun. Math. Phys. 333 (2015) 187 [arXiv:1105.5117] [INSPIRE].
M. Aganagic and S. Shakirov, Refined Chern-Simons theory and topological string, arXiv:1210.2733 [INSPIRE].
V. Mitev, E. Pomoni, M. Taki and F. Yagi, Fiber-base duality and global symmetry enhancement, JHEP 04 (2015) 052 [arXiv:1411.2450] [INSPIRE].
M.A. Olshanetsky and V.B.K. Rogov, Liouville quantum mechanics on a lattice from geometry of quantum Lorentz group, J. Phys. A 27 (1994) 4669 [hep-th/9310084] [INSPIRE].
M. Chaichian, A. Demichev and P.P. Kulish, Quasiclassical limit in q-deformed systems, noncommutativity and the q-path integral, Phys. Lett. A 233 (1997) 251 [q-alg/9702023].
A. Lavagno, A. Scarfone and P. Narayana Swamy, Classical and quantum q-deformed physical systems, Eur. Phys. J. C 47 (2006) 253 [quant-ph/0605026]
P. Bowcock and G.M.T. Watts, Null vectors, three point and four point functions in conformal field theory, Theor. Math. Phys. 98 (1994) 350 [hep-th/9309146] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Integrable structure, W-symmetry and AGT relation, JHEP 01 (2012) 051 [arXiv:1109.4042] [INSPIRE].
V. Mitev, E. Pomoni, M. Taki and F. Yagi, work in progress.
H. Awata, B. Feigin, A. Hoshino, M. Kanai, J. Shiraishi and S. Yanagida, Notes on Ding-Iohara algebra and AGT conjecture, arXiv:1106.4088 [INSPIRE].
S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, hep-th/0612073 [INSPIRE].
J. Gomis and S. Matsuura, Bubbling surface operators and S-duality, JHEP 06 (2007) 025 [arXiv:0704.1657] [INSPIRE].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
D. Gaiotto, Surface operators in N = 2 4D gauge theories, JHEP 11 (2012) 090 [arXiv:0911.1316] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
A. Gadde and S. Gukov, 2D index and surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].
J. Gomis and B. Le Floch, M 2-brane surface operators and gauge theory dualities in Toda, JHEP 04 (2016) 183 [arXiv:1407.1852] [INSPIRE].
D. Gaiotto and H.-C. Kim, Surface defects and instanton partition functions, arXiv:1412.2781 [INSPIRE].
G. Bonelli, A. Tanzini and J. Zhao, Vertices, vortices and interacting surface operators, JHEP 06 (2012) 178 [arXiv:1102.0184] [INSPIRE].
Y. Nakayama, Liouville field theory: a decade after the revolution, Int. J. Mod. Phys. A 19 (2004) 2771 [hep-th/0402009] [INSPIRE].
M. Nishizawa, An elliptic analogue of the multiple gamma function, J. Phys. A 34 (2001) 7411.
S.O. Warnaar, A selberg integral for the Lie algebra A n , Acta Math. 203 (2009) 269 [arXiv:0708.1193].
J. Kaneko, q-Selberg integrals and Macdonald polynomials, Ann. Sci. École Norm. Sup. 29 (1996) 583.
I. G. Macdonald, Hypergeometric series II, unpublished manuscript, http://math.mit.edu/∼plamen/files/mh2.pdf.
T.H. Baker and P.J. Forrester, Transformation formulae for multivariable basic hypergeometric series, math/9803146.
I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Oxford U.K. (1998).
H. Nakajima and K. Yoshioka, Instanton counting on blowup. 1., Invent. Math. 162 (2005) 313 [math/0306198] [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: 1412.3395
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
Isachenkov, M., Mitev, V. & Pomoni, E. Toda 3-point functions from topological strings II. J. High Energ. Phys. 2016, 66 (2016). https://doi.org/10.1007/JHEP08(2016)066
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2016)066