Abstract
We investigate the relation between 3d \( \mathcal{N} \) = 2 theories and 2d free field correlators or Dotsenko-Fateev (DF) integrals for Liouville CFT. We show that the S2 × S1 partition functions of some known 3d Seiberg-like dualities reduce, in a suitable 2d limit, to known basic duality identities for DF correlators. These identities are applied in a variety of contexts in CFT, as for example in the derivation of the DOZZ 3-point function. Reversing the logic, we can try to guess new 3d IR dualities which reduce to more intricate duality relations for the DF correlators. For example, we show that a recently proposed duality relating the U(N) theory with one flavor and one adjoint to a WZ model can be regarded as the 3d ancestor of the evaluation formula for the DF integral representation of the 3-point correlator. We are also able to interpret the analytic continuation in the number of screening charges, which is performed on the CFT side to reconstruct the DOZZ 3-point function, as the geometric transition relating the 3d U(N) theory to the 5d T2 theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Aganagic, N. Haouzi, C. Kozcaz and S. Shakirov, Gauge/Liouville Triality, arXiv:1309.1687 [INSPIRE].
M. Aganagic, N. Haouzi and S. Shakirov, An -Triality, arXiv:1403.3657 [INSPIRE].
A. Nedelin, F. Nieri and M. Zabzine, q-Virasoro modular double and 3d partition functions, Commun. Math. Phys. 353 (2017) 1059 [arXiv:1605.07029] [INSPIRE].
T. Kimura and V. Pestun, Quiver W-algebras, Lett. Math. Phys. 108 (2018) 1351 [arXiv:1512.08533] [INSPIRE].
A. Nedelin, S. Pasquetti and Y. Zenkevich, T [SU(N)] duality webs: mirror symmetry, spectral duality and gauge/CFT correspondences, JHEP 02 (2019) 176 [arXiv:1712.08140] [INSPIRE].
O. Aharony, S.S. Razamat and B. Willett, From 3d duality to 2d duality, JHEP 11 (2017) 090 [arXiv:1710.00926] [INSPIRE].
M. Aganagic, K. Hori, A. Karch and D. Tong, Mirror symmetry in (2+1)-dimensions and (1+1)-dimensions, JHEP 07 (2001) 022 [hep-th/0105075] [INSPIRE].
F. Aprile, S. Pasquetti and Y. Zenkevich, Flipping the head of T [SU(N)]: mirror symmetry, spectral duality and monopoles, JHEP 04 (2019) 138 [arXiv:1812.08142] [INSPIRE].
O. Aharony, IR duality in d = 3 N = 2 supersymmetric USp(2Nc) and U(Nc) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].
F. Benini, S. Benvenuti and S. Pasquetti, SUSY monopole potentials in 2+1 dimensions, JHEP 08 (2017) 086 [arXiv:1703.08460] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Multipoint correlation functions in Liouville field theory and minimal Liouville gravity, Theor. Math. Phys. 154 (2008) 454 [arXiv:0707.1664] [INSPIRE].
P. Baseilhac and V.A. Fateev, Expectation values of local fields for a two-parameter family of integrable models and related perturbed conformal field theories, Nucl. Phys. B 532 (1998) 567 [hep-th/9906010] [INSPIRE].
S. Pasquetti and M. Sacchi, 3d dualities from 2d free field correlators: recombination and rank stabilization, arXiv:1905.05807 [INSPIRE].
M. Goulian and M. Li, Correlation functions in Liouville theory, Phys. Rev. Lett. 66 (1991) 2051 [INSPIRE].
V.S. Dotsenko and V.A. Fateev, Four Point Correlation Functions and the Operator Algebra in the Two-Dimensional Conformal Invariant Theories with the Central Charge c < 1, Nucl. Phys. B 251 (1985) 691 [INSPIRE].
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, A lecture on the Liouville vertex operators, Int. J. Mod. Phys. A 19S2 (2004) 436 [hep-th/0303150] [INSPIRE].
S. Benvenuti, A tale of exceptional 3d dualities, JHEP 03 (2019) 125 [arXiv:1809.03925] [INSPIRE].
O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
S. Benvenuti and S. Giacomelli, Supersymmetric gauge theories with decoupled operators and chiral ring stability, Phys. Rev. Lett. 119 (2017) 251601 [arXiv:1706.02225] [INSPIRE].
S. Benvenuti and S. Giacomelli, Abelianization and sequential confinement in 2 + 1 dimensions, JHEP 10 (2017) 173 [arXiv:1706.04949] [INSPIRE].
D. Kutasov and A. Schwimmer, On duality in supersymmetric Yang-Mills theory, Phys. Lett. B 354 (1995) 315 [hep-th/9505004] [INSPIRE].
S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d \( \mathcal{N} \) = 4 gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [INSPIRE].
D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
F.J. van de Bult, Hyperbolic Hypergeometric Functions, Ph.D. Thesis, Thomas Stieltjes Institute For Mathematics, (2007).
N. Aghaei, A. Amariti and Y. Sekiguchi, Notes on Integral Identities for 3d Supersymmetric Dualities, JHEP 04 (2018) 022 [arXiv:1709.08653] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Contact Terms, Unitarity and F-Maximization in Three-Dimensional Superconformal Theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Comments on Chern-Simons Contact Terms in Three Dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].
Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
A. Kapustin and B. Willett, Generalized Superconformal Index for Three Dimensional Field Theories, arXiv:1106.2484 [INSPIRE].
A. Narukawa, The modular properties and the integral representations of the multiple elliptic gamma functions, math/0306164.
Y. Imamura, Perturbative partition function for squashed S5 , PTEP 2013 (2013) 073B01 [arXiv:1210.6308] [INSPIRE].
G. Lockhart and C. Vafa, Superconformal Partition Functions and Non-perturbative Topological Strings, JHEP 10 (2018) 051 [arXiv:1210.5909] [INSPIRE].
H.-C. Kim, J. Kim and S. Kim, Instantons on the 5-sphere and M5-branes, arXiv:1211.0144 [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [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].
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].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [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].
J. Qiu, L. Tizzano, J. Winding and M. Zabzine, Gluing Nekrasov partition functions, Commun. Math. Phys. 337 (2015) 785 [arXiv:1403.2945] [INSPIRE].
I. Coman, E. Pomoni and J. Teschner, Trinion Conformal Blocks from Topological strings, arXiv:1906.06351 [INSPIRE].
F. Nieri, Y. Pan and M. Zabzine, 3d Mirror Symmetry from S-duality, Phys. Rev. D 98 (2018) 126002 [arXiv:1809.00736] [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].
H.-C. Kim, S.-S. Kim and K. Lee, 5-dim Superconformal Index with Enhanced En Global Symmetry, JHEP 10 (2012) 142 [arXiv:1206.6781] [INSPIRE].
S. Terashima, Supersymmetric gauge theories on S4 × S1 , Phys. Rev. D 89 (2014) 125001 [arXiv:1207.2163] [INSPIRE].
V. Mitev and E. Pomoni, Toda 3-Point Functions From Topological Strings, JHEP 06 (2015) 049 [arXiv:1409.6313] [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].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1903.10817
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
Pasquetti, S., Sacchi, M. From 3d dualities to 2d free field correlators and back. J. High Energ. Phys. 2019, 81 (2019). https://doi.org/10.1007/JHEP11(2019)081
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2019)081