Abstract
We show that the CFT with symmetry group \( {G}_{k_1}\times {G}_{k_2}\times \cdots \times {G}_{k_n} \) consisting of WZW models based on the same group G, but at arbitrary integer levels, admits an integrable deformation depending on 2(n − 1) continuous parameters. We derive the all-loop effective action of the deformed theory and prove integrability. We also calculate the exact in the deformation parameters RG flow equations which can be put in a particularly simple compact form. This allows a full determination and classification of the fixed points of the RG flow, in particular of those that are IR stable. The models under consideration provide concrete realizations of integrable flows between CFTs. We also consider non-Abelian T-duality type limits.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
M. Staudacher, The Factorized S-matrix of CFT/AdS, JHEP 05 (2005) 054 [hep-th/0412188] [INSPIRE].
J. Ambjørn, R.A. Janik and C. Kristjansen, Wrapping interactions and a new source of corrections to the spin-chain/string duality, Nucl. Phys. B 736 (2006) 288 [hep-th/0510171] [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
K. Sfetsos, Integrable interpolations: From exact CFTs to non-Abelian T-duals, Nucl. Phys. B 880 (2014) 225 [arXiv:1312.4560] [INSPIRE].
G. Georgiou, K. Sfetsos and K. Siampos, Double and cyclic λ-deformations and their canonical equivalents, Phys. Lett. B 771 (2017) 576 [arXiv:1704.07834] [INSPIRE].
G. Georgiou and K. Sfetsos, A new class of integrable deformations of CFTs, JHEP 03 (2017) 083 [arXiv:1612.05012] [INSPIRE].
G. Georgiou and K. Sfetsos, Integrable flows between exact CFTs, JHEP 11 (2017) 078 [arXiv:1707.05149] [INSPIRE].
G. Georgiou and K. Sfetsos, Novel all loop actions of interacting CFTs: Construction, integrability and RG flows, Nucl. Phys. B 937 (2018) 371 [arXiv:1809.03522] [INSPIRE].
G. Itsios, K. Sfetsos and K. Siampos, The all-loop non-Abelian Thirring model and its RG flow, Phys. Lett. B 733 (2014) 265 [arXiv:1404.3748] [INSPIRE].
K. Sfetsos and K. Siampos, Gauged WZW-type theories and the all-loop anisotropic non-Abelian Thirring model, Nucl. Phys. B 885 (2014) 583 [arXiv:1405.7803] [INSPIRE].
D. Kutasov, Duality Off the Critical Point in Two-dimensional Systems With Nonabelian Symmetries, Phys. Lett. B 233 (1989) 369 [INSPIRE].
G. Georgiou, E. Sagkrioti, K. Sfetsos and K. Siampos, Quantum aspects of doubly deformed CFTs, Nucl. Phys. B 919 (2017) 504 [arXiv:1703.00462] [INSPIRE].
E. Sagkrioti, K. Sfetsos and K. Siampos, RG flows for λ-deformed CFTs, Nucl. Phys. B 930 (2018) 499 [arXiv:1801.10174] [INSPIRE].
D. Kutasov, String Theory and the Nonabelian Thirring Model, Phys. Lett. B 227 (1989) 68 [INSPIRE].
B. Gerganov, A. LeClair and M. Moriconi, On the β-function for anisotropic current interactions in 2-D, Phys. Rev. Lett. 86 (2001) 4753 [hep-th/0011189] [INSPIRE].
A. LeClair, Chiral stabilization of the renormalization group for flavor and color anisotropic current interactions, Phys. Lett. B 519 (2001) 183 [hep-th/0105092] [INSPIRE].
C. Appadu and T.J. Hollowood, β-function of k deformed AdS 5 × S 5 string theory, JHEP 11 (2015) 095 [arXiv:1507.05420] [INSPIRE].
G. Georgiou, K. Sfetsos and K. Siampos, All-loop anomalous dimensions in integrable λ-deformed σ-models, Nucl. Phys. B 901 (2015) 40 [arXiv:1509.02946] [INSPIRE].
G. Georgiou, K. Sfetsos and K. Siampos, All-loop correlators of integrable λ-deformed σ-models, Nucl. Phys. B 909 (2016) 360 [arXiv:1604.08212] [INSPIRE].
G. Georgiou, K. Sfetsos and K. Siampos, λ-Deformations of left-right asymmetric CFTs, Nucl. Phys. B 914 (2017) 623 [arXiv:1610.05314] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
G. Georgiou, P. Panopoulos, E. Sagkrioti, K. Sfetsos and K. Siampos, The exact C-function in integrable λ-deformed theories, Phys. Lett. B 782 (2018) 613 [arXiv:1805.03731] [INSPIRE].
E. Sagkrioti, K. Sfetsos and K. Siampos, Weyl anomaly and the C-function in λ-deformed CFTs, Nucl. Phys. B 938 (2019) 426 [arXiv:1810.04189] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, Integrable Deformations of Strings on Symmetric Spaces, JHEP 11 (2014) 009 [arXiv:1407.2840] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, An Integrable Deformation of the AdS 5 × S 5 Superstring, J. Phys. A 47 (2014) 495402 [arXiv:1409.1538] [INSPIRE].
K. Sfetsos and K. Siampos, Integrable deformations of the \( {G}_{k_1}\times {G}_{k_2}/{G}_{k_1+{k}_2} \) coset CFTs, Nucl. Phys. B 927 (2018) 124 [arXiv:1710.02515] [INSPIRE].
J. Balog, P. Forgacs, Z. Horvath and L. Palla, A New family of SU(2) symmetric integrable σ-models, Phys. Lett. B 324 (1994) 403 [hep-th/9307030] [INSPIRE].
K. Sfetsos and K. Siampos, The anisotropic λ-deformed SU(2) model is integrable, Phys. Lett. B 743 (2015) 160 [arXiv:1412.5181] [INSPIRE].
K. Sfetsos, K. Siampos and D.C. Thompson, Generalised integrable λ- and η-deformations and their relation, Nucl. Phys. B 899 (2015) 489 [arXiv:1506.05784] [INSPIRE].
K. Sfetsos and D.C. Thompson, Spacetimes for λ-deformations, JHEP 12 (2014) 164 [arXiv:1410.1886] [INSPIRE].
S. Demulder, K. Sfetsos and D.C. Thompson, Integrable λ-deformations: Squashing Coset CFTs and AdS 5 × S 5, JHEP 07 (2015) 019 [arXiv:1504.02781] [INSPIRE].
R. Borsato, A.A. Tseytlin and L. Wulff, Supergravity background of λ-deformed model for AdS 2 × S 2 supercoset, Nucl. Phys. B 905 (2016) 264 [arXiv:1601.08192] [INSPIRE].
Y. Chervonyi and O. Lunin, Supergravity background of the λ-deformed AdS 3 × S 3 supercoset, Nucl. Phys. B 910 (2016) 685 [arXiv:1606.00394] [INSPIRE].
B. Vicedo, Deformed integrable σ-models, classical R-matrices and classical exchange algebra on Drinfel’d doubles, J. Phys. A 48 (2015) 355203 [arXiv:1504.06303] [INSPIRE].
B. Hoare and A.A. Tseytlin, On integrable deformations of superstring σ-models related to AdS n × S n supercosets, Nucl. Phys. B 897 (2015) 448 [arXiv:1504.07213] [INSPIRE].
C. Klimčík, η and λ deformations as ℰ-models, Nucl. Phys. B 900 (2015) 259 [arXiv:1508.05832] [INSPIRE].
C. Klimčík, Poisson-Lie T-duals of the bi-Yang-Baxter models, Phys. Lett. B 760 (2016) 345 [arXiv:1606.03016] [INSPIRE].
B. Hoare and F.K. Seibold, Poisson-Lie duals of the η-deformed AdS 2 × S 2 × T 6 superstring, JHEP 08 (2018) 107 [arXiv:1807.04608] [INSPIRE].
C. Klimčík and P. Ševera, Dual nonAbelian duality and the Drinfeld double, Phys. Lett. B 351 (1995) 455 [hep-th/9502122] [INSPIRE].
K. Sfetsos, Duality invariant class of two-dimensional field theories, Nucl. Phys. B 561 (1999) 316 [hep-th/9904188] [INSPIRE].
C. Klimčík, Yang-Baxter σ-models and dS/AdS T duality, JHEP 12 (2002) 051 [hep-th/0210095] [INSPIRE].
C. Klimčík, On integrability of the Yang-Baxter σ-model, J. Math. Phys. 50 (2009) 043508 [arXiv:0802.3518] [INSPIRE].
C. Klimčík, Integrability of the bi-Yang-Baxter σ-model, Lett. Math. Phys. 104 (2014) 1095 [arXiv:1402.2105] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, On classical q-deformations of integrable σ-models, JHEP 11 (2013) 192 [arXiv:1308.3581] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, An integrable deformation of the AdS 5 × S 5 superstring action, Phys. Rev. Lett. 112 (2014) 051601 [arXiv:1309.5850] [INSPIRE].
G. Arutyunov, R. Borsato and S. Frolov, S-matrix for strings on η-deformed AdS 5 × S 5, JHEP 04 (2014) 002 [arXiv:1312.3542] [INSPIRE].
O. Lunin and W. Tian, Scalar fields on λ-deformed cosets, Nucl. Phys. B 938 (2019) 671 [arXiv:1808.02971] [INSPIRE].
D.M. Schmidtt, Integrable Lambda Models And Chern-Simons Theories, JHEP 05 (2017) 012 [arXiv:1701.04138] [INSPIRE].
D.M. Schmidtt, Lambda Models From Chern-Simons Theories, JHEP 11 (2018) 111 [arXiv:1808.05994] [INSPIRE].
S. Driezen, A. Sevrin and D.C. Thompson, D-branes in λ-deformations, JHEP 09 (2018) 015 [arXiv:1806.10712] [INSPIRE].
P. Bowcock, Canonical Quantization of the Gauged Wess-Zumino Model, Nucl. Phys. B 316 (1989) 80 [INSPIRE].
F. Delduc, S. Lacroix, M. Magro and B. Vicedo, Integrable Coupled σ Models, Phys. Rev. Lett. 122 (2019) 041601 [arXiv:1811.12316] [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: 1812.04033
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Georgiou, G., Sfetsos, K. The most general λ-deformation of CFTs and integrability. J. High Energ. Phys. 2019, 94 (2019). https://doi.org/10.1007/JHEP03(2019)094
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2019)094