Abstract
The Yang-Baxter σ-model is an integrable deformation of the principal chiral model on a Lie group G. The deformation breaks the G × G symmetry to U(1)rank(G) × G. It is known that there exist non-local conserved charges which, together with the unbroken U(1)rank(G) local charges, form a Poisson algebra , which is the semiclassical limit of the quantum group \( {U}_q\left(\mathfrak{g}\right) \), with \( \mathfrak{g} \) the Lie algebra of G. For a general Lie group G with rank(G) > 1, we extend the previous result by constructing local and non-local conserved charges satisfying all the defining relations of the infinite-dimensional Poisson algebra , the classical analogue of the quantum loop algebra \( {U}_q\left(L\mathfrak{g}\right) \), where \( L\mathfrak{g} \) is the loop algebra of \( \mathfrak{g} \). Quite unexpectedly, these defining relations are proved without encountering any ambiguity related to the non-ultralocality of this integrable σ-model.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
F. Delduc, M. Magro and B. Vicedo, On classical q-deformations of integrable σ-models, JHEP 11 (2013) 192 [arXiv:1308.3581] [INSPIRE].
K. Sfetsos, Integrable interpolations: From exact CFTs to non-Abelian T-duals, Nucl. Phys. B 880 (2014) 225 [arXiv:1312.4560] [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].
F. Delduc, M. Magro and B. Vicedo, Derivation of the action and symmetries of the q-deformed AdS 5 × S 5 superstring, JHEP 10 (2014) 132 [arXiv:1406.6286] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, Jordanian deformations of the AdS 5 xS 5 superstring, JHEP 04 (2014) 153 [arXiv:1401.4855] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, Integrable double deformation of the principal chiral model, Nucl. Phys. B 891 (2015) 312 [arXiv:1410.8066] [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].
B. Hoare, Towards a two-parameter q-deformation of AdS 3 × S 3 × M 4 superstrings, Nucl. Phys. B 891 (2015) 259 [arXiv:1411.1266] [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].
N.J. MacKay, On the classical origins of Yangian symmetry in integrable field theory, Phys. Lett. B 281 (1992) 90 [Erratum ibid. B 308 (1993) 444] [INSPIRE].
D. Bernard, An introduction to Yangian symmetries, International Journal of Modern Physics B 7 (1993) 3517 [hep-th/9211133].
I. Kawaguchi and K. Yoshida, Hidden Yangian symmetry in σ-model on squashed sphere, JHEP 11 (2010) 032 [arXiv:1008.0776] [INSPIRE].
G. Itsios, K. Sfetsos, K. Siampos and A. Torrielli, The classical Yang-Baxter equation and the associated Yangian symmetry of gauged WZW-type theories, Nucl. Phys. B 889 (2014) 64 [arXiv:1409.0554] [INSPIRE].
D. Orlando, S. Reffert and L.I. Uruchurtu, Classical Integrability of the Squashed Three-sphere, Warped AdS3 and Schroedinger Spacetime via T-duality, J. Phys. A 44 (2011) 115401 [arXiv:1011.1771] [INSPIRE].
I. Kawaguchi, D. Orlando and K. Yoshida, Yangian symmetry in deformed WZNW models on squashed spheres, Phys. Lett. B 701 (2011) 475 [arXiv:1104.0738] [INSPIRE].
D. Orlando and L.I. Uruchurtu, Integrable Superstrings on the Squashed Three-sphere, JHEP 10 (2012) 007 [arXiv:1208.3680] [INSPIRE].
I. Kawaguchi and K. Yoshida, A deformation of quantum affine algebra in squashed Wess-Zumino-Novikov-Witten models, J. Math. Phys. 55 (2014) 062302 [arXiv:1311.4696] [INSPIRE].
A. Ballesteros, E. Celeghini and M.A. del Olmo, Poisson-Hopf limit of quantum algebras, J. Phys. A 42 (2009) 275202 [arXiv:0903.2178].
I. Kawaguchi and K. Yoshida, Hybrid classical integrability in squashed σ-models, Phys. Lett. B 705 (2011) 251 [arXiv:1107.3662] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, On the classical equivalence of monodromy matrices in squashed σ-model, JHEP 06 (2012) 082 [arXiv:1203.3400] [INSPIRE].
I. Kawaguchi, T. Matsumoto and K. Yoshida, The classical origin of quantum affine algebra in squashed σ-models, JHEP 04 (2012) 115 [arXiv:1201.3058] [INSPIRE].
V. Tolstoy and S. Khoroshkin, The universal R-matrix for quantum untwisted affine Lie algebras, Funct. Anal. Appl. 26 (1992) 69.
J.M. Maillet, Hamiltonian Structures for Integrable Classical Theories From Graded Kac-Moody Algebras, Phys. Lett. B 167 (1986) 401 [INSPIRE].
J.M. Maillet, New Integrable Canonical Structures in Two-dimensional Models, Nucl. Phys. B 269 (1986) 54 [INSPIRE].
N.J. MacKay, Introduction to Yangian symmetry in integrable field theory, Int. J. Mod. Phys. A 20 (2005) 7189 [hep-th/0409183] [INSPIRE].
T. Kameyama and K. Yoshida, Anisotropic Landau-Lifshitz σ-models from q-deformed AdS 5× S 5 superstrings, JHEP 08 (2014) 110 [arXiv:1405.4467] [INSPIRE].
V.G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985) 254 [INSPIRE].
M. Jimbo, A q difference analog of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985) 63 [INSPIRE].
V.G. Drinfeld, Quantum groups, J. Sov. Math. 41 (1988) 898 [INSPIRE].
F. Delduc, S. Lacroix, M. Magro and B. Vicedo, On q-deformed symmetries as Poisson-Lie symmetries and application to Yang-Baxter type models, J. Phys. A 49 (2016) 415402 [arXiv:1606.01712] [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: 1701.03691
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
Delduc, F., Kameyama, T., Magro, M. et al. Affine q-deformed symmetry and the classical Yang-Baxter σ-model. J. High Energ. Phys. 2017, 126 (2017). https://doi.org/10.1007/JHEP03(2017)126
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2017)126