Abstract
Integrable deformation of SU(2) sigma and lambda models are considered at the classical and quantum levels. These are the Yang-Baxter and XXZ-type anisotropic deformations. The XXZ type deformations are UV safe in one regime, while in another regime, like the Yang-Baxter deformations, they exhibit cyclic RG behaviour. The associ-ated affine quantum group symmetry, realized classically at the Poisson bracket level, has q a complex phase in the UV safe regime and q real in the cyclic RG regime, where q is an RG invariant. Based on the symmetries and RG flow we propose exact factorizable S-matrices to describe the scattering of states in the lambda models, from which the sigma models follow by taking a limit and non-abelian T-duality. In the cyclic RG regimes, the S-matrices are periodic functions of rapidity, at large rapidity, and in the Yang-Baxter case violate parity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N.J. MacKay, Introduction to Yangian symmetry in integrable field theory, Int. J. Mod. Phys. A 20 (2005) 7189 [hep-th/0409183] [INSPIRE].
M. Lüscher, Quantum nonlocal charges and absence of particle production in the two-dimensional nonlinear σ-model, Nucl. Phys. B 135 (1978) 1 [INSPIRE].
Y.Y. Goldschmidt and E. Witten, Conservation laws in some two-dimensional models, Phys. Lett. B 91 (1980) 392.
E. Ogievetsky, P. Wiegmann and N. Reshetikhin, The principal chiral field in two-dimensions on classical Lie algebras: the Bethe ansatz solution and factorized theory of scattering, Nucl. Phys. B 280 (1987) 45 [INSPIRE].
A.M. Polyakov and P.B. Wiegmann, Theory of nonabelian Goldstone bosons, Phys. Lett. B 131 (1983)121.
P. Wiegmann, Exact factorized S matrix of the chiral field in two-dimensions, Phys. Lett. B 142 (1984) 173.
P.B. Wiegmann, On the theory of nonabelian Goldstone bosons in two-dimensions: exact solution of the SU(N) × SU(N) nonlinear sigma model, Phys. Lett. B 141 (1984) 217.
P. Hasenfratz, M. Maggiore and F. Niedermayer, The exact mass gap of the O(3) and O(4) nonlinear σ-models in D = 2, Phys. Lett. B 245 (1990) 522 [INSPIRE].
J. Balog, S. Naik, F. Niedermayer and P. Weisz, Exact mass gap of the chiral SU(N) × SU(N) model, Phys. Rev. Lett. 69 (1992) 873 [INSPIRE].
V.A. Fateev, V.A. Kazakov and P.B. Wiegmann, Principal chiral field at large-N , Nucl. Phys. B 424 (1994) 505 [hep-th/9403099] [INSPIRE].
T.J. Hollowood, The exact mass gaps of the principal chiral models, Phys. Lett. B 329 (1994) 450 [hep-th/9402084] [INSPIRE].
P.B. Wiegmann, Exact solution of the O(3) nonlinear sigma model, Phys. Lett. B 152 (1985) 209.
L.D. Faddeev and N.Yu. Reshetikhin, Integrability of the principal chiral field model in (1 + 1)-dimension, Annals Phys. 167 (1986) 227 [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].
B. Hoare, R. Roiban and A.A. Tseytlin, On deformations of AdS n × S n supercosets, JHEP 06 (2014) 002 [arXiv:1403.5517] [INSPIRE].
V.A. Fateev, The σ-model (dual) representation for a two-parameter family of integrable quantum field theories, Nucl. Phys. B 473 (1996) 509 [INSPIRE].
J.M. Evans and T.J. Hollowood, Integrable theories that are asymptotically CFT, Nucl. Phys. B 438 (1995) 469 [hep-th/9407113] [INSPIRE].
K. Sfetsos, Integrable interpolations: from exact CFTs to non-Abelian T-duals, Nucl. Phys. B 880 (2014) 225 [arXiv:1312.4560] [INSPIRE].
E. Witten, Nonabelian bosonization in two-dimensions, Commun. Math. Phys. 92 (1984) 455 [INSPIRE].
K. Bardakci, E. Rabinovici and B. Saering, String models with c < 1 components, Nucl. Phys. B 299 (1988) 151 [INSPIRE].
K. Gawedzki and A. Kupiainen, Coset construction from functional integrals, Nucl. Phys. B 320 (1989) 625 [INSPIRE].
D. Karabali, Q.-H. Park, H.J. Schnitzer and Z. Yang, A GKO construction based on a path integral formulation of gauged Wess-Zumino-Witten actions, Phys. Lett. B 216 (1989) 307 [INSPIRE].
D. Karabali and H.J. Schnitzer, BRST quantization of the gauged WZW action and coset conformal field theories, Nucl. Phys. B 329 (1990) 649 [INSPIRE].
C. Appadu, T.J. Hollowood and D. Price, Quantum inverse scattering and the Lambda deformed principal chiral model, J. Phys. A 50 (2017) 305401 [arXiv:1703.06699] [INSPIRE].
Y. Lozano, NonAbelian duality and canonical transformations, Phys. Lett. B 355 (1995) 165 [hep-th/9503045] [INSPIRE].
A. LeClair, Restricted sine-Gordon theory and the minimal conformal series, Phys. Lett. B 230 (1989) 103 [INSPIRE].
D. Bernard and A. Leclair, Quantum group symmetries and nonlocal currents in 2D QFT, Commun. Math. Phys. 142 (1991) 99 [INSPIRE].
D. Bernard and A. Leclair, Residual quantum symmetries of the restricted sine-Gordon theories, Nucl. Phys. B 340 (1990) 721 [INSPIRE].
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, The classical origin of quantum affine algebra in squashed σ-models, JHEP 04 (2012) 115 [arXiv:1201.3058] [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].
C. Ahn, D. Bernard and A. LeClair, Fractional supersymmetries in perturbed coset CFTs and integrable soliton theory, Nucl. Phys. B 346 (1990) 409 [INSPIRE].
C. Appadu, T.J. Hollowood, D. Price and D.C. Thompson, to appear.
F. Delduc, T. Kameyama, M. Magro and B. Vicedo, Affine q-deformed symmetry and the classical Yang-Baxter σ-model, JHEP 03 (2017) 126 [arXiv:1701.03691] [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].
O. Babelon, D. Bernard and M. Talon, Introduction to classical integrable systems, Cambridge University Press, Cambridge U.K. (2003).
J.M. Maillet, New integrable canonical structures in two-dimensional models, Nucl. Phys. B 269 (1986) 54 [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].
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].
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].
A.A. Tseytlin, Conformal anomaly in two-dimensional σ-model on curved background and strings, Phys. Lett. B 178 (1986) 34 [INSPIRE].
A.A. Tseytlin, σ model Weyl invariance conditions and string equations of motion, Nucl. Phys. B 294 (1987) 383 [INSPIRE].
G.M. Shore, A local renormalization group equation, diffeomorphisms and conformal invariance in σ models, Nucl. Phys. B 286 (1987) 349 [INSPIRE].
A.A. Tseytlin, On a ‘universal’ class of WZW type conformal models, Nucl. Phys. B 418 (1994) 173 [hep-th/9311062] [INSPIRE].
P. Bowcock, Canonical quantization of the gauged Wess-Zumino model, Nucl. Phys. B 316 (1989) 80 [INSPIRE].
D. Bernard, Hidden Yangians in 2D massive current algebras, Commun. Math. Phys. 137 (1991) 191 [INSPIRE].
A. Leclair, J.M. Roman and G. Sierra, Russian doll renormalization group, Kosterlitz-Thouless flows and the cyclic sine-Gordon model, Nucl. Phys. B 675 (2003) 584 [hep-th/0301042] [INSPIRE].
A. LeClair, J.M. Roman and G. Sierra, Log periodic behavior of finite size effects in field theories with RG limit cycles, Nucl. Phys. B 700 (2004) 407 [hep-th/0312141] [INSPIRE].
A. LeClair and G. Sierra, Renormalization group limit cycles and field theories for elliptic S matrices, J. Stat. Mech. 0408 (2004) P08004 [hep-th/0403178] [INSPIRE].
G. Mussardo and S. Penati, A quantum field theory with infinite resonance states, Nucl. Phys. B 567 (2000) 454 [hep-th/9907039] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized s matrices in two-dimensions as the exact solutions of certain relativistic quantum field models, Annals Phys. 120 (1979) 253 [INSPIRE].
P. Dorey, Exact S matrices, hep-th/9810026 [INSPIRE].
K.M. Bulycheva and A.S. Gorsky, Limit cycles in renormalization group dynamics, Phys. Usp. 57 (2014) 171 [arXiv:1402.2431] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 43 (1986) 730 [Pisma Zh. Eksp. Teor. Fiz. 43 (1986) 565] [INSPIRE].
T.L. Curtright, X. Jin and C.K. Zachos, RG flows, cycles and c-theorem folklore, Phys. Rev. Lett. 108 (2012) 131601 [arXiv:1111.2649] [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].
B. Gerganov, A. LeClair and M. Moriconi, On the β-function for anisotropic current interactions in 2D, Phys. Rev. Lett. 86 (2001) 4753 [hep-th/0011189] [INSPIRE].
D. Bernard and A. LeClair, Strong weak coupling duality in anisotropic current interactions, Phys. Lett. B 512 (2001) 78 [hep-th/0103096] [INSPIRE].
A.A. Tseytlin, Conditions of Weyl invariance of two-dimensional σ model from equations of stationarity of ‘central charge’ action, Phys. Lett. B 194 (1987) 63 [INSPIRE].
A.A. Tseytlin, On σ-model RG flow, ‘central charge’ action and Perelman’s entropy, Phys. Rev. D 75 (2007) 064024 [hep-th/0612296] [INSPIRE].
V.A. Fateev and A.B. Zamolodchikov, Parafermionic currents in the two-dimensional conformal quantum field theory and selfdual critical points in Z(n) invariant statistical systems, Sov. Phys. JETP 62 (1985) 215 [Zh. Eksp. Teor. Fiz. 89 (1985) 380] [INSPIRE].
D. Bernard and A. Leclair, The fractional supersymmetric sine-Gordon models, Phys. Lett. B 247 (1990) 309 [INSPIRE].
M. Jimbo, T. Miwa and M. Okado, Solvable lattice models related to the vector representation of classical simple Lie algebras, Commun. Math. Phys. 116 (1988) 507 [INSPIRE].
T.J. Hollowood, Quantizing SL(N) solitons and the Hecke algebra, Int. J. Mod. Phys. A 8 (1993) 947 [hep-th/9203076] [INSPIRE].
A.B. Zamolodchikov, Z(4) symmetric factorized S matrix in two space-time dimensions, Commun. Math. Phys. 69 (1979) 165 [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: 1706.05322
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
Appadu, C., Hollowood, T.J., Price, D. et al. Yang Baxter and anisotropic sigma and lambda models, cyclic RG and exact S-matrices. J. High Energ. Phys. 2017, 35 (2017). https://doi.org/10.1007/JHEP09(2017)035
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2017)035