Abstract
We consider a class of 2d σ-models on products of group spaces that provide new examples of a close connection between integrability and stability under the RG flow. We first study the integrable G × G model derived from the affine Gaudin construction (for which the 1-loop β-functions were found in arXiv:2010.07879) and show that its condition of integrability is preserved also by the 2-loop RG flow. We then investigate the RG flow in the gauged G × G/H model, in particular the integrable T1,1 model found in arXiv:2010.05573. We also construct a new class of integrable G × G/H models in the case when the subgroup H is abelian. In the simplest case of G = SU2, H = U1 this leads to an integrable σ-model on the T1,q space (with a particular B-field). This model is also shown to be stable under the 2-loop RG flow, and we relate this property to its invariance under T-duality in an isometric U1 direction. This T1,q model may be interpreted as an integrable deformation of the GMM model (of two coupled WZW theories with generic levels) away from the conformal point.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V.A. Fateev, E. Onofri and A.B. Zamolodchikov, Integrable deformations of the O(3) sigma model. The sausage model, Nucl. Phys. B 406 (1993) 521 [INSPIRE].
V.A. Fateev, Classical and Quantum Integrable Sigma Models. Ricci Flow, “Nice Duality” and Perturbed Rational Conformal Field Theories, J. Exp. Theor. Phys. 129 (2019) 566 [arXiv:1902.02811] [INSPIRE].
S.L. Lukyanov, The integrable harmonic map problem versus Ricci flow, Nucl. Phys. B 865 (2012) 308 [arXiv:1205.3201] [INSPIRE].
B. Hoare, N. Levine and A.A. Tseytlin, Integrable 2d sigma models: quantum corrections to geometry from RG flow, Nucl. Phys. B 949 (2019) 114798 [arXiv:1907.04737] [INSPIRE].
B. Hoare, N. Levine and A.A. Tseytlin, Integrable sigma models and 2-loop RG flow, JHEP 12 (2019) 146 [arXiv:1910.00397] [INSPIRE].
F. Delduc, S. Lacroix, M. Magro and B. Vicedo, Integrable Coupled σ Models, Phys. Rev. Lett. 122 (2019) 041601 [arXiv:1811.12316] [INSPIRE].
F. Delduc, S. Lacroix, M. Magro and B. Vicedo, Assembling integrable σ-models as affine Gaudin models, JHEP 06 (2019) 017 [arXiv:1903.00368] [INSPIRE].
G. Arutyunov, C. Bassi and S. Lacroix, New integrable coset sigma models, JHEP 03 (2021) 062 [arXiv:2010.05573] [INSPIRE].
C. Klimčík, Yang-Baxter sigma models and dS/AdS T duality, JHEP 12 (2002) 051 [hep-th/0210095] [INSPIRE].
C. Klimčík, On integrability of the Yang-Baxter sigma-model, J. Math. Phys. 50 (2009) 043508 [arXiv:0802.3518] [INSPIRE].
J. Balog, P. Forgacs, Z. Horvath and L. Palla, A New family of SU(2) symmetric integrable sigma models, Phys. Lett. B 324 (1994) 403 [hep-th/9307030] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, On classical q-deformations of integrable sigma-models, JHEP 11 (2013) 192 [arXiv:1308.3581] [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].
F. Delduc, S. Lacroix, K. Sfetsos and K. Siampos, RG flows of integrable σ-models and the twist function, JHEP 02 (2021) 065 [arXiv:2010.07879] [INSPIRE].
R.R. Metsaev and A.A. Tseytlin, Order α′ (two loop) equivalence of the string equations of motion and the σ-model Weyl invariance conditions: dependence on the dilaton and the antisymmetric tensor, Nucl. Phys. B 293 (1987) 385 [INSPIRE].
R.R. Metsaev and A.A. Tseytlin, Two loop β-function for the generalized bosonic sigma model, Phys. Lett. B 191 (1987) 354 [INSPIRE].
M. Bos, Dimensional Regularization in the Wess-Zumino-Witten Model, Phys. Lett. B 189 (1987) 435 [INSPIRE].
C.M. Hull and P.K. Townsend, The Two Loop β-function for σ Models With Torsion, Phys. Lett. B 191 (1987) 115 [INSPIRE].
S.V. Ketov, Two Loop Calculations in the Nonlinear σ Model With Torsion, Nucl. Phys. B 294 (1987) 813 [INSPIRE].
D. Zanon, Two Loop β-functions and Low-energy String Effective Action for the Two-dimensional Bosonic Nonlinear σ Model With a Wess-Zumino-Witten Term, Phys. Lett. B 191 (1987) 363 [INSPIRE].
M. Bos, An Example of Dimensional Regularization With Antisymmetric Tensors, Annals Phys. 181 (1988) 177 [INSPIRE].
V.G. Knizhnik and A.B. Zamolodchikov, Current Algebra and Wess-Zumino Model in Two-Dimensions, Nucl. Phys. B 247 (1984) 83 [INSPIRE].
D. Schubring and M. Shifman, Sigma model on a squashed sphere with a Wess-Zumino term, Phys. Rev. D 103 (2021) 025016 [arXiv:2002.04696] [INSPIRE].
D.N. Page and C.N. Pope, Which Compactifications of D = 11 Supergravity Are Stable?, Phys. Lett. B 144 (1984) 346 [INSPIRE].
L.J. Romans, New Compactifications of Chiral N = 2, d = 10 Supergravity, Phys. Lett. B 153 (1985) 392 [INSPIRE].
P. Candelas and X.C. de la Ossa, Comments on Conifolds, Nucl. Phys. B 342 (1990) 246 [INSPIRE].
I.R. Klebanov and E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199 [hep-th/9807080] [INSPIRE].
E. Guadagnini, M. Martellini and M. Mintchev, Scale invariance sigma models on homogeneous spaces, Phys. Lett. B 194 (1987) 69 [INSPIRE].
E. Guadagnini, Current Algebra in σ Models on Homogeneous Spaces, Nucl. Phys. B 290 (1987) 417 [INSPIRE].
L.A. Pando Zayas and A.A. Tseytlin, Conformal sigma models for a class of T(p,q) spaces, Class. Quant. Grav. 17 (2000) 5125 [hep-th/0007086] [INSPIRE].
J.M. Maillet, Kac-Moody Algebra and Extended Yang-Baxter Relations in the O(N) Nonlinear σ Model, Phys. Lett. B 162 (1985) 137 [INSPIRE].
J.M. Maillet, New Integrable Canonical Structures in Two-dimensional Models, Nucl. Phys. B 269 (1986) 54 [INSPIRE].
S. Lacroix, M. Magro and B. Vicedo, Local charges in involution and hierarchies in integrable sigma-models, JHEP 09 (2017) 117 [arXiv:1703.01951] [INSPIRE].
K. Sfetsos, Integrable interpolations: From exact CFTs to non-Abelian T-duals, Nucl. Phys. B 880 (2014) 225 [arXiv:1312.4560] [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].
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, E. Sagkrioti, K. Sfetsos and K. Siampos, Quantum aspects of doubly deformed CFTs, Nucl. Phys. B 919 (2017) 504 [arXiv:1703.00462] [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, E. Sagkrioti, K. Sfetsos and K. Siampos, An exact symmetry in λ-deformed CFTs, JHEP 01 (2020) 083 [arXiv:1911.02027] [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].
C. Appadu and T.J. Hollowood, β-function of k deformed AdS5 × S5 string theory, JHEP 11 (2015) 095 [arXiv:1507.05420] [INSPIRE].
C.M. Hull and B.J. Spence, The Gauged Nonlinear σ Model With Wess-Zumino Term, Phys. Lett. B 232 (1989) 204 [INSPIRE].
E. Witten, On Holomorphic factorization of WZW and coset models, Commun. Math. Phys. 144 (1992) 189 [INSPIRE].
V.V. Belokurov and P.M. de Barrush Pasheku Seara de Sa, Ultraviolet finiteness of the Wess-Zumino-Witten gauge model on homogeneous manifolds, Moscow Univ. Phys. Bull. 45N3 (1990) 13 [Vestn. Mosk. Univ. Fiz. Astron. 31N3 (1990) 13] [INSPIRE].
K. Bardakci, L.M. Bernardo and N. Sochen, Integrable generalized Thirring model, Nucl. Phys. B 487 (1997) 513 [hep-th/9607018] [INSPIRE].
P. Basu and L.A. Pando Zayas, Chaos rules out integrability of strings on AdS5 × T1,1, Phys. Lett. B 700 (2011) 243 [arXiv:1103.4107] [INSPIRE].
P. Basu and L.A. Pando Zayas, Analytic Non-integrability in String Theory, Phys. Rev. D 84 (2011) 046006 [arXiv:1105.2540] [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].
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].
S. Demulder, S. Driezen, A. Sevrin and D.C. Thompson, Classical and Quantum Aspects of Yang-Baxter Wess-Zumino Models, JHEP 03 (2018) 041 [arXiv:1711.00084] [INSPIRE].
A.A. Tseytlin, Duality and dilaton, Mod. Phys. Lett. A 6 (1991) 1721 [INSPIRE].
P.E. Haagensen and K. Olsen, T duality and two loop renormalization flows, Nucl. Phys. B 504 (1997) 326 [hep-th/9704157] [INSPIRE].
N. Kaloper and K.A. Meissner, Duality beyond the first loop, Phys. Rev. D 56 (1997) 7940 [hep-th/9705193] [INSPIRE].
T.H. Buscher, Path Integral Derivation of Quantum Duality in Nonlinear Sigma Models, Phys. Lett. B 201 (1988) 466 [INSPIRE].
A.S. Schwarz and A.A. Tseytlin, Dilaton shift under duality and torsion of elliptic complex, Nucl. Phys. B 399 (1993) 691 [hep-th/9210015] [INSPIRE].
F. Hassler and T.B. Rochais, O(D, D)-covariant two-loop β-functions and Poisson-Lie T-duality, arXiv:2011.15130 [INSPIRE].
F. Hassler, RG flow of integrable ε-models, arXiv:2012.10451 [INSPIRE].
A.A. Tseytlin, String vacuum backgrounds with covariantly constant null Killing vector and 2D quantum gravity, Nucl. Phys. B 390 (1993) 153 [hep-th/9209023] [INSPIRE].
A.A. Tseytlin, Finite σ-models and exact string solutions with Minkowski signature metric, Phys. Rev. D 47 (1993) 3421 [hep-th/9211061] [INSPIRE].
B. Hoare, N. Levine and A.A. Tseytlin, Sigma models with local couplings: a new integrability — RG flow connection, JHEP 11 (2020) 020 [arXiv:2008.01112] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2103.10513
Also at the Institute of Theoretical and Mathematical Physics, MSU and Lebedev Institute, Moscow. (Arkady A. Tseytlin)
Supplementary Information
ESM 1
(TGZ 14 kb)
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
Levine, N., Tseytlin, A.A. Integrability vs. RG flow in G × G and G × G/H sigma models. J. High Energ. Phys. 2021, 76 (2021). https://doi.org/10.1007/JHEP05(2021)076
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2021)076