Abstract
We study the regularization scheme dependence of Kähler (N = 2) supersymmetric sigma models. At the one-loop order the metric β function is the same as in the non-supersymmetric case and it coincides with the Ricci tensor. The first correction in the MS scheme is known to appear in the fourth loop [1, 2]. We show that for certain integrable Kähler backgrounds, such as the complete T-dual of η-deformed \( \mathbbm{CP}(n) \) sigma models, there is a scheme in which the fourth loop contribution vanishes.
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M.T. Grisaru, A.E.M. van de Ven and D. Zanon, Four Loop beta Function for the N = 1 and N = 2 Supersymmetric Nonlinear Sigma Model in Two-Dimensions, Phys. Lett. B 173 (1986) 423 [INSPIRE].
M.T. Grisaru, A.E.M. van de Ven and D. Zanon, Two-Dimensional Supersymmetric Sigma Models on Ricci Flat Kahler Manifolds Are Not Finite, Nucl. Phys. B 277 (1986) 388 [INSPIRE].
G. Ecker and J. Honerkamp, Application of invariant renormalization to the nonlinear chiral invariant pion lagrangian in the one-loop approximation, Nucl. Phys. B 35 (1971) 481 [INSPIRE].
D.H. Friedan, Nonlinear models in 2 + ϵ dimensions, Annals Phys. 163 (1985) 318 [INSPIRE].
R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982) 255.
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math/0211159 [INSPIRE].
G. Perelman, Ricci flow with surgery on three-manifolds, math/0303109 [INSPIRE].
B. Hoare, Integrable deformations of sigma models, J. Phys. A 55 (2022) 093001 [arXiv:2109.14284] [INSPIRE].
V.A. Fateev and A.V. Litvinov, Integrability, Duality and Sigma Models, JHEP 11 (2018) 204 [arXiv:1804.03399] [INSPIRE].
A.V. Litvinov and L.A. Spodyneiko, On dual description of the deformed O(N) sigma model, JHEP 11 (2018) 139 [arXiv:1804.07084] [INSPIRE].
V. 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].
A.V. Litvinov, Integrable \( \mathfrak{gl}\left(n|n\right) \) Toda field theory and its sigma-model dual, Pisma Zh. Eksp. Teor. Fiz. 110 (2019) 723 [arXiv:1901.04799] [INSPIRE].
M. Alfimov, B. Feigin, B. Hoare and A. Litvinov, Dual description of η-deformed OSP sigma models, JHEP 12 (2020) 040 [arXiv:2010.11927] [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].
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].
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].
M. Alfimov and A. Litvinov, On loop corrections to integrable 2D sigma model backgrounds, JHEP 01 (2022) 043 [arXiv:2110.05418] [INSPIRE].
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].
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].
Y. Kazama and H. Suzuki, New N = 2 Superconformal Field Theories and Superstring Compactification, Nucl. Phys. B 321 (1989) 232 [INSPIRE].
L. Alvarez-Gaume and D.Z. Freedman, Kähler Geometry and the Renormalization of Supersymmetric Sigma Models, Phys. Rev. D 22 (1980) 846 [INSPIRE].
S.J. Graham, Three Loop Beta Function for the Bosonic Nonlinear σ Model, Phys. Lett. B 197 (1987) 543 [INSPIRE].
A.P. Foakes and N. Mohammedi, Three Loop Calculation of the Beta Function for the Purely Metric Nonlinear σ Model, Phys. Lett. B 198 (1987) 359 [INSPIRE].
I. Jack, D.R.T. Jones and N. Mohammedi, A Four Loop Calculation of the Metric Beta Function for the Bosonic σ Model and the String Effective Action, Nucl. Phys. B 322 (1989) 431 [INSPIRE].
L. Alvarez-Gaume, D.Z. Freedman and S. Mukhi, The Background Field Method and the Ultraviolet Structure of the Supersymmetric Nonlinear Sigma Model, Annals Phys. 134 (1981) 85 [INSPIRE].
M.T. Grisaru, D.I. Kazakov and D. Zanon, Five Loop Divergences for the N = 2 Supersymmetric Nonlinear σ Model, Nucl. Phys. B 287 (1987) 189 [INSPIRE].
B. Zumino, Supersymmetry and Kähler Manifolds, Phys. Lett. B 87 (1979) 203 [INSPIRE].
M.T. Grisaru, A.E.M. van de Ven and D. Zanon, Four Loop Divergences for the N = 1 Supersymmetric Nonlinear Sigma Model in Two-Dimensions, Nucl. Phys. B 277 (1986) 409 [INSPIRE].
R.R. Metsaev and A.A. Tseytlin, Order alpha-prime (Two Loop) Equivalence of the String Equations of Motion and the Sigma Model Weyl Invariance Conditions: Dependence on the Dilaton and the Antisymmetric Tensor, Nucl. Phys. B 293 (1987) 385 [INSPIRE].
R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, String propagation in a black hole geometry, Nucl. Phys. B 371 (1992) 269 [INSPIRE].
S.A. Fulling, R.C. King, B.G. Wybourne and C.J. Cummins, Normal forms for tensor polynomials. 1: The Riemann tensor, Class. Quant. Grav. 9 (1992) 1151 [INSPIRE].
C.N. Pope, M.F. Sohnius and K.S. Stelle, Counterterm counterexamples, Nucl. Phys. B 283 (1987) 192 [INSPIRE].
C. Klimcik, On integrability of the Yang-Baxter sigma-model, J. Math. Phys. 50 (2009) 043508 [arXiv:0802.3518] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, On classical q-deformations of integrable sigma-models, JHEP 11 (2013) 192 [arXiv:1308.3581] [INSPIRE].
D. Bykov and D. Lust, Deformed σ-models, Ricci flow and Toda field theories, Lett. Math. Phys. 111 (2021) 150 [arXiv:2005.01812] [INSPIRE].
T. Nutma, xTras: A field-theory inspired xAct package for mathematica, Comput. Phys. Commun. 185 (2014) 1719 [arXiv:1308.3493] [INSPIRE].
E. Abdalla, M.C.B. Abdalla and M. Gomes, Anomaly in the Nonlocal Quantum Charge of the CP(n−1) Model, Phys. Rev. D 23 (1981) 1800 [INSPIRE].
M.A. Olshanetsky, Supersymmetric two-dimensional Toda lattice, Commun. Math. Phys. 88 (1983) 63 [INSPIRE].
J. Evans and T.J. Hollowood, Supersymmetric Toda field theories, Nucl. Phys. B 352 (1991) 723 [INSPIRE].
K. Ito, N = 2 superconformal \( \mathbbm{CP}(n) \) model, Nucl. Phys. B 370 (1992) 123 [INSPIRE].
B. Hoare and F.K. Seibold, Poisson-Lie duals of the η deformed symmetric space sigma model, JHEP 11 (2017) 014 [arXiv:1709.01448] [INSPIRE].
B. Hoare and A.A. Tseytlin, On integrable deformations of superstring sigma models related to AdSn × Sn supercosets, Nucl. Phys. B 897 (2015) 448 [arXiv:1504.07213] [INSPIRE].
S.J. Gates Jr., C.M. Hull and M. Rocek, Twisted Multiplets and New Supersymmetric Nonlinear Sigma Models, Nucl. Phys. B 248 (1984) 157 [INSPIRE].
S. Demulder, F. Hassler, G. Piccinini and D.C. Thompson, Integrable deformation of \( {\mathbbm{CP}}^n \) and generalised Kähler geometry, JHEP 10 (2020) 086 [arXiv:2002.11144] [INSPIRE].
D.-Y. Xu, Two important invariant identities, Phys. Rev. D 35 (1987) 769 [INSPIRE].
Acknowledgments
We acknowledge discussions with Alexey Rosly. A.L. has been supported by the Russian Science Foundation under the grant 22-22-00991 and by Basis foundation. A.L. is also grateful to Abubakir Koshek for collaboration at the early stages of this project. M.A. thanks Anna Popova for her help in linguistic editing of the text. I.K. is supported by Basis Foundation under the grant 23-2-1-53-1.
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Alfimov, M., Kalinichenko, I. & Litvinov, A. On β-function of N = 2 supersymmetric integrable sigma-models. J. High Energ. Phys. 2024, 297 (2024). https://doi.org/10.1007/JHEP05(2024)297
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DOI: https://doi.org/10.1007/JHEP05(2024)297