Abstract
We consider λ-deformed current algebra CFTs at level k, interpolating between an exact CFT in the UV and a PCM in the IR. By employing gravitational techniques, we derive the two-loop, in the large k expansion, β-function. We find that this is covariant under a remarkable exact symmetry involving the coupling λ, the level k and the adjoint quadratic Casimir of the group. Using this symmetry and CFT techniques, we are able to compute the Zamolodchikov metric, the anomalous dimension of the bilinear operator and the Zamolodchikov C -function at two-loops in the large k expansion, as exact func- tions of the deformation parameter. Finally, we extend the above results to λ-deformed parafermionic algebra coset CFTs which interpolate between exact coset CFTs in the UV and a symmetric coset space in the IR.
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References
C. Montonen and D.I. Olive, Magnetic Monopoles as Gauge Particles?, Phys. Lett.72B (1977) 117 [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys.B 125 (1977) 1 [INSPIRE].
E. Witten and D.I. Olive, Supersymmetry Algebras That Include Topological Charges, Phys. Lett.78B (1978) 97 [INSPIRE].
E. Witten, Nonabelian Bosonization in Two-Dimensions, Commun. Math. Phys.92 (1984) 455 [INSPIRE].
D. Kutasov, Duality Off the Critical Point in Two-dimensional Systems With Nonabelian Symmetries, Phys. Lett.B 233 (1989) 369 [INSPIRE].
K. Sfetsos, Integrable interpolations: From exact CFTs to non-Abelian T-duals, Nucl. Phys.B 880 (2014) 225 [arXiv:1312.4560] [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].
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].
G. Georgiou and K. Sfetsos, A new class of integrable deformations of CFTs, JHEP03 (2017) 083 [arXiv:1612.05012] [INSPIRE].
G. Georgiou and K. Sfetsos, Integrable flows between exact CFTs, JHEP11 (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. Georgiou and K. Sfetsos, The most general λ-deformation of CFTs and integrability, JHEP03 (2019) 094 [arXiv:1812.04033] [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 K. Siampos, Integrable deformations of the Gk1× Gk2/Gk1 +k2coset CFTs, Nucl. Phys.B 927 (2018) 124 [arXiv:1710.02515] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, Integrable Deformations of Strings on Symmetric Spaces, JHEP11 (2014) 009 [arXiv:1407.2840] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, An Integrable Deformation of the AdS5× S5Superstring, J. Phys.A 47 (2014) 495402 [arXiv:1409.1538] [INSPIRE].
K. Sfetsos and D.C. Thompson, Spacetimes for λ-deformations, JHEP12 (2014) 164 [arXiv:1410.1886] [INSPIRE].
S. Demulder, K. Sfetsos and D.C. Thompson, Integrable λ-deformations: Squashing Coset CFTs and AdS5× S5 , JHEP07 (2015) 019 [arXiv:1504.02781] [INSPIRE].
R. Borsato, A.A. Tseytlin and L. Wulff, Supergravity background of λ-deformed model for AdS2× S2supercoset, Nucl. Phys.B 905 (2016) 264 [arXiv:1601.08192] [INSPIRE].
Y. Chervonyi and O. Lunin, Supergravity background of the λ-deformed AdS3× S3supercoset, Nucl. Phys.B 910 (2016) 685 [arXiv:1606.00394] [INSPIRE].
R. Borsato and L. Wulff, Target space supergeometry of η and λ-deformed strings, JHEP10 (2016) 045 [arXiv:1608.03570] [INSPIRE].
S. Driezen, A. Sevrin and D.C. Thompson, Integrable asymmetric λ-deformations, JHEP04 (2019) 094 [arXiv:1902.04142] [INSPIRE].
K. Siampos, An exact symmetry in λ-deformed σ-models, in Recent Developments in Strings and Gravity, Corfu, Greece, 10–16 September 2019, [http://www.physics.ntua.gr/corfu2019/Talks/konstantinos siampos@cern ch 01.pdf].
K. Siampos, An exact symmetry in λ-deformed σ-models, in 10thCrete regional meeting in String Theory, Kolymbari, Greece, 15–22 September 2019 [http://hep.physics.uoc.gr/mideast10/talks/friday/siampos.pdf ].
B. Hoare, N. Levine and A.A. Tseytlin, Integrable σ-models and 2-loop RG flow, JHEP12 (2019) 146 [arXiv:1910.00397] [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].
A.M. Polyakov and P.B. Wiegmann, Theory of Nonabelian Goldstone Bosons, Phys. Lett.131B (1983) 121 [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].
T. Curtright and C.K. Zachos, Currents, charges and canonical structure of pseudodual chiral models, Phys. Rev.D 49 (1994) 5408 [hep-th/9401006] [INSPIRE].
E. Alvarez, L. Álvarez-Gaumé and Y. Lozano, A Canonical approach to duality transformations, Phys. Lett.B 336 (1994) 183 [hep-th/9406206] [INSPIRE].
C.R. Nappi, Some Properties of an Analog of the Nonlinear σ Model, Phys. Rev.D 21 (1980) 418 [INSPIRE].
D. Kutasov, String Theory and the Nonabelian Thirring Model, Phys. Lett.B 227 (1989) 68 [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].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett.43 (1986) 730 [INSPIRE].
G. Georgiou and K. Sfetsos, Field theory and λ-deformations: Expanding around the identity, Nucl. Phys.B 950 (2020) 114855 [arXiv:1910.01056] [INSPIRE].
E. Guadagnini, M. Martellini and M. Mintchev, Scale invariance σ-models on homogeneous spaces, Phys. Lett.B 194 (1987) 69 [INSPIRE].
L.A. Pando Zayas and A.A. Tseytlin, Conformal σ-models for a class of Tp,qspaces, Class. Quant. Grav.17 (2000) 5125 [hep-th/0007086] [INSPIRE].
K. Bardakci, M.J. Crescimanno and E. Rabinovici, Parafermions From Coset Models, Nucl. Phys.B 344 (1990) 344 [INSPIRE].
K. Bardakci, M.J. Crescimanno and S. Hotes, Parafermions from nonabelian coset models, Nucl. Phys.B 349 (1991) 439 [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. JETP62 (1985) 215 [INSPIRE].
B. Hoare, N. Levine and A.A. Tseytlin, Integrable 2d σ-models: quantum corrections to geometry from RG flow, Nucl. Phys.B 949 (2019) 114798 [arXiv:1907.04737] [INSPIRE].
G. Georgiou, P. Panopoulos, E. Sagkrioti and K. Sfetsos, Exact results from the geometry of couplings and the effective action, Nucl. Phys.B 948 (2019) 114779 [arXiv:1906.00984] [INSPIRE].
T.L. Curtright and C.K. Zachos, Geometry, Topology and Supersymmetry in Nonlinear Models, Phys. Rev. Lett.53 (1984) 1799 [INSPIRE].
E. Braaten, T.L. Curtright and C.K. Zachos, Torsion and Geometrostasis in Nonlinear σ-models, Nucl. Phys.B 260 (1985) 630 [Erratum ibid.B 266 (1986) 748] [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].
C.M. Hull and P.K. Townsend, String Effective Actions From σ Model Conformal Anomalies, Nucl. Phys.B 301 (1988) 197 [INSPIRE].
R.R. Metsaev and A.A. Tseytlin, Two loop β-function for the generalized bosonic σ-model, Phys. Lett.B 191 (1987) 354 [INSPIRE].
R.R. Metsaev and A.A. Tseytlin, Order alpha-prime (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].
H. Osborn, General Bosonic σ Models and String Effective Actions, Annals Phys.200 (1990) 1 [INSPIRE].
E. Sagkrioti, K. Sfetsos and K. Siampos, RG flows for λ-deformed CFTs, Nucl. Phys.B 930 (2018) 499 [arXiv:1801.10174] [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].
J. Honerkamp, Chiral multiloops, Nucl. Phys.B 36 (1972) 130 [INSPIRE].
D. Friedan, Nonlinear Models in Two Epsilon Dimensions, Phys. Rev. Lett.45 (1980) 1057 [INSPIRE].
D.H. Friedan, Nonlinear Models in Two + Epsilon Dimensions, Annals Phys.163 (1985) 318 [INSPIRE].
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Georgiou, G., Sagkrioti, E., Sfetsos, K. et al. An exact symmetry in λ-deformed CFTs. J. High Energ. Phys. 2020, 83 (2020). https://doi.org/10.1007/JHEP01(2020)083
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DOI: https://doi.org/10.1007/JHEP01(2020)083