Complex structure-induced deformations of σ-models

Open Access
Regular Article - Theoretical Physics

Abstract

We describe a deformation of the principal chiral model (with an evendimensional target space G) by a B-field proportional to the Kähler form on the target space. The equations of motion of the deformed model admit a zero-curvature representation. As a simplest example, we consider the case of G = S1 × S3. We also apply a variant of the construction to a deformation of the AdS3 × S3 × S1 (super-)σ-model.

Keywords

Integrable Field Theories Sigma Models AdS-CFT Correspondence Differential and Algebraic Geometry 

References

  1. [1]
    D. Bykov, Integrable properties of σ-models with non-symmetric target spaces, Nucl. Phys. B 894 (2015) 254 [arXiv:1412.3746] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    D. Bykov, Classical solutions of a flag manifold σ-model, Nucl. Phys. B 902 (2016) 292 [arXiv:1506.08156] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    D. Bykov, Complex structures and zero-curvature equations for σ-models, Phys. Lett. B 760 (2016) 341 [arXiv:1605.01093] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    C. Klimčík, On integrability of the Yang-Baxter σ-model, J. Math. Phys. 50 (2009) 043508 [arXiv:0802.3518] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    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].ADSCrossRefGoogle Scholar
  6. [6]
    G. Arutyunov, R. Borsato and S. Frolov, S-matrix for strings on η-deformed AdS 5 × S 5, JHEP 04 (2014) 002 [arXiv:1312.3542] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    F. Delduc, M. Magro and B. Vicedo, On classical q-deformations of integrable σ-models, JHEP 11 (2013) 192 [arXiv:1308.3581] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Commun. Math. Phys. 46 (1976) 207 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    C. Klimčík, Integrability of the bi-Yang-Baxter σ-model, Lett. Math. Phys. 104 (2014) 1095 [arXiv:1402.2105] [INSPIRE].
  10. [10]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    C.A.S. Young, Non-local charges, Z m gradings and coset space actions, Phys. Lett. B 632 (2006) 559 [hep-th/0503008] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    D.V. Bykov, Cyclic gradings of Lie algebras and Lax pairs for σ-models, Theor. Math. Phys. 189 (2016) 1734 [Teor. Mat. Fiz. 189 (2016) 380] [INSPIRE].
  13. [13]
    P. Spindel, A. Sevrin, W. Troost and A. Van Proeyen, Extended supersymmetric σ-models on group manifolds. 1. The complex structures, Nucl. Phys. B 308 (1988) 662 [INSPIRE].
  14. [14]
    D. Joyce, Compact hypercomplex and quaternionic manifolds, J. Diff. Geom. 35 (1992) 743.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    T.H. Buscher, Path integral derivation of quantum duality in nonlinear σ-models, Phys. Lett. B 201 (1988) 466 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP 05 (2005) 069 [hep-th/0503201] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    I.V. Cherednik, Relativistically invariant quasiclassical limits of integrable two-dimensional quantum models, Theor. Math. Phys. 47 (1981) 422 [Teor. Mat. Fiz. 47 (1981) 225] [INSPIRE].
  18. [18]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    C. Klimčík, Yang-Baxter σ-models and dS/AdS T duality, JHEP 12 (2002) 051 [hep-th/0210095] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    B. Hoare, R. Roiban and A.A. Tseytlin, On deformations of AdS n × S n supercosets, JHEP 06 (2014) 002 [arXiv:1403.5517] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    D. Orlando and L.I. Uruchurtu, Warped anti-de Sitter spaces from brane intersections in type-II string theory, JHEP 06 (2010) 049 [arXiv:1003.0712] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    D. Orlando, S. Reffert and L.I. Uruchurtu, Classical integrability of the squashed three-sphere, warped AdS 3 and Schrödinger spacetime via T-duality, J. Phys. A 44 (2011) 115401 [arXiv:1011.1771] [INSPIRE].ADSMATHGoogle Scholar
  23. [23]
    D. Osten and S.J. van Tongeren, Abelian Yang-Baxter deformations and TsT transformations, Nucl. Phys. B 915 (2017) 184 [arXiv:1608.08504] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    A. Babichenko, B. Stefanski, Jr. and K. Zarembo, Integrability and the AdS 3 /CF T 2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  26. [26]
    V.G. Kac, A sketch of Lie superalgebra theory, Commun. Math. Phys. 53 (1977) 31 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    M. Rooman and P. Spindel, Gödel metric as a squashed anti-de Sitter geometry, Class. Quant. Grav. 15 (1998) 3241 [gr-qc/9804027] [INSPIRE].
  28. [28]
    K. Gödel, An Example of a new type of cosmological solutions of Einsteins field equations of graviation, Rev. Mod. Phys. 21 (1949) 447 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  29. [29]
    E. Witten, Topological σ-models, Commun. Math. Phys. 118 (1988) 411 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    G. Arutyunov, S. Frolov, B. Hoare, R. Roiban and A.A. Tseytlin, Scale invariance of the η-deformed AdS 5 × S 5 superstring, T-duality and modified type-II equations, Nucl. Phys. B 903 (2016) 262 [arXiv:1511.05795] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  31. [31]
    L. Wulff and A.A. Tseytlin, κ-symmetry of superstring σ-model and generalized 10d supergravity equations, JHEP 06 (2016) 174 [arXiv:1605.04884] [INSPIRE].
  32. [32]
    R. Borsato and L. Wulff, Target space supergeometry of η and λ-deformed strings, JHEP 10 (2016) 045 [arXiv:1608.03570] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-InstitutPotsdam-GolmGermany
  2. 2.Steklov Mathematical Institute of Russ. Acad. Sci.MoscowRussia

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