Some integrable deformations of non-linear sigma models F.D., M. Magro (ENS Lyon) B. Vicedo (Univ. Of Hertfordshire) arxiv:1308.3581, arxiv:1309.5850 Hanover, SIS'13
Classical integrability : rare among non-linear σ models. C. Klimčík proposed a way to deform certain integrable models, while preserving integrability. Principal chiral model Symmetric space sigma model String on In all cases, the method makes essential use of a classical antisymmetric R matrix, solution of the classical modified Yang-Baxter equation (cmyb)
Construction of an antisymmetric R matrix Suppose that Lie(G) is defined by some hermiticity condition ( real diagonal matrix) Then, if one may take And check
Reminder on the principal chiral model The field g(x) belongs to a Lie group G and the action reads It is invariant under left and right translation by G, Right currents read Field equation Zero curvature equation Lax pair : Conservation + zero curvature equations of the currents
Cherednik (1981) : deformation of the SU(2) principal chiral model where is a diagonal matrix. 2-parameters integrable deformation of the principal chiral model Squashed sphere C. Klimčík (2002 and 2009) : Generalization of squashed sphere to arbitrary group G
Action of the deformed model (Yang-Baxter sigma model) : real deformation parameter, Geometry contains a metric and a torsion Explicit symmetry breaking Still invariant under right translations Noether Current :
Field equation Current conservation When field equations are satisfied and obeys the cmyb equation satisfies a zero curvature equation There is a Lax pair, which has the same expression in terms of the currents as in the principal model
Hamiltonian formalism : same hamiltonian as in the principal model Poisson Bracket : where is the canonical bracket of the principal model and is the Faddeev-Reshetikhin ultralocal bracket which are two compatible Poisson brackets. The fields may be taken as canonically conjugated coordinates on the phase space. The space component of the current reads in terms of these phase space fields
Left invariance is broken down to the abelian Cartan subgroup. However there are nonlocal conserved charges which Poisson bracket is a deformation of the Lie algebra bracket of G (Kawaguchi, Matsumoto, Yoshida, arxiv:1203.3400) If are the local charges associated to the Cartan generator and the non- local charges associated to simple roots one finds where A is the Cartan matrix, and
Symmetric space sigma model: order 2 automorphism of G, H invariant subgroup under If, is invariant under changes sign under induces a grading of Lie(G) Action of the symmetric space G/H sigma model Invariant under right gauge transformations Field equations Lax pair Field equations + zero curvature equations
Deformed symmetric space sigma model action It is gauge invariant. Introducing The field equation take the same form as in the undeformed case Moreover, provided the field equation is satisfied and mcyb equation, satifies a zero curvature equation satisfies the Thus, the Lax pair has the form Left symmetry is deformed.
An example : SU(2)/U(1) We use stereographic coordinates on the sphere In two dimensions, there cannot be torsion only the metric is deformed One finds the action symmetric space SU(2)/U(1) symmetric space SU(1,1)/U(1) Ricci tensor :
superstring action Coset grading of fermioniques Metsaev-Tseytlin action Introducing One may rewrite the action as
Lax pair Kappa symmetry
Deformed model action Where One is led to replace the undeformed currents by Then * there is a Lax pair * Moreover, there is a kappa invariance
Arutyunov, Borsato and Frolov (arxiv:1312.3542) compute the 2 2 scattering matrix at tree level and successfully compare with known q-deformed S-matrix found from quantum group symmetry, unitarity and crossing (Beisert, Koroteev arxiv:0802.077, and Hoare, Hollowood, Miramontes arxiv:1112.4485).
There remains * To understand this deformation from a Hamiltonian point of view, and find the conserved charges. * To understand the geometry which is encoded in the deformed action. Since it is a superstring action with kappa symmetry, it should correspond to a solution of type IIB supergravity. Metric and antisymmetric field have been derived in (Arutyunov, Borsato, Frolov) but there remains to find the 5-form, the dilaton