Yang–Baxter deformations of the principal chiral model plus Wess–Zumino term

Abstract

A large class of integrable deformations of the Principal Chiral Model, known as the Yang-Baxter deformations, are governed by skew-symmetric R-matrices solving the (modified) classical Yang-Baxter equation. We carry out a systematic investigation of these deformations in the presence of the Wess-Zumino term for simple Lie groups, working in a framework that treats both inhomogeneous and homogeneous deformations on the same footing. After analysing the cohomological conditions under which such a deformation is admissible, we consider an action for the general Yang-Baxter deformation of the Principal Chiral Model plus Wess-Zumino term and prove its classical integrability. We also show how the model is found from a number of alternative formulations: affine Gaudin models, E-models, 4-dimensional Chern-Simons theory and, for homogeneous deformations, non-abelian T-duality.

RG flows of integrable σ-models and the twist function

“…Note that the Yang-Baxter model with WZ term was recently generalized to a larger class of R-matrices in[29]. It would be interesting to check that the present analysis also holds for this more general model.…”

Supergravity solution-generating techniques and canonical transformations of σ-models from O(D, D)

“…Inhomogeneous and homogeneous YB-deformations of PCM + WZ models were discussed in [79]. The homogenous-type deformations leave the generalised fluxes invariant and therefore can be included in our discussion.…”

“…The homogenous-type deformations leave the generalised fluxes invariant and therefore can be included in our discussion. However we would like to point out that the homogeneous YB-deformations of [79] are always such that the image of the r-matrix (which is a subalgebra f of the full Lie algebra of the group used to construct the models) is a solvable algebra. As a consequence of Cartan's criterion, the components of the H-flux computed from the WZ term (e.g.…”

“…H ijk = αf ij l κ kl for some constant α) are zero when all three legs are along f. Selecting a solvable subalgebra is equivalent to splitting the coordinates x m of the full target-space into spectator coordinatesẋμ and coordinates y µ such that H µνρ = 0. Therefore in our language the homogeneous deformations of [79] should be viewed as a deformation of the simpler (F )-orbit. Notice that the solvability condition can be seen as a possible solution to (3.62).…”

“…Let us now turn to the generically inhomogeneous YB-deformation of the PCM + WZ model of [79][80][81]. To compare to those results let us consider the O(d, d) matrix…”