Integrable Lorentz-breaking deformations and RG flows

We construct the all loop effective action for WZW models perturbed by current-bilinear terms of the type $J_+J_- $, $J_+J_+ $ and $J_-J_- $, the last two of which explicitly break Lorentz invariance. For isotropic couplings we prove integrability. For the case in which only the first two terms are present we identify a non-perturbative symmetry of the effective action and we compute the exact beta-functions. These become identically zero outside a bounded region in the parametric space.

💡 Research Summary

The paper investigates a class of two‑dimensional quantum field theories obtained by deforming a Wess–Zumino–Witten (WZW) model with all possible bilinear current operators. Starting from a level‑k WZW action for a semi‑simple group G, the authors add three types of current‑current interactions: the Lorentz‑invariant (J_{+}J_{-}) term and the Lorentz‑breaking (J_{+}J_{+}) and (J_{-}J_{-}) terms. By employing the well‑established gauging procedure (originally used for λ‑deformations), they construct the exact all‑loop effective action. The gauge fields appear quadratically and can be integrated out, yielding a σ‑model action that depends on the group element g, the matrix D (the adjoint action of g), and the deformation matrices λ₁, λ₂, λ (the latter mixing the chiralities).

The authors first prove classical integrability for isotropic couplings, i.e. when all deformation matrices are proportional to the identity. They write the equations of motion in terms of the gauge fields A₊, A₋ and propose a Lax pair of the form (L_{+}=c_{1}A_{+}+c_{2}A_{-}), (L_{-}=c_{3}A_{+}+c_{4}A_{-}). By demanding the flatness condition ∂₊L₋−∂₋L₊−