Abstract
Following arXiv:1907.04737, we continue our investigation of the relation between the renormalizability (with finitely many couplings) and integrability in 2d σ- models. We focus on the “λ-model,” an integrable model associated to a group or symmetric space and containing as special limits a (gauged) WZW model and an “interpolating model” for non-abelian duality. The parameters are the WZ level k and the coupling λ, and the fields are g, valued in a group G, and a 2d vector A± in the corresponding algebra. We formulate the λ-model as a σ-model on an extended G × G × G configuration space (g, h, overline{h} ), defining h and overline{h} by A+ = h∂+h−1, A_ = overline{h} ∂− overline{h} −1. Our central observation is that the model on this extended configuration space is renormalizable without any deformation, with only λ running. This is in contrast to the standard σ-model found by integrating out A±, whose 2-loop renormalizability is only obtained after the addition of specific finite local counterterms, resulting in a quantum deformation of the target space geometry. We compute the 2-loop β-function of the λ-model for general group and symmetric spaces, and illustrate our results on the examples of SU(2)/U(1) and SU(2). Similar conclusions apply in the non-abelian dual limit implying that non-abelian duality commutes with the RG flow. We also find the 2-loop β-function of a “squashed” principal chiral model.
Highlights
The λ-model for a group G is a deformation of the G/G gauged WZW model LG/G(g, A) by the term γA+A−
We focus on the “λ-model,” an integrable model associated to a group or symmetric space and containing as special limits a WZW model and an “interpolating model” for non-abelian duality
Using the parametrization for hand hgiven in eq (2.41), we find that the corresponding target space background depends on ψ and φonly through χ = ψ + φ, which is a manifestation of the U(1) gauge symmetry
Summary
We shall study renormalization of the λ-model on the extended configuration space. Using the Polyakov-Wiegmann identity [46], we can rewrite the Lagrangian in (1.6) as a combination of WZW models (cf (1.5)). In the special cases γ = 0 (gWZW model) and γ = −1 (cWZW model), we can choose q such that (2.4) is a sum of WZW models [10, 11]. L = −(k + 2cH ) LH (h) + LH (h) + k(1 + γ)Tr[h−1∂+h ∂−hh−1] This model is defined on the extended configuration space (g, h, h) ∈ G × H × H. When H/F is a symmetric space this model should again be renormalizable on the extended configuration space (g, h, h) ∈ G × H × H due to chiral gauge symmetry
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