Integrable sigma models and 2-loop RG flow

Abstract

Following arXiv:1907.04737, we continue our investigation of the relation between the renormalizability (with finitely many couplings) and integrability in 2d $\sigma$-models. We focus on the "$\lambda$-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 $\lambda$, and the fields are $g$, valued in a group $G$, and a 2d vector $A_{\pm }$ in the corresponding algebra. We formulate the $\lambda$-model as a $\sigma$-model on an extended $G\times G\times G$ configuration space $(g,h,\bar{h})$, defining $h$ and $\bar{h}$ by $A_{+}=h{\partial }_{+}{h}^{-1}$, $A_{-}=\bar{h}{\partial }_{-}{\bar{h}}^{-1}$. Our central observation is that the model on this extended configuration space is renormalizable without any deformation, with only $\lambda$ running. This is in contrast to the standard $\sigma$-model found by integrating out $A_{\pm }$, 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 $\beta$-function of the $\lambda$-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 $\beta$-function of a "squashed" principal chiral model.

28 pages; v3: minor comments and references added

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• Integrable Deformations of Sigma Models
• Poisson-Lie U-duality in Exceptional Field Theory
• E${}_{6(6)}$ Exceptional Drinfel'd Algebras
• Generalized Dualities and Higher Derivatives
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