On $\beta$-function of $\mathcal{N}=2$ supersymmetric integrable sigma models II

Abstract

We continue studying regularization scheme dependence of the $\mathcal{N}=2$ supersymmetric sigma models. In the present work the previous result for the four loop $\beta$-function is extended to the five loop order. Namely, we find the renormalization scheme, in which the fifth loop contribution is completely eliminated, while the fourth loop contribution is represented by the certain invariant, which is coordinate independent for the metrics of some models. These models include complete $T$-duals of the $\eta$-deformed $SU(n)/U(n-1)$ models, as well as $\eta$- and $\lambda$-deformed $SU(2)/U(1)$ models, whose metrics solve the RG flow equation up to the fifth loop order. We also comment on the $\lambda$-deformed $SU(2)/U(1)$ and $SU(3)/U(2)$ case, showing that they satisfy five-loop RG flow equation, and discuss their Kähler structure.

26 pages, v2: typos fixed, references added, extended the discussion on the Kaehler structure of the lambda-deformed models and CFT limit of eta-deformed ones, submitted to JHEP, v3: Acknowledgements section modified, v4: added additional motivation to the Introduction, clarified the domain of applicability of the result, more details of the scheme change for arbitrary D and the dilaton