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

TL;DR

This work advances the understanding of regularization-scheme dependence in ${\mathcal N}=2$ supersymmetric sigma models by extending the beta-function analysis to five loops and identifying a covariant scheme in which the ${5}$-loop contribution vanishes. Central to this construction is the invariant ${\Delta \tilde{K}}$, which encodes the ${4}$-loop structure and remains coordinate-independent for several integrable backgrounds, including complete ${T}$-duals of the ${\eta}$-deformed ${\mathbb{CP}}^{n-1}$ and certain ${\lambda}$-deformed models. The authors verify that the sausage model family and ${\lambda}$-deformed ${SU(2)}/{U(1)}$ (and some ${SU(3)}/{U(2)}$ cases) satisfy the five-loop RG flow in this scheme, with explicit results for the ${n=2}$ Kähler potential and partial evidence for higher ${n}$. They also illuminate a web of connections between ${\lambda=0}$ CFT limits, eta-deformed backgrounds, and Kähler structure, outlining open questions on extending to broader classes of theories and establishing Kählerity for all cases.

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 $β$-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 $η$-deformed $SU(n)/U(n-1)$ models, as well as $η$- and $λ$-deformed $SU(2)/U(1)$ models, whose metrics solve the RG flow equation up to the fifth loop order. We also comment on the $λ$-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.