Scale without conformal invariance from integrable deformations of coset CFTs

TL;DR

The paper addresses scale invariance without conformal invariance in two-dimensional sigma models by constructing lambda-deformations of coset CFTs based on unitary groups, notably SU(3)_k/U(2)_k and its non-compact SU(2,1)_{-k}/U(2)_{-k}. It derives the one-loop beta function for the deformation parameter and analyzes asymptotic double-scaling limits, revealing a scale-invariant but non-conformal 2D theory that is not of Kerr-Schild type, with a regime where conformal invariance is restored; it also provides Type-II supergravity embeddings for the asymptotic model. The main contributions include explicit lambda-deformed backgrounds, the RG flow and C-function characterization, the identification of scale without conformal invariance in a controlled integrable setting, and holographically relevant supergravity embeddings. The findings illuminate how integrable deformations of coset CFTs can realize scale-invariant but non-conformal dynamics and pave the way for broader explorations in higher-dimensional unitary cosets and generalized supergravity frameworks.

Abstract

We construct the $λ$-model on $SU(3)_k/U(2)_k$ and we compute the one-loop $β$-function for the deformation parameter $λ$. Its non-compact version for $SU(2,1)_{-k}/U(2)_{-k}$ is also considered, whose target space admits an asymptotic region that can be reached when one of the coordinates becomes large. The asymptotic model can be seen as an integrable deformation of the $SU(2)_k/U(1)$ WZW model together with a linear dilaton and a free boson, where integrability is inherited from the parental $λ$-model. By taking an asymptotic double-scaling limit of the $SU(2,1)_{-k}/U(2)_{-k}$ model, we obtain a two-dimensional field theory that is scale-invariant but not conformally invariant at one-loop order. Crucially, this deformation does not admit a Kerr-Schild form, unlike the cases studied in arXiv:2109.05040. However, in a suitable asymptotic regime, scale invariance is enhanced to full conformal invariance. Finally, we construct Type-II supergravity embeddings of the asymptotic model for specific values of the deformation parameter.