RG flows for $\lambda$-deformed CFTs

TL;DR

This paper derives the exact renormalization group equations for anisotropic $ ext{lambda}$-deformed conformal field theories involving two current algebras, providing insights into their quantum behavior and geometric structure.

Contribution

It presents the first exact $eta$-function for fully anisotropic $ ext{lambda}$-deformed CFTs with two current algebras at different levels, using both effective action and gravitational methods.

Findings

Exact $eta$-function derived for anisotropic $ ext{lambda}$-deformed CFTs.

Agreement with known results in symmetric coupling and equal level cases.

Analysis of two-coupling case involving subgroup and coset geometries.

Abstract

We study the renormalization group equations of the fully anisotropic $\lambda$-deformed CFTs involving the direct product of two current algebras at different levels $k_{1,2}$ for general semi-simple groups. The exact, in the deformation parameters, $\beta$-function is found via the effective action of the quantum fluctuations around a classical background as well as from gravitational techniques. Furthermore, agreement with known results for symmetric couplings and/or for equal levels, is demonstrated. We study in detail the two coupling case arising by splitting the group into a subgroup and the corresponding coset manifold which consistency requires to be either a symmetric-space one or a non-symmetric Einstein-space.