Weyl anomaly and the $C$-function in $\lambda$-deformed CFTs

TL;DR

This paper calculates the exact Weyl anomaly and $C$-function in $\lambda$-deformed current algebra CFTs, demonstrating that the anomaly acts as an exact $C$-function interpolating between UV and IR fixed points.

Contribution

It provides the exact Weyl anomaly coefficient and $C$-function for $\lambda$-deformed CFTs, linking them with the exact $eta$-function and demonstrating their role as an interpolating $C$-function.

Findings

Weyl anomaly matches the Zamolodchikov $C$-function.

Explicit example with anisotropic $SU(2)$ case.

Agreement with previous literature on special cases.

Abstract

For a general $\lambda$-deformation of current algebra CFTs we compute the exact Weyl anomaly coefficient and the corresponding metric in the couplings space geometry. By incorporating the exact $\beta$-function found in previous works we show that the Weyl anomaly is in fact the exact Zamolodchikov's $C$-function interpolating between exact CFTs occurring in the UV and in the IR. We provide explicit examples with the anisotropic $SU(2)$ case presented in detail. The anomalous dimension of the operator driving the deformation is also computed in general. Agreement is found with special cases existing already in the literature.