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

For a general λ -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 β -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.