Defects, Rigid Holography and $C$-theorems

We consider a general scalar QFT with a linear defect in $D=4-\epsilon$ and a surface defect in $D=6-\epsilon$. Using holography and the Hamilton-Jacobi formalism, we show that the $\beta$ functions controlling the defect RG flow are the gradient of the entropy function. In the case of conformal field theories, this allows the proof that the relevant $C$-functions decrease monotonically along the RG flow. We provide evidence that this property also holds in the full quantum theory for general scalar field theories. An obstruction to the gradient property seems to appear at two loop order when fermions are added.